80 files changed, 37202 insertions, 0 deletions
diff --git a/.gitattributes b/.gitattributes
new file mode 100644
index 0000000..6833f05
--- /dev/null
+++ b/.gitattributes
@@ -0,0 +1,3 @@
+* text=auto
+*.txt text
+*.md text
diff --git a/39300-8.txt b/39300-8.txt
new file mode 100644
index 0000000..2677078
--- /dev/null
+++ b/39300-8.txt
@@ -0,0 +1,11725 @@
+Project Gutenberg's The Psychology of Arithmetic, by Edward L. Thorndike
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: The Psychology of Arithmetic
+
+Author: Edward L. Thorndike
+
+Release Date: March 29, 2012 [EBook #39300]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC ***
+
+
+
+
+Produced by Jonathan Ingram and the Online Distributed
+Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+
+
+
+                         THE MACMILLAN COMPANY
+               NEW YORK  �  BOSTON  �  CHICAGO  �  DALLAS
+                       ATLANTA  �  SAN FRANCISCO
+
+                        MACMILLAN & CO., LIMITED
+                     LONDON  �  BOMBAY  �  CALCUTTA
+                               MELBOURNE
+
+                         THE MACMILLAN COMPANY
+                           OF CANADA, LIMITED
+                                TORONTO
+
+
+
+                           THE PSYCHOLOGY OF
+                               ARITHMETIC
+
+
+                                   BY
+                          EDWARD L. THORNDIKE
+
+                       TEACHERS COLLEGE, COLUMBIA
+                               UNIVERSITY
+
+
+                                New York
+                         THE MACMILLAN COMPANY
+                                  1929
+
+                         _All rights reserved_
+
+
+
+                            COPYRIGHT, 1922,
+                       BY THE MACMILLAN COMPANY.
+
+      Set up and electrotyped. Published January, 1922. Reprinted
+         October, 1924; May, 1926; August, 1927; October, 1929.
+
+
+              � PRINTED IN THE UNITED STATES OF AMERICA �
+
+
+
+
+PREFACE
+
+
+Within recent years there have been three lines of advance in psychology
+which are of notable significance for teaching. The first is the new
+point of view concerning the general process of learning. We now
+understand that learning is essentially the formation of connections or
+bonds between situations and responses, that the satisfyingness of the
+result is the chief force that forms them, and that habit rules in the
+realm of thought as truly and as fully as in the realm of action.
+
+The second is the great increase in knowledge of the amount, rate, and
+conditions of improvement in those organized groups or hierarchies of
+habits which we call abilities, such as ability to add or ability to
+read. Practice and improvement are no longer vague generalities, but
+concern changes which are definable and measurable by standard tests and
+scales.
+
+The third is the better understanding of the so-called "higher
+processes" of analysis, abstraction, the formation of general notions,
+and reasoning. The older view of a mental chemistry whereby sensations
+were compounded into percepts, percepts were duplicated by images,
+percepts and images were amalgamated into abstractions and concepts, and
+these were manipulated by reasoning, has given way to the understanding
+of the laws of response to elements or aspects of situations and to many
+situations or elements thereof in combination. James' view of reasoning
+as "selection of essentials" and "thinking things together" in a
+revised and clarified form has important applications in the teaching of
+all the school subjects.
+
+This book presents the applications of this newer dynamic psychology to
+the teaching of arithmetic. Its contents are substantially what have
+been included in a course of lectures on the psychology of the
+elementary school subjects given by the author for some years to
+students of elementary education at Teachers College. Many of these
+former students, now in supervisory charge of elementary schools, have
+urged that these lectures be made available to teachers in general. So
+they are now published in spite of the author's desire to clarify and
+reinforce certain matters by further researches.
+
+A word of explanation is necessary concerning the exercises and problems
+cited to illustrate various matters, especially erroneous pedagogy.
+These are all genuine, having their source in actual textbooks, courses
+of study, state examinations, and the like. To avoid any possibility of
+invidious comparisons they are not quotations, but equivalent problems
+such as represent accurately the spirit and intent of the originals.
+
+I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn
+and Company, D. C. Heath and Company, The Macmillan Company, The Oxford
+University Press, Rand, McNally and Company, Dr. C. W. Stone, The
+Teachers College Bureau of Publications, and The World Book Company, in
+permitting various quotations.
+
+                                                EDWARD L. THORNDIKE.
+
+   TEACHERS COLLEGE
+  COLUMBIA UNIVERSITY
+     April 1, 1920
+
+
+
+
+CONTENTS
+
+
+ CHAPTER                                                          PAGE
+
+    INTRODUCTION: THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS  xi
+
+ I. THE NATURE OF ARITHMETICAL ABILITIES                             1
+
+        Knowledge of the Meanings of Numbers
+        Arithmetical Language
+        Problem Solving
+        Arithmetical Reasoning
+        Summary
+        The Sociology of Arithmetic
+
+ II. THE MEASUREMENT OF ARITHMETICAL ABILITIES                      27
+
+        A Sample Measurement of an Arithmetical Ability
+        Ability to Add Integers
+        Measurements of Ability in Computation
+        Measurements of Ability in Applied Arithmetic:
+          the Solution of Problems
+
+ III. THE CONSTITUTION OF ARITHMETICAL ABILITIES                    51
+
+        The Elementary Functions of Arithmetical Learning
+        Knowledge of the Meaning of a Fraction
+        Learning the Processes of Computation
+
+ IV. THE CONSTITUTION OF ARITHMETICAL ABILITIES (_continued_)       70
+
+        The Selection of the Bonds to Be Formed
+        The Importance of Habit Formation
+        Desirable Bonds Now Often Neglected
+        Wasteful and Harmful Bonds
+        Guiding Principles
+
+ V. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS   102
+
+        The Need of Stronger Elementary Bonds
+        Early Mastery
+        The Strength of Bonds for Temporary Service
+        The Strength of Bonds with Technical Facts and Terms
+        The Strength of Bonds Concerning the Reasons for
+          Arithmetical Processes
+        Prop�deutic Bonds
+
+ VI. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF
+          PRACTICE AND THE ORGANIZATION OF ABILITIES               122
+
+        The Amount of Practice
+        Under-learning and Over-learning
+        The Organization of Abilities
+
+ VII. THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS      141
+
+        Conventional _versus_ Effective Orders
+        Decreasing Interference and Increasing Facilitation
+        Interest
+        General Principles
+
+ VIII. THE DISTRIBUTION OF PRACTICE                                156
+
+        The Problem
+        Sample Distributions
+        Possible Improvements
+
+ IX. THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL
+          NOTIONS IN ARITHMETIC                                    169
+
+        Responses to Elements and Classes
+        Facilitating the Analysis of Elements
+        Systematic and Opportunistic Stimuli to Analysis
+        Adaptations to Elementary-school Pupils
+
+ X. THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC            185
+
+        The Essentials of Arithmetical Reasoning
+        Reasoning as the Co�peration of Organized Habits
+
+ XI. ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL            195
+
+        The Utilization of Instinctive Interests
+        The Order of Development of Original Tendencies
+        Inventories of Arithmetical Knowledge and Skill
+        The Perception of Number and Quantity
+        The Early Awareness of Number
+
+ XII. INTEREST IN ARITHMETIC                                       209
+
+        Censuses of Pupils' Interests
+        Relieving Eye Strain
+        Significance for Related Activities
+        Intrinsic Interest in Arithmetical Learning
+
+ XIII. THE CONDITIONS OF LEARNING                                  227
+
+        External Conditions
+        The Hygiene of the Eyes in Arithmetic
+        The Use of Concrete Objects in Arithmetic
+        Oral, Mental, and Written Arithmetic
+
+ XIV. THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE             266
+
+        Illustrative Cases
+        General Principles
+        Difficulty and Success as Stimuli
+        False Inferences
+
+ XV. INDIVIDUAL DIFFERENCES                                        285
+
+        Nature and Amount
+        Differences within One Class
+        The Causes of Individual Differences
+        The Interrelations of Individual Differences
+
+    BIBLIOGRAPHY OF REFERENCES                                     302
+
+    INDEX                                                          311
+
+
+
+
+GENERAL INTRODUCTION
+
+THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS
+
+
+The psychology of the elementary school subjects is concerned with the
+connections whereby a child is able to respond to the sight of printed
+words by thoughts of their meanings, to the thought of "six and eight"
+by thinking "fourteen," to certain sorts of stories, poems, songs, and
+pictures by appreciation thereof, to certain situations by acts of
+skill, to certain others by acts of courtesy and justice, and so on and
+on through the series of situations and responses which are provided by
+the systematic training of the school subjects and the less systematic
+training of school life during their study. The aims of elementary
+education, when fully defined, will be found to be the production of
+changes in human nature represented by an almost countless list of
+connections or bonds whereby the pupil thinks or feels or acts in
+certain ways in response to the situations the school has organized and
+is influenced to think and feel and act similarly to similar situations
+when life outside of school confronts him with them.
+
+We are not at present able to define the work of the elementary school
+in detail as the formation of such and such bonds between certain
+detached situations and certain specified responses. As elsewhere in
+human learning, we are at present forced to think somewhat vaguely in
+terms of mental functions, like "ability to read the vernacular,"
+"ability to spell common words," "ability to add, subtract, multiply,
+and divide with integers," "knowledge of the history of the United
+States," "honesty in examinations," and "appreciation of good music,"
+defined by some general results obtained rather than by the elementary
+bonds which constitute them.
+
+The psychology of the school subjects begins where our common sense
+knowledge of these functions leaves off and tries to define the
+knowledge, interest, power, skill, or ideal in question more adequately,
+to measure improvement in it, to analyze it into its constituent bonds,
+to decide what bonds need to be formed and in what order as means to the
+most economical attainment of the desired improvement, to survey the
+original tendencies and the tendencies already acquired before entrance
+to school which help or hinder progress in the elementary school
+subjects, to examine the motives that are or may be used to make the
+desired connections satisfying, to examine any other special conditions
+of improvement, and to note any facts concerning individual differences
+that are of special importance to the conduct of elementary school work.
+
+Put in terms of problems, the task of the psychology of the elementary
+school subjects is, in each case:--
+
+(1) _What is the function?_ For example, just what is "ability to read"?
+Just what does "the understanding of decimal notation" mean? Just what
+are "the moral effects to be sought from the teaching of literature"?
+
+(2) _How are degrees of ability or attainment, and degrees of progress
+or improvement in the function or a part of the function measured?_ For
+example, how can we determine how well a pupil should write, or how hard
+words we expect him to spell, or what good taste we expect him to show?
+How can we define to ourselves what knowledge of the meaning of a
+fraction we shall try to secure in grade 4?
+
+(3) _What can be done toward reducing the function to terms of
+particular situation-response connections, whose formation can be more
+surely and easily controlled?_ For example, how far does ability to
+spell involve the formation one by one of bonds between the thought of
+almost every word in the language and the thought of that word's letters
+in their correct order; and how far does, say, the bond leading from the
+situation of the sound of _ceive_ in _receive_ and _deceive_ to their
+correct spelling insure the correct spelling of that part of _perceive_?
+Does "ability to add" involve special bonds leading from "27 and 4" to
+"31," from "27 and 5" to "32," and "27 and 6" to "33"; or will the bonds
+leading from "7 and 4" to "11," "7 and 5" to "12" and "7 and 6" to "13"
+(each plus a simple inference) serve as well? What are the situations
+and responses that represent in actual behavior the quality that we call
+school patriotism?
+
+(4) _In almost every case a certain desired change of knowledge or skill
+or power can be attained by any one of several sets of bonds. Which of
+them is the best? What are the advantages of each?_ For example,
+learning to add may include the bonds "0 and 0 are 0," "0 and 1 are 1,"
+"0 and 2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these may be
+all left unformed, the pupil being taught the habits of entering 0 as
+the sum of a column that is composed of zeros and otherwise neglecting 0
+in addition. Are the rules of usage worth teaching as a means toward
+correct speech, or is the time better spent in detailed practice in
+correct speech itself?
+
+(5) _A bond to be formed may be formed in any one of many degrees of
+strength. Which of these is, at any given stage of learning the subject,
+the most desirable, all things considered?_ For example, shall the dates
+of all the early settlements of North America be learned so that the
+exact year will be remembered for ten years, or so that the exact date
+will be remembered for ten minutes and the date with an error plus or
+minus of ten years will be remembered for a year or two? Shall the
+tables of inches, feet, and yards, and pints, quarts, and gallons be
+learned at their first appearance so as to be remembered for a year, or
+shall they be learned only well enough to be usable in the work of that
+week, which in turn fixes them to last for a month or so? Should a pupil
+in the first year of study of French have such perfect connections
+between the sounds of French words and their meanings that he can
+understand simple sentences containing them spoken at an ordinary rate
+of speaking? Or is slow speech permissible, and even imperative, on the
+part of the teacher, with gradual increase of rate?
+
+(6) _In almost every case, any set of bonds may produce the desired
+change when presented in any one of several orders. Which is the best
+order? What are the advantages of each?_ Certain systems for teaching
+handwriting perfect the elementary movements one at a time and then
+teach their combination in words and sentences. Others begin and
+continue with the complex movement-series that actual words require.
+What do the latter lose and gain? The bonds constituting knowledge of
+the metric system are now formed late in the pupil's course. Would it be
+better if they were formed early as a means of facilitating knowledge of
+decimal fractions?
+
+(7) _What are the original tendencies and pre-school acquisitions upon
+which the connection-forming of the elementary school may be based or
+which it has to counteract?_ For example, if a pupil knows the meaning
+of a heard word, he may read it understandingly from getting its sound,
+as by phonic reconstruction. What words does the average beginner so
+know? What are the individual differences in this respect? What do the
+instincts of gregariousness, attention-getting, approval, and
+helpfulness recommend concerning group-work _versus_ individual-work,
+and concerning the size of a group that is most desirable? The original
+tendency of the eyes is certainly not to move along a line from left to
+right of a page, then back in one sweep and along the next line. What is
+their original tendency when confronted with the printed page, and what
+must we do with it in teaching reading?
+
+(8) _What armament of satisfiers and annoyers, of positive and negative
+interests and motives, stands ready for use in the formation of the
+intrinsically uninteresting connections between black marks and
+meanings, numerical exercises and their answers, words and their
+spelling, and the like?_ School practice has tried, more or less at
+random, incentives and deterrents from quasi-physical pain to the most
+sentimental fondling, from sheer cajolery to philosophical argument,
+from appeals to assumed savage and primitive traits to appeals to the
+interest in automobiles, flying-machines, and wireless telegraphy. Can
+not psychology give some rules for guidance, or at least limit
+experimentation to its more hopeful fields?
+
+(9) _The general conditions of efficient learning are described in
+manuals of educational psychology. How do these apply in the case of
+each task of the elementary school?_ For example, the arrangement of
+school drills in addition and in short division in the form of practice
+experiments has been found very effective in producing interest in the
+work and in improvement at it. In what other arithmetical functions may
+we expect the same?
+
+(10) _Beside the general principles concerning the nature and causation
+of individual differences, there must obviously be, in existence or
+obtainable as a possible result of proper investigation, a great fund
+of knowledge of special differences relevant to the learning of reading,
+spelling, geography, arithmetic, and the like. What are the facts as far
+as known? What are the means of learning more of them?_ Courtis finds
+that a child may be specially strong in addition and yet be specially
+weak in subtraction in comparison with others of his age and grade. It
+even seems that such subtle and intricate tendencies are inherited. How
+far is such specialization the rule? Is it, for example, the case that a
+child may have a special gift for spelling certain sorts of words, for
+drawing faces rather than flowers, for learning ancient history rather
+than modern?
+
+       *       *       *       *       *
+
+Such are our problems: this volume discusses them in the case of
+arithmetic. The student who wishes to relate the discussion to the
+general pedagogy of arithmetic may profitably read, in connection with
+this volume: The Teaching of Elementary Mathematics, by D. E. Smith
+['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to
+Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching
+of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic,
+by the author ['21].
+
+
+
+
+THE PSYCHOLOGY OF ARITHMETIC
+
+
+
+
+CHAPTER I
+
+THE NATURE OF ARITHMETICAL ABILITIES
+
+
+According to common sense, the task of the elementary school is to
+teach:--(1) the meanings of numbers, (2) the nature of our system
+of decimal notation, (3) the meanings of addition, subtraction,
+multiplication, and division, and (4) the nature and relations of
+certain common measures; to secure (5) the ability to add, subtract,
+multiply, and divide with integers, common and decimal fractions, and
+denominate numbers, (6) the ability to apply the knowledge and power
+represented by (1) to (5) in solving problems, and (7) certain specific
+abilities to solve problems concerning percentage, interest, and other
+common occurrences in business life.
+
+This statement of the functions to be developed and improved is sound
+and useful so far as it goes, but it does not go far enough to make the
+task entirely clear. If teachers had nothing but the statement above as
+a guide to what changes they were to make in their pupils, they would
+often leave out important features of arithmetical training, and put in
+forms of training that a wise educational plan would not tolerate. It is
+also the case that different leaders in arithmetical teaching, though
+they might all subscribe to the general statement of the previous
+paragraph, certainly do not in practice have identical notions of what
+arithmetic should be for the elementary school pupil.
+
+The ordinary view of the nature of arithmetical learning is obscure or
+inadequate in four respects. It does not define what 'knowledge of the
+meanings of numbers' is; it does not take account of the very large
+amount of teaching of _language_ which is done and should be done as a
+part of the teaching of arithmetic; it does not distinguish between the
+ability to meet certain quantitative problems as life offers them and
+the ability to meet the problems provided by textbooks and courses of
+study; it leaves 'the ability to apply arithmetical knowledge and power'
+as a rather mystical general faculty to be improved by some educational
+magic. The four necessary amendments may be discussed briefly.
+
+
+KNOWLEDGE OF THE MEANINGS OF NUMBERS
+
+Knowledge of the meanings of the numbers from one to ten may mean
+knowledge that 'one' means a single thing of the sort named, that two
+means one more than one, that three means one more than two, and so on.
+This we may call the _series_ meaning. To know the meaning of 'six' in
+this sense is to know that it is one more than five and one less than
+seven--that it is between five and seven in the number series. Or we may
+mean by knowledge of the meanings of numbers, knowledge that two fits a
+collection of two units, that three fits a collection of three units,
+and so on, each number being a name for a certain sized collection of
+discrete things, such as apples, pennies, boys, balls, fingers, and the
+other customary objects of enumeration in the primary school. This we
+may call the _collection_ meaning. To know the meaning of six in this
+sense is to be able to name correctly any collection of six separate,
+easily distinguishable individual objects. In the third place, knowledge
+of the numbers from one to ten may mean knowledge that two is twice
+whatever is called one, that three is three times whatever is one, and
+so on. This is, of course, the _ratio_ meaning. To know the meaning of
+six in this sense is to know that if ___________ is one, a line half a
+foot long is six, that if [___] is one, [____________] is about six,
+while if [__] is one, [______] is about six, and the like. In the fourth
+place, the meaning of a number may be a smaller or larger fraction of
+its _implications_--its numerical relations, facts about it. To know six
+in this sense is to know that it is more than five or four, less than
+seven or eight, twice three, three times two, the sum of five and one,
+or of four and two, or of three and three, two less than eight--that
+with four it makes ten, that it is half of twelve, and the like. This we
+may call the '_nucleus of facts_' or _relational_ meaning of a number.
+
+Ordinary school practice has commonly accepted the second meaning as
+that which it is the task of the school to teach beginners, but each of
+the other meanings has been alleged to be the essential one--the series
+idea by Phillips ['97], the ratio idea by McLellan and Dewey ['95] and
+Speer ['97], and the relational idea by Grube and his followers.
+
+This diversity of views concerning what the function is that is to be
+improved in the case of learning the meanings of the numbers one to ten
+is not a trifling matter of definition, but produces very great
+differences in school practice. Consider, for example, the predominant
+value assigned to counting by Phillips in the passage quoted below, and
+the samples of the sort of work at which children were kept employed
+for months by too ardent followers of Speer and Grube.
+
+THE SERIES IDEA OVEREMPHASIZED
+
+  "This is essentially the counting period, and any words that can be
+  arranged into a series furnish all that is necessary. Counting is
+  fundamental, and counting that is spontaneous, free from sensible
+  observation, and from the strain of reason. A study of these original
+  methods shows that multiplication was developed out of counting, and
+  not from addition as nearly all textbooks treat it. Multiplication is
+  counting. When children count by 4's, etc., they accent the same as
+  counting gymnastics or music. When a child now counts on its fingers
+  it simply reproduces a stage in the growth of the civilization of all
+  nations.
+
+  I would emphasize again that during the counting period there is a
+  somewhat spontaneous development of the number series-idea which
+  Preyer has discussed in his Arithmogenesis; that an immense momentum
+  is given by a systematic series of names; and that these names are
+  generally first learned and applied to objects later. A lady teacher
+  told me that the Superintendent did not wish the teachers to allow the
+  children to count on their fingers, but she failed to see why counting
+  with horse-chestnuts was any better. Her children could hardly avoid
+  using their fingers in counting other objects yet they followed the
+  series to 100 without hesitation or reference to their fingers. This
+  spontaneous counting period, or naming and following the series,
+  should precede its application to objects." [D.E. Phillips, '97,
+  p. 238.]
+
+THE RATIO IDEA OVEREMPHASIZED
+
+    [Illustration: FIG. 1.]
+
+  "Ratios.--1. Select solids having the relation, or ratio, of _a_, _b_,
+     _c_, _d_, _o_, _e_.
+
+  2. Name the solids, _a_, _b_, _c_, _d_, _o_, _e_.
+
+     The means of expressing must be as freely supplied as the means of
+     discovery. The pupil is not expected to invent terms.
+
+  3. Tell all you can about the relation of these units.
+
+  4. Unite units and tell what the sum equals.
+
+  5. Make statements like this: _o_ less _e_ equals _b_.
+
+  6. _c_ can be separated into how many _d_'s? into how many _b_'s?
+
+  7. _c_ can be separated into how many _b_'s? What is the name of the
+     largest unit that can be found in both _c_ and _d_ an exact number
+     of times?
+
+  8. Each of the other units equals what part of _c_?
+
+  9. If _b_ is 1, what is each of the other units?
+
+ 10. If _a_ is 1, what is each of the other units?
+
+ 11. If _b_ is 1, how many 1's are there in each of the other units?
+
+ 12. If _d_ is 1, how many 1's and parts of 1 in each of the other
+     units?
+
+ 13. 2 is the relation of what units?
+
+ 14. 3 is the relation of what units?
+
+ 15. 1/2 is the relation of what units?
+
+ 16. 2/3 is the relation of what units?
+
+ 17. Which units have the relation 3/2?
+
+ 18. Which unit is 3 times as large as 1/2 of _b_?
+
+ 19. _c_ equals 6 times 1/3 of what unit?
+
+ 20. 1/3 of what unit equals 1/6 of _c_?
+
+ 21. What equals 1/2 of _c_? _d_ equals how many sixths of _c_?
+
+ 22. _o_ equals 5 times 1/3 of what unit?
+
+ 23. 1/3 of what unit equals 1/5 of _o_?
+
+ 24. 2/3 of _d_ equals what unit? _b_ equals how many thirds of _d_?
+
+ 25. 2 is the ratio of _d_ to 1/3 of what unit? 3 is the ratio of _d_
+     to 1/2 of what unit?
+
+ 26. _d_ equals 3/4 of what unit? 3/4 is the ratio of what units?"
+     [Speer, '97, p. 9f.]
+
+THE RELATIONAL IDEA OVEREMPHASIZED
+
+  An inspection of books of the eighties which followed the "Grube
+  method" (for example, the _New Elementary Arithmetic_ by E.E. White
+  ['83]) will show undue emphasis on the relational ideas. There will be
+  over a hundred and fifty successive tasks all, or nearly all, on +7
+  and -7. There will be much written work of the sort shown below:
+
+  _Add:_
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    1    2
+          --   --   --
+
+  which must have sorely tried the eyes of all concerned. Pupils are
+  taught to "give the analysis and synthesis of each of the nine
+  digits." Yet the author states that he does not carry the principle
+  of the Grube method "to the extreme of useless repetition and
+  mechanism."
+
+It should be obvious that all four meanings have claims upon the
+attention of the elementary school. Four is the thing between three and
+five in the number series; it is the name for a certain sized collection
+of discrete objects; it is also the name for a continuous magnitude
+equal to four units--for four quarts of milk in a gallon pail as truly
+as for four separate quart-pails of milk; it is also, if we know it
+well, the thing got by adding one to three or subtracting six from ten
+or taking two two's or half of eight. To know the meaning of a number
+means to know somewhat about it in all of these respects. The difficulty
+has been the narrow vision of the extremists. A child must not be left
+interminably counting; in fact the one-more-ness of the number series
+can almost be had as a by-product. A child must not be restricted to
+exercises with collections objectified as in Fig. 2 or stated in words
+as so many apples, oranges, hats, pens, etc., when work with measurement
+of continuous quantities with varying units--inches, feet, yards,
+glassfuls, pints, quarts, seconds, minutes, hours, and the like--is
+so easy and so significant. On the other hand, the elaboration of
+artificial problems with fictitious units of measure just to have
+relative magnitudes as in the exercises on page 5 is a wasteful
+sacrifice. Similarly, special drills emphasizing the fact that eighteen
+is eleven and seven, twelve and six, three less than twenty-one, and the
+like, are simply idolatrous; these facts about eighteen, so far as they
+are needed, are better learned in the course of actual column-addition
+and -subtraction.
+
+    [Illustration: FIG. 2.]
+
+
+ARITHMETICAL LANGUAGE
+
+The second improvement to be made in the ordinary notion of what the
+functions to be improved are in the case of arithmetic is to include
+among these functions the knowledge of certain words. The understanding
+of such words as _both_, _all_, _in all_, _together_, _less_,
+_difference_, _sum_, _whole_, _part_, _equal_, _buy_, _sell_, _have
+left_, _measure_, _is contained in_, and the like, is necessary in
+arithmetic as truly as is the understanding of numbers themselves. It
+must be provided for by the school; for pre-school and extra-school
+training does not furnish it, or furnishes it too late. It can be
+provided for much better in connection with the teaching of arithmetic
+than in connection with the teaching of English.
+
+It has not been provided for. An examination of the first fifty pages of
+eight recent textbooks for beginners in arithmetic reveals very slight
+attention to this matter at the best and no attention at all in some
+cases. Three of the books do not even use the word _sum_, and one uses
+it only once in the fifty pages. In all the four hundred pages the word
+_difference_ occurs only twenty times. When the words are used, no great
+ingenuity or care appears in the means of making sure that their
+meanings are understood.
+
+The chief reason why it has not been provided for is precisely that the
+common notion of what the functions are that arithmetic is to develop
+has left out of account this function of intelligent response to
+quantitative terms, other than the names of the numbers and processes.
+
+Knowledge of language over a much wider range is a necessary element in
+arithmetical ability in so far as the latter includes ability to solve
+verbally stated problems. As arithmetic is now taught, it does include
+that ability, and a large part of the time of wise teaching is given to
+improving the function 'knowing what a problem states and what it asks
+for.' Since, however, this understanding of verbally stated problems may
+not be an absolutely necessary element of arithmetic, it is best to
+defer its consideration until we have seen what the general function of
+problem-solving is.
+
+
+PROBLEM-SOLVING
+
+The third respect in which the function, 'ability in arithmetic,' needs
+clearer definition, is this 'problem-solving.' The aim of the elementary
+school is to provide for correct and economical response to genuine
+problems, such as knowing the total due for certain real quantities at
+certain real prices, knowing the correct change to give or get, keeping
+household accounts, calculating wages due, computing areas, percentages,
+and discounts, estimating quantities needed of certain materials to make
+certain household or shop products, and the like. Life brings these
+problems usually either with a real situation (as when one buys and
+counts the cost and his change), or with a situation that one imagines
+or describes to himself (as when one figures out how much money he must
+save per week to be able to buy a forty-dollar bicycle before a certain
+date). Sometimes, however, the problem is described in words to the
+person who must solve it by another person (as when a life insurance
+agent says, 'You pay only 25 cents a week from now till--and you get
+$250 then'; or when an employer says, 'Your wages would be 9 dollars a
+week, with luncheon furnished and bonuses of such and such amounts').
+Sometimes also the problem is described in printed or written words to
+the person who must solve it (as in an advertisement or in the letter of
+a customer asking for an estimate on this or that). The problem may be
+in part real, in part imagined or described to oneself, and in part
+described to one orally or in printed or written words (as when the
+proposed articles for purchase lie before one, the amount of money one
+has in the bank is imagined, the shopkeeper offers 10 percent discount,
+and the printed price list is there to be read).
+
+To fit pupils to solve these real, personally imagined, or
+self-described problems, and 'described-by-another' problems, schools
+have relied almost exclusively on training with problems of the last
+sort only. The following page taken almost at random from one of the
+best recent textbooks could be paralleled by thousands of others; and
+the oral problems put by teachers have, as a rule, no real situation
+supporting them.
+
+  1. At 70 cents per 100 pounds, what will be the amount of duty on an
+     invoice of 3622 steel rails, each rail being 27 feet long and
+     weighing 60 pounds to the yard?
+
+  2. A man had property valued at $6500. What will be his taxes at the
+     rate of $10.80 per $1000?
+
+  3. Multiply seventy thousand fourteen hundred-thousandths by one
+     hundred nine millionths, and divide the product by five hundred
+     forty-five.
+
+  4. What number multiplied by 43-3/4 will produce 265-5/8?
+
+  5. What decimal of a bushel is 3 quarts?
+
+  6. A man sells 5/8 of an acre of land for $93.75. What would be the
+     value of his farm of 150-3/4 acres at the same rate?
+
+  7. A coal dealer buys 375 tons coal at $4.25 per ton of 2240 pounds.
+     He sells it at $4.50 per ton of 2000 pounds. What is his profit?
+
+  8. Bought 60 yards of cloth at the rate of 2 yards for $5, and 80
+     yards more at the rate of 4 yards for $9. I immediately sold the
+     whole of it at the rate of 5 yards for $12. How much did I gain?
+
+  9. A man purchased 40 bushels of apples at $1.50 per bushel.
+     Twenty-five hundredths of them were damaged, and he sold them
+     at 20 cents per peck. He sold the remainder at 50 cents per
+     peck. How much did he gain or lose?
+
+ 10. If oranges are 37-1/2 cents per dozen, how many boxes, each
+     containing 480, can be bought for $60?
+
+ 11. A man can do a piece of work in 18-3/4 days. What part of it can
+     he do in 6-2/3 days?
+
+ 12. How old to-day is a boy that was born Oct. 29, 1896?
+     [Walsh, '06, Part I, p. 165.]
+
+As a result, teachers and textbook writers have come to think of the
+functions of solving arithmetical problems as identical with the
+function of solving the described problems which they give in school in
+books, examination papers, and the like. If they do not think explicitly
+that this is so, they still act in training and in testing pupils as if
+it were so.
+
+It is not. Problems should be solved in school to the end that pupils
+may solve the problems which life offers. To know what change one should
+receive after a given real purchase, to keep one's accounts accurately,
+to adapt a recipe for six so as to make enough of the article for four
+persons, to estimate the amount of seed required for a plot of a given
+size from the statement of the amount required per acre, to make with
+surety the applications that the household, small stores, and ordinary
+trades require--such is the ability that the elementary school should
+develop. Other things being equal, the school should set problems in
+arithmetic which life then and later will set, should favor the
+situations which life itself offers and the responses which life itself
+demands.
+
+Other things are not always equal. The same amount of time and effort
+will often be more productive toward the final end if directed during
+school to 'made-up' problems. The keeping of personal financial accounts
+as a school exercise is usually impracticable, partly because some of
+the children have no earnings or allowance--no accounts to keep, and
+partly because the task of supervising work when each child has a
+different problem is too great for the teacher. The use of real
+household and shop problems will be easy only when the school program
+includes the household arts and industrial education, and when these
+subjects themselves are taught so as to improve the functions used by
+real life. Very often the most efficient course is to make sure that
+arithmetical procedures are applied to the real and personally initiated
+problems which they fit, by having a certain number of such problems
+arise and be solved; then to make sure that the similarity between these
+real problems and certain described problems of the textbook or
+teacher's giving is appreciated; and then to give the needed drill work
+with described problems. In many cases the school practice is fairly
+well justified in assuming that solving described problems will prepare
+the pupil to solve the corresponding real problems actually much better
+than the same amount of time spent on the real problems themselves.
+
+All this is true, yet the general principle remains that, other things
+being equal, the school should favor real situations, should present
+issues as life will present them.
+
+Where other things make the use of verbally described problems of the
+ordinary type desirable, these should be chosen so as to give a maximum
+of preparation for the real applications of arithmetic in life. We
+should not, for example, carelessly use any problem that comes to mind
+in applying a certain principle, but should stop to consider just what
+the situations of life really require and show clearly the application
+of that principle. For example, contrast these two problems applying
+cancellation:--
+
+  A. A man sold 24 lambs at $18 apiece on each of six days, and
+     bought 8 pounds of metal with the proceeds. How much did he
+     pay per ounce for the metal?
+
+  B. How tall must a rectangular tank 16" long by 8" wide be to
+     hold as much as a rectangular tank 24" by 18" by 6"?
+
+The first problem not only presents a situation that would rarely or
+never occur, but also takes a way to find the answer that would not, in
+that situation, be taken since the price set by another would determine
+the amount.
+
+Much thought and ingenuity should in the future be expended in
+eliminating problems whose solution does not improve the real function
+to be improved by applied arithmetic, or improves it at too great cost,
+and in devising problems which prepare directly for life's demands and
+still can fit into a curriculum that can be administered by one teacher
+in charge of thirty or forty pupils, under the limitations of school
+life.
+
+The following illustrations will to some extent show concretely what the
+ability to apply the knowledge and power represented by abstract or pure
+arithmetic--the so-called fundamentals--in solving problems should mean
+and what it should not mean.
+
+  _Samples of Desirable Applications of Arithmetic in Problems where
+    the Situation is Actually Present to Sense in Whole or in Part_
+
+Keeping the scores and deciding which side beat and by how much in
+appropriate classroom games, spelling matches, and the like.
+
+Computing costs, making and inspecting change, taking inventories, and
+the like with a real or play store.
+
+Mapping the school garden, dividing it into allotments, planning for the
+purchase of seeds, and the like.
+
+Measuring one's own achievement and progress in tests of word-knowledge,
+spelling, addition, subtraction, speed of writing, and the like.
+Measuring the rate of improvement per hour of practice or per week of
+school life, and the like.
+
+Estimating costs of food cooked in the school kitchen, articles made in
+the school shops, and the like.
+
+Computing the cost of telegrams, postage, expressage, for a real message
+or package, from the published tariffs.
+
+Computing costs from mail order catalogues and the like.
+
+  _Samples of Desirable Applications of Arithmetic where the Situation
+    is Not Present to Sense_
+
+The samples given here all concern the subtraction of fractions.
+Samples concerning any other arithmetical principle may be found in the
+appropriate pages of any text which contains problem-material selected
+with consideration of life's needs.
+
+A
+
+  1. Dora is making jelly. The recipe calls for 24 cups of sugar and
+     she has only 21-1/2. She has no time to go to the store so she
+     has to borrow the sugar from a neighbor. How much must she get?
+
+      _Subtract_
+          24        _Think "1/2 and 1/2 = 1." Write 1/2._
+          21-1/2    _Think "2 and 2 = 4." Write the 2._
+        --------
+           2-1/2
+
+  2. A box full of soap weighs 29-1/2 lb. The empty box weighs 3-1/2 lb.
+     How much does the soap alone weigh?
+
+  3. On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt. On July
+     15 there were just 11-1/2 lb. left. How much had he used in the two
+     weeks?
+
+  4. Grace promised to pick 30 qt. blueberries for her mother. So far
+     she has picked 18-1/2 qt. How many more quarts must she pick?
+
+B
+
+  This table of numbers tells                Weight of Mary Adams
+  what Nell's baby sister Mary            When born        7-3/8 lb.
+  weighed every two months from           2 months old    11-1/4 lb.
+  the time she was born till she          4 months old    14-1/8 lb.
+  was a year old.                         6 months old    15-3/4 lb.
+                                          8 months old    17-5/8 lb.
+                                         10 months old    19-1/2 lb.
+                                         12 months old    21-3/8 lb.
+
+  1. How much did the Adams baby gain in the first two months?
+  2. How much did the Adams baby gain in the second two months?
+  3. In the third two months?
+  4. In the fourth two months?
+  5. From the time it was 8 months old till it was 10 months old?
+  6. In the last two months?
+  7. From the time it was born till it was 6 months old?
+
+C
+
+  1. Helen's exact average for December was 87-1/3. Kate's was 84-1/2.
+     How much higher was Helen's than Kate's?
+
+      87-1/3   How do you think of 1/2 and 1/3?
+      84-1/2   How do you think of 1-2/6?
+      ------
+               How do you change the 4?
+
+  2. Find the exact average for each girl in the following list. Write
+     the answers clearly so that you can see them easily. You will use
+     them in solving problems 3, 4, 5, 6, 7, and 8.
+
+                 Alice  Dora  Emma  Grace  Louise  Mary  Nell  Rebecca
+      Reading      91    87    83     81     79     77    76     73
+      Language     88    78    82     79     73     78    73     75
+      Arithmetic   89    85    79     75     84     87    89     80
+      Spelling     90    79    75     80     82     91    68     81
+      Geography    91    87    83     75     78     85    73     79
+      Writing      90    88    75     72     93     92    95     78
+
+  3. Which girl had the highest average?
+
+  4. How much higher was her average than the next highest?
+
+  5. How much difference was there between the highest and the lowest
+     girl?
+
+  6. Was Emma's average higher or lower than Louise's? How much?
+
+  7. How much difference was there between Alice's average and Dora's?
+
+  8. How much difference was there between Mary's average and Nell's?
+
+  9. Write five other problems about these averages, and solve each of
+     them.
+
+_Samples of Undesirable Applications of Arithmetic_[1]
+
+  Will has XXI marbles, XII jackstones, XXXVI pieces of string. How many
+  things had he?
+
+  George's kite rose CDXXXV feet and Tom's went LXIII feet higher. How
+  high did Tom's kite rise?
+
+  If from DCIV we take CCIV the result will be a number IV times as
+  large as the number of dollars Mr. Dane paid for his horse. How much
+  did he pay for his horse?
+
+  Hannah has 5/8 of a dollar, Susie 7/25, Nellie 3/4, Norah 13/16. How
+  much money have they all together?
+
+  A man saves 3-17/80 dollars a week. How much does he save in a year?
+
+  A tree fell and was broken into 4 pieces, 13-1/6 feet, 10-3/7 feet,
+  8-1/2 feet, and 7-16/21 feet long. How tall was the tree?
+
+  Annie's father gave her 20 apples to divide among her friends. She
+  gave each one 2-2/9 apples apiece. How many playmates had she?
+
+  John had 17-2/5 apples. He divided his whole apples into fifths. How
+  many pieces had he in all?
+
+  A landlady has 3-3/7 pies to be divided among her 8 boarders. How much
+  will each boarder receive?
+
+  There are twenty columns of spelling words in Mary's lesson and 16
+  words in each column. How many words are in her lesson?
+
+  There are 9 nuts in a pint. How many pints in a pile of 5,888,673
+  nuts?
+
+  The Adams school contains eight rooms; each room contains 48 pupils;
+  if each pupil has eight cents, how much have they together?
+
+  A pile of wood in the form of a cube contains 15-1/2 cords. What are
+  the dimensions to the nearest inch?
+
+  A man 6 ft. high weighs 175 lb. How tall is his wife who is of similar
+  build, and weighs 125 lb.?
+
+  A stick of timber is in the shape of the frustum of a square pyramid,
+  the lower base being 22 in. square and the upper 14 in. square. How
+  many cubic feet in the log, if it is 22 ft. long?
+
+  Mr. Ames, being asked his age, replied: "If you cube one half of my
+  age and add 41,472 to the result, the sum will be one half the cube of
+  my age. How old am I?"
+
+  [1] The following and later problems are taken from actual textbooks
+      or courses of study or state examinations; to avoid invidious
+      comparisons, they are not exact quotations, but are equivalents
+      in principle and form, as stated in the preface.
+
+These samples, just given, of the kind of problem-solving that should
+not be emphasized in school training refer in some cases to books of
+forty years back, but the following represent the results of a
+collection made in 1910 from books then in excellent repute. It required
+only about an hour to collect them; and I am confident that a thousand
+such problems describing situations that the pupil will never encounter
+in real life, or putting questions that he will never be asked in real
+life, could easily be found in any ten textbooks of the decade from 1900
+to 1910.
+
+  If there are 250 kernels of corn on one ear, how many are there on 24
+  ears of corn the same size?
+
+  Maud is four times as old as her sister, who is 4 years old. What is
+  the sum of their ages?
+
+  If the first century began with the year 1, with what year does it
+  end?
+
+  Every spider has 8 compound eyes. How many eyes have 21 spiders?
+
+  A nail 4 inches long is driven through a board so that it projects
+  1.695 inches on one side and 1.428 on the other. How thick is the
+  board?
+
+  Find the perimeter of an envelope 5 in. by 3-1/4 in.
+
+  How many minutes in 5/9 of 9/4 of an hour?
+
+  Mrs. Knox is 3/4 as old as Mr. Knox, who is 48 years old. Their son
+  Edward is 4/9 as old as his mother. How old is Edward?
+
+  Suppose a pie to be exactly round and 10-1/2 miles in diameter. If it
+  were cut into 6 equal pieces, how long would the curved edge of each
+  piece be?
+
+  8-1/3% of a class of 36 boys were absent on a rainy day. 33-1/3% of
+  those present went out of the room to the school yard. How many were
+  left in the room?
+
+  Just after a ton of hay was weighed in market, a horse ate one pound
+  of it. What was the ratio of what he ate to what was left?
+
+  If a fan having 15 rays opens out so that the outer rays form a
+  straight line, how many degrees are there between any two adjacent
+  rays?
+
+  One half of the distance between St. Louis and New Orleans is 280
+  miles more than 1/10 of the distance; what is the distance between
+  these places?
+
+  If the pressure of the atmosphere is 14.7 lb. per square inch what is
+  the pressure on the top of a table 1-1/4 yd. long and 2/3 yd. wide?
+
+  13/28 of the total acreage of barley in 1900 was 100,000 acres; what
+  was the total acreage?
+
+  What is the least number of bananas that a mother can exactly divide
+  between her 2 sons, or among her 4 daughters, or among all her
+  children?
+
+  If Alice were two years older than four times her actual age she would
+  be as old as her aunt, who is 38 years old. How old is Alice?
+
+  Three men walk around a circular island, the circumference of which is
+  360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a
+  day. If they start together and walk in the same direction, how many
+  days will elapse before they will be together again?
+
+With only thirty or forty dollars a year to spend on a pupil's
+education, of which perhaps eight dollars are spent on improving his
+arithmetical abilities, the immediate guidance of his responses to real
+situations and personally initiated problems has to be supplemented
+largely by guidance of his responses to problems described in words,
+diagrams, pictures, and the like. Of these latter, words will be used
+most often. As a consequence the understanding of the words used in
+these descriptions becomes a part of the ability required in arithmetic.
+Such word knowledge is also required in so far as the problems to be
+solved in real life are at times described, as in advertisements,
+business letters, and the like.
+
+This is recognized by everybody in the case of words like _remainder_,
+_profit_, _loss_, _gain_, _interest_, _cubic capacity_, _gross_, _net_,
+and _discount_, but holds equally of _let_, _suppose_, _balance_,
+_average_, _total_, _borrowed_, _retained_, and many such semi-technical
+words, and may hold also of hundreds of other words unless the textbook
+and teacher are careful to use only words and sentence structures which
+daily life and the class work in English have made well known to the
+pupils. To apply arithmetic to a problem a pupil must understand what
+the problem is; problem-solving depends on problem-reading. In actual
+school practice training in problem-reading will be less and less
+necessary as we get rid of problems to be solved simply for the sake of
+solving them, unnecessarily unreal problems, and clumsy descriptions,
+but it will remain to some extent as an important joint task for the
+'arithmetic' and 'reading' of the elementary school.
+
+
+ARITHMETICAL REASONING
+
+The last respect in which the nature of arithmetical abilities requires
+definition concerns arithmetical reasoning. An adequate treatment of the
+reasoning that may be expected of pupils in the elementary school and of
+the most efficient ways to encourage and improve it cannot be given
+until we have studied the formation of habits. For reasoning is
+essentially the organization and control of habits of thought. Certain
+matters may, however, be decided here. The first concerns the use of
+computation and problems merely for discipline,--that is, the emphasis
+on training in reasoning regardless of whether the problem is otherwise
+worth reasoning about. It used to be thought that the mind was a set of
+faculties or abilities or powers which grew strong and competent by
+being exercised in a certain way, no matter on what they were exercised.
+Problems that could not occur in life, and were entirely devoid of any
+worthy interest, save the intellectual interest in solving them, were
+supposed to be nearly or quite as useful in training the mind to reason
+as the genuine problems of the home, shop, or trade. Anything that gave
+the mind a chance to reason would do; and pupils labored to find when
+the minute hand and hour hand would be together, or how many sheep a
+shepherd had if half of what he had plus ten was one third of twice what
+he had!
+
+We now know that the training depends largely on the particular data
+used, so that efficient discipline in reasoning requires that the pupil
+reason about matters of real importance. There is no magic essence or
+faculty of reasoning that works in general and irrespective of the
+particular facts and relations reasoned about. So we should try to find
+problems which not only stimulate the pupil to reason, but also direct
+his reasoning in useful channels and reward it by results that are of
+real significance. We should replace the purely disciplinary problems by
+problems that are also valuable as special training for important
+particular situations of life. Reasoning sought for reasoning's sake
+alone is too wasteful an expenditure of time and is also likely to be
+inferior as reasoning.
+
+The second matter concerns the relative merits of 'catch' problems,
+where the pupil has to go against some customary habit of thinking, and
+what we may call 'routine' problems, where the regular ways of thinking
+that have served him in the past will, except for some blunder, guide
+him rightly.
+
+Consider, for example, these four problems:
+
+  1. "A man bought ten dozen eggs for $2.50 and sold them for 30 cents
+     a dozen. How many cents did he lose?"
+
+  2. "I went into Smith's store at 9 A.M. and remained until 10 A.M.
+     I bought six yards of gingham at 40 cents a yard and three yards
+     of muslin at 20 cents a yard and gave a $5.00 bill. How long was
+     I in the store?"
+
+  3. "What must you divide 48 by to get half of twice 6?"
+
+  4. "What must you add to 19 to get 30?"
+
+The 'catch' problem is now in disrepute, the wise teacher feeling by a
+sort of intuition that to willfully require a pupil to reason to a
+result sharply contrary to that to which previous habits lead him is
+risky. The four illustrations just given show, however, that mere
+'catchiness' or 'contra-previous-habit-ness' in a problem is not enough
+to condemn it. The fourth problem is a catch problem, but so useful a
+one that it has been adopted in many modern books as a routine drill!
+The first problem, on the contrary, all, save those who demand no higher
+criterion for a problem than that it make the pupil 'think,' would
+reject. It demands the reversal of fixed habits _to no valid purpose_;
+for in life the question in such case would never (or almost never) be
+'How many cents did he lose?' but 'What was the result?' or simply 'What
+of it?' This problem weakens without excuse the child's confidence in
+the training he has had. Problems like (2) are given by teachers of
+excellent reputation, but probably do more harm than good. If a pupil
+should interrupt his teacher during the recitation in arithmetic by
+saying, "I got up at 7 o'clock to multiply 9 by 2-3/4 and got 24-3/4 for
+my answer; was that the right time to get up?" the teacher would not
+thank fortune for the stimulus to thought but would think the child a
+fool. Such catch questions may be fairly useful as an object lesson on
+the value of search for the essential element in a situation if a great
+variety of them are given one after another with routine problems
+intermixed and with warning of the general nature of the exercise at the
+beginning. Even so, it should be remembered that reasoning should be
+chiefly a force organizing habits, not opposing them; and also that
+there are enough bad habits to be opposed to give all necessary
+training. Fabricated puzzle situations wherein a peculiar hidden element
+of the situation makes the good habits called up by the situation
+misleading are useful therefore rather as a relief and amusing variation
+in arithmetical work than as stimuli to thought.
+
+Problems like the third quoted above we might call puzzling rather than
+'catch' problems. They have value as drills in analysis of a situation
+into its elements that will amuse the gifted children, and as tests of
+certain abilities. They also require that of many confusing habits, the
+right one be chosen, rather than that ordinary habits be set aside by
+some hidden element in the situation. Not enough is known about their
+effect to enable us to decide whether or not the elementary school
+should include special facility with them as one of the arithmetical
+functions that it specially trains.
+
+The fourth 'catch' quoted above, which all would admit is a good
+problem, is good because it opposes a good habit for the sake of another
+good habit, forces the analysis of an element whose analysis life very
+much requires, and does it with no obvious waste. It is not safe to
+leave a child with the one habit of responding to 'add, 19, 30' by 49,
+for in life the 'have 19, must get .... to have 30' situation is very
+frequent and important.
+
+On the whole, the ordinary problems which ordinary life proffers seem to
+be the sort that should be reasoned out, though the elementary school
+may include the less noxious forms of pure mental gymnastics for those
+pupils who like them.
+
+
+SUMMARY
+
+These discussions of the meanings of numbers, the linguistic demands of
+arithmetic, the distinction between scholastic and real applications of
+arithmetic, and the possible restrictions of training in reasoning,--may
+serve as illustrations of the significance of the question, "What are
+the functions that the elementary school tries to improve in its
+teaching of arithmetic?" Other matters might well be considered in this
+connection, but the main outline of the work of the elementary school is
+now fairly clear. The arithmetical functions or abilities which it seeks
+to improve are, we may say:--
+
+(1) Working knowledge of the meanings of numbers as names for certain
+sized collections, for certain relative magnitudes, the magnitude of
+unity being known, and for certain centers or nuclei of relations to
+other numbers.
+
+(2) Working knowledge of the system of decimal notation.
+
+(3) Working knowledge of the meanings of addition, subtraction,
+multiplication, and division.
+
+(4) Working knowledge of the nature and relations of certain common
+measures.
+
+(5) Working ability to add, subtract, multiply, and divide with
+integers, common and decimal fractions, and denominate numbers, all
+being real positive numbers.
+
+(6) Working knowledge of words, symbols, diagrams, and the like as
+required by life's simpler arithmetical demands or by economical
+preparation therefor.
+
+(7) The ability to apply all the above as required by life's simpler
+arithmetical demands or by economical preparation therefor, including
+(7 _a_) certain specific abilities to solve problems concerning areas
+of rectangles, volumes of rectangular solids, percents, interest, and
+certain other common occurrences in household, factory, and business
+life.
+
+
+THE SOCIOLOGY OF ARITHMETIC
+
+  The phrase 'life's simpler arithmetical demands' is necessarily left
+  vague. Just what use is being made of arithmetic in this country in
+  1920 by each person therein, we know only very roughly. What may be
+  called a 'sociology' of arithmetic is very much needed to investigate
+  this matter. For rare or difficult demands the elementary school
+  should not prepare; there are too many other desirable abilities that
+  it should improve.
+
+  A most interesting beginning at such an inventory of the actual uses
+  of arithmetic has been made by Wilson ['19] and Mitchell.[2] Although
+  their studies need to be much extended and checked by other methods of
+  inquiry, two main facts seem fairly certain.
+
+  First, the great majority of people in the great majority of their
+  doings use only very elementary arithmetical processes. In 1737 cases
+  of addition reported by Wilson, seven eighths were of five numbers or
+  less. Over half of the multipliers reported were one-figure numbers.
+  Over 95 per cent of the fractions operated with were included in this
+  list: 1/2 1/4 3/4 1/3 2/3 1/8 3/8 1/5 2/5 4/5. Three fourths of all
+  the cases reported were simple one-step computations with integers or
+  United States money.
+
+  Second, they often use these very elementary processes, not because
+  such are the quickest and most convenient, but because they have lost,
+  or maybe never had, mastery of the more advanced processes which would
+  do the work better. The 5 and 10 cent stores, the counter with
+  "Anything on this counter for 25�," and the arrangements for payments
+  on the installment plan are familiar instances of human avoidance of
+  arithmetic. Wilson found very slight use of decimals; and Mitchell
+  found men computing with 49ths as common fractions when the use of
+  decimals would have been more efficient. If given 120 seconds to
+  do a test like that shown below, leading lawyers, physicians,
+  manufacturers, and business men and their wives will, according to my
+  experience, get only about half the work right. Many women, finding on
+  their meat bill "7-3/8 lb. roast beef $2.36," will spend time and
+  money to telephone the butcher asking how much roast beef was per
+  pound, because they have no sure power in dividing by a mixed number.
+
+  [2] The work of Mitchell has not been published, but the author has
+      had the privilege of examining it.
+
+  Test
+
+  Perform the operations indicated. Express all fractions in answers in
+  lowest terms.
+
+      _Add:_
+
+      3/4 + 1/6 + .25       4 yr. 6 mo.
+                            1 yr. 2 mo.
+                            6 yr. 9 mo.
+                            3 yr. 6 mo.
+                            4 yr. 5 mo.
+                            -----------
+
+      _Subtract:_
+
+      8.6 - 6.05007     7/8 - 2/3 =    5-7/16 - 2-3/16 =
+
+
+      _Multiply:_
+
+      29 ft. 6 in.     7 � 8 � 4-1/2 =
+             8
+      ------------
+
+    _Divide:_
+
+    4-1/2 � 7 =
+
+  It seems probable that the school training in arithmetic of the past
+  has not given enough attention to perfecting the more elementary
+  abilities. And we shall later find further evidence of this. On the
+  other hand, the fact that people in general do not at present use a
+  process may not mean that they ought not to use it.
+
+  Life's simpler arithmetical demands certainly do not include matters
+  like the rules for finding cube root or true discount, which no
+  sensible person uses. They should not include matters like computing
+  the lateral surface or volume of pyramids and cones, or knowing the
+  customs of plasterers and paper hangers, which are used only by highly
+  specialized trades. They should not include matters like interest on
+  call loans, usury, exact interest, and the rediscounting of notes,
+  which concern only brokers, bank clerks, and rich men. They should not
+  include the technique of customs which are vanishing from efficient
+  practice, such as simple interest on amount for times longer than a
+  year, days of grace, or extremes and means in proportions. They should
+  not include any elaborate practice with very large numbers, or
+  decimals beyond thousandths, or the addition and subtraction of
+  fractions which not one person in a hundred has to add or subtract
+  oftener than once a year.
+
+  When we have an adequate sociology of arithmetic, stating accurately
+  just who should use each arithmetical ability and how often, we shall
+  be able to define the task of the elementary school in this respect.
+  For the present, we may proceed by common sense, guided by two
+  limiting rules. The first is,--"It is no more desirable for the
+  elementary school to teach all the facts of arithmetic than to teach
+  all the words in the English language, or all the topography of the
+  globe, or all the details of human physiology." The second is,--"It is
+  not desirable to eliminate any element of arithmetical training until
+  you have something better to put in its place."
+
+
+
+
+CHAPTER II
+
+THE MEASUREMENT OF ARITHMETICAL ABILITIES
+
+
+One of the best ways to clear up notions of what the functions are which
+schools should develop and improve is to get measures of them. If any
+given knowledge or skill or power or ideal exists, it exists in some
+amount. A series of amounts of it, varying from less to more, defines
+the ability itself in a way that no general verbal description can do.
+Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to
+tell us what we mean by weight. By finding a series of words like
+_only_, _smoke_, _another_, _pretty_, _answer_, _tailor_, _circus_,
+_telephone_, _saucy_, and _beginning_, which are spelled correctly by
+known and decreasing percentages of children of the same age, or of the
+same school grade, we know better what we mean by 'spelling-difficulty.'
+Indeed, until we can measure the efficiency and improvement of a
+function, we are likely to be vague and loose in our ideas of what the
+function is.
+
+
+A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: THE ABILITY TO ADD
+INTEGERS
+
+Consider first, as a sample, the measurement of ability to add integers.
+
+The following were the examples used in the measurements made by Stone
+['08]:
+
+           596     4695
+           428      872
+  2375      94     7948
+  4052      75     6786
+  6354     304      567
+   260     645      858
+  5041     984     9447
+  1543     897     7499
+  ----     ---     ----
+
+The scoring was as follows: Credit of 1 for each column added correctly.
+Stone combined measures of other abilities with this in a total score
+for amount done correctly in 12 minutes. Stone also scored the
+correctness of the additions in certain work in multiplication.
+
+Courtis uses a sheet of twenty-four tasks or 'examples,' each consisting
+of the addition of nine three-place numbers as shown below. Eight
+minutes is allowed. He scores the amount done by the number of examples,
+and also scores the number of examples done correctly, but does not
+suggest any combination of these two into a general-efficiency score.
+
+  927
+  379
+  756
+  837
+  924
+  110
+  854
+  965
+  344
+  ---
+
+The author long ago proposed that pupils be measured also with series
+like _a_ to _g_ shown below, in which the difficulty increases step by
+step.
+
+  _a._           3  2  2  3  2  2  1  2
+                 2  3  1  2  4  5  5  1
+                 4  2  3  3  3  2  2  2
+                 -  -  -  -  -  -  -  -
+
+  _b._    21    32    12    24    34    34    22    12
+          23    12    52    31    33    12    23    13
+          24    25    15    14    32    23    43    61
+          --    --    --    --    --    --    --    --
+
+  _c._    22     3     4    35    32    83    22     3
+           3    31     3     2    33    11     3    21
+          38    45    52    52     2     4    33    64
+          --    --    --    --    --    --    --    --
+
+  _d._    30    20    10    22    10    20    52    12
+          20    50    40    43    30     4     6    22
+          40    17    24    13    40    23    30    44
+          --    --    --    --    --    --    --    --
+
+  _e._           4     5    20    12    12    20    10
+          20    30     3    40     4    11    20    20
+          10    30    20     4     1    23     7     2
+          20     2    40    23    40    11    10    30
+          20    20    10    11    20    22    30    25
+          --    --    --    --    --    --    --    --
+
+  _f._                19     9                 9
+          14     2    19    24     9     4    13
+           9    14    13    12    13    13     9    14
+          17    23    13    15    15    34    12    25
+          26    29    18    19    25    28    18    39
+          --    --    --    --    --    --    --    --
+
+  _g._                                  13
+                13           9    14    12     9
+                 9          13    12     9    14    24
+          23    19    19    29     9     9    13    21
+          28    26    26    14     8     8    29    23
+          29    16    15    19    17    19    19    22
+          --    --    --    --    --    --    --    --
+
+Woody ['16] has constructed his well-known tests on this principle,
+though he uses only one example at each step of difficulty instead of
+eight or ten as suggested above. His test, so far as addition of
+integers goes, is:--
+
+SERIES A. ADDITION SCALE (in part)
+
+By Clifford Woody
+
+  (1)   (2)    (3)    (4)   (5)   (6)   (7)       (8)           (9)
+   2     2     17     53    72    60   3 + 1 =   2 + 5 + 1 =    20
+   3     4      2     45    26    37                            10
+  --     3     --     --    --    --                             2
+        --                                                      30
+                                                                25
+                                                                --
+
+  (10) (11)   (12)   (13)    (14)        (15)   (16)  (17)    (18)
+   21   32     43     23    25 + 42 =    100      9   199     2563
+   33   59      1     25                  33     24   194     1387
+   35   17      2     16                  45     12   295     4954
+   --   --     13     --                 201     15   156     2065
+               --                         46     19   ---     ----
+                                         ---     --
+
+            (19)        (20)            (21)       (22)
+           $ .75       $12.50          $8.00        547
+            1.25        16.75           5.75        197
+             .49        15.75           2.33        685
+           -----       ------           4.16        678
+                                         .94        456
+                                        6.32        393
+                                       -----        525
+                                                    240
+                                                    152
+                                                    ---
+
+In his original report, Woody gives no scheme for scoring an individual,
+wisely assuming that, with so few samples at each degree of difficulty,
+a pupil's score would be too unreliable for individual diagnosis. The
+test is reliable for a class; and for a class Woody used the degree of
+difficulty such that a stated fraction of the class can do the work
+correctly, if twenty minutes is allowed for the thirty-eight examples of
+the entire test.
+
+The measurement of even so simple a matter as the efficiency of a
+pupil's responses to these tests in adding integers is really rather
+complex. There is first of all the problem of combining speed and
+accuracy into some single estimate. Stone gives no credit for a column
+unless it is correctly added. Courtis evades the difficulty by reporting
+both number done and number correct. The author's scheme, which gives
+specified weights to speed and accuracy at each step of the series,
+involves a rather intricate computation.
+
+This difficulty of equating speed and accuracy in adding means precisely
+that we have inadequate notions of what the ability is that the
+elementary school should improve. Until, for example, we have decided
+whether, for a given group of pupils, fifteen Courtis attempts with ten
+right, is or is not a better achievement than ten Courtis attempts with
+nine right, we have not decided just what the business of the teacher of
+addition is, in the case of that group of pupils.
+
+There is also the difficulty of comparing results when short and long
+columns are used. Correctness with a short column, say of five figures,
+testifies to knowledge of the process and to the power to do four
+successive single additions without error. Correctness with a long
+column, say of ten digits, testifies to knowledge of the process and to
+the power to do nine successive single additions without error. Now if a
+pupil's precision was such that on the average he made one mistake in
+eight single additions, he would get about half of his five-digit
+columns right and almost none of his ten-digit columns right. (He would
+do this, that is, if he added in the customary way. If he were taught to
+check results by repeated addition, by adding in half-columns and the
+like, his percentages of accurate answers might be greatly increased in
+both cases and be made approximately equal.) Length of column in a test
+of addition under ordinary conditions thus automatically overweights
+precision in the single additions as compared with knowledge of the
+process, and ability at carrying.
+
+Further, in the case of a column of whatever size, the result as
+ordinarily scored does not distinguish between one, two, three, or more
+(up to the limit) errors in the single additions. Yet, obviously, a
+pupil who, adding with ten-digit columns, has half of his answer-figures
+wrong, probably often makes two or more errors within a column, whereas
+a pupil who has only one column-answer in ten wrong, probably almost
+never makes more than one error within a column. A short-column test is
+then advisable as a means of interpreting the results of a long-column
+test.
+
+Finally, the choice of a short-column or of a long-column test is
+indicative of the measurer's notion of the kind of efficiency the world
+properly demands of the school. Twenty years ago the author would have
+been readier to accept a long-column test than he now is. In the world
+at large, long-column addition is being more and more done by machine,
+though it persists still in great frequency in the bookkeeping of weekly
+and monthly accounts in local groceries, butcher shops, and the like.
+
+The search for a measure of ability to add thus puts the problem of
+speed _versus_ precision, and of short-column _versus_ long-column
+additions clearly before us. The latter problem has hardly been
+realized at all by the ordinary definitions of ability to add.
+
+It may be said further that the measurement of ability to add gives the
+scientific student a shock by the lack of precision found everywhere in
+schools. Of what value is it to a graduate of the elementary school to
+be able to add with examples like those of the Courtis test, getting
+only eight out of ten right? Nobody would pay a computer for that
+ability. The pupil could not keep his own accounts with it. The supposed
+disciplinary value of habits of precision runs the risk of turning
+negative in such a case. It appears, at least to the author, imperative
+that checking should be taught and required until a pupil can add single
+columns of ten digits with not over one wrong answer in twenty columns.
+Speed is useful, especially indirectly as an indication of control of
+the separate higher-decade additions, but the social demand for addition
+below a certain standard of precision is _nil_, and its disciplinary
+value is _nil_ or negative. This will be made a matter of further study
+later.
+
+
+MEASUREMENTS OF ABILITIES IN COMPUTATION
+
+Measurements of these abilities may be of two sorts--(1) of the speed
+and accuracy shown in doing one same sort of task, as illustrated by the
+Courtis test for addition shown on page 28; and (2) of how hard a task
+can be done perfectly (or with some specified precision) within a
+certain assigned time or less, as illustrated by the author's rough test
+for addition shown on pages 28 and 29, and by the Woody tests, when
+extended to include alternative forms.
+
+The Courtis tests, originated as an improvement on the Stone tests and
+since elaborated by the persistent devotion of their author, are a
+standard instrument of the first sort for measuring the so-called
+'fundamental' arithmetical abilities with integers. They are shown on
+this and the following page.
+
+Tests of the second sort are the Woody tests, which include operations
+with integers, common and decimal fractions, and denominate numbers, the
+Ballou test for common fractions ['16], and the "Ladder" exercises of
+the Thorndike arithmetics. Some of these are shown on pages 36 to 41.
+
+
+Courtis Test
+
+Arithmetic. Test No. 1. Addition
+
+Series B
+
+  You will be given eight minutes to find the answers to as many
+  of these addition examples as possible. Write the answers on this
+  paper directly underneath the examples. You are not expected
+  to be able to do them all. You will be marked for both speed and
+  accuracy, but it is more important to have your answers right than
+  to try a great many examples.
+
+    927    297    136    486    384    176    277    837
+    379    925    340    765    477    783    445    882
+    756    473    988    524    881    697    682    959
+    837    983    386    140    266    200    594    603
+    924    315    353    812    679    366    481    118
+    110    661    904    466    241    851    778    781
+    854    794    547    355    796    535    849    756
+    965    177    192    834    850    323    157    222
+    344    124    439    567    733    229    953    525
+    ---    ---    ---    ---    ---    ---    ---    ---
+
+  and sixteen more addition examples of nine three-place numbers.
+
+
+Courtis Test
+
+Arithmetic. Test No. 2. Subtraction
+
+Series B
+
+  You will be given four minutes to find the answers to as many
+  of these subtraction examples as possible. Write the answers
+  on this paper directly underneath the examples. You are not
+  expected to be able to do them all. You will be marked for both
+  speed and accuracy, but it is more important to have your answers
+  right than to try a great many examples.
+
+    107795491      75088824      91500053      87939983
+     77197029      57406394      19901563      72207316
+    ---------      --------      --------      --------
+
+  and twenty more tasks of the same sort.
+
+
+Courtis Test
+
+Arithmetic. Test No. 3. Multiplication
+
+Series B
+
+  You will be given six minutes to work as many of these multiplication
+  examples as possible. You are not expected to be able to do them all.
+  Do your work directly on this paper; use no other. You will be marked
+  for both speed and accuracy, but it is more important to get correct
+  answers than to try a large number of examples.
+
+    8246      7843      4837      3478      6482
+      29       702        83        15        46
+    ----      ----      ----      ----      ----
+
+  and twenty more multiplication examples of the same sort.
+
+
+Courtis Test
+
+Arithmetic. Test No. 4. Division
+
+Series B
+
+  You will be given eight minutes to work as many of these division
+  examples as possible. You are not expected to be able to do them all.
+  Do your work directly on this paper; use no other. You will be marked
+  for both speed and accuracy, but it is more important to get correct
+  answers than to try a large number of examples.
+      _____        ______        _____        ______
+    25)6775      94)85352      37)9990      86)80066
+
+  and twenty more division examples of the same sort.
+
+
+SERIES B. MULTIPLICATION SCALE
+
+By Clifford Woody
+
+   (1)               (3)               (4)               (5)
+   3 � 7 =           2 � 3 =           4 � 8 =            23
+                                                           3
+                                                          --
+
+   (8)               (9)              (11)              (12)
+    50               254              1036              5096
+     3                 6                 8                 6
+    --               ---              ----              ----
+
+  (13)              (16)              (18)              (20)
+  8754              7898                24               287
+     8                 9               234               .05
+  ----              ----               ---               ---
+
+  (24)              (26)              (27)              (29)
+    16              9742              6.25             1/8 � 2 =
+     2-5/8            59               3.2
+    ---             ----              ----
+
+  (33)                (35)              (37)               (38)
+  2-1/2 � 3-1/2 =     987-3/4    2-1/4 � 4-1/2 � 1-1/2 =   .0963-1/8
+                       25                                  .084
+                     ----                                  -----
+
+
+SERIES B. DIVISION  SCALE
+
+By Clifford Woody
+
+  (1)               (2)              (7)               (8)
+   __               ___                                 ___
+  3)6              9)27             4 � 2 =            9)0
+
+  (11)              (14)              (15)              (17)
+   ___              _____
+  2)13             8)5856          1/4 of 128 =        50 � 7 =
+
+  (19)              (23)              (27)              (28)
+                    ____                                 ______
+  248 � 7 =       23)469          7/8 of 624 =       .003).0936
+
+      (30)                (34)                (36)
+                                            ______________
+   3/4 � 5 =         62.50 � 1-1/4 =       9)69 lbs. 9 oz.
+
+
+Ballou Test
+
+Addition of Fractions
+
+            _Test 1_                                _Test 2_
+  (1)  1/4            (2) 3/14             (1) 1/3            (2) 2/7
+       1/4                1/14                 1/6                3/14
+       ---                ----                 ---                ----
+
+
+           _Test 3_                                 _Test 4_
+  (1)  3/5            (2) 5/6              (1) 1/7            (2) 7/9
+      11/15               1/2                  9/10               1/4
+      -----               ---                  ----               ---
+
+
+           _Test 5_                                 _Test 6_
+  (1)  1/10           (2) 4/9              (1) 1/6            (2) 5/6
+       1/6                5/12                 9/10               3/8
+       ----               ----                 ----               ---
+
+
+An Addition Ladder [Thorndike, '17, III, 5]
+
+Begin at the bottom of the ladder. See if you can climb to the top
+without making a mistake. Be sure to copy the numbers correctly.
+
+  #Step 6.#
+    _a._ Add 1-1/3 yd., 7/8 yd., 1-1/4 yd., 3/4 yd., 7/8 yd.,
+          and 1-1/2 yd.
+    _b._ Add 62-1/2�, 66-2/3�, 56-1/4�, 60�, and 62-1/2�.
+    _c._ Add 1-5/16, 1-9/32, 1-3/8, 1-11/32, and 1-7/16.
+    _d._ Add 1-1/3 yd., 1-1/4 yd., 1-1/2 yd., 2 yd., 3/4 yd.,
+          and 2/3 yd.
+
+  #Step 5.#
+    _a._ Add 4 ft. 6-1/2 in., 53-1/4 in., 5 ft. 1/2 in., 56-3/4 in.,
+          and 5 ft.
+    _b._ Add 7 lb., 6 lb. 11 oz., 7-1/2 lb., 6 lb. 4-1/2 oz.,
+          and 8-1/2 lb.
+    _c._ Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min.,
+          and 55 min.
+    _d._ Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes,
+          and 19 nickels.
+
+  #Step 4.#
+    _a._ Add .05-1/2, .06, .04-3/4, .02-3/4, and .05-1/4.
+    _b._ Add .33-1/3, .12-1/2, .18, .16-2/3, .08-1/3 and .15.
+    _c._ Add .08-1/3, .06-1/4, .21, .03-3/4, and .16-2/3.
+    _d._ Add .62, .64-1/2, .66-2/3, .10-1/4, and .68.
+
+  #Step 3.#
+    _a._ Add 7-1/4, 6-1/2, 8-3/8, 5-3/4, 9-5/8 and 3-7/8.
+    _b._ Add 4-5/8, 12, 7-1/2, 8-3/4, 6 and 5-1/4.
+    _c._ Add 9-3/4, 5-7/8, 4-1/8, 6-1/2, 7, 3-5/8.
+    _d._ Add 12, 8-1/2, 7-1/3, 5, 6-2/3, and 9-1/2.
+
+  #Step 2.#
+    _a._ Add 12.04, .96, 4.7, 9.625, 3.25, and 20.
+    _b._ Add .58, 6.03, .079, 4.206, 2.75, and 10.4.
+    _c._ Add 52, 29.8, 41.07, 1.913, 2.6, and 110.
+    _d._ Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05.
+
+  #Step 1.#
+    _a._ Add 10-3/5, 11-1/5, 10-4/5, 11, 11-2/5, 10-3/5, and 11.
+    _b._ Add 7-3/8, 6-5/8, 8, 9-1/8, 7-7/8, 5-3/8, and 8-1/8.
+    _c._ Add 21-1/2, 18-3/4, 31-1/2, 19-1/4, 17-1/4, 22, and 16-1/2.
+    _d._ Add 14-5/12, 12-7/12, 9-11/12, 6-1/12, and 5.
+
+
+A Subtraction Ladder [Thorndike, '17, III, 11]
+
+  #Step 9.#
+    _a._  2.16 mi. - 1-3/4 mi.
+    _b._  5.72 ft. - 5 ft. 3 in.
+    _c._  2 min. 10-1/2 sec. - 93.4 sec.
+    _d._  30.28 A. - 10-1/5 A.
+    _e._  10 gal. 2-1/2 qt. - 4.623 gal.
+
+  #Step 8.#
+     _a_             _b_            _c_            _d_            _e_
+    25-7/12         10-1/4          9-5/16         5-7/16         4-2/3
+    12-3/4           7-1/3          6-3/8          2-3/4          1-3/4
+    -------         ------          ------         ------         -----
+
+  #Step 7.#
+     _a_             _b_            _c_            _d_            _e_
+    28-3/4          40-1/2         10-1/4         24-1/3         37-1/2
+    16-1/8          14-3/8          6-1/2         11-1/2         14-3/4
+    ------          ------         ------         ------         ------
+
+  #Step 6.#
+     _a_             _b_            _c_            _d_            _e_
+    10-1/3           7-1/4         15-1/8         12-1/5          4-1/16
+     4-2/3           2-3/4          6-3/8         11-4/5          2-7/16
+    ------           -----         ------         ------          ------
+
+  #Step 5.#
+     _a_             _b_            _c_            _d_            _e_
+    58-4/5          66-2/3         28-7/8         62-1/2          9-7/12
+    52-1/5          33-1/3          7-5/8         37-1/2          4-5/12
+    ------          ------         ------         ------          ------
+
+  #Step 4.#
+    _a._  4 hr. - 2 hr. 17 min.
+    _b._  4 lb. 7 oz. - 2 lb. 11 oz.
+    _c._  1 lb. 5 oz. - 13 oz.
+    _d._  7 ft. - 2 ft. 8 in.
+    _e._  1 bu. - 1 pk.
+
+  #Step 3.#
+      _a_             _b_         _c_        _d_            _e_
+    92    mi.       6735 mi.    $3 - 89�    28.4  mi.     $508.40
+    84.15 mi.       6689 mi.                18.04 mi.      208.62
+    ---------       --------    --------    ---------     -------
+
+  #Step 2.#
+      _a_        _b_        _c_        _d_            _e_
+    $25.00     $100.00    $750.00   6124 sq. mi.   7846 sq. mi.
+      9.36       71.28     736.50   2494 sq. mi.   2789 sq. mi.
+    ------     -------    -------   ------------   ------------
+
+  #Step 1.#
+      _a_        _b_         _c_        _d_         _e_
+    $18.64     $25.39      $56.70     819.4 mi.    67.55 mi.
+      7.40      13.37       45.60     209.2 mi.    36.14 mi.
+    ------     ------      ------     ---------    ---------
+
+
+An Average Ladder [Thorndike, '17, III, 132]
+
+Find the average of the quantities on each line. Begin with #Step 1#.
+Climb to the top without making a mistake. Be sure to copy the numbers
+correctly. Extend the division to two decimal places if necessary.
+
+  #Step 6.#
+    _a_. 2-2/3, 1-7/8, 2-3/4, 4-1/4, 3-5/8, 3-1/2
+    _b_. 62-1/2�, 66-2/3�, 40�, 83-1/3�, $1.75, $2.25
+    _c_. 3-11/16, 3-9/32, 3-3/8, 3-17/32, 3-7/16
+    _d_. .17, 19, .16-2/3, .15-1/2, .23-1/4, .18
+
+  #Step 5.#
+    _a_. 5 ft. 3-1/2 in., 61-1/4 in., 58-3/4 in., 4 ft. 11 in.
+    _b_. 6 lb. 9 oz., 6 lb. 11 oz., 7-1/4 lb., 7-3/8 lb.
+    _c_. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1-1/4 hr.
+    _d_. 2.8 miles, 3-1/2 miles, 2.72 miles
+
+  #Step 4.#
+    _a._ .03-1/2, .06, .04-3/4, .05-1/2, .05-1/4
+    _b._ .043, .045, .049, .047, .046, .045
+    _c._ 2.20, .87-1/2, 1.18, .93-3/4, 1.2925, .80
+    _d._ .14-1/2, .12-1/2, .33-1/3, .16-2/3, .15, .17
+
+  #Step 3.#
+    _a._ 5-1/4, 4-1/2, 8-3/8, 7-3/4, 6-5/8, 9-3/8
+    _b._ 9-5/8, 12, 8-1/2, 8-3/4, 6, 5-1/4, 9
+    _c._ 9-3/8, 5-3/4, 4-1/8, 7-1/2, 6
+    _d._ 11, 9-1/2, 10-1/3, 13, 16-2/3, 9-1/2
+
+  #Step 2.#
+    _a._ 13.05, .97, 4.8, 10.625, 3.37
+    _b._ 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4
+    _c._ 68, 71.4, 59.8, 112, 96.1, 79.8
+    _d._ 2.079, 3.908, 4.165, 2.74
+
+  #Step 1.#
+    _a._ 4, 9-1/2, 6, 5, 7-1/2, 8, 10, 9
+    _b._ 6, 5, 3.9, 7.1, 8
+    _c._ 1086, 1141, 1059, 1302, 1284
+    _d._ $100.82, $206.49, $317.25, $244.73
+
+As such tests are widened to cover the whole task of the elementary
+school in respect to arithmetic, and accepted by competent authorities
+as adequate measures of achievement in computing, they will give, as has
+been said, a working definition of the task. The reader will observe,
+for example, that work such as the following, though still found in many
+textbooks and classrooms, does not, in general, appear in the modern
+tests and scales.
+
+Reduce the following improper fractions to mixed numbers:--
+
+          19/13    43/21   176/25   198/14
+
+Reduce to integral or mixed numbers:--
+
+          61381/37   2134/67   413/413   697/225
+
+Simplify:--
+
+          3/4 of 8/9 of 3/5 of 15/22
+
+Reduce to lowest terms:--
+
+  357/527  264/312  492/779  418/874  854/1769  30/735  44/242
+
+  77/847 18/243  96/224
+
+Find differences:--
+
+     6-2/7         8-5/11        8-4/13        5-1/4          7-1/8
+     3-1/14        5-1/7         3-7/13        2-11/14        2-1/7
+     ------        ------        ------        -------        ------
+
+Square:--
+
+  2/3  4/5  5/7  6/9  10/11  12/13  2/7  15/16  19/20  17/18
+
+  25/30  41/53
+
+Multiply:--
+
+     2/11 � 33    32 � 3/14    39 � 2/13    60 � 11/28    77 � 4/11
+
+     63 � 2/27    54 � 8/45    65 � 3/13     344-16/21   432-2/7
+
+
+MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE SOLUTION OF PROBLEMS
+
+Stone ['08] measured achievement with the following problems, fifteen
+minutes being the time allowed.
+
+"Solve as many of the following problems as you have time for; work them
+in order as numbered:
+
+  1. If you buy 2 tablets at 7 cents each and a book for 65 cents, how
+     much change should you receive from a two-dollar bill?
+
+  2. John sold 4 Saturday Evening Posts at 5 cents each. He kept 1/2
+     the money and with the other 1/2 he bought Sunday papers at 2 cents
+     each. How many did he buy?
+
+  3. If James had 4 times as much money as George, he would have $16.
+     How much money has George?
+
+  4. How many pencils can you buy for 50 cents at the rate of 2 for 5
+     cents?
+
+  5. The uniforms for a baseball nine cost $2.50 each. The shoes cost
+     $2 a pair. What was the total cost of uniforms and shoes for the
+     nine?
+
+  6. In the schools of a certain city there are 2200 pupils; 1/2 are
+     in the primary grades, 1/4 in the grammar grades, 1/8 in the high
+     school, and the rest in the night school. How many pupils are there
+     in the night school?
+
+  7. If 3-1/2 tons of coal cost $21, what will 5-1/2 tons cost?
+
+  8. A news dealer bought some magazines for $1. He sold them for
+     $1.20, gaining 5 cents on each magazine. How many magazines were
+     there?
+
+  9. A girl spent 1/8 of her money for car fare, and three times as
+     much for clothes. Half of what she had left was 80 cents. How much
+     money did she have at first?
+
+ 10. Two girls receive $2.10 for making buttonholes. One makes 42,
+     the other 28. How shall they divide the money?
+
+ 11. Mr. Brown paid one third of the cost of a building; Mr. Johnson
+     paid 1/2 the cost. Mr. Johnson received $500 more annual rent than
+     Mr. Brown. How much did each receive?
+
+ 12. A freight train left Albany for New York at 6 o'clock. An
+     express left on the same track at 8 o'clock. It went at the rate of
+     40 miles an hour. At what time of day will it overtake the freight
+     train if the freight train stops after it has gone 56 miles?"
+
+The criteria he had in mind in selecting the problems were as follows:--
+
+"The main purpose of the reasoning test is the determination of the
+ability of VI A children to reason in arithmetic. To this end, the
+problems, as selected and arranged, are meant to embody the following
+conditions:--
+
+  1. Situations equally concrete to all VI A children.
+
+  2. Graduated difficulties.
+      _a._ As to arithmetical thinking.
+      _b._ As to familiarity with the situation presented.
+
+  3. The omission of
+      _a._ Large numbers.
+      _b._ Particular memory requirements.
+      _c._ Catch problems.
+      _d._ All subject matter except whole numbers, fractions, and
+             United States money.
+
+The test is purposely so long that only very rarely did any pupil fully
+complete it in the fifteen minute limit."
+
+Credits were given of 1, for each of the first five problems, 1.4, 1.2,
+and 1.6 respectively for problems 6, 7, and 8, and of 2 for each of the
+others.
+
+Courtis sought to improve the Stone test of problem-solving, replacing
+it by the two tests reproduced below.
+
+
+ARITHMETIC--Test No. 6. Speed Test--Reasoning
+
+#Do not work# the following examples. Read each example through, make
+up your mind what operation you would use if you were going to work it,
+then write the name of the operation selected in the blank space after
+the example. Use the following abbreviations:--"Add." for addition,
+"Sub." for subtraction, "Mul." for multiplication, and "Div." for
+division.
+
+                                                      +-----------+----+
+                                                      | OPERATION |    |
+                                                      |-----------+----|
+  1. A girl brought a collection of 37 colored postal |           |    |
+  cards to school one day, and gave away 19 cards to  |           |    |
+  her friends. How many cards did she have left to    |           |    |
+  take home?                                          |           |    |
+                                                      |-----------+----|
+  2. Five boys played marbles. When the game was      |           |    |
+  over, each boy had the same number of marbles. If   |           |    |
+  there were 45 marbles altogether, how many did each |           |    |
+  boy have?                                           |           |    |
+                                                      |-----------+----|
+  3. A girl, watching from a window, saw 27           |           |    |
+  automobiles pass the school the first hour, and     |           |    |
+  33 the second. How many autos passed by the         |           |    |
+  school in the two hours?                            |           |    |
+                                                      |-----------+----|
+  4. In a certain school there were eight rooms and   |           |    |
+  each room had seats for 50 children. When all the   |           |    |
+  places were taken, how many children were there in  |           |    |
+  the school?                                         |           |    |
+                                                      |-----------+----|
+  5. A club of boys sent their treasurer to buy       |           |    |
+  baseballs. They gave him $3.15 to spend. How many   |           |    |
+  balls did they expect him to buy, if the balls cost |           |    |
+  45�. apiece?                                        |           |    |
+                                                      |-----------+----|
+  6. A teacher weighed all the girls in a certain     |           |    |
+  grade. If one girl weighed 79 pounds and another    |           |    |
+  110 pounds, how many pounds heavier was one girl    |           |    |
+  than the other?                                     |           |    |
+                                                      |-----------+----|
+  7. A girl wanted to buy a 5-pound box of candy to   |           |    |
+  give as a present to a friend. She decided to get   |           |    |
+  the kind worth 35�. a pound. What did she pay for   |           |    |
+  the present?                                        |           |    |
+                                                      |-----------+----|
+  8. One day in vacation a boy went on a fishing trip |           |    |
+  and caught 12 fish in the morning, and 7 in the     |           |    |
+  afternoon. How many fish did he catch altogether?   |           |    |
+                                                      |-----------+----|
+  9. A boy lived 15 blocks east of a school; his chum |           |    |
+  lived on the same street, but 11 blocks west of the |           |    |
+  school. How many blocks apart were the two boys'    |           |    |
+  houses?                                             |           |    |
+                                                      |-----------+----|
+  10. A girl was 5 times as strong as her small       |           |    |
+  sister. If the little girl could lift a weight of   |           |    |
+  20 pounds, how large a weight could the older girl  |           |    |
+  lift?                                               |           |    |
+                                                      |-----------+----|
+  11. The children of a school gave a sleigh-ride     |           |    |
+  party. There were 270 children to go on the ride    |           |    |
+  and each sleigh held 30 children. How many sleighs  |           |    |
+  were needed?                                        |           |    |
+                                                      |-----------+----|
+  12. In September there were 43 children in the      |           |    |
+  eighth grade of a certain school; by June there     |           |    |
+  were 59. How many children entered the grade        |           |    |
+  during the year?                                    |           |    |
+                                                      |-----------+----|
+  13. A girl who lived 17 blocks away walked to       |           |    |
+  school and back twice a day. What was the total     |           |    |
+  number of blocks the girl walked each day in        |           |    |
+  going to and from school?                           |           |    |
+                                                      |-----------+----|
+  14. A boy who made 67�. a day carrying papers, was  |           |    |
+  hired to run on a long errand for which he received |           |    |
+  50�. What was the total amount the boy earned that  |           |    |
+  day?                                                |           |    |
+                                                      |-----------+----|
+                                          Total Right |           |    |
+                                                      +-----------+----+
+
+(Two more similar problems follow.)
+
+Test 6 and Test 8 are from the Courtis Standard Test. Used by permission
+of S. A. Courtis.
+
+
+ARITHMETIC--Test No. 8. Reasoning
+
+In the blank space below, work as many of the following examples as
+possible in the time allowed. Work them in order as numbered, entering
+each answer in the "answer" column before commencing a new example. Do
+not work on any other paper.
+
+                                                            +--------+-+
+                                                            | ANSWER | |
+                                                            |--------+-|
+  1. The children in a certain school gave a Christmas      |        | |
+  party. One of the presents was a box of candy. In filling |        | |
+  the boxes, one grade used 16 pounds of candy, another 17  |        | |
+  pounds, a third 12 pounds, and a fourth 13 pounds. What   |        | |
+  did the candy cost at 26�. a pound?                       |        | |
+                                                            |--------+-|
+  2. A school in a certain city used 2516 pieces of chalk   |        | |
+  in 37 school days. Three new rooms were opened, each      |        | |
+  room holding 50 children, and the school was then found   |        | |
+  to use 84 sticks of chalk per day. How many more sticks   |        | |
+  of chalk were used per day than at first?                 |        | |
+                                                            |--------+-|
+  3. Several boys went on a bicycle trip of 1500 miles.     |        | |
+  The first week they rode 374 miles, the second week 264   |        | |
+  miles, the third 423 miles, the fourth 401 miles. They    |        | |
+  finished the trip the next week. How many miles did they  |        | |
+  ride the last week?                                       |        | |
+                                                            |--------+-|
+  4. Forty-five boys were hired to pick apples from 15      |        | |
+  trees in an apple orchard. In 50 minutes each boy had     |        | |
+  picked 48 choice apples. If all the apples picked were    |        | |
+  packed away carefully in 8 boxes of equal size, how many  |        | |
+  apples were put in each box?                              |        | |
+                                                            |--------+-|
+  5. In a certain school 216 children gave a sleigh-ride    |        | |
+  party. They rented 7 sleighs at a cost of $30.00 and paid |        | |
+  $24.00 for the refreshments. The party travelled 15 miles |        | |
+  in 2-1/2 hours and had a very pleasant time. What was     |        | |
+  each child's share of the expense?                        |        | |
+                                                            |--------+-|
+  6. A girl found, by careful counting, that there were     |        | |
+  2400 letters on one page of her history, and only 2295    |        | |
+  letters on a page of her reader. How many more letters    |        | |
+  had she read in one book than in the other if she had     |        | |
+  read 47 pages in each of the books?                       |        | |
+                                                            |--------+-|
+  7. Each of 59 rooms in the schools of a certain city      |        | |
+  contributed 25 presents to a Christmas entertainment for  |        | |
+  poor children. The stores of the city gave 1986 other     |        | |
+  articles for presents. What was the total number of       |        | |
+  presents given away at the entertainment?                 |        | |
+                                                            |--------+-|
+  8. Forty-eight children from a certain school paid 10�.   |        | |
+  apiece to ride 7 miles on the cars to a woods. There in a |        | |
+  few hours they gathered 2765 nuts. 605 of these were bad, |        | |
+  but the rest were shared equally among the children. How  |        | |
+  many good nuts did each one get?                          |        | |
+                                                            |--------+-|
+                                                      Total |        | |
+                                                            +--------+-+
+
+These proposed measures of ability to apply arithmetic illustrate very
+nicely the differences of opinion concerning what applied arithmetic and
+arithmetical reasoning should be. The thinker who emphasizes the fact
+that in life out of school the situation demanding quantitative
+treatment is usually real rather than described, will condemn a test all
+of whose constituents are _described_ problems. Unless we are
+excessively hopeful concerning the transfer of ideas of method and
+procedure from one mental function to another we shall protest against
+the artificiality of No. 3 of the Stone series, and of the entire
+Courtis Test 8 except No. 4. The Courtis speed-reasoning test (No. 6) is
+a striking example of the mixture of ability to understand quantitative
+relations with the ability to understand words. Consider these five, for
+example, in comparison with the revised versions attached.[3]
+
+  [3] The form of Test 6 quoted here is that given by Courtis ['11-'12,
+      p. 20]. This differs a little from the other series of Test 6,
+      shown on pages 43 and 44.
+
+  1. The children of a school gave a sleigh-ride party. There were 9
+  sleighs, and each sleigh held 30 children. How many children were
+  there in the party?
+
+  REVISION. _If one sleigh holds 30 children, 9 sleighs hold ....
+  children._
+
+  2. Two school-girls played a number-game. The score of the girl
+  that lost was 57 points and she was beaten by 16 points. What was
+  the score of the girl that won?
+
+  REVISION. _Mary and Nell played a game. Mary had a score of 57.
+  Nell beat Mary by 16. Nell had a score of ...._
+
+  3. A girl counted the automobiles that passed a school. The total
+  was 60 in two hours. If the girl saw 27 pass the first hour how
+  many did she see the second?
+
+  REVISION. _In two hours a girl saw 60 automobiles. She saw 27 the
+  first hour. She saw .... the second hour._
+
+  4. On a playground there were five equal groups of children each
+  playing a different game. If there were 75 children all together,
+  how many were there in each group?
+
+  REVISION. _75 pounds of salt just filled five boxes. The boxes were
+  exactly alike. There were .... pounds in a box._
+
+  5. A teacher weighed all the children in a certain grade. One girl
+  weighed 70 pounds. Her older sister was 49 pounds heavier. How many
+  pounds did the sister weigh?
+
+  REVISION. _Mary weighs 70 lb. Jane weighs 49 pounds more than Mary.
+  Jane weighs .... pounds._
+
+The distinction between a problem described as clearly and simply as
+possible and the same problem put awkwardly or in ill-known words or
+willfully obscured should be regarded; and as a rule measurements of
+ability to apply arithmetic should eschew all needless obscurity or
+purely linguistic difficulty. For example,
+
+  _A boy bought a two-cent stamp. He gave the man in the store 10
+  cents. The right change was .... cents._
+
+is better as a test than
+
+  _If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in
+  payment, what change should he be expected to receive in return?_
+
+The distinction between the description of a _bona fide_ problem that a
+human being might be called on to solve out of school and the
+description of imaginary possibilities or puzzles should also be
+considered. Nos. 3 and 9 of Stone are bad because to frame the problems
+one must first know the answers, so that in reality there could never be
+any point in solving them. It is probably safe to say that nobody in the
+world ever did or ever will or ever should find the number of apples in
+a box by the task of No. 4 of the Courtis Test 8.
+
+This attaches no blame to Dr. Stone or to Mr. Courtis. Until very
+recently we were all so used to the artificial problems of the
+traditional sort that we did not expect anything better; and so blind to
+the language demands of described problems that we did not see their
+very great influence. Courtis himself has been active in reform and has
+pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.
+
+"Tests Nos. 6 and 8, the so-called reasoning tests, have proved the
+least satisfactory of the series. The judgments of various teachers and
+superintendents as to the inequalities of the units in any one test, and
+of the differences between the different editions of the same test, have
+proved the need of investigating these questions. Tests of adults in
+many lines of commercial work have yielded in many cases lower scores
+than those of the average eighth grade children. At the same time the
+scores of certain individuals of marked ability have been high, and
+there appears to be a general relation between ability in these tests
+and accuracy in the abstract work. The most significant facts, however,
+have been the difficulties experienced by teachers in attempting to
+remedy the defects in reasoning. It is certain that the tests measure
+abilities of value but the abilities are probably not what they seem to
+be. In an attempt to measure the value of different units, for instance,
+as many problems as possible were constructed based upon a single
+situation. Twenty-one varieties were secured by varying the relative
+form of the question and the relative position of the different phrases.
+One of these proved nineteen times as hard as another as measured by the
+number of mistakes made by the children; yet the cause of the difference
+was merely the changes in the phrasing. This and other facts of the same
+kind seem to show that Tests 6 and 8 measure mainly the ability to
+read."
+
+The scientific measurement of the abilities and achievements concerned
+with applied arithmetic or problem-solving is thus a matter for the
+future. In the case of described problems a beginning has been made in
+the series which form a part of the National Intelligence Tests ['20],
+one of which is shown on page 49 f. In the case of problems with real
+situations, nothing in systematic form is yet available.
+
+Systematic tests and scales, besides defining the abilities we are to
+establish and improve, are of very great service in measuring the status
+and improvement of individuals and of classes, and the effects of
+various methods of instruction and of study. They are thus helpful to
+pupils, teachers, supervisors, and scientific investigators; and are
+being more and more widely used every year. Information concerning the
+merits of the different tests, the procedure to follow in giving and
+scoring them, the age and grade standards to be used in interpreting
+results, and the like, is available in the manuals of Educational
+Measurement, such as Courtis, _Manual of Instructions for Giving and
+Scoring the Courtis Standard Tests in the Three R's_ ['14]; Starch,
+_Educational Measurements_ ['16]; Chapman and Rush, _Scientific
+Measurement of Classroom Products_ ['17]; Monroe, DeVoss, and Kelly,
+_Educational Tests and Measurements_ ['17]; Wilson and Hoke, _How to
+Measure_ ['20]; and McCall, _How to Measure in Education_ ['21].
+
+TEST 1
+
+  National Intelligence Tests.
+  Scale A. Form 1, Edition 1
+
+  Find the answers as quickly as you can.
+  Write the answers on the dotted lines.
+  Use the side of the page to figure on.
+
+  #Begin here#
+
+  1  Five cents make 1 nickel. How many nickels make a
+     dime?                                                _Answer_ .....
+
+  2  John paid 5 dollars for a watch and 3 dollars for a chain.
+     How many dollars did he pay for the watch and chain? _Answer_ .....
+
+  3  Nell is 13 years old. Mary is 9 years old. How much
+     younger is Mary than Nell?                           _Answer_ .....
+
+  4  One quart of ice cream is enough for 5 persons. How
+     many quarts of ice cream are needed for 25 persons?  _Answer_ .....
+
+  5  John's grandmother is 86 years old. If she lives, in
+     how many years will she be 100 years old?            _Answer_ .....
+
+  6  If a man gets $2.50 a day, what will he be paid for six
+     days' work?                                          _Answer_ .....
+
+  7  How many inches are there in a foot and a half?      _Answer_ .....
+
+  8  What is the cost of 12 cakes at 6 for 5 cents?       _Answer_ .....
+
+  9  The uniforms for a baseball team of nine boys cost $2.50
+     each. The shoes cost $2 a pair. What was the total
+     cost of uniforms and shoes for the nine?             _Answer_ .....
+
+  10 A train that usually arrives at half-past ten was 17
+     minutes late. When did it arrive?                    _Answer_ .....
+
+  11 At 10� a yard, what is the cost of a piece 10-1/2 ft. long?
+                                                          _Answer_ .....
+
+  12 A man earns $6 a day half the time, $4.50 a day one
+     fourth of the time, and nothing on the remaining days
+     for a total period of 40 days. What did he earn in all
+     in the 40 days?                                      _Answer_ .....
+
+  13 What per cent of $800 is 4% of $1000?                _Answer_ .....
+
+  14 If 60 men need 1500 lb. flour per month, what is the
+     requirement per man per day counting a month as 30
+     days?                                                _Answer_ .....
+
+  15 A car goes at the rate of a mile a minute. A truck goes
+     20 miles an hour. How many times as far will the car
+     go as the truck in 10 seconds?                       _Answer_ .....
+
+  16 The area of the base (inside measure) of a cylindrical
+     tank is 90 square feet. How tall must it be to hold
+     100 cubic yards?                                     _Answer_ .....
+
+  From National Intelligence Tests by National Research Council.
+
+  Copyright, 1920, by The World Book Company, Yonkers-on-Hudson,
+  New York.
+
+  Used by permission of the publishers.
+
+
+
+
+CHAPTER III
+
+THE CONSTITUTION OF ARITHMETICAL ABILITIES
+
+
+THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING
+
+It would be a useful work for some one to try to analyze arithmetical
+learning into the unitary abilities which compose it, showing just what,
+in detail, the mind has to do in order to be prepared to pass a thorough
+test on the whole of arithmetic. These unitary abilities would make a
+very long list. Examination of a well-planned textbook will show that
+such an ability as multiplication is treated as a composite of the
+following: knowledge of the multiplications up to 9 � 9; ability to
+multiply two (or more)-place numbers by 2, 3, and 4 when 'carrying' is
+not required and no zeros occur in the multiplicand; ability to multiply
+by 2, 3, ... 9, with carrying; the ability to handle zeros in the
+multiplicand; the ability to multiply with two-place numbers not ending
+in zero; the ability to handle zero in the multiplier as last number;
+the ability to multiply with three (or more)-place numbers not including
+a zero; the ability to multiply with three- and four-place numbers with
+zero in second or third, or second and third, as well as in last place;
+the ability to save time by annexing zeros; and so on and on through a
+long list of further abilities required to multiply with United States
+money, decimal fractions, common fractions, mixed numbers, and
+denominate numbers.
+
+The units or 'steps' thus recognized by careful teaching would make a
+long list, but it is probable that a still more careful study of
+arithmetical ability as a hierarchy of mental habits or connections
+would greatly increase the list. Consider, for example, ordinary column
+addition. The majority of teachers probably treat this as a simple
+application of the knowledge of the additions to 9 + 9, plus
+understanding of 'carrying.' On the contrary there are at least seven
+processes or minor functions involved in two-place column addition, each
+of which is psychologically distinct and requires distinct educational
+treatment.
+
+These are:--
+
+  A. Learning to keep one's place in the column as one adds.
+
+  B. Learning to keep in mind the result of each addition until the
+        next number is added to it.
+
+  C. Learning to add a seen to a thought-of number.
+
+  D. Learning to neglect an empty space in the columns.
+
+  E. Learning to neglect 0s in the columns.
+
+  F. Learning the application of the combinations to higher decades
+        may for the less gifted pupils involve as much time and labor
+        as learning all the original addition tables. And even for
+        the most gifted child the formation of the connection
+        '8 and 7 = 15' probably never quite insures the presence
+        of the connections '38 and 7 = 45' and '18 + 7 = 25.'
+
+  G. Learning to write the figure signifying units rather than the
+        total sum of a column. In particular, learning to write 0 in
+        the cases where the sum of the column is 10, 20, etc. Learning
+        to 'carry' also involves in itself at least two distinct
+        processes, by whatever way it is taught.
+
+We find evidence of such specialization of functions in the results with
+such tests as Woody's. For example, 2 + 5 + 1 = .... surely involves
+abilities in part different from
+
+    2
+    4
+    3
+    -
+
+because only 77 percent of children in grade 3 do the former correctly,
+whereas 95 percent of children in that grade do the latter correctly. In
+grade 2 the difference is even more marked. In the case of subtraction
+
+    4
+    4
+    -
+
+involves abilities different from those involved in
+
+    9
+    3,
+    -
+
+being much less often solved correctly in grades 2 and 4.
+
+    6
+    0
+    -
+
+is much harder than either of the above.
+
+   43
+    1                       21
+    2                       33
+   13  is much harder than  35.
+   --                       --
+
+It may be said that these differences in difficulty are due to different
+amounts of practice. This is probably not true, but if it were, it would
+not change the argument; if the two abilities were identical, the
+practice of one would improve the other equally.
+
+I shall not undertake here this task of listing and describing the
+elementary functions which constitute arithmetical learning, partly
+because what they are is not fully known, partly because in many cases a
+final ability may be constituted in several different ways whose
+descriptions become necessarily tedious, and partly because an adequate
+statement of what is known would far outrun the space limits of this
+chapter. Instead, I shall illustrate the results by some samples.
+
+
+KNOWLEDGE OF THE MEANING OF A FRACTION
+
+As a first sample, consider knowledge of the meaning of a fraction. Is
+the ability in question simply to understand that a fraction is a
+statement of the number of parts, each of a certain size, the upper
+number or numerator telling how many parts are taken and the lower
+number or denominator telling what fraction of unity each part is? And
+is the educational treatment required simply to describe and illustrate
+such a statement and have the pupils apply it to the recognition of
+fractions and the interpretation of each of them? And is the learning
+process (1) the formation of the notions of part, size of part, number
+of part, (2) relating the last two to the numbers in a fraction, and, as
+a necessary consequence, (3) applying these notions adequately whenever
+one encounters a fraction in operation?
+
+Precisely this was the notion a few generations ago. The nature of
+fractions was taught as one principle, in one step, and the habits of
+dealing with fractions were supposed to be deduced from the general law
+of a fraction's nature. As a result the subject of fractions had to be
+long delayed, was studied at great cost of time and effort, and, even
+so, remained a mystery to all save gifted pupils. These gifted pupils
+probably of their own accord built up the ability piecemeal out of
+constituent insights and habits.
+
+At all events, scientific teaching now does build up the total ability
+as a fusion or organization of lesser abilities. What these are will be
+seen best by examining the means taken to get them. (1) First comes the
+association of 1/2 of a pie, 1/2 of a cake, 1/2 of an apple, and such
+like with their concrete meanings so that a pupil can properly name a
+clearly designated half of an obvious unit like an orange, pear, or
+piece of chalk. The same degree of understanding of 1/4, 1/8, 1/3, 1/6,
+and 1/5 is secured. The pupil is taught that 1 pie = 2 1/2s, 3 1/3s, 4
+1/4s, 5 1/5s, 6 1/6s, and 8 1/8s; similarly for 1 cake, 1 apple, and the
+like.
+
+So far he understands 1/_x_ of _y_ in the sense of certain simple parts
+of obviously unitary _y_s.
+
+(2) Next comes the association with 1/2 of an inch, 1/2 of a foot, 1/2
+of a glassful and other cases where _y_ is not so obviously a unitary
+object whose pieces still show their derivation from it. Similarly for
+1/4, 1/3, etc.
+
+(3) Next comes the association with 1/2 of a collection of eight pieces
+of candy, 1/3 of a dozen eggs, 1/5 of a squad of ten soldiers, etc.,
+until 1/2, 1/3, 1/4, 1/5, 1/6, and 1/8 are understood as names of
+certain parts of a collection of objects.
+
+(4) Next comes the similar association when the nature of the collection
+is left undefined, the pupil responding to
+
+  1/2 of 6 is ..., 1/4 of 8 is ..., 2 is 1/5 of ...,
+  1/3 of 6 is ..., 1/3 of 9 is ..., 2 is 1/3 of ..., and the like.
+
+Each of these abilities is justified in teaching by its intrinsic
+merits, irrespective of its later service in helping to constitute the
+general understanding of the meaning of a fraction. The habits thus
+formed in grades 3 or 4 are of constant service then and thereafter in
+and out of school.
+
+(5) With these comes the use of 1/5 of 10, 15, 20, etc., 1/6 of 12, 18,
+42, etc., as a useful variety of drill on the division tables, valuable
+in itself, and a means of making the notion of a unit fraction more
+general by adding 1/7 and 1/9 to the scheme.
+
+(6) Next comes the connection of 3/4, 2/5, 3/5, 4/5, 2/3, 1/6, 5/6, 3/8,
+5/8, 7/8, 3/10, 7/10, and 9/10, each with its meaning as a certain part
+of some conveniently divisible unit, and, (7) and (8), connections
+between these fractions and their meanings as parts of certain
+magnitudes (7) and collections (8) of convenient size, and (9)
+connections between these fractions and their meanings when the nature
+of the magnitude or collection is unstated, as in 4/5 of 15 = ...,
+5/8 of 32 = ....
+
+(10) That the relation is general is shown by using it with
+numbers requiring written division and multiplication, such as
+7/8 of 1736 = ..., and with United States money.
+
+Elements (6) to (10) again are useful even if the pupil never goes
+farther in arithmetic. One of the commonest uses of fractions is in
+calculating the cost of fractions of yards of cloth, and fractions of
+pounds of meat, cheese, etc.
+
+The next step (11) is to understand to some extent the principle that
+the value of any of these fractions is unaltered by multiplying or
+dividing the numerator and denominator by the same number. The drills in
+expressing fractions in lower and higher terms which accomplish this are
+paralleled by (12) and (13) simple exercises in adding and subtracting
+fractions to show that fractions are quantities that can be operated on
+like any quantities, and by (14) simple work with mixed numbers
+(addition and subtraction and reductions), and (15) improper fractions.
+All that is done with improper fractions is (_a_) to have the pupil use
+a few of them as he would any fractions and (_b_) to note their
+equivalent mixed numbers. In (12), (13), and (14) only fractions of the
+same denominators are added or subtracted, and in (12) (13), (14), and
+(15) only fractions with 2, 3, 4, 5, 6, 8, or 10 in the denominator are
+used. As hitherto, the work of (11) to (15) is useful in and of itself.
+(16) Definitions are given of the following type:--
+
+Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called whole numbers.
+
+Numbers like 7/8, 1/5, 2/3, 3/4, 11/8, 7/6, 1/3, 4/3, 1/8, 1/6 are
+called fractions.
+
+Numbers like 5-1/4, 7-3/8, 9-1/2, 16-4/5, 315-7/8, 1-1/3, 1-2/3 are
+called mixed numbers.
+
+(17) The terms numerator and denominator are connected with the upper
+and lower numbers composing a fraction.
+
+Building this somewhat elaborate series of minor abilities seems to be a
+very roundabout way of getting knowledge of the meaning of a fraction,
+and is, if we take no account of what is got along with this knowledge.
+Taking account of the intrinsically useful habits that are built up, one
+might retort that the pupil gets his knowledge of the meaning of a
+fraction at zero cost.
+
+
+KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES
+
+Consider next the knowledge of the subtraction and division 'Tables.'
+The usual treatment presupposes that learning them consists of forming
+independently the bonds:--
+
+   3 - 1 = 2                4 � 2 = 2
+   3 - 2 = 1                6 � 2 = 3
+   4 - 1 = 3                6 � 3 = 2
+     .                        .
+     .                        .
+     .                        .
+     .                        .
+     .                        .
+     .                        .
+  18 - 9 = 9               81 � 9 = 9
+
+In fact, however, these 126 bonds are not formed independently. Except
+perhaps in the case of the dullest twentieth of pupils, they are
+somewhat facilitated by the already learned additions and
+multiplications. And by proper arrangement of the learning they may be
+enormously facilitated thereby. Indeed, we may replace the independent
+memorizing of these facts by a set of instructive exercises wherein the
+pupil derives the subtractions from the corresponding additions by
+simple acts of reasoning or selective thinking. As soon as the additions
+giving sums of 9 or less are learned, let the pupil attack an exercise
+like the following:--
+
+Write the missing numbers:--
+
+         A                 B                 C                 D
+  3 and ... are 5.  5 and ... are 8.  4 and ... are 5.  4 and ... are 8.
+  3 and ... are 9.  3 and ... are 6.  5 and ... are 6.  1 and ... are 7.
+  4 and ... are 7.  4 and ... are 9.  6 and ... are 9.  6 and ... are 7.
+  5 and ... are 7.  2 and ...  =  6.  1 and ... are 8.  8 and ... are 9.
+  6 and ... are 8.  5 and ...  =  9.  3 and ... are 7.  3  +  ... are 4.
+  4 and ... are 6.  2 and ...  =  7.  1  +  ... are 3.  7  +  ... are 8.
+  2 and ... are 5.  3 and ...  =  8.  1  +  ... are 5.  4  +  ... are 9.
+  2 and ...  =  8.  1 and ...  =  4.  4  +  ... are 8.  2  +  ... are 3.
+  3 and ...  =  6.  2 and ...  =  4.  7  +  ... are 9.  1  +  ... are 9.
+  6 and ...  =  9.  3 and ...  =  8.  2  +  ...  =  4.  3  +  ...  =  6.
+  4 and ...  =  6.  6 and ...  =  7.  3  +  ...  =  8.  5  +  ...  =  9.
+  4 and ...  =  7.  2 and ...  =  5.  4  +  ...  =  5.  1  +  ...  =  3.
+
+The task for reasoning is only to try, one after another, numbers that
+seem promising and to select the right one when found. With a little
+stimulus and direction children can thus derive the subtractions up to
+those with 9 as the larger number. Let them then be taught to do the
+same with the printed forms:--
+
+Subtract
+
+  9   7   8   5   8   6
+  3   5   6   2   2   4   etc.
+  -   -   -   -   -   -
+
+and 9 - 7 = ..., 9 - 5 = ..., 7 - 5 = ..., etc.
+
+In the case of the divisions, suppose that the pupil has learned his
+first table and gained surety in such exercises as:--
+
+  4 5s = ....    6 � 5 = ....    9 nickels = .... cents.
+  8 5s = ....    4 � 5 = ....    6    "    = ....   "
+  3 5s = ....    2 � 5 = ....    5    "    = ....   "
+  7 5s = ....    9 � 5 = ....    7    "    = ....   "
+
+  If one ball costs 5 cents,
+                two balls cost .... cents,
+                three balls cost .... cents, etc.
+
+He may then be set at once to work at the answers to exercises like the
+following:--
+
+Write the answers and the missing numbers:--
+
+       A            B                 C                 D
+  .... 5s = 15   40 = .... 5s   .... � 5 = 25   20 cents = .... nickels.
+  .... 5s = 20   20 = .... 5s   .... � 5 = 50   30 cents = .... nickels.
+  .... 5s = 40   15 = .... 5s   .... � 5 = 35   15 cents = .... nickels.
+  .... 5s = 25   45 = .... 5s   .... � 5 = 10   40 cents = .... nickels.
+  .... 5s = 30   50 = .... 5s   .... � 5 = 40
+  .... 5s = 35   25 = .... 5s   .... � 5 = 45
+
+                          E
+  For  5 cents you can buy 1 small loaf of bread.
+  For 10 cents you can buy 2 small loaves of bread.
+  For 25 cents you can buy .... small loaves of bread.
+  For 45 cents you can buy .... small loaves of bread.
+  For 35 cents you can buy .... small loaves of bread.
+
+               F
+   5 cents pays 1 car fare.
+  15 cents pays .... car fares.
+  10 cents pays .... car fares.
+  20 cents pays .... car fares.
+
+                           G
+  How many 5 cent balls can you buy with 30 cents? ....
+  How many 5 cent balls can you buy with 35 cents? ....
+  How many 5 cent balls can you buy with 25 cents? ....
+  How many 5 cent balls can you buy with 15 cents? ....
+
+In the case of the meaning of a fraction, the ability, and so the
+learning, is much more elaborate than common practice has assumed; in
+the case of the subtraction and division tables the learning is much
+less so. In neither case is the learning either mere memorizing of facts
+or the mere understanding of a principle _in abstracto_ followed by its
+application to concrete cases. It is (and this we shall find true of
+almost all efficient learning in arithmetic) the formation of
+connections and their use in such an order that each helps the others to
+the maximum degree, and so that each will do the maximum amount for
+arithmetical abilities other than the one specially concerned, and for
+the general competence of the learner.
+
+
+LEARNING THE PROCESSES OF COMPUTATION
+
+As another instructive topic in the constitution of arithmetical
+abilities, we may take the case of the reasoning involved in
+understanding the manipulations of figures in two (or more)-place
+addition and subtraction, multiplication and division involving a two
+(or more)-place number, and the manipulations of decimals in all four
+operations. The psychology of these is of special interest and
+importance. For there are two opposite explanations possible here,
+leading to two opposite theories of teaching.
+
+The common explanation is that these methods of manipulation, if
+understood at all, are understood as deductions from the properties of
+our system of decimal notation. The other is that they are understood
+partly as inductions from the experience that they always give the right
+answer. The first explanation leads to the common preliminary deductive
+explanations of the textbooks. The other leads to explanations by
+verification; _e.g._, of addition by counting, of subtraction by
+addition, of multiplication by addition, of division by multiplication.
+Samples of these two sorts of explanation are given below.
+
+
+SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION
+
+MULTIPLICATION is the process of taking one number as many times as
+there are units in another number.
+
+The PRODUCT is the result of the multiplication.
+
+The MULTIPLICAND is the number to be taken.
+
+The MULTIPLIER is the number denoting how many times the multiplicand is
+to be taken.
+
+The multiplier and multiplicand are the FACTORS.
+
+    Multiply 623 by 3
+
+    OPERATION
+
+      _Multiplicand_    623
+      _Multiplier_        3
+                       ----
+      _Product_        1869
+
+    EXPLANATION.--For convenience we write the multiplier under the
+    multiplicand, and begin with units to multiply. 3 times 3 units are
+    9 units. We write the nine units in units' place in the product. 3
+    times 2 tens are 6 tens. We write the 6 tens in tens' place in the
+    product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8
+    hundreds. The 1 thousand we write in thousands' place and the 8
+    hundreds in hundreds' place in the product. Therefore, the product
+    is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869.
+
+
+SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION
+
+  1. The children of the third grade are to have a picnic. 32 are going.
+  How many sandwiches will they need if each of the 32 children has four
+  sandwiches?
+
+      _Here is a quick way to find out_:--
+
+       32   _Think "4 � 2," write 8 under the 2 in the ones column._
+        4   _Think "4 � 3," write 12 under the 3 in the tens column._
+       --
+
+  2. How many bananas will they need if each of the 32 children has
+  two bananas? 32 � 2 or 2 � 32 will give the answer.
+
+  3. How many little cakes will they need if each child has three
+  cakes? 32 � 3 or 3 � 32 will give the answer.
+
+    32           3 � 2 = ....  _Where do you write the 6?_
+     3           3 � 3 = ....  _Where do you write the 9?_
+    --
+
+  4.  Prove that 128, 64, and 96 are right by adding four 32s, two 32s,
+  and three 32s.
+
+      32
+      32  32
+      32  32  32
+      32  32  32
+      --  --  --
+
+
+Multiplication
+
+  You #multiply# when you find the answers to questions like
+
+    How many are 9 � 3?
+    How many are 3 � 32?
+    How many are 8 � 5?
+    How many are 4 � 42?
+
+  1. Read these lines. Say the right numbers where the dots are:
+
+     If you #add# 3 to 32, you have .... 35 is the #sum#.
+     If you #subtract# 3 from 32, the result is .... 29 is the
+          #difference# or #remainder#.
+     If you #multiply# 3 by 32 or 32 by 3, you have .... 96 is the
+          #product#.
+
+  Find the products. Check your answers to the first line by adding.
+
+   2.    3.    4.    5.    6.    7.    8.    9.
+
+   41    33    42    44    53    43    34    24
+    3     2     4     2     3     2     2     2
+   --    --    --    --    --    --    --    --
+
+      10.   11.   12.   13.   14.   15.   16.
+
+      43    52    32    23    41    51    14
+       3     3     3     3     2     4     2
+      --    --    --    --    --    --    --
+
+  17.     213  _Write the 9 in the ones column._
+            3  _Write the 3 in the tens column._
+          ---  _Write the 6 in the hundreds column._
+
+  _Check your answer by adding._    Add
+                                    213
+                                    213
+                                    213
+                                    ---
+
+   18.    19.    20.    21.    22.    23.    24.
+
+   214    312    432    231    132    314    243
+     2      3      2      3      3      2      2
+   ---    ---    ---    ---    ---    ---    ---
+
+
+SHORT DIVISION: DEDUCTIVE EXPLANATION
+
+Divide 1825 by 4
+
+  Divisor  4 |1825    Dividend
+             --------
+             456-1/4
+             Quotient
+
+    EXPLANATION.--For convenience we write the divisor at the left of
+    the dividend, and the quotient below it, and begin at the left to
+    divide. 4 is not contained in 1 thousand any thousand times,
+    therefore the quotient contains no unit of any order higher than
+    hundreds. Consequently we find how many times 4 is contained in the
+    hundreds of the dividend. 1 thousand and 8 hundreds are 18
+    hundreds. 4 is contained in 18 hundreds 4 hundred times and 2
+    hundreds remaining. We write the 4 hundreds in the quotient. The 2
+    hundreds we consider as united with the 2 tens, making 22 tens. 4
+    is contained in 22 tens 5 tens times, and 2 tens remaining. We
+    write the 5 tens in the quotient, and the remaining 2 tens we
+    consider as united with the 5 units, making 25 units. 4 is
+    contained in 25 units 6 units times and 1 unit remaining. We write
+    the 6 units in the quotient and indicate the division of the
+    remainder, 1 unit, by the divisor 4.
+
+    Therefore the quotient of 1825 divided by 4 is 456-1/4, or 456 and
+    1 remainder.
+
+
+SHORT DIVISION: INDUCTIVE EXPLANATION
+
+Dividing Large Numbers
+
+  1. Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent
+  bag of marbles. There are 128 marbles in it. How many should each boy
+  have, if they divide the marbles equally among the four boys?
+
+      -----
+    4 |128
+
+    _Think "12 = three 4s." Write the 3 over the 2 in the tens column._
+    _Think "8 = two 4s." Write the 2 over the 8 in the ones column._
+    _32 is right, because 4 � 32 = 128._
+
+  2. Mary, Nell, and Alice are going to buy a book as a present for
+  their Sunday-school teacher. The present costs 69 cents. How much
+  should each girl pay, if they divide the cost equally among the
+  three girls?
+     ----
+    3|69
+
+    _Think "6 = .... 3s." Write the 2 over the 6 in the tens column._
+    _Think "9 = .... 3s." Write the 3 over the 9 in the ones column._
+    _23 is right, for 3 � 23 = 69._
+
+  3. Divide the cost of a 96-cent present equally among three girls. How
+  much should each girl pay?
+     ------
+    3|96
+
+  4. Divide the cost of an 84-cent present equally among 4 girls. How
+  much should each girl pay?
+
+  5. Learn this: (Read � as "_divided by_.")
+
+      12 + 4 = 16.   16 is the sum.
+      12 - 4 =  8.    8 is the difference or remainder.
+      12 � 4 = 48.   48 is the product.
+      12 � 4 =  3.    3 is the quotient.
+
+  6. Find the quotients. Check your answers by multiplying.
+     ----     ----     ----     -----     -----     -----
+    3|99     2|86     5|155    6|246     4|168     3|219
+
+[Uneven division is taught by the same general plan, extended.]
+
+
+LONG DIVISION: DEDUCTIVE EXPLANATION
+
+To Divide by Long Division
+
+1. Let it be required to divide 34531 by 15.
+
+              _Operation_
+
+               Divided
+  Divisor  15)34531(2302-1/15  Quotient
+              30
+              --
+               45
+               45
+               --
+                 31
+                 30
+                 --
+                  1 Remainder
+
+For convenience we write the divisor at the left and the quotient at the
+right of the dividend, and begin to divide as in Short Division.
+
+15 is contained in 3 ten-thousands 0 ten-thousands times; therefore,
+there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is
+contained in 34 thousands 2 thousands times; we write the 2 thousands in
+the quotient. 15 � 2 thousands = 30 thousands, which, subtracted from 34
+thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we
+have 45 hundreds.
+
+15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the
+quotient. 15 � 3 hundreds = 45 hundreds, which subtracted from 45
+hundreds, leaves nothing. Adding the 3 tens, we have 3 tens.
+
+15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to
+the three tens, which equal 30 units, the 1 unit, we have 31 units.
+
+15 in 31 units 2 units times; we write the 2 units in the quotient. 15 �
+2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a
+remainder. Indicating the division of the 1 unit, we annex the
+fractional expression, 1/15 unit, to the integral part of the quotient.
+
+Therefore, 34531 divided by 15 is equal to 2302-1/15.
+
+[B. Greenleaf, _Practical Arithmetic_, '73, p. 49.]
+
+
+LONG DIVISION: INDUCTIVE EXPLANATION
+
+Dividing by Large Numbers
+
+  1. Just before Christmas Frank's father sent 360 oranges to be divided
+     among the children in Frank's class. There are 29 children. How
+     many oranges should each child receive? How many oranges will be
+     left over?
+
+  _Here is the best way to find out:_
+
+       12    and 12    _Think how many 29s there are in 36. 1 is right._
+     ______ remainder  _Write 1 over the 6 of 36. Multiply 29 by 1._
+  29 )360              _Write the 29 under the 36. Subtract 29 from 36._
+       29              _Write the 0 of 360 after the 7._
+      ---              _Think how many 29s there are in 70. 2 is right._
+       70              _Write 2 over the 0 of 360. Multiply 29 by 2._
+       58              _Write the 58 under 70. Subtract 58 from 70._
+       --              _There is 12 remainder._
+       12              _Each child gets 12 oranges, and there are 12
+                         left over. This is right, for 12 multiplied
+                         by 29 = 348, and 348 + 12 = 360._
+
+       *       *       *       *       *
+
+  8.                  _In No. 8, keep on dividing by 31 until you have
+    ________            used the 5, the 8, and the 7, and have four
+  31)99,587             figures in the quotient._
+
+  9.         10.         11.         12.        13.
+    _____      _____       _____       ____       _______
+  22)253     22)2895     21)8891     22)290     32)16,368
+
+Check your results for 9, 10, 11, 12, and 13.
+
+  1. The boys and girls of the Welfare Club plan to earn money to buy
+     a victrola. There are 23 boys and girls. They can get a good
+     second-hand victrola for $5.75. How much must each earn if they
+     divide the cost equally?
+
+     _Here is the best way to find out_:
+
+      $.25     _Think how many 23s there are in 57. 2 is right._
+     -----     _Write 2 over the 7 of 57. Multiply 23 by 2._
+  23|$5.75     _Write 46 under 57 and subtract. Write the 5 of 575
+        46       after the 11._
+      ----     _Think how many 23s there are in 115. 5 is right._
+       115     _Write 5 over the 5 of 575. Multiply 23 by 5._
+       115     _Write the 115 under the 115 that is there and subtract._
+      ----     _There is no remainder._
+               _Put $ and the decimal point where they belong._
+               _Each child must earn 25 cents. This is right, for $.25
+                 multiplied by 23 = $5.75._
+
+  2. Divide $71.76 equally among 23 persons. How much is each person's
+     share?
+
+  3. Check your result for No. 2 by multiplying the quotient by the
+     divisor.
+
+  Find the quotients. Check each quotient by multiplying it by the
+  divisor.
+
+  4.           5.           6.            7.           8.
+    _______      _______      ________      _______      _______
+  23)$99.13    25)$18.50    21)$129.15    13)$29.25    32)$73.92
+
+    1 bushel = 32 qt.
+
+  9. How many bushels are there in 288 qt.?
+
+  10. In 192 qt.?
+
+  11. In 416 qt.?
+
+Crucial experiments are lacking, but there are several lines of
+well-attested evidence. First of all, there can be no doubt that the
+great majority of pupils learn these manipulations at the start from the
+placing of units under units, tens under tens, etc., in adding, to the
+placing of the decimal point in division with decimals, by imitation and
+blind following of specific instructions, and that a very large
+proportion of the pupils do not to the end, that is to the fifth
+school-year, understand them as necessary deductions from decimal
+notation. It also seems probable that this proportion would not be much
+reduced no matter how ingeniously and carefully the deductions were
+explained by textbooks and teachers. Evidence of this fact will appear
+abundantly to any one who will observe schoolroom life. It also appears
+in the fact that after the properties of the decimal notation have been
+thus used again and again; _e.g._, for deducing 'carrying' in addition,
+'borrowing' in subtraction, 'carrying' in multiplication, the value of
+the digits in the partial product, the value of each remainder in short
+division, the value of the quotient figures in division, the addition,
+subtraction, multiplication, and division of United States money, and
+the placing of the decimal point in multiplication, no competent teacher
+dares to rely upon the pupil, even though he now has four or more years'
+experience with decimal notation, to deduce the placing of the decimal
+point in division with decimals. It may be an illusion, but one seems to
+sense in the better textbooks a recognition of the futility of the
+attempt to secure deductive derivations of those manipulations. I refer
+to the brevity of the explanations and their insertion in such a form
+that they will influence the pupils' thinking as little as possible. At
+any rate the fact is sure that most pupils do not learn the
+manipulations by deductive reasoning, or understand them as necessary
+consequences of abstract principles.
+
+It is a common opinion that the only alternative is knowing them by
+rote. This, of course, is one common alternative, but the other
+explanation suggests that understanding the manipulations by inductive
+reasoning from their results is another and an important alternative.
+The manipulations of 'long' multiplication, for instance, learned by
+imitation or mechanical drill, are found to give for 25 � _A_ a result
+about twice as large as for 13 � _A_, for 38 or 39 � _A_ a result about
+three times as large; for 115 � _A_ a result about ten times as large as
+for 11 � _A_. With even the very dull pupils the procedure is verified
+at least to the extent that it gives a result which the scientific
+expert in the case--the teacher--calls right. With even the very bright
+pupils, who can appreciate the relation of the procedure to decimal
+notation, this relation may be used not as the sole deduction of the
+procedure beforehand, but as one partial means of verifying it
+afterward. Or there may be the condition of half-appreciation of the
+relation in which the pupil uses knowledge of the decimal notation to
+convince himself that the procedure _does_, but not that it _must_ give
+the right answer, the answer being 'right' because the teacher, the
+answer-list, and collateral evidence assure him of it.
+
+I have taken the manipulation of the partial products as an illustration
+because it is one of the least favored cases for the explanation I am
+presenting. If we take the first case where a manipulation may be
+deduced from decimal notation, known merely by rote, or verified
+inductively, namely, the addition of two-place numbers, it seems sure
+that the mental processes just described are almost the universal rule.
+
+Surely in our schools at present children add the 3 of 23 to the 3 of 53
+and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten
+because they see the teacher do so and are told to do so. They are
+protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but
+because they do not know how to add 8 and 5, because they have been
+taught the habit of adding figures that stand one above the other, or
+with a + between them; and because they are shown or told what they are
+to do. They are protected from adding 3 + 5 and 2 + 3, again, by no
+deductive reasoning but for the second and third reasons just given.
+In nine cases out of ten they do not even think of the possibility
+of adding in any other way than the '3 + 3, 2 + 5' way, much less
+do they select that way on account of the facts that 53 = 50 + 3
+and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that
+(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_)!
+
+Just as surely all but the very dullest twentieth or so of children come
+in the end to something more than rote knowledge,--to _understand_, to
+_know_ that the procedure in question is right.
+
+Whether they know _why_ 76 is right depends upon what is meant by
+_why_. If it means that 76 is the result which competent people
+agree upon, they do. If it means that 76 is the result which would
+come from accurate counting they perhaps know why as well as they
+would have, had they been given full explanations of the relation
+of the procedure in two-place addition to decimal notation.
+If _why_ means because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and
+(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_), they do not.
+Nor, I am tempted to add, would most of them by any sort of teaching
+whatever.
+
+I conclude, therefore, that school children may and do reason about and
+understand the manipulations of numbers in this inductive, verifying way
+without being able to, or at least without, under present conditions,
+finding it profitable to derive them deductively. I believe, in fact,
+that pure arithmetic _as it is learned and known_ is largely an
+_inductive science_. At one extreme is a minority to whom it is a series
+of deductions from principles; at the other extreme is a minority to
+whom it is a series of blind habits; between the two is the great
+majority, representing every gradation but centering about the type of
+the inductive thinker.
+
+
+
+
+CHAPTER IV
+
+THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED): THE SELECTION OF
+THE BONDS TO BE FORMED
+
+
+When the analysis of the mental functions involved in arithmetical
+learning is made thorough it turns into the question, 'What are the
+elementary bonds or connections that constitute these functions?' and
+when the problem of teaching arithmetic is regarded, as it should be in
+the light of present psychology, as a problem in the development of a
+hierarchy of intellectual habits, it becomes in large measure a problem
+of the choice of the bonds to be formed and of the discovery of the best
+order in which to form them and the best means of forming each in that
+order.
+
+
+THE IMPORTANCE OF HABIT-FORMATION
+
+The importance of habit-formation or connection-making has been grossly
+underestimated by the majority of teachers and writers of textbooks.
+For, in the first place, mastery by deductive reasoning of such matters
+as 'carrying' in addition, 'borrowing' in subtraction, the value of the
+digits in the partial products in multiplication, the manipulation of
+the figures in division, the placing of the decimal point after
+multiplication or division with decimals, or the manipulation of the
+figures in the multiplication and division of fractions, is impossible
+or extremely unlikely in the case of children of the ages and experience
+in question. They do not as a rule deduce the method of manipulation
+from their knowledge of decimal notation. Rather they learn about
+decimal notation by carrying, borrowing, writing the last figure of each
+partial product under the multiplier which gives that product, etc. They
+learn the method of manipulating numbers by seeing them employed, and by
+more or less blindly acquiring them as associative habits.
+
+In the second place, we, who have already formed and long used the right
+habits and are thereby protected against the casual misleadings of
+unfortunate mental connections, can hardly realize the force of mere
+association. When a child writes sixteen as 61, or finds 428 as the sum
+of
+
+  15
+  19
+  16
+  18
+  --
+
+or gives 642 as an answer to 27 � 36, or says that 4 divided by 1/4 = 1,
+we are tempted to consider him mentally perverse, forgetting or perhaps
+never having understood that he goes wrong for exactly the same general
+reason that we go right; namely, the general law of habit-formation. If
+we study the cases of 61 for 16, we shall find them occurring in the
+work of pupils who after having been drilled in writing 26, 36, 46, 62,
+63, and so on, in which the order of the six in writing is the same as
+it is in speech, return to writing the 'teen numbers. If our language
+said onety-one for eleven and onety-six for sixteen, we should probably
+never find such errors except as 'lapses' or as the results of
+misperception or lack of memory. They would then be more frequent
+_before_ the 20s, 30s, etc., were learned.
+
+If pupils are given much drill on written single column addition
+involving the higher decades (each time writing the two-figure sum),
+they are forming a habit of writing 28 after the sum of 8, 6, 9, and 5
+is reached; and it should not surprise us if the pupil still
+occasionally writes the two-figure sum for the first column though a
+second column is to be added also. On the contrary, unless some counter
+force influences him, he is absolutely sure to make this mistake.
+
+The last mistake quoted (4 � 1/4 = 1) is interesting because here we
+have possibly one of the cases where deduction from psychology alone can
+give constructive aid to teaching. Multiplication and division by
+fractions have been notorious for their difficulty. The former is now
+alleviated by using _of_ instead of � until the new habit is fixed. The
+latter is still approached with elaborate caution and with various means
+of showing why one must 'invert and multiply' or 'multiply by the
+reciprocal.'
+
+But in the author's opinion it seems clear that the difficulty in
+multiplying and dividing by a fraction was not that children felt any
+logical objections to canceling or inverting. I fancy that the majority
+of them would cheerfully invert any fraction three times over or cancel
+numbers at random in a column if they were shown how to do so. But if
+you are a youngster inexperienced in numerical abstractions and if you
+have had _divide_ connected with 'make smaller' three thousand times and
+never once connected with 'make bigger,' you are sure to be somewhat
+impelled to make the number smaller the three thousand and first time
+you are asked to divide it. Some of my readers will probably confess
+that even now they feel a slight irritation or doubt in saying or
+writing that 16/1 � 1/8 = 128.
+
+The habits that have been confirmed by every multiplication and division
+by integers are, in this particular of '_the ratio of result to number
+operated upon_,' directly opposed to the formation of the habits
+required with fractions. And that is, I believe, the main cause of the
+difficulty. Its treatment then becomes easy, as will be shown later.
+
+These illustrations could be added to almost indefinitely, especially in
+the case of the responses made to the so-called 'catch' problems. The
+fact is that the learner rarely can, and almost never does, survey and
+analyze an arithmetical situation and justify what he is going to do by
+articulate deductions from principles. He usually feels the situation
+more or less vaguely and responds to it as he has responded to it or
+some situation like it in the past. Arithmetic is to him not a logical
+doctrine which he applies to various special instances, but a set of
+rather specialized habits of behavior toward certain sorts of quantities
+and relations. And in so far as he does come to know the doctrine it is
+chiefly by doing the will of the master. This is true even with the
+clearest expositions, the wisest use of objective aids, and full
+encouragement of originality on the pupil's part.
+
+Lest the last few paragraphs be misunderstood, I hasten to add that the
+psychologists of to-day do not wish to make the learning of arithmetic a
+mere matter of acquiring thousands of disconnected habits, nor to
+decrease by one jot the pupil's genuine comprehension of its general
+truths. They wish him to reason not less than he has in the past, but
+more. They find, however, that you do not secure reasoning in a pupil by
+demanding it, and that his learning of a general truth without the
+proper development of organized habits back of it is likely to be, not a
+rational learning of that general truth, but only a mechanical
+memorizing of a verbal statement of it. They have come to know that
+reasoning is not a magic force working in independence of ordinary
+habits of thought, but an organization and co�peration of those very
+habits on a higher level.
+
+The older pedagogy of arithmetic stated a general law or truth or
+principle, ordered the pupil to learn it, and gave him tasks to do which
+he could not do profitably unless he understood the principle. It left
+him to build up himself the particular habits needed to give him
+understanding and mastery of the principle. The newer pedagogy is
+careful to help him build up these connections or bonds ahead of and
+along with the general truth or principle, so that he can understand it
+better. The older pedagogy commanded the pupil to reason and let him
+suffer the penalty of small profit from the work if he did not. The
+newer provides instructive experiences with numbers which will stimulate
+the pupil to reason so far as he has the capacity, but will still be
+profitable to him in concrete knowledge and skill, even if he lacks the
+ability to develop the experiences into a general understanding of the
+principles of numbers. The newer pedagogy secures more reasoning in
+reality by not pretending to secure so much.
+
+The newer pedagogy of arithmetic, then, scrutinizes every element of
+knowledge, every connection made in the mind of the learner, so as to
+choose those which provide the most instructive experiences, those which
+will grow together into an orderly, rational system of thinking about
+numbers and quantitative facts. It is not enough for a problem to be a
+test of understanding of a principle; it must also be helpful in and of
+itself. It is not enough for an example to be a case of some rule; it
+must help review and consolidate habits already acquired or lead up to
+and facilitate habits to be acquired. Every detail of the pupil's work
+must do the maximum service in arithmetical learning.
+
+
+DESIRABLE BONDS NOW OFTEN NEGLECTED
+
+As hitherto, I shall not try to list completely the elementary bonds
+that the course of study in arithmetic should provide for. The best
+means of preparing the student of this topic for sound criticism and
+helpful invention is to let him examine representative cases of bonds
+now often neglected which should be formed and representative cases of
+useless, or even harmful, bonds now often formed at considerable waste
+of time and effort.
+
+(1) _Numbers as measures of continuous quantities._--The numbers one,
+two, three, 1, 2, 3, etc., should be connected soon after the beginning
+of arithmetic each with the appropriate amount of some continuous
+quantity like length or volume or weight, as well as with the
+appropriate sized collection of apples, counters, blocks, and the like.
+Lines should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two
+inches, three inches, etc.; weights should be lifted and called one
+pound, two pounds, etc.; things should be measured in glassfuls,
+handfuls, pints, and quarts. Otherwise the pupil is likely to limit the
+meaning of, say, _four_ to four sensibly discrete things and to have
+difficulty in multiplication and division. Measuring, or counting by
+insensibly marked off repetitions of a unit, binds each number name to
+its meaning as ---- _times whatever 1 is_, more surely than mere
+counting of the units in a collection can, and should re�nforce the
+latter.
+
+(2) _Additions in the higher decades._--In the case of all save the very
+gifted children, the additions with higher decades--that is, the bonds,
+16 + 7 = 23, 26 + 7 = 33, 36 + 7 = 43, 14 + 8 = 22, 24 + 8 = 32, and the
+like--need to be specifically practiced until the tendency becomes
+generalized. 'Counting' by 2s beginning with 1, and with 2, counting by
+3s beginning with 1, with 2, and with 3, counting by 4s beginning with
+1, with 2, with 3, and with 4, and so on, make easy beginnings in the
+formation of the decade connections. Practice with isolated bonds should
+soon be added to get freer use of the bonds. The work of column addition
+should be checked for accuracy so that a pupil will continually get
+beneficial practice rather than 'practice in error.'
+
+(3) _The uneven divisions._--The quotients with remainders for the
+divisions of every number to 19 by 2, every number to 29 by 3, every
+number to 39 by 4, and so on should be taught as well as the even
+divisions. A table like the following will be found a convenient means
+of making these connections:--
+
+  10 = .... 2s
+  10 = .... 3s and .... rem.
+  10 = .... 4s and .... rem.
+  10 = .... 5s
+  11 = .... 2s and .... rem.
+  11 = .... 3s and .... rem.
+  .
+  .
+  .
+  89 = .... 9s and .... rem.
+
+These bonds must be formed before short division can be efficient, are
+useful as a partial help toward selection of the proper quotient figures
+in long division, and are the chief instruments for one of the important
+problem series in applied arithmetic,--"How many _x_s can I buy for _y_
+cents at _z_ cents per _x_ and how much will I have left?" That these
+bonds are at present sadly neglected is shown by Kirby ['13], who found
+that pupils in the last half of grade 3 and the first half of grade 4
+could do only about four such examples per minute (in a ten-minute
+test), and even at that rate made far from perfect records, though they
+had been taught the regular division tables. Sixty minutes of practice
+resulted in a gain of nearly 75 percent in number done per minute, with
+an increase in accuracy as well.
+
+(4) _The equation form._--The equation form with an unknown quantity to
+be determined, or a missing number to be found, should be connected with
+its meaning and with the problem attitude long before a pupil begins
+algebra, and in the minds of pupils who never will study algebra.
+
+Children who have just barely learned to add and subtract learn easily
+to do such work as the following:--
+
+Write the missing numbers:--
+
+  4 + 8 = ....
+  5 + .... = 14
+  .... + 3 = 11
+  .... = 5 + 2
+  16 = 7 + ....
+  12 = .... + 5
+
+The equation form is the simplest uniform way yet devised to state a
+quantitative issue. It is capable of indefinite extension if certain
+easily understood conventions about parentheses and fraction signs are
+learned. It should be employed widely in accounting and the treatment of
+commercial problems, and would be except for outworn conventions. It is
+a leading contribution of algebra to business and industrial life.
+Arithmetic can make it nearly as well. It saves more time in the case of
+drills on reducing fractions to higher and lower terms alone than is
+required to learn its meaning and use. To rewrite a quantitative
+problem as an equation and then make the easy selection of the
+necessary technique to solve the equation is one of the most universally
+useful intellectual devices known to man. The words 'equals,' 'equal,'
+'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer
+equivalents should therefore early give way on many occasions to the '='
+which so far surpasses them in ultimate convenience and simplicity.
+
+(5) _Addition and subtraction facts in the case of fractions._--In the
+case of adding and subtracting fractions, certain specific
+bonds--between the situation of halves and thirds to be added and the
+responses of thinking of the numbers as equal to so many sixths, between
+the situation thirds and fourths to be added and thinking of them as so
+many twelfths, between fourths and eighths to be added and thinking of
+them as eighths, and the like--should be formed separately. The general
+rule of thinking of fractions as their equivalents with some convenient
+denominator should come as an organization and extension of such special
+habits, not as an edict from the textbook or teacher.
+
+(6) _Fractional equivalents._--Efficiency requires that in the end the
+much used reductions should be firmly connected with the situations
+where they are needed. They may as well, therefore, be so connected from
+the beginning, with the gain of making the general process far easier
+for the dull pupils to master. We shall see later that, for all save the
+very gifted pupils, the economical way to get an understanding of
+arithmetical principles is not, usually, to learn a rule and then apply
+it, but to perform instructive operations and, in the course of
+performing them, to get insight into the principles.
+
+(7) _Protective habits in multiplying and dividing with fractions._--In
+multiplying and dividing with fractions special bonds should be formed
+to counteract the now harmful influence of the 'multiply = get a larger
+number' and 'divide = get a smaller number' bonds which all work with
+integers has been re�nforcing.
+
+For example, at the beginning of the systematic work with multiplication
+by a fraction, let the following be printed clearly at the top of every
+relevant page of the textbook and displayed on the blackboard:--
+
+_When you multiply a number by anything more than 1 the result is larger
+than the number._
+
+_When you multiply a number by 1 the result is the same as the number._
+
+_When you multiply a number by anything less than 1 the result is
+smaller than the number._
+
+Let the pupils establish the new habit by many such exercises as:--
+
+   18 � 4 = ....          9 � 2 = ....
+    4 � 4 = ....          6 � 2 = ....
+    2 � 4 = ....          3 � 2 = ....
+    1 � 4 = ....          1 � 2 = ....
+  1/2 � 4 = ....        1/3 � 2 = ....
+  1/4 � 4 = ....        1/6 � 2 = ....
+  1/8 � 4 = ....        1/9 � 2 = ....
+
+In the case of division by a fraction the old harmful habit should be
+counteracted and refined by similar rules and exercises as follows:--
+
+_When you divide a number by anything more than 1 the result is smaller
+than the number._
+
+_When you divide a number by 1 the result is the same as the number._
+
+_When you divide a number by anything less than 1 the result is larger
+than the number._
+
+State the missing numbers:--
+
+    8 = .... 4s      12 = .... 6s      9 = .... 9s
+    8 = .... 2s      12 = .... 4s      9 = .... 3s
+    8 = .... 1s      12 = .... 3s      9 = .... 1s
+    8 = .... 1/2s    12 = .... 2s      9 = .... 1/3s
+    8 = .... 1/4s    12 = .... 1s      9 = .... 1/9s
+    8 = .... 1/8s    12 = .... 1/2s
+                     12 = .... 1/3s
+                     12 = .... 1/4s
+
+  16 � 16  =         9 �  9  =      10 � 10   =    12 �  6  =
+  16 �  8  =         9 �  3  =      10 �  5   =    12 �  4  =
+  16 �  4  =         9 �  1  =      10 �  1   =    12 �  3  =
+  16 �  2  =         9 � 1/3 =      10 � 1/5  =    12 �  2  =
+  16 �  1  =         9 � 1/9 =      10 � 1/10 =    12 �  1  =
+  16 � 1/2 =                                       12 � 1/2 =
+  16 � 1/4 =                                       12 � 1/3 =
+  16 � 1/8 =                                       12 � 1/4 =
+                                                   12 � 1/6 =
+
+(8) _'% of' means 'hundredths times'._--In the case of percentage a
+series of bonds like the following should be formed:--
+
+   5 percent of = .05 times
+  20  "   "   " = .20   "
+   6  "   "   " = .06   "
+  25%         " = .25 �
+  12%         " = .12 �
+   3%         " = .03 �
+
+Four five-minute drills on such connections between '_x_ percent of' and
+'its decimal equivalent times' are worth an hour's study of verbal
+definitions of the meaning of percent as per hundred or the like. The
+only use of the study of such definitions is to facilitate the later
+formation of the bonds, and, with all save the brighter pupils, the
+bonds are more needed for an understanding of the definitions than the
+definitions are needed for the formation of the bonds.
+
+(9) _Habits of verifying results._--Bonds should early be formed between
+certain manipulations of numbers and certain means of checking, or
+verifying the correctness of, the manipulation in question. The
+additions to 9 + 9 and the subtractions to 18 - 9 should be verified by
+objective addition and subtraction and counting until the pupil has sure
+command; the multiplications to 9 � 9 should be verified by objective
+multiplication and counting of the result (in piles of tens and a pile
+of ones) eight or ten times,[4] and by addition eight or ten times;[4]
+the divisions to 81 � 9 should be verified by multiplication and
+occasionally objectively until the pupil has sure command; column
+addition should be checked by adding the columns separately and adding
+the sums so obtained, and by making two shorter tasks of the given task
+and adding the two sums; 'short' multiplication should be verified eight
+or ten times by addition; 'long' multiplication should be checked by
+reversing multiplier and multiplicand and in other ways; 'short' and
+'long' division should be verified by multiplication.
+
+  [4] Eight or ten times _in all_, not eight or ten times for each fact
+      of the tables.
+
+These habits of testing an obtained result are of threefold value. They
+enable the pupil to find his own errors, and to maintain a standard of
+accuracy by himself. They give him a sense of the relations of the
+processes and the reasons why the right ways of adding, subtracting,
+multiplying, and dividing are right, such as only the very bright pupils
+can get from verbal explanations. They put his acquisition of a certain
+power, say multiplication, to a real and intelligible use, in checking
+the results of his practice of a new power, and so instill a respect for
+arithmetical power and skill in general. The time spent in such
+verification produces these results at little cost; for the practice in
+adding to verify multiplications, in multiplying to verify divisions,
+and the like is nearly as good for general drill and review of the
+addition and multiplication themselves as practice devised for that
+special purpose.
+
+Early work in adding, subtracting, and reducing fractions should be
+verified by objective aids in the shape of lines and areas divided
+in suitable fractional parts. Early work with decimal fractions
+should be verified by the use of the equivalent common fractions
+for .25, .75, .125, .375, and the like. Multiplication and division
+with fractions, both common and decimal, should in the early stages
+be verified by objective aids. The placing of the decimal point in
+multiplication and division with decimal fractions should be verified
+by such exercises as:--
+
+           20  It cannot be 200; for 200 � 1.23 is much more than 24.6.
+       ______  It cannot be 2; for 2 � 1.23 is much less than 24.6.
+  1.23 )24.60
+        246
+       ----
+
+The establishment of habits of verifying results and their use is very
+greatly needed. The percentage of wrong answers in arithmetical work in
+schools is now so high that the pupils are often being practiced in
+error. In many cases they can feel no genuine and effective confidence
+in the processes, since their own use of the processes brings wrong
+answers as often as right. In solving problems they often cannot decide
+whether they have done the right thing or the wrong, since even if they
+have done the right thing, they may have done it inaccurately. A wrong
+answer to a problem is therefore too often ambiguous and uninstructive
+to them.[5]
+
+  [5] The facts concerning the present inaccuracy of school work in
+      arithmetic will be found on pages 102 to 105.
+
+These illustrations of the last few pages are samples of the procedures
+recommended by a consideration of all the bonds that one might form and
+of the contribution that each would make toward the abilities that the
+study of arithmetic should develop and improve. It is by doing more or
+less at haphazard what psychology teaches us to do deliberately and
+systematically in this respect that many of the past advances in the
+teaching of arithmetic have been made.
+
+
+WASTEFUL AND HARMFUL BONDS
+
+A scrutiny of the bonds now formed in the teaching of arithmetic with
+questions concerning the exact service of each, results in a list of
+bonds of small value or even no value, so far as a psychologist can
+determine. I present here samples of such psychologically unjustifiable
+bonds with some of the reasons for their deficiencies.
+
+(1) _Arbitrary units._--In drills intended to improve the ability to see
+and use the meanings of numbers as names for ratios or relative
+magnitudes, it is unwise to employ entirely arbitrary units. The
+procedure in II (on page 84) is better than that in I. Inches,
+half-inches, feet, and centimeters are better as units of length than
+arbitrary As. Square inches, square centimeters, and square feet are
+better for areas. Ounces and pounds should be lifted rather than
+arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and
+cubic inches are better for volume.
+
+All the real merit in the drills on relative magnitude advocated by
+Speer, McLellan and Dewey, and others can be secured without spending
+time in relating magnitudes for the sake of relative magnitude alone.
+The use of units of measure in drills which will never be used in _bona
+fide_ measuring is like the use of fractions like sevenths, elevenths,
+and thirteenths. A very little of it is perhaps desirable to test the
+appreciation of certain general principles, but for regular training it
+should give place to the use of units of practical significance.
+
+    [Illustration: FIG. 3.
+
+      A ----------
+      B ------------------------------
+      C --------------------
+      D ----------------------------------------
+
+    I. If _A_ is 1 which line is 2? Which line is 4? Which line is 3?
+    _A_ and _C_ together equal what line? _A_ and _B_ together equal
+    what line? How much longer is _B_ than _A_? How much longer is _B_
+    than _C_? How much longer is _D_ than _A_?]
+
+    [Illustration: FIG. 4.
+
+      A ----------
+      B ------------------------------
+      C --------------------
+      D ----------------------------------------
+
+    II. _A_ is 1 inch long. Which line is 2 inches long? Which line is
+    4 inches long? Which line is 3 inches long? _A_ and _C_ together
+    make ... inches? _A_ and _B_ together make ... inches? _B_ is ...
+    ... longer than _A_? _B_ is ... ... longer than _C_? _D_ is ...
+    ... longer than _A_?]
+
+(2) _Multiples of 11._--The multiplications of 2 to 12 by 11 and 12 as
+single connections should be left for the pupil to acquire by himself as
+he needs them. These connections interfere with the process of learning
+two-place multiplication. The manipulations of numbers there required
+can be learned much more easily if 11 and 12 are used as multipliers in
+just the same way that 78 or 96 would be. Later the 12 � 2, 12 � 3,
+etc., may be taught. There is less reason for knowing the multiples
+of 11 than for knowing the multiples of 15, 16, or 25.
+
+(3) _Abstract and concrete numbers._--The elaborate emphasis of the
+supposed fact that we cannot multiply 726 by 8 dollars and the still
+more elaborate explanations of why nevertheless we find the cost of 726
+articles at $8 each by multiplying 726 by 8 and calling the answer
+dollars are wasteful. The same holds of the corresponding pedantry about
+division. These imaginary difficulties should not be raised at all. The
+pupil should not think of multiplying or dividing men or dollars, but
+simply of the necessary equation and of the sort of thing that the
+missing number represents. "8 � 726 = .... Answer is dollars," or
+"8, 726, multiply. Answer is dollars," is all that he needs to think,
+and is in the best form for his thought. Concerning the distinction
+between abstract and concrete numbers, both logic and common sense as
+well as psychology support the contention of McDougle ['14, p. 206f.],
+who writes:--
+
+"The most elementary counting, even that stage when the counts were not
+carried in the mind, but merely in notches on a stick or by DeMorgan's
+stones in a pot, requires some thought; and the most advanced counting
+implies memory of things. The terms, therefore, abstract and concrete
+number, have long since ceased to be used by thinking people.
+
+"Recently the writer visited an arithmetic class in a State Normal
+School and saw a group of practically adult students confused about this
+very question concerning abstract and concrete numbers, according to
+their previous training in the conventionalities of the textbook. Their
+teacher diverted the work of the hour and she and the class spent almost
+the whole period in re�stablishing the requirements 'that the product
+must always be the same kind of unit as the multiplicand,' and 'addends
+must all be alike to be added.' This is not an exceptional case.
+Throughout the whole range of teaching arithmetic in the public schools
+pupils are obfuscated by the philosophical encumbrances which have been
+imposed upon the simplest processes of numerical work. The time is
+surely ripe, now that we are readjusting our ideas of the subject of
+arithmetic, to revise some of these wasteful and disheartening
+practices. Algebra historically grew out of arithmetic, yet it has not
+been laden with this distinction. No pupil in algebra lets _x_ equal the
+horses; he lets _x_ equal the _number_ of horses, and proceeds to drop
+the idea of horses out of his consideration. He multiplies, divides, and
+extracts the root of the _number_, sometimes handling fractions in the
+process, and finally interprets the result according to the conditions
+of his problem. Of course, in the early number work there have been the
+sense-objects from which number has been perceived, but the mind
+retreats naturally from objectivity to the pure conception of number,
+and then to the number symbol. The following is taken from the appendix
+to Horn's thesis, where a seventh grade girl gets the population of the
+United States in 1820:--
+
+  7,862,166 whites
+    233,634 free negroes
+  1,538,022 slaves
+  ---------
+  9,633,822
+
+In this problem three different kinds of addends are combined, if we
+accept the usual distinction. Some may say that this is a mistake,--that
+the pupil transformed the 'whites,' 'free negroes,' and 'slaves' into a
+common unit, such as 'people' of 'population' and then added these
+common units. But this 'explanation' is entirely gratuitous, as one will
+find if he questions the pupil about the process. It will be found that
+the child simply added the figures as numbers only and then interpreted
+the result, according to the statement of the problem, without so much
+mental gymnastics. The writer has questioned hundreds of students in
+Normal School work on this point, and he believes that the ordinary
+mind-movement is correctly set forth here, no matter how well one may
+maintain as an academic proposition that this is not logical. Many
+classes in the Eastern Kentucky State Normal have been given this
+problem to solve, and they invariably get the same result:--
+
+'In a garden on the Summit are as many cabbage-heads as the total number
+of ladies and gentlemen in this class. How many cabbage-heads in the
+garden?'
+
+And the blackboard solution looks like this each time:--
+
+  29 ladies
+  15 gentlemen
+  --
+  44 cabbage-heads
+
+So, also, one may say: I have 6 times as many sheep as you have cows. If
+you have 5 cows, how many sheep have I? Here we would multiply the
+number of cows, which is 5, by 6 and call the result 30, which must be
+linked with the idea of sheep because the conditions imposed by the
+problem demand it. The mind naturally in this work separates the pure
+number from its situation, as in algebra, handles it according to the
+laws governing arithmetical combinations, and labels the result as the
+statement of the problem demands. This is expressed in the following,
+which is tacitly accepted in algebra, and should be accepted equally in
+arithmetic:
+
+'In all computations and operations in arithmetic, all numbers are
+essentially abstract and should be so treated. They are concrete only in
+the thought process that attends the operation and interprets the
+result.'"
+
+(4) _Least common multiple._--The whole set of bonds involved in
+learning 'least common multiple' should be left out. In adding and
+subtracting fractions the pupil should _not_ find the least common
+multiple of their denominators but should find any common multiple that
+he can find quickly and correctly. No intelligent person would ever
+waste time in searching for the least common multiple of sixths, thirds,
+and halves except for the unfortunate traditions of an oversystematized
+arithmetic, but would think of their equivalents in sixths or twelfths
+or twenty-fourths or _any other convenient common multiple_. The process
+of finding the least common multiple is of such exceedingly rare
+application in science or business or life generally that the textbooks
+have to resort to purely fantastic problems to give drill in its use.
+
+(5) _Greatest common divisor._--The whole set of bonds involved in
+learning 'greatest common divisor' should also be left out. In reducing
+fractions to lowest terms the pupil should divide by anything that he
+sees that he can divide by, favoring large divisors, and continue doing
+so until he gets the fraction in terms suitable for the purpose in hand.
+The reader probably never has had occasion to compute a greatest common
+divisor since he left school. If he has computed any, the chances are
+that he would have saved time by solving the problem in some other way!
+
+The following problems are taken at random from those given by one of
+the best of the textbooks that make the attempt to apply the facts of
+Greatest Common Divisor and Least Common Multiple to problems.[6] Most
+of these problems are fantastic. The others are trivial, or are better
+solved by trial and adaptation.
+
+  1. A certain school consists of 132 pupils in the high school, 154
+     in the grammar, and 198 in the primary grades. If each group is
+     divided into sections of the same number containing as many pupils
+     as possible, how many pupils will there be in each section?
+
+  2. A farmer has 240 bu. of wheat and 920 bu. of oats, which he
+     desires to put into the least number of boxes of the same capacity,
+     without mixing the two kinds of grain. Find how many bushels each
+     box must hold.
+
+  3. Four bells toll at intervals of 3, 7, 12, and 14 seconds
+     respectively, and begin to toll at the same instant. When will
+     they next toll together?
+
+  4. A, B, C, and D start together, and travel the same way around an
+     island which is 600 mi. in circuit. A goes 20 mi. per day, B 30,
+     C 25, and D 40. How long must their journeying continue, in order
+     that they may all come together again?
+
+  5. The periods of three planets which move uniformly in circular
+     orbits round the sun, are respectively 200, 250, and 300 da.
+     Supposing their positions relatively to each other and the sun
+     to be given at any moment, determine how many da. must elapse
+     before they again have exactly the same relative positions.
+
+  [6] McLellan and Ames, _Public School Arithmetic_ [1900].
+
+(6) _Rare and unimportant words._--The bonds between rare or unimportant
+words and their meanings should not be formed for the mere sake of
+verbal variety in the problems of the textbook. A pupil should not be
+expected to solve a problem that he cannot read. He should not be
+expected in grades 2 and 3, or even in grade 4, to read words that he
+has rarely or never seen before. He should not be given elaborate drill
+in reading during the time devoted to the treatment of quantitative
+facts and relations.
+
+All this is so obvious that it may seem needless to relate. It is not.
+With many textbooks it is now necessary to give definite drill in
+reading the words in the printed problems intended for grades 2, 3, and
+4, or to replace them by oral statements, or to leave the pupils in
+confusion concerning what the problems are that they are to solve. Many
+good teachers make a regular reading-lesson out of every page of
+problems before having them solved. There should be no such necessity.
+
+To define _rare_ and _unimportant_ concretely, I will say that for
+pupils up to the middle of grade 3, such words as the following are rare
+and unimportant (though each of them occurs in the very first fifty
+pages of some well-known beginner's book in arithmetic).
+
+  absentees
+  account
+  Adele
+  admitted
+  Agnes
+  agreed
+  Albany
+  Allen
+  allowed
+  alternate
+  Andrew
+  Arkansas
+  arrived
+  assembly
+  automobile
+  baking powder
+  balance
+  barley
+  beggar
+  Bertie
+  Bessie
+  bin
+  Boston
+  bouquet
+  bronze
+  buckwheat
+  Byron
+  camphor
+  Carl
+  Carrie
+  Cecil
+  Charlotte
+  charity
+  Chicago
+  cinnamon
+  Clara
+  clothespins
+  collect
+  comma
+  committee
+  concert
+  confectioner
+  cranberries
+  crane
+  currants
+  dairyman
+  Daniel
+  David
+  dealer
+  debt
+  delivered
+  Denver
+  department
+  deposited
+  dictation
+  discharged
+  discover
+  discovery
+  dish-water
+  drug
+  due
+  Edgar
+  Eddie
+  Edwin
+  election
+  electric
+  Ella
+  Emily
+  enrolled
+  entertainment
+  envelope
+  Esther
+  Ethel
+  exceeds
+  explanation
+  expression
+  generally
+  gentlemen
+  Gilbert
+  Grace
+  grading
+  Graham
+  grammar
+  Harold
+  hatchet
+  Heralds
+  hesitation
+  Horace Mann
+  impossible
+  income
+  indicated
+  inmost
+  inserts
+  installments
+  instantly
+  insurance
+  Iowa
+  Jack
+  Jennie
+  Johnny
+  Joseph
+  journey
+  Julia
+  Katherine
+  lettuce-plant
+  library
+  Lottie
+  Lula
+  margin
+  Martha
+  Matthew
+  Maud
+  meadow
+  mentally
+  mercury
+  mineral
+  Missouri
+  molasses
+  Morton
+  movements
+  muslin
+  Nellie
+  nieces
+  Oakland
+  observing
+  obtained
+  offered
+  office
+  onions
+  opposite
+  original
+  package
+  packet
+  palm
+  Patrick
+  Paul
+  payments
+  peep
+  Peter
+  perch
+  phaeton
+  photograph
+  piano
+  pigeons
+  Pilgrims
+  preserving
+  proprietor
+  purchased
+  Rachel
+  Ralph
+  rapidity
+  rather
+  readily
+  receipts
+  register
+  remanded
+  respectively
+  Robert
+  Roger
+  Ruth
+  rye
+  Samuel
+  San Francisco
+  seldom
+  sheared
+  shingles
+  skyrockets
+  sloop
+  solve
+  speckled
+  sponges
+  sprout
+  stack
+  Stephen
+  strap
+  successfully
+  suggested
+  sunny
+  supply
+  Susan
+  Susie's
+  syllable
+  talcum
+  term
+  test
+  thermometer
+  Thomas
+  torpedoes
+  trader
+  transaction
+  treasury
+  tricycle
+  tube
+  two-seated
+  united
+  usually
+  vacant
+  various
+  vase
+  velocipede
+  votes
+  walnuts
+  Walter
+  Washington
+  watched
+  whistle
+  woodland
+  worsted
+
+(7) _Misleading facts and procedures._--Bonds should not be formed
+between articles of commerce and grossly inaccurate prices therefor,
+between events and grossly improbable consequences, or causes or
+accompaniments thereof, nor between things, qualities, and events which
+have no important connections one with another in the real world. In
+general, things should not be put together in the pupil's mind that do
+not belong together.
+
+If the reader doubts the need of this warning let him examine problems 1
+to 5, all from reputable books that are in common use, or have been
+within a few years, and consider how addition, subtraction, and the
+habits belonging with each are confused by exercise 6.
+
+  1. If a duck flying 3/5 as fast as a hawk flies 90 miles in an hour,
+     how fast does the hawk fly?
+
+  2. At 5/8 of a cent apiece how many eggs can I buy for $60?
+
+  3. At $.68 a pair how many pairs of overshoes can you buy for $816?
+
+  4. At $.13 a dozen how many dozen bananas can you buy for $3.12?
+
+  5. How many pecks of beans can be put into a box that will hold just
+     21 bushels?
+
+  6. Write answers:
+
+     537    Beginning at the bottom say 11, 18, and 2 (writing it in
+     365    its place) are 20. 5, 11, 14, and 6 (writing it) are 20,
+       ?    5, 10. The number, omitted, is 62.
+      36
+    ----
+    1000
+
+  _a._  581     _b._  625     _c._  752     _d._  314     _e._    ?
+         97             ?           414           429           845
+        364            90           130             ?           223
+          ?           417             ?            76            95
+       ----          ----          ----          ----          ----
+       1758          2050          2460          1000          2367
+
+(8) _Trivialities and absurdities._--Bonds should not be formed between
+insignificant or foolish questions and the labor of answering them,
+nor between the general arithmetical work of the school and such
+insignificant or foolish questions. The following are samples from
+recent textbooks of excellent standing:--
+
+  On one side of George's slate there are 32 words, and on the other
+  side 26 words. If he erases 6 words from one side, and 8 from the
+  other, how many words remain on his slate?
+
+  A certain school has 14 rooms, and an average of 40 children in a
+  room. If every one in the school should make 500 straight marks on
+  each side of his slate, how many would be made in all?
+
+  8 times the number of stripes in our flag is the number of years
+  from 1800 until Roosevelt was elected President. In what year was
+  he elected President?
+
+  From the Declaration of Independence to the World's Fair in Chicago
+  was 9 times as many years as there are stripes in the flag. How
+  many years was it?
+
+(9) _Useless methods._--Bonds should not be formed between a described
+situation and a method of treating the situation which would not be a
+useful one to follow in the case of the real situation. For example, "If
+I set 96 trees in rows, sixteen trees in a row, how many rows will I
+have?" forms the habit of treating by division a problem that in reality
+would be solved by counting the rows. So also "I wish to give 25 cents
+to each of a group of boys and find that it will require $2.75. How many
+boys are in the group?" forms the habit of answering a question by
+division whose answer must already have been present to give the data of
+the problem.
+
+(10) _Problems whose answers would, in real life, be already
+known._--The custom of giving problems in textbooks which could not
+occur in reality because the answer has to be known to frame the problem
+is a natural result of the lazy author's tendency to work out a problem
+to fit a certain process and a certain answer. Such bogus problems are
+very, very common. In a random sampling of a dozen pages of "General
+Review" problems in one of the most widely used of recent textbooks, I
+find that about 6 percent of the problems are of this sort. Among the
+problems extemporized by teachers these bogus problems are probably
+still more frequent. Such are:--
+
+  A clerk in an office addressed letters according to a given list.
+  After she had addressed 2500, 4/9 of the names on the list had not
+  been used; how many names were in the entire list?
+
+  The Canadian power canal at Sault Ste. Marie furnished 20,000
+  horse power. The canal on the Michigan side furnished 2-1/2
+  times as much. How many horse power does the latter furnish?
+
+It may be asserted that the ideal of giving as described problems only
+problems that might occur and demand the same sort of process for
+solution with a real situation, is too exacting. If a problem is
+comprehensible and serves to illustrate a principle or give useful
+drill, that is enough, teachers may say. For really scientific teaching
+it is not enough. Moreover, if problems are given merely as tests of
+knowledge of a principle or as means to make some fact or principle
+clear or emphatic, and are not expected to be of direct service in the
+quantitative work of life, it is better to let the fact be known. For
+example, "I am thinking of a number. Half of this number is twice six.
+What is the number?" is better than "A man left his wife a certain sum
+of money. Half of what he left her was twice as much as he left to his
+son, who receives $6000. How much did he leave his wife?" The former is
+better because it makes no false pretenses.
+
+(11) _Needless linguistic difficulties._--It should be unnecessary to
+add that bonds should not be formed between the pupil's general attitude
+toward arithmetic and needless, useless difficulty in language or
+needless, useless, wrong reasoning. Our teaching is, however, still
+tainted by both of these unfortunate connections, which dispose the
+pupil to think of arithmetic as a mystery and folly.
+
+Consider, for example, the profitless linguistic difficulty of problems
+1-6, whose quantitative difficulties are simply those of:--
+
+  1.  5 + 8 + 3 + 7
+  2.  64 � 8, and knowledge that 1 peck = 8 quarts
+  3.  12 � 4
+  4.  6 � 2
+  5.  3 � 2
+  6.  4 � 4
+
+  1. What amount should you obtain by putting together 5 cents, 8
+     cents, 3 cents, and 7 cents? Did you find this result by adding or
+     multiplying?
+
+  2. How many times must you empty a peck measure to fill a basket
+     holding 64 quarts of beans?
+
+  3. If a girl commits to memory 4 pages of history in one day, in
+     how many days will she commit to memory 12 pages?
+
+  4. If Fred had 6 chickens how many times could he give away 2
+     chickens to his companions?
+
+  5. If a croquet-player drove a ball through 2 arches at each
+     stroke, through how many arches will he drive it by 3 strokes?
+
+  6. If mamma cut the pie into 4 pieces and gave each person a piece,
+     how many persons did she have for dinner if she used 4 whole pies
+     for dessert?
+
+Arithmetically this work belongs in the first or second years of
+learning. But children of grades 2 and 3, save a few, would be utterly
+at a loss to understand the language.
+
+We are not yet free from the follies illustrated in the lessons of pages
+96 to 99, which mystified our parents.
+
+LESSON I
+
+    [Illustration: FIG. 5.]
+
+  1. In this picture, how many girls are in the swing?
+
+  2. How many girls are pulling the swing?
+
+  3. If you count both girls together, how many are they?
+     _One_ girl and _one_ other girl are how many?
+
+  4. How many kittens do you see on the stump?
+
+  5. How many on the ground?
+
+  6. How many kittens are in the picture? One kitten and one other
+     kitten are how many?
+
+  7. If you should ask me how many girls are in the swing, or how
+     many kittens are on the stump, I could answer aloud, _One_; or I
+     could write _One_; or thus, _1_.
+
+  8. If I write _One_, this is called the _word One_.
+
+  9. This, _1_, is named a _figure One_, because it means the same as
+     the word _One_, and stands for _One_.
+
+ 10. Write 1. What is this named? Why?
+
+ 11. A figure 1 may stand for _one_ girl, _one_ kitten, or _one_
+     anything.
+
+ 12. When children first attend school, what do they begin to learn?
+     _Ans._ Letters and words.
+
+ 13. Could you read or write before you had learned either letters
+     or words?
+
+ 14. If we have all the _letters_ together, they are named the
+     Alphabet.
+
+ 15. If we write or speak _words_, they are named Language.
+
+ 16. You are commencing to study Arithmetic; and you can read and
+     write in Arithmetic only as you learn the Alphabet and Language
+     of Arithmetic. But little time will be required for this purpose.
+
+LESSON II
+
+    [Illustration: FIG. 6.]
+
+  1. If we speak or write words, what do we name them, when taken
+     together?
+
+  2. What are you commencing to study? _Ans._ Arithmetic.
+
+  3. What Language must you now learn?
+
+  4. What do we name this, 1? Why?
+
+  5. This figure, 1, is part of the Language of Arithmetic.
+
+  6. If I should write something to stand for _Two_--_two_ girls,
+     _two_ kittens, or _two_ things of any kind--what do you think we
+     would name it?
+
+  7. A _figure Two_ is written thus: _2._ Make a _figure two_.
+
+  8. Why do we name this a _figure two_?
+
+  9. This figure two (2) is part of the Language of Arithmetic.
+
+ 10. In this picture one boy is sitting, playing a flageolet. What
+     is the other boy doing? If the boy standing should sit down by the
+     other, how many boys would be sitting together? One boy and one
+     other boy are how many boys?
+
+ 11. You see a flageolet and a violin. They are musical instruments.
+     One musical instrument and one other musical instrument are how
+     many?
+
+ 12. I will write thus: 1 1 2. We say that 1 boy and 1 other boy,
+     counted together, are 2 boys; or are equal to 2 boys. We will now
+     write something to show that the first 1 and the other 1 are to be
+     counted together.
+
+ 13. We name a line drawn thus, -, a _horizontal line_. Draw such a
+     line. Name it.
+
+ 14. A line drawn thus, |, we name a _vertical line_. Draw such a
+     line. Name it.
+
+ 15. Now I will put two such lines together; thus, +. What kind of a
+     line do we name the first (-)? And what do we name the last? (|)?
+     Are these lines long or short? Where do they cross each other?
+
+ 16. Each of you write thus: -, |, +.
+
+ 17. This, +, is named _Plus_. _Plus_ means _more_; and + also means
+     _more_.
+
+ 18. I will write.
+
+        _One and One More Equal Two._
+
+ 19. Now I will write part of this in the Language of Arithmetic.
+     I write the first _One_ thus, 1; then the other _One_ thus, 1.
+     Afterward I write, for the word _More_, thus, +, placing
+     the + between 1 and 1, so that the whole stands thus: 1 + 1.
+     As I write, I say, _One and One more_.
+
+ 20. Each of you write 1 + 1. Read what you have written.
+
+ 21. This +, when written between the 1s, shows that they are to be
+     put together, or counted together, so as to make 2.
+
+ 22. Because + shows what is to be done, it is called a _Sign_. If
+     we take its name, _Plus_, and the word _Sign_, and put both words
+     together, we have _Sign Plus_, or _Plus Sign_. In speaking of this
+     we may call it _Sign Plus_, or _Plus Sign_, or _Plus_.
+
+ 23. 1, 2, +, are part of the Language of Arithmetic.
+
+        _Write the following in the Language of Arithmetic_:
+
+ 24. One and one more.
+
+ 25. One and two more.
+
+ 26. Two and one more.
+
+(12) _Ambiguities and falsities._--Consider the ambiguities and false
+reasoning of these problems.
+
+  1. If you can earn 4 cents a day, how much can you earn in 6 weeks?
+     (Are Sundays counted? Should a child who earns 4 cents some day
+     expect to repeat the feat daily?)
+
+  2. How many lines must you make to draw ten triangles and five
+     squares? (I can do this with 8 lines, though the answer the book
+     requires is 50.)
+
+  3. A runner ran twice around an 1/8 mile track in two minutes. What
+     distance did he run in 2/3 of a minute? (I do not know, but I do
+     know that, save by chance, he did not run exactly 2/3 of 1/8 mile.)
+
+  4. John earned $4.35 in a week, and Henry earned $1.93. They put
+     their money together and bought a gun. What did it cost? (Maybe $5,
+     maybe $10. Did they pay for the whole of it? Did they use all their
+     earnings, or less, or more?)
+
+  5. Richard has 12 nickels in his purse. How much more than 50 cents
+     would you give him for them? (Would a wise child give 60 cents to a
+     boy who wanted to swap 12 nickels therefor, or would he suspect a
+     trick and hold on to his own coins?)
+
+  6. If a horse trots 10 miles in one hour how far will he travel in
+     9 hours?
+
+  7. If a girl can pick 3 quarts of berries in 1 hour how many quarts
+     can she pick in 3 hours?
+
+       (These last two, with a teacher insisting on the 90 and 9, might
+       well deprive a matter-of-fact boy of respect for arithmetic for
+       weeks thereafter.)
+
+     The economics and physics of the next four problems speak for
+     themselves.
+
+  8. I lost $15 by selling a horse for $85. What was the value of the
+     horse?
+
+  9. If floating ice has 7 times as much of it under the surface of
+     the water as above it, what part is above water? If an iceberg is
+     50 ft. above water, what is the entire height of the iceberg? How
+     high above water would an iceberg 300 ft. high have to be?
+
+ 10. A man's salary is $1000 a year and his expenses $625. How many
+     years will elapse before he is worth $10,000 if he is worth $2500
+     at the present time?
+
+ 11. Sound travels 1120 ft. a second. How long after a cannon is
+     fired in New York will the report be heard in Philadelphia, a
+     distance of 90 miles?
+
+
+GUIDING PRINCIPLES
+
+The reader may be wearied of these special details concerning bonds now
+neglected that should be formed and useless or harmful bonds formed for
+no valid reason. Any one of them by itself is perhaps a minor matter,
+but when we have cured all our faults in this respect and found all the
+possibilities for wiser selection of bonds, we shall have enormously
+improved the teaching of arithmetic. The ideal is such choice of bonds
+(and, as will be shown later, such arrangement of them) as will most
+improve the functions in question at the least cost of time and effort.
+The guiding principles may be kept in mind in the form of seven simple
+but golden rules:--
+
+1. Consider the situation the pupil faces.
+
+2. Consider the response you wish to connect with it.
+
+3. Form the bond; do not expect it to come by a miracle.
+
+4. Other things being equal, form no bond that will have to be broken.
+
+5. Other things being equal, do not form two or three bonds when one
+will serve.
+
+6. Other things being equal, form bonds in the way that they are
+required later to act.
+
+7. Favor, therefore, the situations which life itself will offer, and
+the responses which life itself will demand.
+
+
+
+
+CHAPTER V
+
+THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS
+
+
+An inventory of the bonds to be formed in learning arithmetic should be
+accompanied by a statement of how strong each bond is to be made and
+kept year by year. Since, however, the inventory itself has been
+presented here only in samples, the detailed statement of desired
+strength for each bond cannot be made. Only certain general facts will
+be noted here.
+
+
+THE NEED OF STRONGER ELEMENTARY BONDS
+
+The constituent bonds involved in the fundamental operations with
+numbers need to be much stronger than they now are. Inaccuracy in these
+operations means weakness of the constituent bonds. Inaccuracy exists,
+and to a degree that deprives the subject of much of its possible
+disciplinary value, makes the pupil's achievements of slight value for
+use in business or industry, and prevents the pupil from verifying his
+work with new processes by some previously acquired process.
+
+The inaccuracy that exists may be seen in the measurements made by the
+many investigators who have used arithmetical tasks as tests of fatigue,
+practice, individual differences and the like, and in the special
+studies of arithmetical achievements for their own sake made by Courtis
+and others.
+
+Burgerstein ['91], using such examples as
+
+    28704516938276546397
+  + 35869427359163827263
+  ----------------------
+
+and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5
+or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred
+answer-figures, or 3/5 of an error per example. The children were 9-1/2
+to 15 years old. Laser ['94], using the same sort of addition and
+multiplication, found somewhat over 3 errors per hundred answer-figures
+in the case of boys and girls averaging 11-1/2 years, during the period
+of their most accurate work. Holmes ['95], using addition of the sort
+just described, found 346 errors in 23,713 answer-figures or about 1-1/2
+per hundred. The children were from all grades from the third to the
+eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained
+per minute. Friedrich ['97] with similar examples, giving the very long
+time of 20 minutes for obtaining about 200 answer-figures, found from 1
+to 2 per hundred wrong. King ['07] had children in grade 5 do sums, each
+consisting of 5 two-place numbers. In the most accurate work-period,
+they made 1 error per 20 columns. In multiplying a four-place by a
+four-place number they had less than one total answer right out of
+three. In New York City Courtis found ['11-'12] with his Test 7 that in
+12 minutes the average achievement of fourth-grade children is 8.8 units
+attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with
+5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15
+attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right.
+These results are near enough to those obtained from the country at
+large to serve as a text here.
+
+The following were set as official standards, in an excellent school
+system, Courtis Series B being used:--
+
+
+                            SPEED        PERCENT OF
+                  GRADE.    ATTEMPTS.    CORRECT ANSWERS.
+  Addition          8         12           80
+                    7         11           80
+                    6         10           70
+                    5          9           70
+                    4          8           70
+
+  Subtraction       8         12           90
+                    7         11           90
+                    6         10           90
+                    5          9           80
+                    4          7           80
+
+  Multiplication    8         11           80
+                    7         10           80
+                    6          9           80
+                    5          7           70
+                    4          6           60
+
+  Division          8         11           90
+                    7         10           90
+                    6          8           80
+                    5          6           70
+                    4          4           60
+
+Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like
+those printed below, children in grade 4 got on the average less than 80
+percent of correct answers. Their average speed was about 2 columns per
+minute. In doing division of the sort printed below children of grades 3
+_B_ and 4 _A_ got less than 95 percent of correct answers, the average
+speed being 4 divisions per minute. In both cases the slower computers
+were no more accurate than the faster ones. Practice improved the speed
+very rapidly, but the accuracy remained substantially unchanged. Brown
+['11 and '12] found a similar low status of ability and notable
+improvement from a moderate amount of special practice.
+
+   3    5    6    2    3    8    9    7    4    9
+   7    9    6    5    5    6    4    5    8    2
+   3    4    7    8    7    3    7    9    3    7
+   8    8    4    8    2    6    8    2    9    8
+   2    2    4    7    6    9    8    5    6    2
+   6    9    5    7    8    5    2    3    2    4
+   9    6    4    2    7    2    9    4    4    5
+   3    3    7    9    9    9    2    8    9    7
+   6    8    9    6    4    7    7    9    2    4
+   8    4    6    9    9    2    6    9    8    9
+  --   --   --   --   --   --   --   --   --   --
+
+  20 = .... 5s
+  56 = .... 9s and .... _r_.
+  30 = .... 7s and .... _r_.
+  89 = .... 9s and .... _r_.
+  20 = .... 8s and .... _r_.
+  56 = .... 6s and .... _r_.
+  31 = .... 4s and .... _r_.
+  86 = .... 9s and .... _r_.
+
+It is clear that numerical work as inaccurate as this has little or no
+commercial or industrial value. If clerks got only six answers out of
+ten right as in the Courtis tests, one would need to have at least four
+clerks make each computation and would even then have to check many of
+their discrepancies by the work of still other clerks, if he wanted his
+accounts to show less than one error per hundred accounting units of the
+Courtis size.
+
+It is also clear that the "habits of ... absolute accuracy, and
+satisfaction in truth as a result" which arithmetic is supposed to
+further must be largely mythical in pupils who get right answers only
+from three to nine times out of ten!
+
+
+EARLY MASTERY
+
+The bonds in question clearly must be made far stronger than they now
+are. They should in fact be strong enough to abolish errors in
+computation, except for those due to temporary lapses. It is much
+better for a child to know half of the multiplication tables, and to
+know that he does not know the rest, than to half-know them all; and
+this holds good of all the elementary bonds required for computation.
+Any bond should be made to work perfectly, though slowly, very soon
+after its formation is begun. Speed can easily be added by proper
+practice.
+
+The chief reasons why this is not done now seem to be the following:
+(1) Certain important bonds (like the additions with higher decades)
+are not given enough attention when they are first used. (2) The special
+training necessary when a bond is used in a different connection (as
+when the multiplications to 9 � 9 are used in examples like
+
+  729
+    8
+  ---
+
+where the pupil has also to choose the right number to multiply, keep in
+mind what is carried, use it properly, and write the right figure in the
+right place, and carry a figure, or remember that he carries none) is
+neglected. (3) The pupil is not taught to check his work. (4) He is not
+made responsible for substantially accurate results. Furthermore, the
+requirement of (4) without the training of (1), (2), and (3) will
+involve either a fruitless failure on the part of many pupils, or an
+utterly unjust requirement of time. The common error of supposing that
+the task of computation with integers consists merely in learning the
+additions to 9 + 9, the subtractions to 18 - 9, the multiplications to
+8 � 9, and the divisions to 81 � 9, and in applying this knowledge in
+connection with the principles of decimal notation, has had a large
+share in permitting the gross inaccuracy of arithmetical work. The bonds
+involved in 'knowing the tables' do not make up one fourth of the bonds
+involved in real adding, subtracting, multiplying, and dividing (with
+integers alone).
+
+It should be noted that if the training mentioned in (1) and (2) is
+well cared for, the checking of results as recommended in (3) becomes
+enormously more valuable than it is under present conditions, though
+even now it is one of our soundest practices. If a child knows the
+additions to higher decades so that he can add a seen one-place number
+to a thought-of two-place number in three seconds or less with a correct
+answer 199 times out of 200, there is only an infinitesimal chance that
+a ten-figure column twice added (once up, once down) a few minutes apart
+with identical answers will be wrong. Suppose that, in long
+multiplication, a pupil can multiply to 9 � 9 while keeping his place
+and keeping track of what he is 'carrying' and of where to write the
+figure he writes, and can add what he carries without losing track of
+what he is to add it to, where he is to write the unit figure, what he
+is to multiply next and by what, and what he will then have to carry, in
+each case to a surety of 99 percent of correct responses. Then two
+identical answers got by multiplying one three-place number by another a
+few minutes apart, and with reversal of the numbers, will not be wrong
+more than twice in his entire school career. Checks approach proofs when
+the constituent bonds are strong.
+
+If, on the contrary, the fundamental bonds are so weak that they do not
+work accurately, checking becomes much less trustworthy and also very
+much more laborious. In fact, it is possible to show that below a
+certain point of strength of the fundamental bonds, the time required
+for checking is so great that part of it might better be spent in
+improving the fundamental bonds.
+
+For example, suppose that a pupil has to find the sum of five numbers
+like $2.49, $5.25, $6.50, $7.89, and $3.75. Counting each act of
+holding in mind the number to be carried and each writing of a column's
+result as equivalent in difficulty to one addition, such a sum equals
+nineteen single additions. On this basis and with certain additional
+estimates[7] we can compute the practical consequences for a pupil's use
+of addition in life according to the mastery of it that he has gained in
+school.
+
+  [7] These concern allowances for two errors occurring in the same
+      example and for the same wrong answer being obtained in both
+      original work and check work.
+
+I have so computed the amount of checking a pupil will have to do to
+reach two agreeing numbers (out of two, or three, or four, or five, or
+whatever the number before he gets two that are alike), according to his
+mastery of the elementary processes. The facts appear in Table 1.
+
+It is obvious that a pupil whose mastery of the elements is that denoted
+by getting them right 96 times out of 100 will require so much time for
+checking that, even if he were never to use this ability for anything
+save a few thousand sums in addition, he would do well to improve this
+ability before he tried to do the sums. An ability of 199 out of 200, or
+995 out of 1000, seems likely to save much more time than would be taken
+to acquire it, and a reasonable defense could be made for requiring 996
+or 997 out of 1000.
+
+A precision of from 995 to 997 out of 1000 being required, and ordinary
+sagacity being used in the teaching, speed will substantially take care
+of itself. Counting on the fingers or in words will not give that
+precision. Slow recourse to memory of serial addition tables will not
+give that precision. Nothing save sure memory of the facts operating
+under the conditions of actual examples will give it. And such memories
+will operate with sufficient speed.
+
+TABLE 1
+
+THE EFFECT OF MASTERY OF THE ELEMENTARY FACTS OF ADDITION UPON THE LABOR
+REQUIRED TO SECURE TWO AGREEING ANSWERS WHEN ADDING FIVE THREE-FIGURE
+NUMBERS
+
+  ======================================================================
+  MASTERY OF   |APPROXIMATE  |APPROXIMATE  |APPROXIMATE  |APPROXIMATE
+  THE          |NUMBER OF    |NUMBER OF    |NUMBER OF    |NUMBER OF
+  ELEMENTARY   |WRONG ANSWERS|AGREEING     |AGREEING     |CHECKINGS
+  ADDITIONS    |IN SUMS OF 5 |ANSWERS,     |ANSWERS,     |REQUIRED (OVER
+  TIMES RIGHT  |THREE-PLACE  |AFTER ONE    |AFTER A      |AND ABOVE THE
+  IN 1000      |NUMBERS PER  |CHECKING,    |CHECKING OF  |FIRST GENERAL
+               |1000         |PER 1000     |THE FIRST    |CHECKING OF
+               |             |             |DISCREPANCIES|THE 1000 SUMS)
+               |             |             |             |TO SECURE TWO
+               |             |             |             |AGREEING
+               |             |             |             |RESULTS
+  -------------+-------------+-------------+-------------+--------------
+       960     |     700     |      90     |     216     |     4500
+       980     |     380     |     384     |     676     |     1200
+       990     |     190     |     656     |     906     |      470
+       995     |      95     |     819     |     975     |      210
+       996     |      76     |     854     |     984     |      165
+       997     |      54     |     895     |     992     |      115
+       998     |      38     |     925     |     996     |       80
+       999     |      19     |     962     |     999     |       40
+  -------------+-------------+-------------+-------------+--------------
+
+There is one intelligent objection to the special practice necessary to
+establish arithmetical connections so fully as to give the accuracy
+which both utilitarian and disciplinary aims require. It may be said
+that the pupils in grades 3, 4, and 5 cannot appreciate the need and
+that consequently the work will be dull, barren, and alien, without
+close personal appropriation by the pupil's nature. It is true that no
+vehement life-purpose is directly involved by the problem of perfecting
+one's power to add 7 to 28 in grade 2, or by the problem of multiplying
+253 by 8 accurately in grade 3 or by precise subtraction in long
+division in grade 4. It is also true, however, that the most humanly
+interesting of problems--one that the pupil attacks most
+whole-heartedly--will not be solved correctly unless the pupil has the
+necessary associative mechanisms in order; and the surer he is of them,
+the freer he is to think out the problem as such. Further, computation
+is not dull if the pupil can compute. He does not himself object to its
+barrenness of vital meaning, so long as the barrenness of failure is
+prevented. We must not forget that pupils like to learn. In teaching
+excessively dull individuals, who has not often observed the great
+interest which they display in anything that they are enabled to master?
+There is pathos in their joy in learning to recognize parts of speech,
+perform algebraic simplifications, or translate Latin sentences, and in
+other accomplishments equally meaningless to all their interests save
+the universal human interest in success and recognition. Still further,
+it is not very hard to show to pupils the imperative need of accuracy in
+scoring games, in the shop, in the store, and in the office. Finally,
+the argument that accurate work of this sort is alien to the pupil in
+these grades is still stronger against _inaccurate_ work of the same
+sort. If we are to teach computation with two- and three- and four-place
+numbers at all, it should be taught as a reliable instrument, not as a
+combination of vague memories and faith. The author is ready to cut
+computation with numbers above 10 out of the curriculum of grades 1-6 as
+soon as more valuable educational instruments are offered in its place,
+but he is convinced that nothing in child-nature makes a large variety
+of inaccurate computing more interesting or educative or germane to felt
+needs, than a smaller variety of accurate computing!
+
+
+THE STRENGTH OF BONDS FOR TEMPORARY SERVICE
+
+The second general fact is that certain bonds are of service for only a
+limited time and so need to be formed only to a limited and slight
+degree of strength. The data of problems set to illustrate a principle
+or improve some habit of computation are, of course, the clearest cases.
+The pupil needs to remember that John bought 3 loaves of bread and that
+they were 5-cent loaves and that he gave 25 cents to the baker only long
+enough to use the data to decide what change John should receive. The
+connections between the total described situation and the answer
+obtained, supposing some considerable computation to intervene, is a
+bond that we let expire almost as soon as it is born.
+
+It is sometimes assumed that the bond between a certain group of
+features which make a problem a 'Buy _a_ things at _b_ per thing, find
+total cost' problem or a 'Buy _a_ things at _b_ per thing, what change
+from _c_' problem or a 'What gain on buying for _a_ and selling for _b_'
+problem or a 'How many things at _a_ each can I buy for _b_ cents'
+problem--it is assumed that the bond between these essential defining
+features and the operation or operations required for solution is as
+temporary as the bonds with the name of the buyer or the price of the
+thing. It is assumed that all problems are and should be solved by some
+pure act of reasoning without help or hindrance from bonds with the
+particular verbal structure and vocabulary of the problems. Whether or
+not they _should_ be, they _are not_. Every time that a pupil solves a
+'bought-sold' problem by subtraction he strengthens the tendency to
+respond to any problem whatsoever that contains the words 'bought for'
+and 'sold for' by subtraction; and he will by no means surely stop and
+survey every such problem in all its elements to make sure that no
+other feature makes inapplicable the tendency to subtract which the
+'bought sold' evokes.
+
+To prevent pupils from responding to the form of statement rather than
+the essential facts, we should then not teach them to forget the form of
+statement, but rather give them all the common forms of statement to
+which the response in question is an appropriate response, and only
+such. If a certain form of statement does in life always signify a
+certain arithmetical procedure, the bond between it and that procedure
+may properly be made very strong.
+
+Another case of the formation of bonds to only a slight degree
+of strength concerns the use of so-called 'crutches' such as
+writing +, -, and � in copying problems like those below:--
+
+  Add       Subtract       Multiply
+  23           79             32
+  61           24              3
+  --           --             --
+
+or altering the figures when 'borrowing' in subtraction, and the like.
+Since it is undesirable that the pupil should regard the 'crutch'
+response as essential to the total procedure, or become so used to
+having it that he will be disturbed by its absence later, it is supposed
+that the bond between the situation and the crutch should not be fully
+formed. There is a better way out of the difficulty, in case crutches
+are used at all. This is to associate the crutch with a special 'set,'
+and its non-use with the general set which is to be the permanent one.
+For example, children may be taught from the start never to write
+the crutch sign or crutch figure unless the work is accompanied by
+"Write ... to help you to...."
+
+  Write - to help you to        Find the differences:--
+  remember that you must        39     67     78     56     45
+  subtract in this row.         23     44     36     26     24
+                                --     --     --     --     --
+
+  Remember that you must        Find the differences:--
+  subtract in this row.         85     27     96     38     78
+                                63     14     51     45     32
+                                --      --    --     --     --
+
+The bond evoking the use of the crutch may then be formed thoroughly
+enough so that there is no hesitation, insecurity, or error, without
+interfering to any harmful extent with the more general bond from the
+situation to work without the crutch.
+
+
+THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS
+
+Another instructive case concerns the bonds between certain words and
+their meanings, and between certain situations of commerce, industry, or
+agriculture and useful facts about these situations. Illustrations of
+the former are the bonds between _cube root_, _hectare_, _brokerage_,
+_commission_, _indorsement_, _vertex_, _adjacent_, _nonagon_, _sector_,
+_draft_, _bill of exchange_, and their meanings. Illustrations of the
+latter are the bonds from "Money being lent 'with interest' at no
+specified rate, what rate is charged?" to "The legal rate of the state,"
+from "$_X_ per M as a rate for lumber" to "Means $_X_ per thousand board
+feet, a board foot being 1 ft. by 1 ft. by 1 in."
+
+It is argued by many that such bonds are valuable for a short time;
+namely, while arithmetical procedures in connection with which they
+serve are learned, but that their value is only to serve as a means for
+learning these procedures and that thereafter they may be forgotten.
+"They are formed only as accessory means to certain more purely
+arithmetical knowledge or discipline; after this is acquired they may
+be forgotten. Everybody does in fact forget them, relearning them later
+if life requires." So runs the argument.
+
+In some cases learning such words and facts only to use them in solving
+a certain sort of problems and then forget them may be profitable. The
+practice is, however, exceedingly risky. It is true that everybody does
+in fact forget many such meanings and facts, but this commonly means
+either that they should not have been learned at all at the time that
+they were learned, or that they should have been learned more
+permanently, or that details should have been learned with the
+expectation that they themselves would be forgotten but that a general
+fact or attitude would remain. For example, duodecagon should not be
+learned at all in the elementary school; indorsement should either not
+be learned at all there, or be learned for permanence of a year or more;
+the details of the metric system should be so taught as to leave for
+several years at least knowledge of the facts that there is a system so
+named that is important, whose tables go by tens, hundreds, or
+thousands, and a tendency (not necessarily strong) to connect meter,
+kilogram, and liter with measurement by the metric system and with
+approximate estimates of their several magnitudes.
+
+If an arithmetical procedure seems to require accessory bonds which are
+to be forgotten, once the procedure is mastered, we should be suspicious
+of the value of the procedure itself. If pupils forget what compound
+interest is, we may be sure that they will usually also have forgotten
+how to compute it. Surely there is waste if they have learned what it is
+only to learn how to compute it only to forget how to compute it!
+
+
+THE STRENGTH OF BONDS CONCERNING THE REASONS FOR ARITHMETICAL PROCESSES
+
+The next case of the formation of bonds to slight strength is the
+problematic one of forming the bonds involved in understanding the
+reasons for certain processes only to forget them after the process has
+become a habit. Should a pupil, that is, learn why he inverts and
+multiplies, only to forget it as soon as he can be trusted to divide by
+a fraction? Should he learn why he puts the units figure of each partial
+product in multiplication under the figure that he multiplies by, only
+to forget the reason as soon as he has command of the process? Should he
+learn why he gets the number of square inches in a rectangle by
+multiplying the length by the width, both being expressed in linear
+inches, and forget why as soon as he is competent to make computations
+of the areas of rectangles?
+
+On general psychological grounds we should be suspicious of forming
+bonds only to let them die of starvation later, and tend to expect that
+elaborate explanations learned only to be forgotten either should not be
+learned at all, or should be learned at such a time and in such a way
+that they would not be forgotten. Especially we should expect that the
+general principles of arithmetic, the whys and wherefores of its
+fundamental ways of manipulating numbers, ought to be the last bonds of
+all to be forgotten. Details of _how_ you arranged numbers to multiply
+might vanish, but the general reasons for the placing would be expected
+to persist and enable one to invent the detailed manipulations that had
+been forgotten.
+
+This suspicion is, I think, justified by facts. The doctrine that the
+customary deductive explanations of why we invert and multiply, or place
+the partial products as we do before adding, may be allowed to be
+forgotten once the actual habits are in working order, has a suspicious
+source. It arose to meet the criticism that so much time and effort were
+required to keep these deductive explanations in memory. The fact was
+that the pupil learned to compute correctly _irrespective of_ the
+deductive explanations. They were only an added burden. His inductive
+learning that the procedure gave the right answer really taught him. So
+he wisely shuffled off the extra burden of facts about the consequences
+of the nature of a fraction or the place values of our decimal notation.
+The bonds weakened because they were not used. They were not used
+because they were not useful in the shape and at the time that they were
+formed, or because the pupil was unable to understand the explanations
+so as to form them at all.
+
+The criticism was valid and should have been met in part by replacing
+the deductive explanations by inductive verifications, and in part by
+using the deductive reasoning as a check after the process itself is
+mastered. The very same discussions of place-value which are futile as
+proof that you must do a certain thing before you have done it, often
+become instructive as an explanation of why the thing that you have
+learned to do and are familiar with and have verified by other tests
+works as well as it does. The general deductive theory of arithmetic
+should not be learned only to be forgotten. Much of it should, by most
+pupils, not be learned at all. What is learned should be learned much
+later than now, as a synthesis and rationale of habits, not as their
+creator. What is learned of such deductive theory should rank among the
+most rather than least permanent of a pupil's stock of arithmetical
+knowledge and power. There are bonds which are formed only to be lost,
+and bonds formed only to be lost _in their first form_, being used in a
+new organization as material for bonds of a higher order; but the bonds
+involved in deductive explanations of why certain processes are right
+are not such: they are not to be formed just to be forgotten, nor as
+mere prop�deutics to routine manipulations.
+
+
+PROP�DEUTIC BONDS
+
+The formation of bonds to a limited strength because they are to be lost
+in their first form, being worked over in different ways in other bonds
+to which they are prop�deutic or contributing is the most important case
+of low strength, or rather low permanence, in bonds.
+
+The bond between four 5s in a column to be added and the response of
+thinking '10, 15, 20' is worth forming, but it is displaced later by the
+multiplication bond or direct connection of 'four 5s to be added' with
+'20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc.,
+forms serial bonds which as series might well be left to disappear.
+Their separate steps are kept as permanent bonds for use in column
+addition, but their serial nature is changed from 2 (and 2) 4, (and 2)
+6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.;
+after playing their part in producing the bonds whereby any multiple of
+2 by 2 to 9, can be got, the original serial bonds are, as series,
+needed no longer. The verbal response of saying 'and' in adding, after
+helping to establish the bonds whereby the general set of the mind
+toward adding co�perates with the numbers seen or thought of to produce
+their sum, should disappear; or remain so slurred in inner speech as to
+offer no bar to speed.
+
+The rule for such bonds is, of course, to form them strongly enough so
+that they work quickly and accurately for the time being and facilitate
+the bonds that are to replace them, but not to overlearn them. There is
+a difference between learning something to be held for a short time, and
+the same amount of energy spent in learning for long retention. The
+former sort of learning is, of course, appropriate with many of these
+prop�deutic bonds.
+
+The bonds mentioned as illustrations are not _purely_ prop�deutic, nor
+formed _only_ to be transmuted into something else. Even the saying of
+'and' in addition has some genuine, intrinsic value in distinguishing
+the process of addition, and may perhaps be usefully reviewed for a
+brief space during the first steps in adding common fractions. Some such
+prop�deutic bonds may be worth while apart from their value in preparing
+for other bonds. Consider, for example, exercises like those shown below
+which are prop�deutic to long division, giving the pupil some basis in
+experience for his selection of the quotient figures. These
+multiplications are intrinsically worth doing, especially the 12s and
+25s. Whatever the pupil remembers of them will be to his advantage.
+
+  1. Count by 11s to 132, beginning 11, 22, 33.
+
+  2. Count by 12s to 144, beginning 12, 24, 36.
+
+  3. Count by 25s to 300, beginning 25, 50, 75.
+
+  4. State the missing numbers:--
+
+          A.          B.             C.                D.
+       3 11s =     5 11s =     8 ft. = .... in.     2 dozen =
+       4 12s =     3 12s =    10 ft. = .... in.     4 dozen =
+       5 12s =     6 12s =     7 ft. = .... in.    10 dozen =
+       6 11s =    12 11s =     4 ft. = .... in.     5 dozen =
+       9 11s =     2 12s =     6 ft. = .... in.     7 dozen =
+       7 12s =     9 12s =     9 ft. = .... in.    12 dozen =
+       8 12s =     7 11s =    11 ft. = .... in.     9 dozen =
+      11 11s =    12 12s =     5 ft. = .... in.     6 dozen =
+
+  5. Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents,
+     one dollar," and so on.
+
+  6. Count by 15s to $1.50.
+
+  7. Find the products. Do not use pencil. Think what they are.
+
+         A.        B.         C.          D.          E.
+       2 � 25    3 � 15     2 � 12      4 � 11      6 � 25
+       3 � 25   10 � 15     2 � 15      4 � 15      6 � 15
+       5 � 25    4 � 15     2 � 25      4 � 12      6 � 12
+      10 � 25    2 � 15     2 � 11      4 � 25      6 � 11
+       4 � 25    7 � 15     3 � 25      5 � 11      7 � 12
+       6 � 25    9 � 15     3 � 15      5 � 12      7 � 15
+       8 � 25    5 � 15     3 � 11      5 � 15      7 � 25
+       7 � 25    8 � 15     3 � 12      5 � 25      7 � 11
+       9 � 25    6 � 15     8 � 12      9 � 12      8 � 25
+
+  State the missing numbers:--
+
+    A.  36 = .... 12s    B.  44 = .... 11s    C.  50 = .... 25s
+        60 = .... 12s        88 = .... 11s       125 = .... 25s
+        24 = .... 12s        77 = .... 11s        75 = .... 25s
+        48 = .... 12s        55 = .... 11s       200 = .... 25s
+       144 = .... 12s        99 = .... 11s       250 = .... 25s
+       108 = .... 12s       110 = .... 11s       175 = .... 25s
+        72 = .... 12s        33 = .... 11s       225 = .... 25s
+        96 = .... 12s        66 = .... 11s       150 = .... 25s
+        84 = .... 12s        22 = .... 11s       100 = .... 25s
+
+  Find the quotients and remainders. If you need to use paper and pencil
+  to find them, you may. But find as many as you can without pencil and
+  paper. Do Row A first. Then do Row B. Then Row C, etc.
+             __      __      __      __      __      __
+  Row A.  11|45   12|45   25|45   15|45   21|45   22|45
+             __      __      __      __      __      __
+  Row B.  25|55   11|55   12|55   15|55   22|55   30|55
+             __      __      __      __      __      __
+  Row C.  12|60   25|60   15|60   11|60   30|60   21|60
+             __      __      __      __      __      __
+  Row D.  12|75   11|75   15|75   25|75   30|75   35|75
+             ___     ___     ___     ___     ___     ___
+  Row E.  11|100  12|100  25|100  15|100  30|100  22|100
+             __      __      __      __      __      __
+  Row F.  11|96   12|96   25|96   15|96   30|96   22|96
+             ___     ___     ___     ___     ___     ___
+  Row G.  25|105  11|105  15|105  12|105  22|105  35|105
+             __      __      __      __      __      __
+  Row H.  12|64   15|64   25|64   11|64   22|64   21|64
+             __      __      __      __      __      __
+  Row I.  11|80   12|80   15|80   25|80   35|80   21|80
+             ___     ___     ___     ___     ___     ___
+  Row J.  25|200  30|200  75|200  63|200  65|200  66|200
+
+  Do this section again. Do all the first column first. Then do
+  the second column, then the third, and so on.
+
+Consider, from the same point of view, exercises like (3 � 4) + 2,
+(7 � 6) + 5, (9 � 4) + 6, given as a preparation for written
+multiplication. The work of
+
+  48   68   47
+   3    7    9
+  --   --   --
+
+and the like is facilitated if the pupil has easy control of the process
+of getting a product, and keeping it in mind while he adds a one-place
+number to it. The practice with (3 � 4) + 2 and the like is also good
+practice intrinsically. So some teachers provide systematic preparatory
+drills of this type just before or along with the beginning of short
+multiplication.
+
+In some cases the bonds are purely prop�deutic or are formed _only_ for
+later reconstruction. They then differ little from 'crutches.' The
+typical crutch forms a habit which has actually to be broken, whereas
+the purely prop�deutic bond forms a habit which is left to rust out from
+disuse.
+
+For example, as an introduction to long division, a pupil may be given
+exercises using one-figure divisors in the long form, as:--
+
+     773 and 5 remainder
+   ______
+  7)5416
+    49
+    --
+     51
+     49
+     --
+      26
+      21
+      --
+       5
+
+The important recommendation concerning these purely prop�deutic bonds,
+and bonds formed only for later reconstruction, is to be very critical
+of them, and not indulge in them when, by the exercise of enough
+ingenuity, some bond worthy of a permanent place in the individual's
+equipment can be devised which will do the work as well. Arithmetical
+teaching has done very well in this respect, tending to err by leaving
+out really valuable preparatory drills rather than by inserting
+uneconomical ones. It is in the teaching of reading that we find the
+formation of prop�deutic bonds of dubious value (with letters,
+phonograms, diacritical marks, and the like) often carried to
+demonstrably wasteful extremes.
+
+
+
+
+CHAPTER VI
+
+THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE
+ORGANIZATION OF ABILITIES
+
+
+THE AMOUNT OF PRACTICE
+
+It will be instructive if the reader will perform the following
+experiment as an introduction to the discussion of this chapter, before
+reading any of the discussion.
+
+Suppose that a pupil does all the work, oral and written, computation
+and problem-solving, presented for grades 1 to 6 inclusive (that is, in
+the first two books of a three-book series) in the average textbook now
+used in the elementary school. How many times will he have exercised
+each of the various bonds involved in the four operations with integers
+shown below? That is, how many times will he have thought, "1 and 1
+are 2," "1 and 2 are 3," etc.? Every case of the action of each bond
+is to be counted.
+
+
+THE FUNDAMENTAL BONDS
+
+  1 + 1                             2 - 1        1 � 1      2 � 1
+  1 + 2                             2 - 2        2 � 1      2 � 2
+  1 + 3                                          3 � 1
+  1 + 4                                          4 � 1
+  1 + 5                             3 - 1        5 � 1      3 � 1
+  1 + 6                             3 - 2        6 � 1      3 � 2
+  1 + 7                             3 - 3        7 � 1      3 � 3
+  1 + 8                                          8 � 1
+  1 + 9                                          9 � 1
+
+                                    4 - 1                   4 � 1
+                                    4 - 2                   4 � 2
+  11 (or 21 or 31, etc.) + 1        4 - 3        1 � 2      4 � 3
+  11        "            + 2        4 - 4        2 � 2      4 � 4
+  11        "            + 3                     3 � 2
+  11        "            + 4                     4 � 2
+  11        "            + 5        5 - 1        5 � 2      5 � 1
+  11        "            + 6        5 - 2        6 � 2      5 � 2
+  11        "            + 7        5 - 3        7 � 2      5 � 3
+  11        "            + 8        5 - 4        8 � 2      5 � 4
+  11        "            + 9        5 - 5        9 � 2      5 � 5
+
+                                    6 - 1        1 � 3      6 � 1
+  2 + 1                             6 - 2        2 � 3      6 � 2
+  2 + 2                             6 - 3        3 � 3      6 � 3
+  2 + 3                             6 - 4        4 � 3      6 � 4
+  2 + 4                             6 - 5        5 � 3      6 � 5
+  2 + 5                             6 - 6        6 � 3      6 � 6
+  2 + 6                                          7 � 3
+  2 + 7                                          8 � 3
+  2 + 8                             7 - 1        9 � 3      7 � 1
+  2 + 9                             7 - 2                   7 � 2
+                                    7 - 3                   7 � 3
+                                    7 - 4        1 � 4      7 � 4
+  12 (or 22 or 32, etc.) + 1        7 - 5        2 � 4      7 � 5
+  12        "            + 2        7 - 6        and so on  7 � 6
+                                    7 - 7        to 9 � 9   7 � 7
+  and so on to                      and so on               and so on to
+  9 + 9                             to 18 - 9               82 � 9
+  19 (or 29 or 39, etc.) + 9                                83 � 9, etc.
+
+If estimating for the entire series is too long a task, it will be
+sufficient to use eight or ten from each, say:--
+
+   3 + 2    13, 23, etc. + 2     7 + 2    17, 27, etc. + 2
+     " 3           "       3       " 3            "      3
+     " 4           "       4       " 4            "      4
+     " 5           "       5       " 5            "      5
+     " 6           "       6       " 6            "      6
+     " 7           "       7       " 7            "      7
+     " 8           "       8       " 8            "      8
+     " 9           "       9       " 9            "      9
+
+   3 - 3     7 - 7     9 � 7    63 � 9
+   4   "     8   "     7 � 9    64   "
+   5   "     9   "     8 � 6    65   "
+   6   "    10   "     6 � 8    66   "
+   7   "    11   "              67   "
+   8   "    12   "              68   "
+   9   "    13   "              69   "
+  10   "    14   "              70   "
+  11   "    15   "              71   "
+  12   "    16   "
+
+
+TABLE 2
+
+ESTIMATES OF THE AMOUNT OF PRACTICE PROVIDED IN BOOKS I AND II OF THE
+AVERAGE THREE-BOOK TEXT IN ARITHMETIC; BY 50 EXPERIENCED TEACHERS
+
+  ======================================================================
+                         | LOWEST | MEDIAN | HIGHEST |RANGE REQUIRED TO
+    ARITHMETICAL FACT    |ESTIMATE|ESTIMATE|ESTIMATE | INCLUDE HALF OF
+                         |        |        |         |  THE ESTIMATES
+  -----------------------+--------+--------+---------+------------------
+  3 or 13 or 23, etc. + 2|   25   |  1500  |1,000,000|    800-5000
+       "         "      3|   24   |  1450  |   80,000|    475-5000
+       "         "      4|   23   |  1150  |   50,000|    750-5000
+       "         "      5|   22   |  1400  |   44,000|    700-5000
+       "         "      6|   21   |  1350  |   41,000|    700-4500
+       "         "      7|   21   |  1500  |   37,000|    600-4000
+       "         "      8|   20   |  1400  |   33,000|    550-4100
+       "         "      9|   20   |  1150  |   28,000|    650-4500
+                         |        |        |         |
+  7 or 17 or 27, etc. + 2|   20   |  1250  |2,000,000|    600-5000
+       "         "      3|   19   |  1100  |1,000,000|    650-4900
+       "         "      4|   18   |  1000  |   80,000|    650-4900
+       "         "      5|   17   |  1300  |   80,000|    650-4400
+       "         "      6|   16   |  1100  |   29,000|    650-4500
+       "         "      7|   15   |  1100  |   25,000|    500-4500
+       "         "      8|   13   |  1100  |   21,000|    650-3800
+       "         "      9|   10   |  1275  |   17,000|    500-4000
+                         |        |        |         |
+          3 - 3          |   25   |  1000  |  100,000|    500-4000
+          4 - 3          |   20   |  1050  |  500,000|    525-3000
+          5 - 3          |   20   |  1100  |2,500,000|    650-4200
+          6 - 3          |   10   |  1050  |   21,000|    650-3250
+          7 - 3          |   22   |  1100  |   15,000|    550-3050
+          8 - 3          |   21   |  1075  |   15,000|    650-3000
+          9 - 3          |   21   |  1000  |   15,000|    700-2600
+         10 - 3          |   20   |  1000  |   20,000|    600-2500
+         11 - 3          |   20   |  1000  |   15,000|    465-2550
+         12 - 3          |   18   |  1000  |   15,000|    650-2100
+                         |        |        |         |
+          7 - 7          |   10   |  1000  |   18,000|    425-3000
+          8 - 7          |   15   |  1000  |   18,000|    413-3100
+          9 - 7          |   15   |   950  |   18,000|    550-3000
+         10 - 7          |   15   |   950  |   18,000|    600-3950
+         11 - 7          |   10   |   900  |   18,000|    550-3000
+         12 - 7          |   10   |   925  |   18,000|    525-3100
+         13 - 7          |   10   |   900  |   18,000|    500-2600
+         14 - 7          |   10   |   900  |   18,000|    500-3100
+         15 - 7          |   10   |   925  |   18,000|    500-3000
+         16 - 7          |   10   |   875  |   18,000|    500-2500
+                         |        |        |         |
+          9 � 7          |   10   |   700  |   20,000|    500-2000
+          7 � 9          |   10   |   700  |   20,000|    500-1750
+          8 � 6          |   10   |   750  |   20,000|    500-2500
+          6 � 8          |    9   |   700  |   20,000|    500-2500
+                         |        |        |         |
+         63 � 9          |    9   |   500  |    4,500|    300-2500
+         64 � 9          |    9   |   200  |    4,000|    100- 700
+         65 � 9          |    8   |   200  |    4,000|    100- 600
+         66 � 9          |    7   |   200  |    4,000|    100- 550
+         67 � 9          |    7   |   200  |    4,000|     75- 450
+         68 � 9          |    6   |   200  |    4,000|     87- 575
+         69 � 9          |    6   |   200  |    4,000|     87- 450
+         70 � 9          |    5   |   200  |    4,000|     75- 575
+         71 � 9          |    5   |   200  |    4,000|     75- 700
+                         |        |        |         |
+           _XX_          |   40   |   550  |1,000,000|    300-2000
+           _XO_          |   20   |   500  |   11,500|    150-2000
+          _XXX_          |   15   |   450  |   12,000|    100-1000
+          _XXO_          |   25   |   400  |   15,000|    150-1000
+          _XOO_          |   15   |   400  |    5,000|    100-1000
+          _XOX_          |   10   |   400  |   10,000|    100- 975
+  ======================================================================
+
+Having made his estimates the reader should compare them first with
+similar estimates made by experienced teachers (shown on page 124 f.),
+and then with the results of actual counts for representative textbooks
+in arithmetic (shown on pages 126 to 132).
+
+It will be observed in Table 2 that even experienced teachers vary
+enormously in their estimates of the amount of practice given by an
+average textbook in arithmetic, and that most of them are in serious
+error by overestimating the amount of practice. In general it is the
+fact that we use textbooks in arithmetic with very vague and erroneous
+ideas of what is in them, and think they give much more practice than
+they do.
+
+The authors of the textbooks as a rule also probably had only very vague
+and erroneous ideas of what was in them. If they had known, they would
+almost certainly have revised their books. Surely no author would
+intentionally provide nearly four times as much practice on 2 + 2 as on
+8 + 8, or eight times as much practice on 2 � 2 as on 9 � 8, or eleven
+times as much practice on 2 - 2 as on 17 - 8, or over forty times as
+much practice on 2 � 2 as on 75 � 8 and 75 � 9, both together. Surely
+no author would have provided intentionally only twenty to thirty
+occurrences each of 16 - 7, 16 - 8, 16 - 9, 17 - 8, 17 - 9, and 18 - 9
+for the entire course through grade 6; or have left the practice on
+60 � 7, 60 � 8, 60 � 9, 61 � 7, 61 � 8, 61 � 9, and the like to occur
+only about once a year!
+
+
+TABLE 3
+
+AMOUNT OF PRACTICE: ADDITION BONDS IN A RECENT TEXTBOOK (A) OF EXCELLENT
+REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY
+MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION
+
+The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times,
+22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.
+
+  ======================================================================
+                  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9 | TOTAL
+  ----------------+-----+-----+-----+-----+-----+-----+-----+----+------
+    2             | 226 | 154 | 162 | 150 |  97 |  87 |  66 |  45|
+   12             |  74 |  53 |  76 |  46 |  51 |  37 |  36 |  33|
+   22, etc.       |  50 |  60 |  68 |  63 |  42 |  50 |  38 |  26|
+                  |     |     |     |     |     |     |     |    |
+    3             | 216 | 141 | 127 |  89 |  82 |  54 |  58 |  40|
+   13             |  43 |  43 |  60 |  70 |  52 |  30 |  22 |  18|
+   23, etc.       |  15 |  30 |  51 |  50 |  42 |  32 |  29 |  30|
+                  |     |     |     |     |     |     |     |    |
+    7             |  85 |  90 | 103 | 103 |  84 |  81 |  61 |  47|
+   17             |  35 |  25 |  42 |  32 |  35 |  21 |  29 |  16|
+   27, etc.       |  30 |  23 |  32 |  29 |  24 |  23 |  25 |  28|
+                  |     |     |     |     |     |     |     |    |
+    8             | 185 | 112 | 146 |  99 |  75 |  71 |  73 |  61|
+   18             |  28 |  35 |  52 |  46 |  28 |  29 |  24 |  14|
+   28, etc.       |  53 |  36 |  34 |  38 |  23 |  36 |  27 |  27|
+                  |     |     |     |     |     |     |     |    |
+    9             | 104 |  81 | 112 |  96 |  63 |  74 |  58 |  57|
+   19             |  13 |  11 |  31 |  38 |  25 |  14 |  22 |  11|
+   29, etc.       |  19 |  17 |  27 |  20 |  32 |  32 |  19 |  18|
+                  |     |     |     |     |     |     |     |    |
+  2, 12, 22, etc. | 350 | 277 | 306 | 260 | 190 | 174 | 140 | 104| 1801
+  3, 13, 23, etc. | 274 | 214 | 230 | 209 | 176 | 116 | 109 |  88| 1406
+                  |     |     |     |     |     |     |     |    |
+  7, 17, 27, etc. | 148 | 138 | 187 | 164 | 141 | 125 | 115 |  91| 1109
+  8, 18, 28, etc. | 266 | 183 | 232 | 185 | 126 | 136 | 124 | 102| 1354
+  9, 19, 29, etc. | 136 | 109 | 170 | 154 | 120 | 120 |  99 |  86|  994
+                  |     |     |     |     |     |     |     |    |
+  Totals          |1164 | 921 |1125 | 972 | 753 | 671 | 687 | 471|
+  ======================================================================
+
+
+TABLE  4
+
+AMOUNT OF PRACTICE: SUBTRACTION BONDS IN A RECENT TEXTBOOK (A)
+OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF
+SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION
+
+  ================================================================
+            |                     SUBTRAHENDS
+   MINUENDS |-----------------------------------------------------
+            |  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----
+    1       | 372 |     |     |     |     |     |     |     |
+    2       | 214 | 311 |     |     |     |     |     |     |
+    3       | 136 | 149 | 189 |     |     |     |     |     |
+    4       | 146 | 142 | 103 | 205 |     |     |     |     |
+    5       | 171 |  91 |  92 | 164 | 136 |     |     |     |
+    6       |  80 |  59 |  69 |  71 |  81 | 192 |     |     |
+    7       | 106 |  57 |  55 |  67 |  59 | 156 |  80 |     |
+    8       |  73 |  50 |  50 |  75 |  50 |  62 |  48 | 152 |
+    9       |  71 |  75 |  54 |  74 |  48 |  55 |  55 | 124 | 133
+   10       | 261 |  84 |  63 | 100 | 193 |  83 |  57 | 124 |  91
+            |     |     |     |     |     |     |     |     |
+   11       |     |  48 |  31 |  50 |  36 |  41 |  32 |  46 |  35
+   12       |     |     |  48 |  77 |  57 |  51 |  35 |  80 |  30
+   13       |     |     |     |  35 |  22 |  40 |  29 |  35 |  28
+   14       |     |     |     |     |  25 |  37 |  36 |  49 |  32
+   15       |     |     |     |     |     |  33 |  19 |  48 |  20
+            |     |     |     |     |     |     |     |     |
+   16       |     |     |     |     |     |     |  16 |  36 |  26
+   17       |     |     |     |     |     |     |     |  27 |  20
+   18       |     |     |     |     |     |     |     |     |  19
+            |     |     |     |     |     |     |     |     |
+  Total     |     |     |     |     |     |     |     |     |
+  excluding |     |     |     |     |     |     |     |     |
+  1-1, 2-2, |     |     |     |     |     |     |     |     |
+  etc.      |1258 | 755 | 565 | 713 | 571 | 558 | 327 | 569 | 301
+  ================================================================
+
+
+TABLE 5
+
+FREQUENCIES OF SUBTRACTIONS NOT INCLUDED IN TABLE 4
+
+These are cases where the pupil would, by reason of his stage of
+advancement, probably operate 35-30, 46-46, etc., as one bond.
+
+  ======================================================================
+                      |                    SUBTRAHENDS
+                      |----+----+----+----+----+----+----+----+----+----
+                      |   1|  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |
+        MINUENDS      |  11| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |  10
+                      |  21| 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |  20
+                      |etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.
+  --------------------+----+----+----+----+----+----+----+----+----+----
+  10, 20, 30, 40, etc.| 11 | 29 | 16 | 52 | 32 | 51 |  7 | 30 | 22 | 60
+  11, 21, 31, 41, etc.| 42 | 14 | 22 | 32 | 12 | 26 | 19 | 52 | 17 | 10
+  12, 22, 32, 42, etc.| 47 | 97 |  5 | 13 |  9 | 21 | 11 | 24 | 19 | 17
+  13, 23, 33, 43, etc.|  7 | 40 |  7 | 14 | 15 | 13 | 19 | 19 | 22 |  3
+  14, 24, 34, 44, etc.|  8 | 28 | 14 | 58 | 13 | 16 | 14 | 26 | 19 |  7
+  15, 25, 35, 45, etc.| 21 | 28 | 29 | 54 | 51 | 15 | 21 | 12 | 24 |  8
+  16, 26, 36, 46, etc.|  5 | 18 | 12 | 27 | 35 | 69 | 13 | 17 | 19 |  2
+  17, 27, 37, 47, etc.|  5 |  9 | 12 | 40 | 32 | 54 | 24 | 12 | 12 |  1
+  18, 28, 38, 48, etc.|  2 | 16 | 10 | 23 | 22 | 36 | 18 | 47 | 16 |  0
+  19, 29, 39, etc.    |  5 |  7 |  7 | 10 | 13 | 28 | 14 | 23 | 16 |  0
+                      |    |    |    |    |    |    |    |    |    |
+  Totals              |153 |286 |134 |323 |234 |329 |160 |262 |186 |108
+  =====================================================================
+
+
+TABLE 6
+
+AMOUNT OF PRACTICE: MULTIPLICATION BONDS IN ANOTHER RECENT TEXTBOOK (B)
+OF EXCELLENT REPUTE. BOOKS I AND II
+
+  ======================================================================
+              |                     MULTIPLICANDS
+  MULTIPLIERS |---------------------------------------------------------
+              |  0 |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |Totals
+  ------------+----+----+----+----+----+----+----+----+----+----+-------
+       1      | 299| 534| 472| 271| 310| 293| 261| 178| 195|  99| 2912
+       2      | 350| 644| 668| 480| 458| 377| 332| 238| 239| 155| 3941
+       3      | 280| 487| 509| 388| 318| 302| 247| 199| 227| 152| 3109
+       4      | 186| 375| 398| 242| 203| 265| 197| 163| 159|  93| 2281
+       5      | 268| 359| 393| 234| 263| 243| 217| 192| 197| 114| 2480
+       6      | 180| 284| 265| 199| 196| 191| 168| 169| 165| 106| 1923
+       7      | 135| 283| 277| 176| 187| 158| 155| 121| 145| 118| 1755
+       8      | 137| 272| 292| 175| 192| 164| 158| 157| 126| 126| 1799
+       9      |  71| 173| 140| 122|  97| 102| 101| 100|  82| 110| 1098
+              |    |    |    |    |    |    |    |    |    |    |
+  Totals      |1906|3411|3414|2287|2224|2095|1836|1517|1535|1073|
+  ======================================================================
+
+
+TABLE  7
+
+AMOUNT OF PRACTICE: DIVISIONS WITHOUT REMAINDER IN TEXTBOOK B,
+PARTS I AND II
+
+  ======================================================================
+                         |                   DIVISORS
+        DIVIDENDS        |----------------------------------------------
+                         |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |Totals
+  -----------------------+----+----+----+----+----+----+----+----+------
+  Integral               | 397| 224| 250| 130|  93|  44|  98|  23| 1259
+  multiples              |    |    |    |    |    |    |    |    |
+  of 2 to 9              | 256| 124| 152|  79|  28|  43|  61|  25|  768
+  in sequence;           |    |    |    |    |    |    |    |    |
+  _i.e._, 4 � 2          | 318| 123| 130|  65|  50|  19|  39|  19|  763
+  occurred               |    |    |    |    |    |    |    |    |
+  397 times,             | 258|  98|  86| 105|  25|  24|  34|  20|  650
+  6 � 2 occurred         |    |    |    |    |    |    |    |    |
+  256 times,             | 198|  49|  76|  27|  22|  30|  33|  16|  451
+  6 � 3, 224 times,      |    |    |    |    |    |    |    |    |
+  9 � 3, 124 times.      |  77|  54|  36|  31|  28|  27|  16|   9|  278
+                         | 180|  91|  50|  38|  17|  13|  22|  16|  427
+                         |  69|  46|  37|  24|  12|  17|  16|  15|  236
+                         |    |    |    |    |    |    |    |    |
+  Totals                 |1753| 809| 817| 499| 275| 217| 319| 142|
+  ======================================================================
+
+
+TABLE 8
+
+DIVISION BONDS, WITH AND WITHOUT REMAINDERS. BOOK B
+
+All work through grade 6, except estimates of quotient figures in long
+division.
+
+  Dividend       2       3           4               5
+  Divisor        1   2   1   2   3   1   2   3   4   1   2   3   4   5
+  Number of
+  Occurrences   41 386  27 189 240  26 397  66 185  23 136  43  53 135
+
+  Dividend       6                       7
+  Divisor        1   2   3   4   5   6   1   2   3   4   5   6   7
+  Number of
+  Occurrences   21 256 224  68  43  83  23  72  55  38  46  32  54
+
+  Dividend       8                            9
+  Divisor        1   2   3   4   5  6  7  8   1  2   3  4  5  6  7  8  9
+  Number of
+  Occurrences   17 318  30 250  22 28 39 91  19 50 124 49 25 15 18 30 38
+
+  Dividend      10                         11
+  Divisor        2  3  4   5  6 7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences  258 38 46 120 19 9 24 24    32 21 16  3  7 11 14  3
+
+  Dividend      12                         13
+  Divisor        2   3   4  5  6 7  8 9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences  198 123 152 29 93 9 16 7    45 16 15 11  7  4  5  3
+
+  Dividend      14                         15
+  Divisor        2  3  4  5  6  7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences   77 20 13  5  8 44  8  6    69 98 16 79  8  8  4  6
+
+  Dividend      16                         17
+  Divisor        2  3   4  5 6  7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences  180 19 130 14 6  9 98  3    61  9 15 14  6  6 12  3
+
+  Dividend      18                         19
+  Divisor        2  3  4  5  6  7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences   69 49 13  6 28  7  7 23    21  6 10  5  3  4 10  4
+
+  Dividend      20                      21
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences   24 86 65 11  3 23  5    54 12  8  5 43 10  5
+
+  Dividend      22                      23
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences   17 16 15  8 13  6 15     7  8 11  8  6  3  2
+
+  Dividend      24                      25
+  Divisor        3  4  5  6  7  8  9     3  4   5 6  7  8  9
+  Number of
+  Occurrences   91 76 18 50  5 61  1    11 13 105 5  6  5  3
+
+  Dividend      26                      27
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences    5  6  3  3  4  6  3    46  8 10  4  2  6 25
+
+  Dividend      28                       29
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences    4 36  8  3 19  3  7     6  8  0  5 11  2  3
+
+  Dividend      30                 31                 32
+  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
+  Number of
+  Occurrences   21 27 25  6  7 13   4  3  1  1  4  2  50 11  3  6 39  5
+
+  Dividend      33                 34                 35
+  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
+  Number of
+  Occurrences    8  7  7  2  6  1   8  3  5  2  1  1  10 31  5 24  5  3
+
+  Dividend      36                 37                 38
+  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
+  Number of
+  Occurrences   37 16 22  2  6 19   12 8  7  5  3  9   7  8  7  1  1  5
+
+  Dividend      39             40            41            42
+  Divisor        4 5 6 7 8 9    5 6 7  8 9    5 6 7 8 9     5  6  7  8 9
+  Number of
+  Occurrences    4 3 7 4 3 1   38 9 2 34 2    6 6 3 7 5     7 28 30 10 3
+
+  Dividend      43             44            45            46
+  Divisor        5 6  7 8 9     5 6 7 8 9     5 6 7  8  9   5 6 7 8 9
+  Number of
+  Occurrences    7 5 10 3 2     7 6 4 5 0    24 6 7 10 20   3 3 2 2 2
+
+  Dividend      47             48            49            50
+  Divisor        5 6 7 8 9      5  6 7  8 9   5 6  7 8 9    6 7 8 9
+  Number of
+  Occurrences    6 2 2 0 3      7 17 4 33 2   4 7 27 9 2    4 6 3 8
+
+  Dividend      51         52           53         54
+  Divisor        6 7 8 9    6 7 8 9      6 7 8 9    6 7 8  9
+  Number of
+  Occurrences    2 3 1 2    5 5 5 3      4 3 2 2    12 5 1 16
+
+  Dividend      55         56           57         58         59
+  Divisor        6 7 8 9    6  7  8 9    6 7 8 9    6 7 8 9    6 7 8 9
+  Number of
+  Occurrences    5 3 4 2    0 13 16 8    0 3 1 3    2 2 3 1    2 3 0 3
+
+  Dividend      60       61       62       63       64         65
+  Divisor        7 8 9    7 8 9    7 8 9    7 8 9    7  8  9    7  8 9
+  Number of
+  Occurrences    3 9 1    1 2 5    4 6 1   17 5 9    5 22  0    1 10 1
+
+  Dividend      66       67       68       69       70     71
+  Divisor        7 8 9    7 8 9    7 8 9    7 8 9    8 9    8 9
+  Number of
+  Occurrences    2 1 4    0 1 1    1 3 2    0 6 1    6 2    1 0
+
+  Dividend      72      73     74     75     76     77     78     79
+  Divisor        8  9    8 9    8 9    8 9    8 9    8 9    8 9    8 9
+  Number of
+  Occurrences   16 10    7 5    3 3    5 3    3 2    3 0    4 1    0 2
+
+  Dividend      80   81   82   83   84   85   86   87   88   89
+  Divisor        9    9    9    9    9    9    9    9    9    9
+  Number of
+  Occurrences    4   15    2    4    1    2    0    3    2    7
+
+Tables 3 to 8 show that even gifted authors make instruments for
+instruction in arithmetic which contain much less practice on certain
+elementary facts than teachers suppose; and which contain relatively
+much more practice on the more easily learned facts than on those which
+are harder to learn.
+
+How much practice should be given in arithmetic? How should it be
+divided among the different bonds to be formed? Below a certain amount
+there is waste because, as has been shown in Chapter VI, the pupil will
+need more time to detect and correct his errors than would have been
+required to give him mastery. Above a certain amount there is waste
+because of unproductive overlearning. If 668 is just enough for 2 � 2,
+82 is not enough for 9 � 8. If 82 is just enough for 9 � 8, 668 is too
+much for 2 � 2.
+
+It is possible to find the answers to these questions for the pupil of
+median ability (or any stated ability) by suitable experiments. The
+amount of practice will, of course, vary according to the ability of
+the pupil. It will also vary according to the interest aroused in him
+and the satisfaction he feels in progress and mastery. It will also vary
+according to the amount of practice of other related bonds; 7 + 7 = 14
+and 60 � 7 = 8 and 4 remainder will help the formation of 7 + 8 = 15
+and 61 � 7 = 8 and 5 remainder. It will also, of course, vary with the
+general difficulty of the bond, 17 - 8 = 9 being under ordinary
+conditions of teaching harder to form than 7 - 2 = 5.
+
+Until suitable experiments are at hand we may estimate for the
+fundamental bonds as follows, assuming that by the end of grade 6 a
+strength of 199 correct out of 200 is to be had, and that the teaching
+is by an intelligent person working in accord with psychological
+principles as to both ability and interest.
+
+For one of the easier bonds, most facilitated by other bonds (such
+as 2 � 5 = 10, or 10 - 2 = 8, or the double bond 7 = two 3s and 1
+remainder) in the case of the median or average pupil, twelve practices
+in the week of first learning, supported by twenty-five practices during
+the two months following, and maintained by thirty practices well spread
+over the later periods should be enough. For the more gifted pupils
+lesser amounts down to six, twelve, and fifteen may suffice. For the
+less gifted pupils more may be required up to thirty, fifty, and a
+hundred. It is to be doubted, however, whether pupils requiring nearly
+two hundred repetitions of each of these easy bonds should be taught
+arithmetic beyond a few matters of practical necessity.
+
+For bonds of ordinary difficulty, with average facilitation from other
+bonds (such as 11 - 3, 4 � 7, or 48 � 8 = 6) in the case of the median
+or average pupil, we may estimate twenty practices in the week of first
+learning, supported by thirty, and maintained by fifty practices well
+spread over the later periods. Gifted pupils may gain and keep mastery
+with twelve, fifteen, and twenty practices respectively. Pupils dull at
+arithmetic may need up to twenty, sixty, and two hundred. Here, again,
+it is to be doubted whether a pupil for whom arithmetical facts, well
+taught and made interesting, are so hard to acquire as this, should
+learn many of them.
+
+For bonds of greater difficulty, less facilitated by other bonds (such
+as 17 - 9, 8 � 7, or 12-1/2% of = 1/8 of), the practice may be from ten
+to a hundred percent more than the above.
+
+
+UNDERLEARNING AND OVERLEARNING
+
+If we accept the above provisional estimates as reasonable, we may
+consider the harm done by giving less and by giving more than these
+reasonable amounts. Giving less is indefensible. The pupil's time is
+wasted in excessive checking to find his errors. He is in danger of
+being practiced in error. His attention is diverted from the learning of
+new facts and processes by the necessity of thinking out these
+supposedly mastered facts. All new bonds are harder to learn than they
+should be because the bonds which should facilitate them are not strong
+enough to do so. Giving more does harm to some extent by using up time
+that could be spent better for other purposes, and (though not
+necessarily) by detracting from the pupil's interest in arithmetic. In
+certain cases, however, such excess practice and overlearning are
+actually desirable. Three cases are of special importance.
+
+The first is the case of a bond operating under a changed mental set or
+adjustment. A pupil may know 7 � 8 adequately as a thing by itself, but
+need more practice to operate it in
+
+  285
+    7
+  ---
+
+where he has to remember that 3 is to be added to the 56 when he
+obtains it, and that only the 9 is to be written down, the 5 to be held
+in mind for later use. The practice required to operate the bond
+efficiently in this new set is desirable, even though it is excess from
+a narrower point of view, and causes the straightforward 'seven eights
+are fifty-six' to be overlearned. So also a pupil's work with 24, 34,
+44, etc., +9 may react to give what would be excess practice from the
+point of view of 4 + 9 alone; his work in estimating approximate
+quotient figures in long division may give excess practice on the
+division tables. There are many such cases. Even adding the 5 and 7 in
+5/12 + 7/12 is not quite the same task as adding 5 and 7 undisturbed by
+the fact that they are twelfths. We know far too little about the amount
+of practice needed to adapt arithmetical bonds to efficient operation in
+these more complicated conditions to estimate even approximately the
+allowances to be made. But some allowance, and often a rather large
+allowance, must be made.
+
+The second is the case where the computation in general should be made
+very easy and sure for the pupil except for some one new element that
+is being learned. For example, in teaching the meaning and uses of
+'Averages' and of uneven division, we may deliberately use 2, 3, and 4
+as divisors rather than 7 and 9, so as to let all the pupil's energy be
+spent in learning the new facts, and so that the fraction in the
+quotient may be something easily understood, real, and significant. In
+teaching the addition of mixed numbers, we may use, in the early steps,
+
+  11-1/2
+  13-1/2
+  24
+  ------
+
+rather than
+
+  79-1/2
+  98-1/2
+  67
+  ------
+
+so as to save attention for the new process itself. In cancellation, we
+may give excess practice to divisions by 2, 3, 4, and 5 in order to make
+the transfer to the new habits of considering two numbers together from
+the point of view of their divisibility by some number. In introducing
+trade discount, we may give excess practice on '5% of' and '10% of'
+deliberately, so that the meaning of discount may not be obscured by
+difficulties in the computation itself. Excess practice on, and
+overlearning of, certain bonds is thus very often justifiable.
+
+The third case concerns bonds whose importance for practical uses in
+life or as notable facilitators of other bonds is so great that they may
+profitably be brought to a greater strength than 199 correct out of 200
+at a speed of 2 sec. or less, or be brought to that degree of strength
+very early. Examples of bonds of such special practical use are the
+subtractions from 10, 1/2 + 1/2, 1/2 + 1/4, 1/2 of 60, 1/4 of 60,
+and the fractional parts of 12 and of $1.00. Examples of notable
+facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like
+2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 � 9.
+
+In consideration of these three modifying cases or principles, a volume
+could well be written concerning just how much practice to give to each
+bond, in each of the types of complex situations where it has to
+operate. There is evidently need for much experimentation to expose the
+facts, and for much sagacity and inventiveness in making sure of
+effective learning without wasteful overlearning.
+
+The facts of primary importance are:--
+
+  (1) The textbook or other instrument of instruction which is a
+  teacher's general guide may give far too little practice on
+  certain bonds.
+
+  (2) It may divide the practice given in ways that are apparently
+  unjustifiable.
+
+  (3) The teacher needs therefore to know how much practice it does
+  give, where to supplement it, and what to omit.
+
+  (4) The omissions, on grounds of apparent excess practice, should
+  be made only after careful consideration of the third principle
+  described above.
+
+  (5) The amount of practice should always be considered in the
+  light of its interest and appeal to the pupil's tendency to work
+  with full power and zeal. Mere repetition of bonds when the
+  learner does not care whether he is improving is rarely
+  justifiable on any grounds.
+
+  (6) Practice that is actually in excess is not a very grave defect
+  if it is enjoyed and improves the pupil's attitude toward
+  arithmetic. Not much time is lost; a hundred practices for each of
+  a thousand bonds after mastery to 199 in 200 at 2 seconds will use
+  up less than 60 hours, or 15 hours per year in grades 3 to 6.
+
+  (7) By the proper division of practice among bonds, the
+  arrangement of learning so that each bond helps the others, the
+  adroit shifting of practice of a bond to each new type of
+  situation requiring it to operate under changed conditions, and
+  the elimination of excess practice where nothing substantial is
+  gained, notable improvements over the past hit-and-miss customs
+  may be expected.
+
+  (8) Unless the material for practice is adequate, well balanced
+  and sufficiently motivated, the teacher must keep close account of
+  the learning of pupils. Otherwise disastrous underlearning of many
+  bonds is almost sure to occur and retard the pupil's development.
+
+
+THE ORGANIZATION OF ABILITIES
+
+There is danger that the need of brevity and simplicity which has made
+us speak so often of a bond or an ability, and of the amount of
+practice it requires, may mislead the reader into thinking that these
+bonds and abilities are to be formed each by itself alone and kept so.
+They should rarely be formed so and never kept so. This we have
+indicated from time to time by references to the importance of forming a
+bond in the way in which it is to be used, to the action of bonds in
+changed situations, to facilitation of one bond by others, to the
+co�peration of abilities, and to their integration into a total
+arithmetical ability.
+
+As a matter of fact, only a small part of drill work in arithmetic
+should be the formation of isolated bonds. Even the very young
+pupil learning 5 and 3 are 8 should learn it with '5 and 5 = 10,'
+'5 and 2 = 7,' at the back of his mind, so to speak. Even so early,
+5 + 3 = 8 should be part of an organized, co�perating system of bonds.
+Later 50 + 30 = 80 should become allied to it. Each bond should be
+considered, not simply as a separate tool to be put in a compartment
+until needed, but also as an improvement of one total tool or machine,
+arithmetical ability.
+
+There are differences of course. Knowledge of square root can be
+regarded somewhat as a separate tool to be sharpened, polished, and used
+by itself, whereas knowledge of the multiplication tables cannot. Yet
+even square root is probably best made more closely a part of the total
+ability, being taught as a special case of dividing where divisor is to
+be the same as quotient, the process being one of estimating and
+correcting.
+
+In general we do not wish the pupil to be a repository of separated
+abilities, each of which may operate only if you ask him the sort of
+questions which the teacher used to ask him, or otherwise indicate to
+him which particular arithmetical tool he is to use. Rather he is to be
+an effective organization of abilities, co�perating in useful ways to
+meet the quantitative problems life offers. He should not as a rule
+have to think in such fashion as: "Is this interest or discount? Is it
+simple interest or compound interest? What did I do in compound
+interest? How do I multiply by 2 percent?" The situation that calls up
+interest should also call up the kind of interest that is appropriate,
+and the technique of operating with percents should be so welded
+together with interest in his mind that the right co�peration will occur
+almost without supervision by him.
+
+As each new ability is acquired, then, we seek to have it take its place
+as an improvement of a thinking being, as a co�perative member of a
+total organization, as a soldier fighting together with others, as an
+element in an educated personality. Such an organization of bonds will
+not form itself any more than any one bond will create itself. If the
+elements of arithmetical ability are to act together as a total
+organized unified force they must be made to act together in the course
+of learning. What we wish to have work together we must put together and
+give practice in teamwork.
+
+We can do much to secure such co�perative action when and where and as
+it is needed by a very simple expedient; namely, to give practice with
+computation and problems such as life provides, instead of making up
+drills and problems merely to apply each fact or principle by itself.
+Though a pupil has solved scores of problems reading, "A triangle has a
+base of _a_ feet and an altitude of _b_ feet, what is its area?" he may
+still be practically helpless in finding the area of a triangular plot
+of ground; still more helpless in using the formula for a triangle which
+is one of two into which a trapezoid is divided. Though a pupil has
+learned to solve problems in trade discount, simple interest, compound
+interest, and bank discount one at a time, stated in a few set forms,
+he may be practically helpless before the actual series of problems
+confronting him in starting in business, and may take money out of the
+savings bank when he ought to borrow on a time loan, or delay payment on
+his bills when by paying cash he could save money as well as improve his
+standing with the wholesaler.
+
+Instead of making up problems to fit the abilities given by school
+instruction, we should preferably modify school instruction so that
+arithmetical abilities will be organized into an effective total ability
+to meet the problems that life will offer. Still more generally, _every
+bond formed should be formed with due consideration of every other bond
+that has been or will be formed; every ability should be practiced in
+the most effective possible relations with other abilities_.
+
+
+
+
+CHAPTER VII
+
+THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS
+
+
+The bonds to be formed having been chosen, the next step is to arrange
+for their most economical order of formation--to arrange to have each
+help the others as much as possible--to arrange for the maximum of
+facilitation and the minimum of inhibition.
+
+The principle is obvious enough and would probably be admitted in theory
+by any intelligent teacher, but in practice we are still wedded to
+conventional usages which arose long before the psychology of arithmetic
+was studied. For example, we inherit the convention of studying addition
+of integers thoroughly, and then subtraction, and then multiplication,
+and then division, and many of us follow it though nobody has ever given
+a proof that this is the best order for arithmetical learning. We
+inherit also the opposite convention of studying in a so-called "spiral"
+plan, a little addition, subtraction, multiplication, and division, and
+then some more of each, and then some more, and many of us follow this
+custom, with an unreasoned faith that changing about from one process to
+another is _per se_ helpful.
+
+Such conventions are very strong, illustrating our common tendency to
+cherish most those customs which we cannot justify! The reductions of
+denominate numbers ascending and descending were, until recently, in
+most courses of study, kept until grade 4 or grade 5 was reached,
+although this material is of far greater value for drills on the
+multiplication and division tables than the customary problems about
+apples, eggs, oranges, tablets, and penholders. By some historical
+accident or for good reasons the general treatment of denominate numbers
+was put late; by our na�ve notions of order and system we felt that any
+use of denominate numbers before this time was heretical; we thus became
+blind to the advantages of quarts and pints for the tables of 2s; yards
+and feet for the tables of 3s; gallons and quarts for the tables of 4s;
+nickels and cents for the 5s; weeks and days for the 7s; pecks and
+quarts for the 8s; and square yards and square feet for the 9s.
+Problems like 5 yards = __ feet or 15 feet = __ yards have not only the
+advantages of brevity, clearness, practical use, real reference, and
+ready variation, but also the very great advantage that part of the data
+have to be _thought of_ in a useful way instead of _read off_ from the
+page. In life, when a person has twenty cents with which to buy tablets
+of a certain sort, he _thinks of_ the price in making his purchase,
+asking it of the clerk only in case he does not know it, and in planning
+his purchases beforehand he _thinks of_ prices as a rule. In spite of
+these and other advantages, not one textbook in ten up to 1900 made
+early use of these exercises with denominate numbers. So strong is mere
+use and wont.
+
+Besides these conventional customs, there has been, in those responsible
+for arithmetical instruction, an admiration for an arrangement of topics
+that is easy for a person, after he knows the subject, to use in
+thinking of its constituent parts and their relations. Such arrangements
+are often called 'logical' arrangements of subject matter, though they
+are often far from logical in any useful sense. Now the easiest order in
+which to think of a hierarchy of habits after you have formed them all
+may be an extremely difficult order in which to form them. The criticism
+of other orders as 'scrappy,' or 'unsystematic,' valid enough if the
+course of study is thought of as an object of contemplation, may be
+foolish if the course of study is regarded as a working instrument for
+furthering arithmetical learning.
+
+We must remember that all our systematizing and labeling is largely
+without meaning to the pupils. They cannot at any point appreciate the
+system as a progression from that point toward this and that, since they
+have no knowledge of the 'this or that.' They do not as a rule think of
+their work in grade 4 as an outcome of their work in grade 3 with
+extensions of a to a_1, and additions of b_2 and b_3 to b and b_1, and
+refinements of c and d by c_4 and d_5. They could give only the vaguest
+account of what they did in grade 3, much less of why it should have
+been done then. They are not much disturbed by a lack of so-called
+'system' and 'logical' progression for the same reason that they are not
+much helped by their presence. What they need and can use is a
+_dynamically_ effective system or order, one that they can learn easily
+and retain long by, regardless of how it would look in a museum of
+arithmetical systems. Unless their actual arithmetical habits are
+usefully related it does no good to see the so-called logical relations;
+and if their habits are usefully related, it does not very much matter
+whether or not they do see these; finally, they can be brought to see
+them best by first acquiring the right habits in a dynamically effective
+order.
+
+
+DECREASING INTERFERENCE AND INCREASING FACILITATION
+
+Psychology offers no single, easy, royal road to discovering this
+dynamically best order. It can only survey the bonds, think what each
+demands as prerequisite and offers as future help, recommend certain
+orders for trial, and measure the efficiency of each order as a means of
+attaining the ends desired. The ingenious thought and careful
+experimentation of many able workers will be required for many years to
+come.
+
+Psychology can, however, even now, give solid constructive help in many
+instances, either by recommending orders that seem almost certainly
+better than those in vogue, or by proposing orders for trial which can
+be justified or rejected by crucial tests.
+
+Consider, for example, the situation, 'a column of one-place numbers to
+be added, whose sum is over 9,' and the response 'writing down the sum.'
+This bond is commonly firmly fixed before addition with two-place
+numbers is undertaken. As a result the pupil has fixed a habit that he
+has to break when he learns two-place addition. If _oral_ answers only
+are given with such single columns until two-place addition is well
+under way, the interference is avoided.
+
+In many courses of study the order of systematic formation of the
+multiplication table bonds is: 1 � 1, 2 � 1, etc., 1 � 2, 2 � 2, etc.,
+1 � 3, 2 � 3, etc., 1 � 9, 2 � 9, etc. This is probably wrong in two
+respects. There is abundant reason to believe that the � 5s should be
+learned first, since they are easier to learn than the 1s or the 2s, and
+give the idea of multiplying more emphatically and clearly. There is
+also abundant reason to believe that the 1 � 5, 1 � 2, 1 � 3, etc.,
+should be put very late--after at least three or four tables are
+learned, since the question "What is 1 times 2?" (or 3 or 5) is
+unnecessary until we come to multiplication of two- and three-place
+numbers, seems a foolish question until then, and obscures the notion of
+multiplication if put early. Also the facts are best learned once for
+all as the habits "1 times _k_ is the same as _k_," and "_k_ times 1 is
+the same as _k_."[8]
+
+  [8] The very early learning of 2 � 2, 2 � 3, 3 � 2, 2 � 4, 4 � 2,
+      3 � 3, and perhaps a few more multiplications is not considered
+      here. It is advisable. The treatment of 0 � 0, 0 � 1, 1 � 0, etc.,
+      is not considered here. It is probably best to defer the '� 0'
+      bonds until after all the others are formed and are being used
+      in short multiplication, and to form them in close connection
+      with their use in short multiplication. The '0 �' bonds may well
+      be deferred until they are needed in 'long' multiplication,
+      0 � 0 coming last of all.
+
+In another connection it was recommended that the divisions to 81 � 9
+be learned by selective thinking or reasoning from the multiplications.
+This determines the order of bonds so far as to place the formation of
+the division bonds soon after the learning of the multiplications. For
+other reasons it is well to make the proximity close.
+
+One of the arbitrary systematizations of the order of formation of bonds
+restricts operations at first to the numbers 1 to 10, then to numbers
+under 100, then to numbers under 1000, then to numbers under 10,000.
+Apart from the avoidance of unreal and pedantic problems in applied
+arithmetic to which work with large numbers in low grades does somewhat
+predispose a teacher, there is little merit in this restriction of the
+order of formation of bonds. Its demerits are many. For example, when
+the pupil is learning to 'carry' in addition he can be given better
+practice by soon including tasks with sums above 100, and can get a
+valuable sense of the general use of the process by being given a few
+examples with three- and four-place numbers to be added. The same holds
+for subtraction. Indeed, there is something to be said in favor of using
+six- or seven-place numbers in subtraction, enforcing the 'borrowing'
+process by having it done again and again in the same example, and
+putting it under control by having the decision between 'borrowing' and
+'not borrowing' made again and again in the same example. When the
+multiplication tables are learned the most important use for them is not
+in tedious reviews or trivial problems with answers under 100, but in
+regular 'short' multiplication of two- and three- and even four-place
+numbers. Just as the addition combinations function mainly in the
+higher-decade modifications of them, so the multiplication combinations
+function chiefly in the cases where the bond has to operate while the
+added tasks of keeping one's place, adding what has been carried,
+writing down the right figure in the right place, and holding the right
+number for later addition, are also taken care of. It seems best to
+introduce such short multiplication as soon as the � 5s, � 2s, � 3s,
+and � 4s are learned and to put the � 6s, � 7s, and the rest to work
+in such short multiplication as soon as each is learned.
+
+Still surer is the need for four-, five-, and six-place numbers when
+two-place numbers are used in multiplying. When the process with a
+two-place multiplier is learned, multiplications by three-place numbers
+should soon follow. They are not more difficult then than later. On the
+contrary, if the pupil gets used to multiplying only as one does with
+two-place multipliers, he will suffer more by the resulting interference
+than he does from getting six- or seven-place answers whose meaning he
+cannot exactly realize. They teach the rationale and the manipulations
+of long multiplication with especial economy because the principles and
+the procedures are used two or three times over and the contrasts
+between the values which the partial products have in adding become
+three instead of one.
+
+The entire matter of long multiplication with integers and United States
+money should be treated as a teaching unit and the bonds formed in close
+organization, even though numbers as large as 900,000 are occasionally
+involved. The reason is not that it is more logical, or less scrappy,
+but that each of the bonds in question thus gets much help from, and
+gives much help to, the others.
+
+In sharp contrast to a topic like 'long multiplication' stands a topic
+like denominate numbers. It most certainly should not be treated as a
+large teaching unit, and all the bonds involved in adding, subtracting,
+multiplying, and dividing with all the ordinary sorts of measures should
+certainly not be formed in close sequence. The reductions ascending and
+descending for many of the measures should be taught as drills on the
+appropriate multiplication and division tables. The reduction of feet
+and inches to inches, yards and feet to yards, gallons and quarts to
+quarts, and the like are admirable exercises in connection with the
+(_a_ � _b_) + _c_ = .... problems,--the 'Bought 3 lbs. of sugar at
+7 cents and 5 cents worth of matches' problems. The reductions of
+inches to feet and inches and the like are admirable exercises in
+the _d_ = (.... � _b_) + _c_ or 'making change' problem, which in its
+small-number forms is an excellent preparatory step for short division.
+They are also of great service in early work with fractions. The
+feet-mile, square-foot-square-inch, and other simple relations give a
+genuine and intelligible demand for multiplication with large numbers.
+
+Knowledge of the metric system for linear and square measure would
+perhaps, as an introduction to decimal fractions, more than save the
+time spent to learn it. It would even perhaps be worth while to invent a
+measure (call it the _twoqua_) midway between the quart and gallon and
+teach carrying in addition and borrowing in subtraction by teaching
+first the addition and subtraction of 'gallon, twoqua, quart, and pint'
+series! Many of the bonds which a system-made tradition huddled together
+uselessly in a chapter on denominate numbers should thus be formed as
+helpful preparations for and applications of other bonds all the way
+from the first to the eighth half-year of instruction in arithmetic.
+
+The bonds involved in the ability to respond correctly to the series:--
+
+   5 = .... 2s and .... remainder
+   5 = .... 3s and .... remainder
+  88 = .... 9s and .... remainder
+
+should be formed before, not during, the training in short division.
+They are admirable at that point as practice on the division tables; are
+of practical service in the making-change problems of the small purchase
+and the like; and simplify the otherwise intricate task of keeping one's
+place, choosing the quotient figure, multiplying by it, subtracting and
+holding in mind the new number to be divided, which is composed half of
+the remainder and half of a figure in the written dividend. This change
+of order is a good illustration of the nearly general rule that "_When
+the practice or review required to perfect or hold certain bonds can, by
+an inexpensive modification, be turned into a useful preparation for new
+bonds, that modification should be made._"
+
+The bonds involved in the four operations with United States money
+should be formed in grades 3 and 4 along with or very soon after the
+corresponding bonds with three-place and four-place integers. This
+statement would have seemed preposterous to the pedagogues of fifty
+years ago. "United States money," they would have said, "is an
+application of decimals. How can it be learned until the essentials of
+decimal fractions are known? How will the child understand when
+multiplying $.75 by 3 that 3 times 5 cents is 1 dime and 5 cents, or
+that 3 times 70 cents is 2 dollars and 1 dime? Why perplex the young
+pupils with the difficulties of placing the decimal point? Why disturb
+the learning of the four operations with integers by adding at each step
+a second 'procedure with United States money'?"
+
+The case illustrates very well the error of the older oversystematic
+treatment of the order of topics and the still more important error of
+confusing the logic of proof with the psychology of learning. To prove
+that 3 � $.75 = $2.25 to the satisfaction of certain arithmeticians, you
+may need to know the theory of decimal fractions; but to do such
+multiplication all a child needs is to do just what he has been doing
+with integers and then "Put a $ before the answer to show that it means
+dollars and cents, and put a decimal point in the answer to show which
+figures mean dollars and which figures mean cents." And this is general.
+The ability to operate with integers plus the two habits of prefixing $
+and separating dollars from cents in the result will enable him to
+operate with United States money.
+
+Consequently good practice came to use United States money not as a
+consequence of decimal fractions, learned by their aid, but as an
+introduction to decimal fractions which aids the pupil to learn them. So
+it has gradually pushed work with United States money further and
+further back, though somewhat timidly.
+
+We need not be timid. The pupil will have no difficulty in adding,
+subtracting, multiplying, and dividing with United States money--unless
+we create it by our explanations! If we simply form the two bonds
+described above and show by proper verification that the procedure
+always gives the right answer, the early teaching of the four operations
+with United States money will in fact actually show a learning profit!
+It will save more time in the work with integers than was spent in
+teaching it! For, in the first place, it will help to make work with
+four-place and five-place numbers more intelligible and vital. A pupil
+can understand $16.75 or $28.79 more easily than 1675 or 2879. The
+former may be the prices of a suit or sewing machine or bicycle. In the
+second place, it permits the use of a large stock of genuine problems
+about spending, saving, sharing, and the like with advertisements and
+catalogues and school enterprises. In the third place, it permits the
+use of common-sense checks. A boy may find one fourth of 3000 as 7050 or
+75 and not be disturbed, but he will much more easily realize that one
+fourth of $30.00 is not over $70 or less than $1. Even the decimal point
+of which we used to be so afraid may actually help the eye to keep its
+place in adding.
+
+
+INTEREST
+
+So far, the illustrations of improvements in the order of bonds so as to
+get less interference and more facilitation than the customary orders
+secure have sought chiefly to improve the mechanical organization of the
+bonds. Any gain in interest which the changes described effected would
+be largely due to the greater achievement itself. Dewey and others have
+emphasized a very different principle of improving the order of
+formation of bonds--the principle of determination of the bonds to be
+formed by some vital, engaging problem which arouses interest enough to
+lighten the labor and which goes beyond or even against cut-and-dried
+plans for sequences in order to get effective problems. For example, the
+work of the first month in grade 2B might sacrifice facilitations of the
+mechanical sort in order to put arithmetic to use in deciding what
+dimensions a rabbit's cage should have to give him 12 square feet of
+floor space, how much bread he should have per meal to get 6 ounces a
+day, how long a ten-cent loaf would last, how many loaves should be
+bought per week, how much it costs to feed the rabbit, how much he has
+gained in weight since he was brought to the school, and so on.
+
+Such sacrifices of the optimal order if interest were equal, in order to
+get greater interest or a healthier interest, are justifiable. Vital
+problems as nuclei around which to organize arithmetical learning are of
+prime importance. It is even safe probably to insist that some genuine
+problem-situation requiring a new process, such as addition with
+carrying, multiplication by two-place numbers, or division with
+decimals, be provided in every case as a part of the introduction to
+that process. The sacrifice should not be too great, however; the search
+for vital problems that fit an economical order of subject matter is as
+much needed as the amendment of that order to fit known interests; and
+the assurance that a problem helps the pupil to learn arithmetic is as
+important as the assurance that arithmetic is used to help the pupil
+solve his personal problems.
+
+Much ingenuity and experimentation will be required to find the order
+that is satisfactory in both quality and quantity of interest or motive
+and helpfulness of the bonds one to another. The difficulty of
+organizing arithmetic around attractive problems is much increased by
+the fact of class instruction. For any one pupil vital, personal
+problems or projects could be found to provide for many arithmetical
+abilities; and any necessary knowledge and technique which these
+projects did not develop could be somehow fitted in along with them. But
+thirty children, half boys and half girls, varying by five years in age,
+coming from different homes, with different native capacities, will not,
+in September, 1920, unanimously feel a vital need to solve any one
+problem, and then conveniently feel another on, say, October 15! In the
+mechanical laws of learning children are much alike, and the gain we
+may hope to make from reducing inhibitions and increasing facilitations
+is, for ordinary class-teaching, probably greater than that to be made
+from the discovery of attractive central problems. We should, however,
+get as much as possible of both.
+
+
+GENERAL PRINCIPLES
+
+The reader may by now feel rather helpless before the problem of the
+arrangement of arithmetical subject matter. "Sometimes you complete a
+topic, sometimes you take it piecemeal months or years apart, often you
+make queer twists and shifts to get a strategic advantage over the
+enemy," he may think, "but are there no guiding principles, no general
+rules?" There is only one that is absolutely general, to _take the order
+that works best for arithmetical learning_. There are particular rules,
+but there are so many and they are so limited by an 'other things being
+equal' clause, that probably a general eagerness to think out the _pros_
+and _cons_ for any given proposal is better than a stiff attempt to
+adhere to these rules. I will state and illustrate some of them, and let
+the reader judge.
+
+_Other things being equal, one new sort of bonds should not be started
+until the previous set is fairly established, and two different sets
+should not be started at once._ Thus, multiplication of two- and
+three-place numbers by 2, 3, 4, and 5 will first use numbers such that
+no carrying is required, and no zero difficulties are encountered, then
+introduce carrying, then introduce multiplicands like 206 and 320.
+If other things were equal, the carrying would be split into two
+steps--first drills with (4 � 6) + 2, (3 � 7) + 3, (5 � 4) + 1, and the
+like, and second the actual use of these habits in the multiplication.
+The objection to this separation of the double habit is that the first
+part of it in isolation is too artificial--that it may be better to
+suffer the extra difficulty of forming the two together than to teach so
+rarely used habits as the (_a_ � _b_) + _c_ series. Experimental tests
+are needed to decide this point.
+
+_Other things being equal, bonds should be formed in such order that
+none will have to be broken later._ For example, there is a strong
+argument for teaching long division first, or very early, with
+remainders, letting the case of zero remainder come in as one of many.
+If the pupils have been familiarized with the remainder notion by the
+drills recommended as preparation for short division,[9] the use of
+remainders in long division will offer little difficulty. The exclusive
+use of examples without remainders may form the habit of not being exact
+in computation, of trusting to 'coming out even' as a sole check, and
+even of writing down a number to fit the final number to be divided
+instead of obtaining it by honest multiplication.
+
+  [9] See page 76.
+
+For similar reasons additions with 2 and 3 as well as 1 to be 'carried'
+have much to recommend them in the very first stages of column addition
+with carrying. There is here the added advantage that a pupil will be
+more likely to remember to carry if he has to think _what_ to carry. The
+present common practice of using small numbers for ease in the addition
+itself teaches many children to think of carrying as adding one.
+
+_Other things being equal, arrange to have variety._ Thus it is
+probably, though not surely, wise to interrupt the monotony of learning
+the multiplication and division tables, by teaching the fundamentals of
+'short' multiplication and perhaps of division after the 5s, 2s, 3s, and
+4s are learned. This makes a break of several weeks. The facts for the
+6s, 7s, 8s, and 9s can then be put to varied use as fast as learned. It
+is almost certainly wise to interrupt the first half-year's work with
+addition and subtraction, by teaching 2 � 2, 2 � 3, 3 � 2, 2 � 4, 4 � 2,
+2 � 5, later by 2 � 10, 3 � 10, 4 � 10, 5 � 10, later by 1/2 + 1/2,
+1-1/2 + 1/2, 1/2 of 2, 1/2 of 4, 1/2 of 6, and at some time by certain
+profitable exercises wherein a pupil tells all he knows about certain
+numbers which may be made nuclei of important facts (say, 5, 8, 10, 12,
+15, and 20).
+
+_Other things being equal, use objective aids to verify an arithmetical
+process or inference after it is made, as well as to provoke it._ It is
+well at times to let pupils do everything that they can with relations
+abstractly conceived, testing their results by objective counting,
+measuring, adding, and the like. For example, an early step in adding
+should be to show three things, put them under a book, show two more,
+put these under the book, and then ask how many there are under the
+book, letting the objective counting come later as the test of the
+correctness of the addition.
+
+_Other things being equal, reserve all explanations of why a process
+must be right until the pupils can use the process accurately, and have
+verified the fact that it is right._ Except for the very gifted pupils,
+the ordinary preliminary deductive explanations of what must be done are
+probably useless as means of teaching the pupils what to do. They use up
+much time and are of so little permanent effect that, as we have seen,
+the very arithmeticians who advocate making them, admit that after a
+pupil has mastered the process he may be allowed to forget the reasons
+for it. I am not sure that the deductive proofs of why we place the
+decimal point as we do in division by a decimal, or invert and multiply
+in dividing by a fraction, and the like, are worth teaching at all. If
+they are to be taught at all, the time to teach them is (except for the
+very gifted) after the pupil has mastered the process and has confidence
+in it. He then at least knows what process he is to prove is right, and
+that it is right, and has had some chance of seeing _why_ it is right
+from his experience with it.
+
+One more principle may be mentioned without illustration. _Arrange the
+order of bonds with due regard for the aims of the other studies of the
+curriculum and the practical needs of the pupil outside of school._
+Arithmetic is not a book or a closed system of exercises. It is the
+quantitative work of the pupils in the elementary school. No narrower
+view of it is adequate.
+
+
+
+
+CHAPTER VIII
+
+THE DISTRIBUTION OF PRACTICE
+
+
+THE PROBLEM
+
+The same amount of practice may be distributed in various ways. Figures
+7 to 10, for example, show 200 practices with division by a fraction
+distributed over three and a half years of 10 months in four different
+ways. In Fig. 7, practice is somewhat equally distributed over the whole
+period. In Fig. 8 the practice is distributed at haphazard. In Fig. 9
+there is a first main learning period, a review after about ten weeks, a
+review at the beginning of the seventh grade, another review at the
+beginning of the eighth grade, and some casual practice rather at
+random. In Fig. 10 there is a main learning period, with reviews
+diminishing in length and separated by wider and wider intervals, with
+occasional practice thereafter to keep the ability alive and healthy.
+
+Plans I and II are obviously inferior to Plans III and IV; and Plan IV
+gives promise of being more effective than Plan III, since there seems
+danger that the pupil working by Plan III might in the ten weeks lose
+too much of what he had gained in the initial practice, and so again in
+the next ten weeks.
+
+It is not wise, however, to try now to make close decisions in the case
+of practice with division by a fraction; or to determine what the best
+distribution of practice is for that or any other ability to be
+improved. The facts of psychology are as yet not adequate for very close
+decisions, nor are the types of distribution of practice that are best
+adapted to different abilities even approximately worked out.
+
+    [Illustration: FIG. 7.--Plan I. 200 practices distributed somewhat
+    evenly over 3-1/2 years of 10 months. In Figs. 7, 8, 9, and 10,
+    each tenth of an inch along the base line represents one month.
+    Each hundredth of a square inch represents four practices, a
+    little square 1/20 of an inch wide and 1/20 inch high representing
+    one practice.]
+
+    [Illustration: FIG. 8.--Plan II. 200 practices distributed
+    haphazard over 3-1/2 years of 10 months.]
+
+    [Illustration: FIG. 9.--Plan III. A learning period, three reviews,
+    and incidental practice.]
+
+    [Illustration: FIG. 10.--Plan IV. A learning period with reviews
+    of decreasing length at increasing intervals.]
+
+
+SAMPLE DISTRIBUTIONS
+
+Let us rather examine some actual cases of distribution of practice
+found in school work and consider, not the attainment of the best
+possible distribution, but simply the avoidance of gross blunders and
+the attainment of reasonable, defensible procedures in this regard.
+
+Figures 11 to 18 show the distribution of examples in multiplication
+with multipliers of various sorts. _X_ stands for any digit except zero.
+_O_ stands for 0. _XXO_ thus means a multiplier like 350 or 270 or 160;
+_XOX_ means multipliers like 407, 905, or 206; _XX_ means multipliers
+like 25, 17, 38. Each of these diagrams covers approximately 3-1/2 years
+of school work, or from about the middle of grade 3 to the end of grade
+6. They are made from counts of four textbooks (A, B, C, and D), the
+count being taken for each successive 8 pages.[10] Each tenth of an inch
+along the base line equals 8 pages of the text in question. Each .01 sq.
+in. equals one example. The books, it will be observed, differ in the
+amount of practice given, as well as in the way in which it is
+distributed.
+
+  [10] At the end of a volume or part, the count may be from as
+       few as 5 or as many as 12 pages.
+
+These distributions are worthy of careful study; we shall note only a
+few salient facts about them here. Of the distributions of
+multiplications with multipliers of the _XX_ type, that of book D (Fig.
+14) is perhaps the best. A (Fig. 11) has too much of the practice too
+late; B (Fig. 12) gives too little practice in the first learning; C
+(Fig. 13) gives too much in the first learning and in grade 6. Among the
+distributions of multiplication with multipliers of the _XOX_ type, that
+of book D (Fig. 18) is again probably the best. A, B, and C (Figs. 15,
+16, and 17) have too much practice early and too long intervals between
+reviews. Book C (Fig. 17) by a careless oversight has one case of this
+very difficult process, without any explanation, weeks before the
+process is taught!
+
+    [Illustration: FIG. 11.--Distribution of practise with multipliers
+    of the _XX_ type in the first two books of the three-book text A.]
+
+    [Illustration: FIG. 12.--Same as Fig. 11, but for text B. Following
+    this period come certain pages of computation to be used by the
+    teacher at her discretion, containing 24 _XX_ multiplications.]
+
+    [Illustration: FIG. 13.--Same as Fig. 11, but for text C.]
+
+    [Illustration: FIG. 14.--Same as Fig. 11, but for text D.]
+
+    [Illustration: FIG. 15.--Distribution of practice with multipliers
+    of the _XOX_ type in the first two books of the three-book text
+    A.]
+
+    [Illustration: FIG. 16.--Same as Fig. 15, but for text B. Following
+    this period come certain pages of computation to be used by the
+    teacher at her discretion, containing 17 _XOX_ multiplications.]
+
+    [Illustration: FIG. 17.--Same as Fig. 16, but for text C.]
+
+    [Illustration: FIG. 18.--Same as Fig. 16, but for text D.]
+
+Figures 19, 20, 21, 22, and 23 all concern the first two books of the
+three-book text E.
+
+Figure 19 shows the distribution of practice on 5 � 5 in the first two
+books of text E. The plan is the same as in Figs. 11 to 18, except that
+each tenth of an inch along the base line represents ten pages. Figure
+20 shows the distribution of practice on 7 � 7; Fig. 21 shows it for
+6 � 7 and 7 � 6 together. In Figs. 20 and 21 also, 0.1 inch along the
+base line equals ten pages.
+
+Figures 22 and 23 show the distribution of practice on the divisions of
+72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or 9, and on the
+divisions of 81, 82 ... 89 by 9. Each tenth of an inch along the base
+line represents ten pages here also.
+
+Figures 19 to 23 show no consistent plan for distributing practice.
+With 5 � 5 (Fig. 19) the amount of practice increases from the first
+treatment in grade 3 to the end of grade 6, so that the distribution
+would be better if the pupil began at the end and went backward! With
+7 � 7 (Fig. 20) the practice is distributed rather evenly and in small
+doses. With 6 � 7 and 7 � 6 (Fig. 21) much of it is in very large doses.
+With the divisions (Figs. 22 and 23) the practice is distributed more
+suitably, though in Fig. 23 there is too much of it given at one time
+in the middle of the period.
+
+    [Illustration: FIG. 19.--Distribution of practice with 5 � 5 in
+    the first two books of the three-book text E.]
+
+    [Illustration: FIG. 20.--Distribution of practice with 7 � 7 in
+    the first two books of text E.]
+
+    [Illustration: FIG. 21.--Distribution of practice with 6 � 7
+    or 7 � 6 in the first two books of text E.]
+
+    [Illustration: FIG. 22.--Distribution of practice with
+    72, 73 ... 79 � 8 or 9 in the first two books of text E.]
+
+    [Illustration: FIG. 23.--Distribution of practice with
+    81, 82 ... 89 � 9 in the first two books of text E.]
+
+
+POSSIBLE IMPROVEMENTS
+
+Even if we knew what the best distribution of practice was for each
+ability of the many to be inculcated by arithmetical instruction, we
+could perhaps not provide it for all of them. For, in the first place,
+the allotments for some of them might interfere with those for others.
+In the second place, there are many other considerations of importance
+in the ordering of topics besides giving the optimal distribution of
+practice to each ability. Such are considerations of interest, of
+welding separate abilities into an integrated total ability, and of the
+limitations due to the school schedule with its Saturdays, Sundays,
+holidays, and vacations.
+
+Improvement can, however, be made over present practice in many
+respects. A scientific examination of the teaching of almost any class
+for a year, or of many of our standard instruments of instruction, will
+reveal opportunities for improving the distribution of practice with no
+sacrifice of interest, and with an actual gain in integrated functioning
+arithmetical power. In particular it will reveal cases where an ability
+is given practice and then, never being used again, left to die of
+inactivity. It will reveal cases where an ability is given practice and
+then left so long without practice that the first effect is nearly lost.
+There will be cases where practice is given and reviews are given, but
+all in such isolation from everything else in arithmetic that the
+ability, though existent, does not become a part of the pupil's general
+working equipment. There will be cases where more practice is given in
+the late than the earlier periods for no apparent extrinsic advantage;
+and cases where the practice is put where it is for no reason that is
+observable save that the teacher or author in question has decided to
+have some drill work at that time!
+
+Each ability has its peculiar needs in this matter, and no set rules are
+at present of much value. It will be enough for the present if we are
+aroused to the problem of distribution, avoid obvious follies like those
+just noted, and exercise what ingenuity we have.
+
+
+
+
+CHAPTER IX
+
+THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL NOTIONS IN
+ARITHMETIC[11]
+
+  [11] Certain paragraphs in this and the following chapter are
+       taken from the author's _Educational Psychology_, with
+       slight modifications.
+
+
+RESPONSES TO ELEMENTS AND CLASSES
+
+The plate which you see, the egg before you at the breakfast table, and
+this page are concrete things, but whiteness, whether of plate, egg, or
+paper, is, we say, an abstract quality. To be able to think of whiteness
+irrespective of any concrete white object is to be able to have an
+abstract idea or notion of white; to be able to respond to whiteness,
+irrespective of whether it is a part of china, eggshell, paper or
+whatever object, is to be able to respond to the abstract element of
+whiteness.
+
+Learning arithmetic involves the formation of very many such ideas, the
+acquisition of very many such powers of response to elements regardless
+of the gross total situations in which they appear. To appreciate the
+fiveness of five boys, five pencils, five inches, five rings of a bell;
+to understand the division into eight equal parts of 40 cents, 32 feet,
+64 minutes, or 16 ones; to respond correctly to the fraction relation in
+2/3, 5/6, 3/4, 7/12, 1/8, or any other; to be sensitive to the common
+element of 9 = 3 � 3, 16 = 4 � 4, 625 = 25 � 25, .04 = .2 � .2, 1/4 =
+1/2 � 1/2,--these are obvious illustrations. All the numbers which the
+pupil learns to understand and manipulate are in fact abstractions; all
+the operations are abstractions; percent, discount, interest, height,
+length, area, volume, are abstractions; sum, difference, product,
+quotient, remainder, average, are facts that concern elements or aspects
+which may appear with countless different concrete surroundings or
+concomitants.
+
+Towser is a particular dog; your house lot on Elm Street is a particular
+rectangle; Mr. and Mrs. I.S. Peterson and their daughter Louise are a
+particular family of three. In contrast to these particulars, we mean
+by a dog, a rectangle, and a family of three, _any_ specimens of these
+classes of facts. The idea of a dog, of rectangles in general, of any
+family of three is a general notion, a concept or idea of a class or
+species. The ability to respond to any dog, or rectangle, or family of
+three, regardless of which particular one it may be, is the general
+notion in action.
+
+Learning arithmetic involves the formation of very many such general
+notions, such powers of response to any member of a certain class. Thus
+a hundred different sized lots may all be responded to as rectangles;
+9/18, 12/27, 15/24, and 27/36 may all be responded to as members of the
+class, 'both members divisible by 3.' The same fact may be responded to
+in different ways according to the class to which it is assigned. Thus 4
+in 3/4, 4/5, 45, 54, and 405 is classed respectively as 'a certain sized
+part of unity,' 'a certain number of parts of the size shown by the 5,'
+'a certain number of tens,' 'a certain number of ones,' and 'a certain
+number of hundreds.' Each abstract quality may become the basis of a
+class of facts. So fourness as a quality corresponds to the class
+'things four in number or size'; the fractional quality or relation
+corresponds to the class 'fractions.' The bonds formed with classes of
+facts and with elements or features by which one whole class of facts is
+distinguished from another, are in fact, a chief concern of arithmetical
+learning.[12]
+
+  [12] It should be noted that just as concretes give rise to
+       abstractions, so these in turn give rise to still more
+       abstract abstractions. Thus fourness, fiveness, twentyness,
+       and the like give rise to 'integral-number-ness.' Similarly
+       just as individuals are grouped into general classes, so
+       classes are grouped into still more general classes.
+       Half, quarter, sixth, and tenth are general notions, but
+       'one ...th' is more general; and 'fraction' is still more
+       general.
+
+
+FACILITATING THE ANALYSIS OF ELEMENTS
+
+Abstractions and generalizations then depend upon analysis and upon
+bonds formed with more or less subtle elements rather than with gross
+total concrete situations. The process involved is most easily
+understood by considering the means employed to facilitate it.
+
+The first of these is having the learner respond to the total situations
+containing the element in question with the attitude of piecemeal
+examination, and with attentiveness to one element after another,
+especially to so near an approximation to the element in question as he
+can already select for attentive examination. This attentiveness to one
+element after another serves to emphasize whatever appropriate minor
+bonds from the element in question the learner already possesses. Thus,
+in teaching children to respond to the 'fiveness' of various
+collections, we show five boys or five girls or five pencils, and say,
+"See how many boys are standing up. Is Jack the only boy that is
+standing here? Are there more than two boys standing? Name the boys
+while I point at them and count them. (Jack) is one, and (Fred) is one
+more, and (Henry) is one more. Jack and Fred make (two) boys. Jack and
+Fred and Henry make (three) boys." (And so on with the attentive
+counting.) The mental set or attitude is directed toward favoring the
+partial and predominant activity of 'how-many-ness' as far as may be;
+and the useful bonds that the 'fiveness,' the 'one and one and one and
+one and one-ness,' already have, are emphasized as far as may be.
+
+The second of the means used to facilitate analysis is having the
+learner respond to many situations each containing the element in
+question (call it A), but with varying concomitants (call these V. C.)
+his response being so directed as, so far as may be, to separate each
+total response into an element bound to the A and an element bound to
+the V. C.
+
+Thus the child is led to associate the responses--'Five boys,' 'Five
+girls,' 'Five pencils,' 'Five inches,' 'Five feet,' 'Five books,' 'He
+walked five steps,' 'I hit my desk five times,' and the like--each with
+its appropriate situation. The 'Five' element of the response is thus
+bound over and over again to the 'fiveness' element of the situation,
+the mental set being 'How many?,' but is bound only once to any one of
+the concomitants. These concomitants are also such as have preferred
+minor bonds of their own (the sight of a row of boys _per se_ tends
+strongly to call up the 'Boys' element of the response). The other
+elements of the responses (boys, girls, pencils, etc.) have each only a
+slight connection with the 'fiveness' element of the situations. These
+slight connections also in large part[13] counteract each other, leaving
+the field clear for whatever uninhibited bond the 'fiveness' has.
+
+  [13] They may, of course, also result in a fusion or an alternation
+       of responses, but only rarely.
+
+The third means used to facilitate analysis is having the learner
+respond to situations which, pair by pair, present the element in a
+certain context and present that same context with _the opposite of the
+element in question_, or with something at least very unlike the
+element. Thus, a child who is being taught to respond to 'one fifth' is
+not only led to respond to 'one fifth of a cake,' 'one fifth of a pie,'
+'one fifth of an apple,' 'one fifth of ten inches,' 'one fifth of an
+army of twenty soldiers,' and the like; he is also led to respond to
+each of these _in contrast with_ 'five cakes,' 'five pies,' 'five
+apples,' 'five times ten inches,' 'five armies of twenty soldiers.'
+Similarly the 'place values' of tenths, hundredths, and the rest are
+taught by contrast with the tens, hundreds, and thousands.
+
+These means utilize the laws of connection-forming to disengage a
+response element from gross total responses and attach it to some
+situation element. The forces of use, disuse, satisfaction, and
+discomfort are so maneuvered that an element which never exists by
+itself in nature can influence man almost as if it did so exist, bonds
+being formed with it that act almost or quite irrespective of the gross
+total situation in which it inheres. What happens can be most
+conveniently put in a general statement by using symbols.
+
+Denote by _a_ + _b_, _a_ + _g_, _a_ + _l_, _a_ + _q_, _a_ + _v_, and
+_a_ + _B_ certain situations alike in the element _a_ and different in
+all else. Suppose that, by original nature or training, a child responds
+to these situations respectively by r_{1} + r_{2}, r_{1} + r_{7},
+r_{1} + r_{12}, r_{1} + r_{17}, r_{1} + r_{22}, r_{1} + r_{27}. Suppose
+that man's neurones are capable of such action that r_{1}, r_{2}, r_{7},
+r_{12}, r_{22}, and r_{27}, can each be made singly.
+
+
+Case I. Varying Concomitants
+
+Suppose that _a_ + _b_, _a_ + _g_, _a_ + _l_, etc., occur once each.
+
+  We have _a_ + _b_ responded to by r_{1} + r_{2},
+          _a_ + _g_      "       "  r_{1} + r_{7},
+          _a_ + _l_      "       "  r_{1} + r_{12},
+          _a_ + _q_      "       "  r_{1} + r_{17},
+          _a_ + _v_      "       "  r_{1} + r_{22}, and
+          _a_ + _B_      "       "  r_{1} + r_{27}, as shown in
+                                      Scheme I.
+
+Scheme I
+
+              _a_   _b_   _g_   _l_   _q_   _v_   _B_
+  r_{1}        6     1     1     1     1     1     1
+  r_{2}        1     1
+  r_{7}        1           1
+  r_{12}       1                 1
+  r_{17}       1                       1
+  r_{22}       1                             1
+  r_{27}       1                                   1
+
+_a_ is thus responded to by r_{1} (that is, connected with r_{1}) each
+time, or six in all, but only once each with _b_, _g_, _l_, _q_, _v_,
+and _B_. _b_, _g_, _l_, _q_, _v_, and _B_ are connected once each with
+r_{1} and once respectively with r_{2}, r_{7}, r_{12}, etc. The bond
+from _a_ to r_{1}, has had six times as much exercise as the bond from
+_a_ to r_{2}, or from _a_ to r_{7}, etc. In any new gross situation, _a_
+0, _a_ will be more predominant in determining response than it would
+otherwise have been; and r_{1} will be more likely to be made than
+r_{2}, r_{7}, r_{12}, etc., the other previous associates in the
+response to a situation containing _a_. That is, the bond from the
+element _a_ to the response r_{1} has been notably strengthened.
+
+
+Case II. Contrasting Concomitants
+
+Now suppose that _b_ and _g_ are very dissimilar elements (_e.g._, white
+and black), that _l_ and _q_ are very dissimilar (_e.g._, long and
+short), and that _v_ and _B_ are also very dissimilar. To be very
+dissimilar means to be responded to very differently, so that r_{7}, the
+response to _g_, will be very unlike r_{2}, the response to _b_. So
+r_{7} may be thought of as r_{not 2} or r_{-2}. In the same way r_{12}
+may be thought of as r_{not 12} or r_{-12}, and r_{27} may be called
+r_{not 22} or r_{-22}.
+
+Then, if the situations _a_ _b_, _a _g_, _a _l_, _a _q_, _a _v_, and
+_a_ _B_ are responded to, each once, we have:--
+
+  _a_ + _b_ responded to by r_{1} + r_{2},
+  _a_ + _g_     "        "  r_{1} + r_{not 2},
+  _a_ + _l_     "        "  r_{1} + r_{12},
+  _a_ + _q_     "        "  r_{1} + r_{not 12},
+  _a_ + _v_     "        "  r_{1} + r_{22}, and
+  _a_ + _B_     "        "  r_{1} + r_{not 22}, as shown in Scheme II.
+
+Scheme II
+
+              _a_  _b_     _g_       _l_     _q_       _v_     _B_
+                       (opp. of _b_)     (opp. of _l_)     (opp. of _v_)
+  r_{1}        6    1       1         1       1         1       1
+  r_{not 1}
+  r_{2}        1    1
+  r_{not 2}    1            1
+  r_{12}       1                      1
+  r_{not 12}   1                               1
+  r_{22}       1                                        1
+  r_{not 22}   1                                                1
+
+r_{1} is connected to _a_ by 6 repetitions. r_{2} and r_{not 2} are each
+connected to _a_ by 1 repetition, but since they interfere, canceling
+each other so to speak, the net result is for _a_ to have zero tendency
+to call up r_{2} or r_{not 2}. r_{12} and r_{not 12} are each connected
+to _a_ by 1 repetition, but they interfere with or cancel each other
+with the net result that _a_ has zero tendency to call up r_{12} or
+r_{not 12}. So with r_{22} and r_{not 22}. Here then the net result of
+the six connections of _a_ _b_, _a_ _g_, _a_ _l_, _a_ _q_, _a_ _v_, and
+_a_ _B_ is to connect _a_ with _r_, and with nothing else.
+
+
+Case III. Contrasting Concomitants and Contrasting Element
+
+Suppose now that the facts are as in Case II, but with the addition of
+six experiences where a certain element which is the opposite of, or
+very dissimilar to, _a_ is connected with the response r_{not 1}, or
+r_{-1} which is opposite to, or very dissimilar to r_{1}. Call this
+opposite of _a_, - _a_.
+
+That is, we have not only
+
+  _a_ + _b_ responded to by r_{1} + r_{2},
+  _a_ + _g_     "       "   r_{1} + r_{not 2},
+  _a_ + _l_     "       "   r_{1} + r_{12},
+  _a_ + _q_     "       "   r_{1} + r_{not 12},
+  _a_ + _v_     "       "   r_{1} + r_{22}, and
+  _a_ + _B_     "       "   r_{1} + r_{not 22},
+
+but also
+
+  - _a_ + _b_ responded to by r_{not 1} + r_{2},
+  - _a_ + _g_     "       "   r_{not 1} + r_{not 2},
+  - _a_ + _l_     "       "   r_{not 1} + r_{12},
+  - _a_ + _q_     "       "   r_{not 1} + r_{not 12},
+  - _a_ + _v_     "       "   r_{not 1} + r_{22}, and
+  - _a_ + _B_     "       "   r_{not 1} + r_{not 22}, as shown in
+                                    Scheme III.
+
+Scheme III
+
+            _a_    opp.     _b_     _g_     _l_     _q_     _v_     _B_
+                  of _a_   (opp. of _b_)   (opp. of _l_)   (opp. of _v_)
+  r_{1}       6              1       1       1       1       1       1
+  r_{not 1}         6        1       1       1       1       1       1
+  r_{2}       1     1        2
+  r_{not 2}   1     1                2
+  r_{12}      1     1                        2
+  r_{not 12}  1     1                                2
+  r_{22}      1     1                                        2
+  r_{not 22}  1     1                                                2
+
+In this series of twelve experiences _a_ connects with r_{1} six times
+and the opposite of _a_ connects with r_{not 1} six times. _a_ connects
+equally often with three pairs of mutual destructives r_{2} and
+r_{not 2}, r_{12} and r_{not 12}, r_{22} and r_{not 22}, and so has zero
+tendency to call them up. - _a_ has also zero tendency to call up any of
+these responses except its opposite, r_{not 1}. _b_, _g_, _l_, _q_, _v_,
+and _B_ are made to connect equally often with r_{1} and r_{not 1}. So,
+of these elements, _a_ is the only one left with a tendency to call up
+r_{1}.
+
+Thus, by the mere action of frequency of connection, r_{1} is connected
+with _a_; the bonds from _a_ to anything except r_{1} are being
+counteracted, and the slight bonds from anything except _a_ to r_{1} are
+being counteracted. The element _a_ becomes predominant in situations
+containing it; and its bond toward r_{1} becomes relatively enormously
+strengthened and freed from competition.
+
+These three processes occur in a similar, but more complicated,
+form if the situations _a_ + _b_, _a_ + _g_, etc., are replaced by
+_a_ + _b_ + _c_ + _d_ + _e_ + _f_, _a_ + _g_ + _h_ + _i_ + _j_ + _k_,
+etc., and the responses r_{1} + r_{2}, r_{1} + r_{7}, r_{1} + r_{12},
+etc., are replaced by r_{1} + r_{2} + r_{3} + r_{4} + r_{5} + r_{6},
+r_{1} + r_{7} + r_{8} + r_{9} + r_{10} + r_{11}, etc.--_provided the_
+r_{1}, r_{2}, r_{3}, r_{4}, etc., _can be made singly_. In so far as any
+one of the responses is necessarily co-active with any one of the others
+(so that, for example, r_{13} always brings r_{26} with it and _vice
+versa_), the exact relations of the numbers recorded in schemes like
+schemes I, II, and III on pages 172 to 174 will change; but, unless
+r_{1} has such an inevitable co-actor, the general results of schemes I,
+II, and III will hold good. If r_{1} does have such an inseparable
+co-actor, say r_{2}, then, of course, _a_ can never acquire bonds with
+r_{1} alone, but everywhere that r_{1} or r_{2} appears in the preceding
+schemes the other element must appear also. r_{1} r_{2} would then have
+to be used as a unit in analysis.
+
+The '_a_ + _b_,' '_a_ + _g_,' '_a_ + _l_,' ... '_a_ + _B_' situations
+may occur unequal numbers of times, altering the exact numerical
+relations of the connections formed and presented in schemes I, II,
+and III; but the process in general remains the same.
+
+So much for the effect of use and disuse in attaching appropriate
+response elements to certain subtle elements of situations. There are
+three main series of effects of satisfaction and discomfort. They
+serve, first, to emphasize, from the start, the desired bonds leading to
+the responses r_{1} + r_{2}, r_{1} + r_{7}, etc., to the total
+situations, and to weed out the undesirable ones. They also act to
+emphasize, in such comparisons and contrasts as have been described,
+every action of the bond from _a_ to r_{1}; and to eliminate every
+tendency of _a_ to connect with aught save r_{1}, and of aught save _a_
+to connect with r_{1}. Their third service is to strengthen the bonds
+produced of appropriate responses to _a_ wherever it occurs, whether or
+not any formal comparisons and contrasts take place.
+
+The process of learning to respond to the difference of pitch in tones
+from whatever instrument, to the 'square-root-ness' of whatever number,
+to triangularity in whatever size or combination of lines, to equality
+of whatever pairs, or to honesty in whatever person or instance, is thus
+a consequence of associative learning, requiring no other forces than
+those of use, disuse, satisfaction, and discomfort. "What happens in
+such cases is that the response, by being connected with many situations
+alike in the presence of the element in question and different in other
+respects, is bound firmly to that element and loosely to each of its
+concomitants. Conversely any element is bound firmly to any one response
+that is made to all situations containing it and very, very loosely to
+each of those responses that are made to only a few of the situations
+containing it. The element of triangularity, for example, is bound
+firmly to the response of saying or thinking 'triangle' but only very
+loosely to the response of saying or thinking white, red, blue, large,
+small, iron, steel, wood, paper, and the like. A situation thus acquires
+bonds not only with some response to it as a gross total, but also with
+responses to any of its elements that have appeared in any other gross
+totals. Appropriate response to an element regardless of its
+concomitants is a necessary consequence of the laws of exercise and
+effect if an animal learns to make that response to the gross total
+situations that contain the element and not to make it to those that do
+not. Such prepotent determination of the response by one or another
+element of the situation is no transcendental mystery, but, given the
+circumstances, a general rule of all learning." Such are at bottom only
+extreme cases of the same learning as a cat exhibits that depresses a
+platform in a certain box whether it faces north or south, whether the
+temperature is 50 or 80 degrees, whether one or two persons are in
+sight, whether she is exceedingly or moderately hungry, whether fish or
+milk is outside the box. All learning is analytic, representing the
+activity of elements within a total situation. In man, by virtue of
+certain instincts and the course of his training, very subtle elements
+of situations can so operate.
+
+       *       *       *       *       *
+
+Learning by analysis does not often proceed in the carefully organized
+way represented by the most ingenious marshaling of comparing and
+contrasting activities. The associations with gross totals, whereby in
+the end an element is elevated to independent power to determine
+response, may come in a haphazard order over a long interval of time.
+Thus a gifted three-year-old boy will have the response element of
+'saying or thinking _two_,' bound to the 'two-ness' element of very many
+situations in connection with the 'how-many' mental set; and he will
+have made this analysis without any formal, systematic training. An
+imperfect and inadequate analysis already made is indeed usually the
+starting point for whatever systematic abstraction the schools direct.
+Thus the kindergarten exercises in analyzing out number, color, size,
+and shape commonly assume that 'one-ness' _versus_ 'more-than-one-ness,'
+black and white, big and little, round and not round are, at least
+vaguely, active as elements responded to in some independence of their
+contexts. Moreover, the tests of actual trial and success in further
+undirected exercises usually co�perate to confirm and extend and refine
+what the systematic drills have given. Thus the ordinary child in school
+is left, by the drills on decimal notation, with only imperfect power
+of response to the 'place-values.' He continues to learn to respond
+properly to them by finding that 4 � 40 = 160, 4 � 400 = 1600,
+800 - 80 = 720, 800 - 8 = 792, 800-800 = 0, 42 � 48 = 2016,
+24 � 48 = 1152, and the like, are satisfying; while 4 � 40 = 16,
+23 � 48 = 832, 800 - 8 = 0, and the like, are not. The process of
+analysis is the same in such casual, unsystematized formation of
+connections with elements as in the deliberately managed, piecemeal
+inspection, comparison, and contrast described above.
+
+
+SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS
+
+The arrangement of a pupil's experiences so as to direct his attention
+to an element, vary its concomitants instructively, stimulate
+comparison, and throw the element into relief by contrast may be by
+fixed, formal, systematic exercises. Or it may be by much less formal
+exercises, spread over a longer time, and done more or less incidentally
+in other connections. We may call these two extremes the 'systematic'
+and 'opportunistic,' since the chief feature of the former is that it
+systematically provides experiences designed to build up the power of
+correct response to the element, whereas the chief feature of the latter
+is that it uses especially such opportunities as occur by reason of the
+pupil's activities and interests.
+
+Each method has its advantages and disadvantages. The systematic method
+chooses experiences that are specially designed to stimulate the
+analysis; it provides these at a certain fixed time so that they may
+work together; it can then and there test the pupils to ascertain
+whether they really have the power to respond to the element or aspect
+or feature in question. Its disadvantages are, first, that many of the
+pupils will feel no need for and attach no interest or motive to these
+formal exercises; second, that some of the pupils may memorize the
+answers as a verbal task instead of acquiring insight into the facts;
+third, that the ability to respond to the element may remain restricted
+to the special cases devised for the systematic training, and not be
+available for the genuine uses of arithmetic.
+
+The opportunistic method is strong just where the systematic is weak.
+Since it seizes upon opportunities created by the pupil's abilities and
+interests, it has the attitude of interest more often. Since it builds
+up the experiences less formally and over a wider space of time, the
+pupils are less likely to learn verbal answers. Since its material comes
+more from the genuine uses of life, the power acquired is more likely to
+be applicable to life.
+
+Its disadvantage is that it is harder to manage. More thought and
+experimentation are required to find the best experiences; greater care
+is required to keep track of the development of an abstraction which is
+taught not in two days, but over two months; and one may forget to test
+the pupils at the end. In so far as the textbook and teacher are able to
+overcome these disadvantages by ingenuity and care, the opportunistic
+method is better.
+
+
+ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS
+
+We may expect much improvement in the formation of abstract and general
+ideas in arithmetic from the application of three principles in addition
+to those already described. They are: (1) Provide enough actual
+experiences before asking the pupil to understand and use an abstract or
+general idea. (2) Develop such ideas gradually, not attempting to give
+complete and perfect ideas all at once. (3) Develop such ideas so far as
+possible from experiences which will be valuable to the pupil in and of
+themselves, quite apart from their merit as aids in developing the
+abstraction or general notion. Consider these three principles in order.
+
+Children, especially the less gifted intellectually, need more
+experiences as a basis for and as applications of an arithmetical
+abstraction or concept than are usually given them. For example, in
+paving the way for the principle, "Any number times 0 equals 0," it is
+not safe to say, "John worked 8 days for 0 minutes per day. How many
+minutes did he work?" and "How much is 0 times 4 cents?" It will be
+much better to spend ten or fifteen minutes as follows:[14] "What does
+zero mean? (Not any. No.) How many feet are there in eight yards?
+In 5 yards? In 3 yards? In 2 yards? In 1 yard? In 0 yard? How many
+inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... qt.
+5 pk. = .... qt. 0 pk. = .... qt. A boy receives 60 cents an hour
+when he works. How much does he receive when he works 3 hr.? 8 hr.?
+6 hr.? 0 hr.? A boy received 60 cents a day for 0 days. How much did he
+receive? How much is 0 times $600? How much is 0 times $5000? How much
+is 0 times a million dollars? 0 times any number equals....
+
+   232  (At the blackboard.)  0 time 232 equals what?
+    30  I write 0 under the 0.[15]  3 times 232 equals what?
+  ----
+  6960  Continue at the blackboard with
+
+        734         321         312          41
+         20          40          30          60  etc."
+        ---         ---         ---          --
+
+  [14] The more gifted children may be put to work using the principle
+       after the first minute or two.
+
+  [15] 232
+        30   If desired this form may be used, with the appropriate
+       ---      difference in the form of the questions and statements.
+       000
+      696
+      ----
+      6960
+
+Pupils in the elementary school, except the most gifted, should not be
+expected to gain mastery over such concepts as _common fraction_,
+_decimal fraction_, _factor_, and _root_ quickly. They can learn a
+definition quickly and learn to use it in very easy cases, where even a
+vague and imperfect understanding of it will guide response correctly.
+But complete and exact understanding commonly requires them to take,
+not one intellectual step, but many; and mastery in use commonly comes
+only as a slow growth. For example, suppose that pupils are taught
+that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03,
+etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean
+1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal
+fractions. They may then respond correctly when asked to write a decimal
+fraction, or to state which of these,--1/4, .4, 3/8, .07, .002,
+5/6,--are common fractions and which are decimal fractions. They may
+be able, though by no means all of them will be, to write decimal
+fractions which equal 1/2 and 1/5, and the common fractions which
+equal .1 and .09. Most of them will not, however, be able to respond
+correctly to "Write a decimal mixed number"; or to state which of
+these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which
+are decimals; or to write the decimal fractions which equal 3/4 and 1/3.
+
+If now the teacher had given all at once the additional experiences
+needed to provide the ability to handle these more intricate and subtle
+features of decimal-fraction-ness, the result would have been confusion
+for most pupils. The general meaning of .32, .14, .99, and the like
+requires some understanding of .30, .10, .90, and .02, .04, .08; but it
+is not desirable to disturb the child with .30 while he is trying to
+master 2.3, 4.3, 6.3, and the like. Decimals in general require
+connection with place value and the contrasts of .41 with 41, 410, 4.1,
+and the like, but if the relation to place values in general is taught
+in the same lesson with the relation to /10s, /100s, /1000s, the mind
+will suffer from violent indigestion.
+
+A wise pedagogy in fact will break up the process of learning the
+meaning and use of decimal fractions into many teaching units, for
+example, as follows:--
+
+(1) Such familiarity with fractions with large denominators as is
+desirable for pupils to have, as by an exercise in reducing to lowest
+terms, 8/10, 36/64, 20/25, 18/24, 24/32, 21/30, 25/100, 40/100, and the
+like. This is good as a review of cancellation, and as an extension of
+the idea of a fraction.
+
+(2) Objective work, showing 1/10 sq. ft., 1/50 sq. ft., 1/100 sq. ft.,
+and 1/1000 sq. ft., and having these identified and the forms 1/10 sq.
+ft., 1/100 sq. ft., and 1/1000 sq. ft. learned. Finding how many
+feet = 1/10 mile and 1/100 mile.
+
+(3) Familiarity with /100s and /1000s by reductions of 750/1000, 50/100,
+etc., to lowest terms and by writing the missing numerators in
+500/1000 = /100 = /10 and the like, and by finding 1/10, 1/100, and
+1/1000 of 3000, 6000, 9000, etc.
+
+(4) Writing 1/10 as .1 and 1/100 as .01, 11/100, 12/100, 13/100, etc.,
+as .11, .12, .13. United States money is used as the introduction.
+Application is made to miles.
+
+(5) Mixed numbers with a first decimal place. The cyclometer or
+speedometer. Adding numbers like 9.1, 14.7, 11.4, etc.
+
+(6) Place value in general from thousands to hundredths.
+
+(7) Review of (1) to (6).
+
+(8) Tenths and hundredths of a mile, subtraction when both numbers
+extend to hundredths, using a railroad table of distances.
+
+(9) Thousandths. The names 'decimal fractions or decimals,' and 'decimal
+mixed numbers or decimals.' Drill in reading any number to thousandths.
+The work will continue with gradual extension and refinement of the
+understanding of decimals by learning how to operate with them in
+various ways.
+
+Such may seem a slow progress, but in fact it is not, and many of these
+exercises whereby the pupil acquires his mastery of decimals are useful
+as organizations and applications of other arithmetical facts.
+
+That, it will be remembered, was the third principle:--"Develop abstract
+and general ideas by experiences which will be intrinsically valuable."
+The reason is that, even with the best of teaching, some pupils will
+not, within any reasonable limits of time expended, acquire ideas that
+are fully complete, rigorous when they should be, flexible when they
+should be, and absolutely exact. Many children (and adults, for that
+matter) could not within any reasonable limits of time be so taught the
+nature of a fraction that they could decide unerringly in original
+exercises like:--
+
+Is 2.75/25 a common fraction?
+
+Is $.25 a decimal fraction?
+
+Is one _x_th of _y_ a fraction?
+
+Can the same words mean both a common fraction and a decimal fraction?
+
+Express 1 as a common fraction.
+
+Express 1 as a decimal fraction.
+
+These same children can, however, be taught to operate correctly with
+fractions in the ordinary uses thereof. And that is the chief value of
+arithmetic to them. They should not be deprived of it because they
+cannot master its subtler principles. So we seek to provide experiences
+that will teach all pupils something of value, while stimulating in
+those who have the ability the growth of abstract ideas and general
+principles.
+
+Finally, we should bear in mind that working with qualities and
+relations that are only partly understood or even misunderstood does
+under certain conditions give control over them. The general process of
+analytic learning in life is to respond as well as one can; to get a
+clearer idea thereby; to respond better the next time; and so on. For
+instance, one gets some sort of notion of what 1/5 means; he then
+answers such questions as 1/5 of 10 = ? 1/5 of 5 = ? 1/5 of 20 = ?;
+by being told when he is right and when he is wrong, he gets from
+these experiences a better idea of 1/5; again he does his best with
+1/5 = _/10, 1/5 = _/15, etc., and as before refines and enlarges his
+concept of 1/5. He adds 1/5 to 2/5, etc., 1/5 to 3/10, etc., 1/5 to 1/2,
+etc., and thereby gains still further, and so on.
+
+What begins as a blind habit of manipulation started by imitation may
+thus grow into the power of correct response to the essential element.
+The pupil who has at the start no notion at all of 'multiplying' may
+learn what multiplying is by his experience that '4 6 multiplying
+gives 24'; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing
+something with numbers and differentiates right results, he will often
+reach in the end the abstractions which he is supposed to need in the
+beginning. It may even be the case with some of the abstractions
+required in arithmetic that elaborate provision for comprehension
+beforehand is not so efficient as the same amount of energy devoted
+partly to provision for analysis itself beforehand and partly to
+practice in response to the element in question without full
+comprehension.
+
+It certainly is not the best psychology and not the best educational
+theory to think that the pupil first masters a principle and then merely
+applies it--first does some thinking and then computes by mere routine.
+On the contrary, the applications should help to establish, extend, and
+refine the principle--the work a pupil does with numbers should be a
+main means of increasing his understanding of the principles of
+arithmetic as a science.
+
+
+
+
+CHAPTER X
+
+THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC
+
+
+THE ESSENTIALS OF ARITHMETICAL REASONING
+
+We distinguish aimless reverie, as when a child dreams of a vacation
+trip, from purposive thinking, as when he tries to work out the answer
+to "How many weeks of vacation can a family have for $120 if the cost is
+$22 a week for board, $2.25 a week for laundry, and $1.75 a week for
+incidental expenses, and if the railroad fares for the round trip are
+$12?" We distinguish the process of response to familiar situations,
+such as five integral numbers to be added, from the process of response
+to novel situations, such as (for a child who has not been trained with
+similar problems):--"A man has four pieces of wire. The lengths are 120
+yd., 132 meters, 160 feet, and 1/8 mile. How much more does he need to
+have 1000 yd. in all?" We distinguish 'thinking things together,' as
+when a diagram or problem or proof is understood, from thinking of one
+thing after another as when a number of words are spelled or a poem in
+an unknown tongue is learned. In proportion as thinking is purposive,
+with selection from the ideas that come up, and in proportion as it
+deals with novel problems for which no ready-made habitual response is
+available, and in proportion as many bonds act together in an organized
+way to produce response, we call it reasoning.
+
+When the conclusion is reached as the effect of many particular
+experiences, the reasoning is called inductive. When some principle
+already established leads to another principle or to a conclusion about
+some particular fact, the reasoning is called deductive. In both cases
+the process involves the analysis of facts into their elements, the
+selection of the elements that are deemed significant for the question
+at hand, the attachment of a certain amount of importance or weight to
+each of them, and their use in the right relations. Thought may fail
+because it has not suitable facts, or does not select from them the
+right ones, or does not attach the right amount of weight to each, or
+does not put them together properly.
+
+In the world at large, many of our failures in thinking are due to not
+having suitable facts. Some of my readers, for example, cannot solve the
+problem--"What are the chances that in drawing a card from an ordinary
+pack of playing-cards four times in succession, the same card will be
+drawn each time?" And it will be probably because they do not know
+certain facts about the theory of probabilities. The good thinkers among
+such would look the matter up in a suitable book. Similarly, if a person
+did not happen to know that there were fifty-two cards in all and that
+no two were alike, he could not reason out the answer, no matter what
+his mastery of the theory of probabilities. If a competent thinker, he
+would first ask about the size and nature of the pack. In the actual
+practice of reasoning, that is, we have to survey our facts to see if we
+lack any that are necessary. If we do, the first task of reasoning is to
+acquire those facts.
+
+This is specially true of the reasoning about arithmetical facts in
+life. "Will 3-1/2 yards of this be enough for a dress?" Reason directs
+you to learn how wide it is, what style of dress you intend to make of
+it, how much material that style normally calls for, whether you are a
+careful or a wasteful cutter, and how big the person is for whom the
+dress is to be made. "How much cheaper as a diet is bread alone, than
+bread with butter added to the extent of 10% of the weight of the
+bread?" Reason directs you to learn the cost of bread, the cost of
+butter, the nutritive value of bread, and the nutritive value of butter.
+
+In the arithmetic of the school this feature of reasoning appears in
+cases where some fact about common measures must be brought to bear, or
+some table of prices or discounts must be consulted, or some business
+custom must be remembered or looked up.
+
+Thus "How many badges, each 9 inches long, can be made from 2-1/2 yd.
+ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches.
+"At Jones' prices, which costs more, 3-3/4 lb. butter or 6-1/2 lb. lard?
+How much more?" is a problem which directs the thinker to ascertain
+Jones' prices.
+
+It may be noted that such problems are, other things being equal,
+somewhat better training in thinking than problems where all the data
+are given in the problem itself (_e.g._, "Which costs more, 3-3/4 lb.
+butter at 48� per lb. or 6-1/2 lb. lard at 27� per lb.? How much
+more?"). At least it is unwise to have so many problems of the latter
+sort that the pupil may come to think of a problem in applied arithmetic
+as a problem where everything is given and he has only to manipulate the
+data. Life does not present its problems so.
+
+The process of selecting the right elements and attaching proper weight
+to them may be illustrated by the following problem:--"Which of these
+offers would you take, supposing that you wish a D.C.K. upright piano,
+have $50 saved, can save a little over $20 per month, and can borrow
+from your father at 6% interest?"
+
+  A
+
+  A Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down
+  and $21 a month for only a year and a half. _No interest_ to pay.
+  We ask you to pay only for the piano and allow you plenty of time.
+
+  B
+
+  We offer the well-known D.C.K. Piano for $390. $50 cash and $20 a
+  month thereafter. Regular interest at 6%. The interest soon is
+  reduced to less than $1 a month.
+
+  C
+
+  The D.C.K. Piano. Special Offer, $375, cash. Compare our prices
+  with those of any reliable firm.
+
+If you consider chiefly the "only," "No interest to pay," "only," and
+"plenty of time" in offer A, attaching much weight to them and little to
+the thought, "How much will $50 plus (18 � $21) be?", you will probably
+decide wrongly.
+
+The situations of life are often complicated by many elements of little
+or even of no relevance to the correct solution. The offerer of A may
+belong to your church; your dearest friend may urge you to accept offer
+B; you may dislike to talk with the dealer who makes offer C; you may
+have a prejudice against owing money to a relative; that prejudice may
+be wise or foolish; you may have a suspicion that the B piano is
+shopworn; that suspicion may be well-founded or groundless; the salesman
+for C says, "You don't want your friends to say that you bought on the
+installment plan. Only low-class persons do that," etc. The statement of
+arithmetical problems in school usually assists the pupil to the extent
+of ruling out all save definitely quantitative elements, and of ruling
+out all quantitative elements except those which should be considered.
+The first of the two simplifications is very beneficial, on the whole,
+since otherwise there might be different correct solutions to a problem
+according to the nature and circumstances of the persons involved. The
+second simplification is often desirable, since it will often produce
+greater improvement in the pupils, per hour of time spent, than would be
+produced by the problems requiring more selection. It should not,
+however, be a universal custom; for in that case the pupils are tempted
+to think that in every problem they must use all the quantities given,
+as one must use all the pieces in a puzzle picture.
+
+It is obvious that the elements selected must not only be right but also
+be in the right relations to one another. For example, in the problems
+below, the 6 must be thought of in relation to a dozen and as being half
+of a dozen, and also as being 6 times 1. 1 must be mentally tied to
+"each." The 6 as half of a dozen must be related to the $1.00, $1.60,
+etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc.
+
+  Buying in Quantity
+
+  These are a grocer's prices for certain things by the dozen and
+  for a single one. He sells a half dozen at half the price of a
+  dozen. Find out how much you save by buying 6 all at one time
+  instead of buying them one at a time.
+
+                              Doz.  Each
+  1. Evaporated Milk         $1.00  $.09
+  2. Puffed Rice              1.60   .14
+  3. Puffed Wheat             1.10   .10
+  4. Canned Soup              1.90   .17
+  5. Sardines                 1.80   .16
+  6. Beans (No. 2 cans)       1.50   .13
+  7. Pork and Beans           1.70   .15
+  8. Peas (No. 2 cans)        1.40   .12
+  9. Tomatoes (extra cans)    3.20   .28
+  10. Ripe olives (qt. cans)  7.20   .65
+
+It is obvious also that in such arithmetical work as we have been
+describing, the pupil, to be successful, must 'think things together.'
+Many bonds must co�perate to determine his final response.
+
+As a preface to reasoning about a problem we often have the discovery of
+the problem and the classification of just what it is, and as a
+postscript we have the critical inspection of the answer obtained to
+make sure that it is verified by experiment or is consistent with known
+facts. During the process of searching for, selecting, and weighting
+facts, there may be similar inspection and validation, item by item.
+
+
+REASONING AS THE CO�PERATION OF ORGANIZED HABITS
+
+The pedagogy of the past made two notable errors in practice based on
+two errors about the psychology of reasoning. It considered reasoning as
+a somewhat magical power or essence which acted to counteract and
+overrule the ordinary laws of habit in man; and it separated too sharply
+the 'understanding of principles' by reasoning from the 'mechanical'
+work of computation, reading problems, remembering facts and the like,
+done by 'mere' habit and memory.
+
+Reasoning or selective, inferential thinking is not at all opposed to,
+or independent of, the laws of habit, but really is their necessary
+result under the conditions imposed by man's nature and training. A
+closer examination of selective thinking will show that no principles
+beyond the laws of readiness, exercise, and effect are needed to explain
+it; that it is only an extreme case of what goes on in associative
+learning as described under the 'piecemeal' activity of situations; and
+that attributing certain features of learning to mysterious faculties of
+abstraction or reasoning gives no real help toward understanding or
+controlling them.
+
+It is true that man's behavior in meeting novel problems goes beyond, or
+even against, the habits represented by bonds leading from gross total
+situations and customarily abstracted elements thereof. One of the two
+reasons therefor, however, is simply that the finer, subtle,
+preferential bonds with subtler and less often abstracted elements go
+beyond, and at times against, the grosser and more usual bonds. One set
+is as much due to exercise and effect as the other. The other reason is
+that in meeting novel problems the mental set or attitude is likely to
+be one which rejects one after another response as their unfitness to
+satisfy a certain desideratum appears. What remains as the apparent
+course of thought includes only a few of the many bonds which did
+operate, but which, for the most part, were unsatisfying to the ruling
+attitude or adjustment.
+
+Successful responses to novel data, associations by similarity and
+purposive behavior are in only apparent opposition to the fundamental
+laws of associative learning. Really they are beautiful examples of it.
+Man's successful responses to novel data--as when he argues that the
+diagonal on a right triangle of 796.278 mm. base and 137.294 mm.
+altitude will be 808.022 mm., or that Mary Jones, born this morning,
+will sometime die--are due to habits, notably the habits of response to
+certain elements or features, under the laws of piecemeal activity and
+assimilation.
+
+Nothing is less like the mysterious operations of a faculty of reasoning
+transcending the laws of connection-forming, than the behavior of men in
+response to novel situations. Let children who have hitherto confronted
+only such arithmetical tasks, in addition and subtraction with one- and
+two-place numbers and multiplication with one-place numbers, as those
+exemplified in the first line below, be told to do the examples shown in
+the second line.
+
+  ADD    ADD     ADD     SUBT.   SUBT.  MULTIPLY   MULTIPLY   MULTIPLY
+   8      37      35       8      37       8          9           6
+   5      24      68       5      24       5          7           3
+  --      --      23      --      --      --         --          --
+                  19
+                  --
+
+          MULTIPLY                MULTIPLY                MULTIPLY
+             32                      43                      34
+             23                      22                      26
+             --                      --                      --
+
+They will add the numbers, or subtract the lower from the upper number,
+or multiply 3 � 2 and 2 � 3, etc., getting 66, 86, and 624, or respond
+to the element of 'Multiply' attached to the two-place numbers by "I
+can't" or "I don't know what to do," or the like; or, if one is a child
+of great ability, he may consider the 'Multiply' element and the bigness
+of the numbers, be reminded by these two aspects of the situation of the
+fact that
+
+  '9
+   9 multiply'
+  --
+
+gave only 81, and that
+
+  '10
+   10 multiply'
+   --
+
+gave only 100, or the like; and so may report an intelligent and
+justified "I can't," or reject the plan of 3 � 2 and 2 � 3, with 66, 86,
+and 624 for answers, as unsatisfactory. What the children will do will,
+in every case, be a product of the elements in the situation that are
+potent with them, the responses which these evoke, and the further
+associates which these responses in turn evoke. If the child were one of
+sufficient genius, he might infer the procedure to be followed as a
+result of his knowledge of the principles of decimal notation and the
+meaning of 'Multiply,' responding correctly to the 'place-value' element
+of each digit and adding his 6 tens and 9 tens, 20 twos and 3 thirties;
+but if he did thus invent the shorthand addition of a collection of
+twenty-three collections, each of 32 units, he would still do it by the
+operation of bonds, subtle but real.
+
+Association by similarity is, as James showed long ago, simply the
+tendency of an element to provoke the responses which have been bound to
+it. _abcde_ leads to _vwxyz_ because _a_ has been bound to _vwxyz_ by
+original nature, exercise, or effect.
+
+Purposive behavior is the most important case of the influence of the
+attitude or set or adjustment of an organism in determining (1) which
+bonds shall act, and (2) which results shall satisfy. James early
+described the former fact, showing that the mechanism of habit can give
+the directedness or purposefulness in thought's products, provided that
+mechanism includes something paralleling the problem, the aim, or need,
+in question.
+
+The second fact, that the set or attitude of the man helps to determine
+which bonds shall satisfy, and which shall annoy, has commonly been
+somewhat obscured by vague assertions that the selection and retention
+is of what is "in point," or is "the right one," or is "appropriate," or
+the like. It is thus asserted, or at least hinted, that "the will," "the
+voluntary attention," "the consciousness of the problem," and other such
+entities are endowed with magic power to decide what is the "right" or
+"useful" bond and to kill off the others. The facts are that in
+purposive thinking and action, as everywhere else, bonds are selected
+and retained by the satisfyingness, and are killed off by the
+discomfort, which they produce; and that the potency of the man's set or
+attitude to make this satisfy and that annoy--to put certain
+conduction-units in readiness to act and others in unreadiness--is in
+every way as important as its potency to set certain conduction-units in
+actual operation.
+
+Reasoning is not a radically different sort of force operating against
+habit but the organization and co�peration of many habits, thinking
+facts together. Reasoning is not the negation of ordinary bonds, but
+the action of many of them, especially of bonds with subtle elements of
+the situation. Some outside power does not enter to select and
+criticize; the pupil's own total repertory of bonds relevant to the
+problem is what selects and rejects. An unsuitable idea is not killed
+off by some _actus purus_ of intellect, but by the ideas which it itself
+calls up, in connection with the total set of mind of the pupil, and
+which show it to be inadequate.
+
+Almost nothing in arithmetic need be taught as a matter of mere
+unreasoning habit or memory, nor need anything, first taught as a
+principle, ever become a matter of mere habit or memory. 5 � 4 = 20
+should not be learned as an isolated fact, nor remembered as we remember
+that Jones' telephone number is 648 J 2. Almost everything in arithmetic
+should be taught as a habit that has connections with habits already
+acquired and will work in an organization with other habits to come. The
+use of this organized hierarchy of habits to solve novel problems is
+reasoning.
+
+
+
+
+CHAPTER XI
+
+ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL
+
+
+THE UTILIZATION OF INSTINCTIVE INTERESTS
+
+The activities essential to acquiring ability in arithmetic can rely on
+little in man's instinctive equipment beyond the purely intellectual
+tendencies of curiosity and the satisfyingness of thought for thought's
+sake, and the general enjoyment of success rather than failure in an
+enterprise to which one sets oneself. It is only by a certain amount of
+artifice that we can enlist other vehement inborn interests of childhood
+in the service of arithmetical knowledge and skill. When this can be
+done at no cost the gain is great. For example, marching in files of
+two, in files of three, in files of four, etc., raising the arms once,
+two times, three times, showing a foot, a yard, an inch with the hands,
+and the like are admirable because learning the meanings of numbers thus
+acquires some of the zest of the passion for physical action. Even in
+late grades chances to make pictures showing the relations of fractional
+parts, to cut strips, to fold paper, and the like will be useful.
+
+Various social instincts can be utilized in matches after the pattern of
+the spelling match, contests between rows, certain number games, and the
+like. The scoring of both the play and the work of the classroom is a
+useful field for control by the teacher of arithmetic.
+
+Hunt ['12] has noted the more important games which have some
+considerable amount of arithmetical training as a by-product and which
+are more or less suitable for class use. Flynn ['12] has described
+games, most of them for home use, which give very definite arithmetical
+drill, though in many cases the drills are rather behind the needs of
+children old enough to understand and like the game itself.
+
+It is possible to utilize the interests in mystery, tricks, and puzzles
+so as to arouse a certain form of respect for arithmetic and also to get
+computational work done. I quote one simple case from Miss Selkin's
+admirable collection ['12, p. 69 f.]:--
+
+  I. ADDITION
+
+  "We must admit that there is nothing particularly interesting in
+  a long column of numbers to be added. Let the teacher, however,
+  suggest that he can write the answer at sight, and the task will
+  assume a totally different aspect.
+
+  "A very simple number trick of this kind can be performed by
+  making use of the principle of complementary addition. The
+  arithmetical complement of a number with respect to a larger
+  number is the difference between these two numbers. Most
+  interesting results can be obtained by using complements with
+  respect to 9.
+
+  "The children may be called upon to suggest several numbers of
+  two, three, or more digits. Below these write an equal number of
+  addends and immediately announce the answer. The children,
+  impressed by this apparently rapid addition, will set to work
+  enthusiastically to test the results of this lightning
+  calculation.
+
+  "Example:--     357 }                          999
+                  682 }  A                       � 3
+                  793 }                         ----
+                                                2997
+
+                  642 }
+                  317 }  B
+                  206 }
+
+  "Explanation:--The addends in group A are written down at
+  random or suggested by the class. Those in group B are their
+  complements. To write the first number in group B we look at the
+  first number in group A and, starting at the left write 6, the
+  complement of 3 with respect to 9; 4, the complement of 5; 2, the
+  complement of 7. The second and third addends in group B are
+  derived in the same way. Since we have three addends in each
+  group, the problem reduces itself to multiplying 999 by 3, or to
+  taking 3000 - 3. Any number of addends may be used and each addend
+  may consist of any number of digits."
+
+Respect for arithmetic as a source of tricks and magic is very much less
+important than respect for its everyday services; and computation to
+test such tricks is likely to be undertaken zealously only by the abler
+pupils. Consequently this source of interest should probably be used
+only sparingly, and perhaps the teacher should give such exhibitions
+only as a reward for efficiency in the regular work. For example, if the
+work for a week is well done in four days the fifth day might be given
+up to some semi-arithmetical entertainment, such as the demonstration of
+an adding machine, the story of primitive methods of counting, team
+races in computation, an exhibition of lightning calculation and
+intellectual sleight-of-hand by the teacher, or the voluntary study of
+arithmetical puzzles.
+
+The interest in achievement, in success, mentioned above is stronger in
+children than is often realized and makes advisable the systematic use
+of the practice experiment as a method of teaching much of arithmetic.
+Children who thus compete with their own past records, keeping an exact
+score from week to week, make notable progress and enjoy hard work in
+making it.
+
+
+THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES
+
+Negatively the difficulty of the work that pupils should be expected to
+do is conditioned by the gradual maturing of their capacities. Other
+things being equal, the common custom of reserving hard things for late
+in the elementary school course is, of course, sound. It seems probable
+that little is gained by using any of the child's time for arithmetic
+before grade 2, though there are many arithmetical facts that he can
+learn in grade 1. Postponement of systematic work in arithmetic to grade
+3 or even grade 4 is allowable if better things are offered. With proper
+textbooks and oral and written exercises, however, a child in grades 2
+and 3 can spend time profitably on arithmetical work. When all children
+can be held in school through the eighth grade it does not much matter
+whether arithmetic is begun early or late. If, however, many children
+are to leave in grades 5 and 6 as now, we may think it wise to provide
+somehow that certain minima of arithmetical ability be given them.
+
+There are, so far as is known, no special times and seasons at which the
+human animal by inner growth is specially ripe for one or another
+section or aspect of arithmetic, except in so far as the general inner
+growth of intellectual powers makes the more abstruse and complex tasks
+suitable to later and later years.
+
+Indeed, very few of even the most enthusiastic devotees of the
+recapitulation theory or culture-epoch theory have attempted to apply
+either to the learning of arithmetic, and Branford is the only
+mathematician, so far as I know, who has advocated such application,
+even tempered by elaborate shiftings and reversals of the racial order.
+He says:--
+
+  "Thus, for each age of the individual life--infancy, childhood,
+  school, college--may be selected from the racial history
+  the most appropriate form in which mathematical experience
+  can be assimilated. Thus the capacity of the infant and early
+  childhood is comparable with the capacity of animal consciousness
+  and primitive man. The mathematics suitable to later childhood
+  and boyhood (and, of course, girlhood) is comparable with Arch�an
+  mathematics passing on through Greek and Hindu to medi�val
+  European mathematics; while the student is become sufficiently
+  mature to begin the assimilation of modern and highly abstract
+  European thought. The filling in of details must necessarily
+  be left to the individual teacher, and also, within some such
+  broadly marked limits, the precise order of the marshalling of the
+  material for each age. For, though, on the whole, mathematical
+  development has gone forward, yet there have been lapses from
+  advances already made. Witness the practical world-loss of much
+  valuable Hindu thought, and, for long centuries, the neglect of
+  Greek thought: witness the world-loss of the invention by the
+  Babylonians of the Zero, until re-invented by the Hindus, passed
+  on by them to the Arabs, and by these to Europe.
+
+  "Moreover, many blunders and false starts and false principles
+  have marked the whole course of development. In a phrase, rivers
+  have their backwaters. But it is precisely the teacher's function
+  to avoid such racial mistakes, to take short cuts ultimately
+  discovered, and to guide the young along the road ultimately found
+  most accessible with such halts and retracings--returns up
+  side-cuts--as the mental peculiarities of the pupils demand.
+
+  "All this, the practical realization of the spirit of the principle,
+  is to be wisely left to the mathematical teacher, familiar with the
+  history of mathematical science and with the particular limitations
+  of his pupils and himself." ['08, p. 245.]
+
+The latitude of modification suggested by Branford reduces the guidance
+to be derived from racial history to almost _nil_. Also it is apparent
+that the racial history in the case of arithmetical achievement is
+entirely a matter of acquisition and social transmission. Man's original
+nature is destitute of all arithmetical ideas. The human germs do not
+know even that one and one make two!
+
+
+INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL
+
+A scientific plan for teaching arithmetic would begin with an exact
+inventory of the knowledge and skill which the pupils already possessed.
+Our ordinary notions of what a child knows at entrance to grade 1, or
+grade 2, or grade 3, and of what a first-grade child or second-grade
+child can do, are not adequate. If they were, we should not find
+reputable textbooks arranging to teach elaborately facts already
+sufficiently well known to over three quarters of the pupils when they
+enter school. Nor should we find other textbooks presupposing in their
+first fifty pages a knowledge of words which not half of the children
+can read even at the end of the 2 B grade.
+
+We do find just such evidence that ordinary ideas about the abilities of
+children at the beginning of systematic school training in arithmetic
+may be in gross error. For example, a reputable and in many ways
+admirable recent book has fourteen pages of exercises to teach the
+meaning of two and the fact that one and one make two! As an example of
+the reverse error, consider putting all these words in the first
+twenty-five pages of a beginner's book:--_absentees, attendance, blanks,
+continue, copy, during, examples, grouped, memorize, perfect, similar,
+splints, therefore, total_!
+
+Little, almost nothing, has been done toward providing an exact
+inventory compared with what needs to be done. We may note here (1) the
+facts relevant to arithmetic found by Stanley Hall, Hartmann, and others
+in their general investigations of the knowledge possessed by children
+at entrance to school, (2) the facts concerning the power of children to
+perceive differences in length, area, size of collection, and
+organization within a collection such as is shown in Fig. 24, and
+certain facts and theories about early awareness of number.
+
+In the Berlin inquiry of 1869, knowledge of the meaning of two, three,
+and four appeared in 74, 74, and 73 percent of the children upon
+entrance to school. Some of those recorded as ignorant probably really
+knew, but failed to understand that they were expected to reply or were
+shy. Only 85 percent were recorded as knowing their fathers' names.
+Seven eighths as many children knew the meanings of two, three, and four
+as knew their fathers' names. In a similar but more careful experiment
+with Boston children in September, 1880, Stanley Hall found that 92
+percent knew three, 83 percent knew four, and 71-1/2 percent knew five.
+Three was known about as well as the color red; four was known about as
+well as the color blue or yellow or green. Hartmann ['90] found that two
+thirds of the children entering school in Annaberg could count from one
+to ten. This is about as many as knew money, or the familiar objects of
+the town, or could repeat words spoken to them.
+
+    [Illustration: FIG. 24.--Objective presentation.]
+
+In the Stanford form of the Binet tests counting four pennies is given
+as an ability of the typical four-year-old. Counting 13 pennies
+correctly in at least one out of two trials, and knowing three of the
+four coins,--penny, nickel, dime, and quarter,--are given as abilities
+of the typical six-year-old.
+
+
+THE PERCEPTION OF NUMBER AND QUANTITY
+
+We know that educated adults can tell how many lines or dots, etc., they
+see in a single glance (with an exposure too short for the eye to move)
+up to four or more, according to the clearness of the objects and their
+grouping. For example, Nanu ['04] reports that when a number of bright
+circles on a dark background are shown to educated adults for only .033
+second, ten can be counted when arranged to form a parallelogram, but
+only five when arranged in a row. With certain groupings, of course,
+their 'perception' involves much inference, even conscious addition and
+multiplication. Similarly they can tell, up to twenty and beyond, the
+number of taps, notes, or other sounds in a series too rapid for single
+counting if the sounds are grouped in a convenient rhythm.
+
+These abilities are, however, the product of a long and elaborate
+learning, including the learning of arithmetic itself. Elementary
+psychology and common experience teach us that the mere observation of
+groups or quantities, no matter how clear their number quality appears
+to the person who already knows the meanings of numbers, does not of
+itself create the knowledge of the meanings of numbers in one who does
+not. The experiments of Messenger ['03] and Burnett ['06] showed that
+there is no direct intuitive apprehension even of two as distinct from
+one. We have to _learn_ to feel the two touches or see the two dots or
+lines as two.
+
+We do not know by exact measurements the growth in children of this
+ability to count or infer the number of elements in a collection seen or
+series heard. Still less do we know what the growth would be without
+the influence of school training in counting, grouping, adding, and
+multiplying. Many textbooks and teachers seem to overestimate it
+greatly. Not all educated adults can, apart from measurement, decide
+with surety which of these lines is the longer, or which of these areas
+is the larger, or whether this is a ninth or a tenth or an eleventh of a
+circle.
+
+    [Illustration]
+
+Children upon entering school have not been tested carefully in respect
+to judgments of length and area, but we know from such studies as
+Gilbert's ['94] that the difference required in their case is probably
+over twice that required for children of 13 or 14. In judging weights,
+for example, a difference of 6 is perceived as easily by children 13 to
+15 years of age as a difference of 15 by six-year-olds.
+
+A teacher who has adult powers of estimating length or area or weight
+and who also knows already which of the two is longer or larger or
+heavier, may use two lines to illustrate a difference which they really
+hide from the child. It is unlikely, for example, that the first of
+these lines ______________ ________________  would be recognized as
+shorter than the second by every child in a fourth-grade class, and it
+is extremely unlikely that it would be recognized as being 7/8 of the
+length of the latter, rather than 3/4 of it or 5/6 of it or 9/10 of it
+or 11/12 of it. If the two were shown to a second grade, with the
+question, "The first line is 7. How long is the other line?" there would
+be very many answers of 7 or 9; and these might be entirely correct
+arithmetically, the pupils' errors being all due to their inability to
+compare the lengths accurately.
+
+  _A_   ______________     ________________
+
+        ______________     ________________
+  _B_  |______________|   |________________|
+
+
+  _C_  |-|-|-|-|-|-|-|
+
+       |-|-|-|-|-|-|-|-|
+
+        __ __ __ __ __ __ __
+  _D_  |__|__|__|__|__|__|__|
+        __ __ __ __ __ __ __ __
+       |__|__|__|__|__|__|__|__|
+
+
+  _E_   .'\##|##/`.      .'\##|##/`.
+       /###\#|#/   \    /###\#|#/###\
+       |-----------|    |-----------|
+       \###/#|#\###/    \###/#|#\###/
+        `./##|##\.'      `./##|##\.'
+
+The quantities used should be such that their mere discrimination offers
+no difficulty even to a child of blunted sense powers. If 7/8 and 1 are
+to be compared, _A_ and _B_ are not allowable. _C_, _D_, and _E_ are
+much better.
+
+Teachers probably often underestimate or neglect the sensory
+difficulties of the tasks they assign and of the material they use to
+illustrate absolute and relative magnitudes. The result may be more
+pernicious when the pupils answer correctly than when they fail. For
+their correct answering may be due to their divination of what the
+teacher wants; and they may call a thing an inch larger to suit her
+which does not really seem larger to them at all. This, of course, is
+utterly destructive of their respect for arithmetic as an exact and
+matter-of-fact instrument. For example, if a teacher drew a series of
+lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this
+form--____ ______ and asked, "This is 20 inches long, how long is
+this?" she might, after some errors and correction thereof, finally
+secure successful response to all the lines by all the children. But
+their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be
+actually damaged by the exercise.
+
+
+THE EARLY AWARENESS OF NUMBER
+
+There has been some disagreement concerning the origin of awareness of
+number in the individual, in particular concerning the relative
+importance of the perception of how-many-ness and that of how-much-ness,
+of the perception of a defined aggregate and the perception of a defined
+ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and
+Decroly and Degand ['12].)
+
+The chief facts of significance for practice seem to be these: (1)
+Children with rare exceptions hear the names _one_, _two_, _three_,
+_four_, _half_, _twice_, _two times_, _more_, _less_, _as many as_,
+_again_, _first_, _second_, and _third_, long before they have analyzed
+out the qualities and relations to which these words refer so as to feel
+them at all clearly. (2) Their knowledge of the qualities and relations
+is developed in the main in close association with the use of these
+words to the child and by the child. (3) The ordinary experiences of the
+first five years so develop in the child awareness of the 'how many
+somethings' in various groups, of the relative magnitudes of two groups
+or quantities of any sort, and of groups and magnitudes as related to
+others in a series. For instance, if fairly gifted, a child comes, by
+the age of five, to see that a row of four cakes is an aggregate of
+four, seeing each cake as a part of the four and the four as the sum of
+its parts, to know that two of them are as many as the other two, that
+half of them would be two, and to think, when it is useful for him to do
+so, of four as a step beyond three on the way to five, or to think of
+hot as a step from warm on the way to very hot. The degree of
+development of these abilities depends upon the activity of the law of
+analysis in the individual and the character of his experiences.
+
+(4) He gets certain bad habits of response from the ambiguity of common
+usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears
+his parents or older children or others count pennies or rolls or eggs
+by saying one, two, three, four, and so on. He himself is perhaps misled
+into so counting. Thus the names properly belonging to a series of
+aggregations varying in amount come to be to him the names of the
+positions of the parts in a counted whole. This happens especially with
+numbers above 3 or 4, where the correct experience of the number as a
+name for the group has rarely been present. This attaching to the
+cardinal numbers above three or four the meanings of the ordinal numbers
+seems to affect many children on entrance to school. The numbering of
+pages in books, houses, streets, etc., and bad teaching of counting
+often prolong this error.
+
+(5) He also gets the habit, not necessarily bad, but often indirectly
+so, of using many names such as eight, nine, ten, eleven, fifteen, a
+hundred, a million, without any meaning.
+
+(6) The experiences of half, twice, three times as many, three times as
+long, etc., are rarer; even if they were not, they would still be less
+easily productive of the analysis of the proper abstract element than
+are the experiences of two, three, four, etc., in connection with
+aggregates of things each of which is usually called one, such as boys,
+girls, balls, apples. Experiences of the names, two, three, and four, in
+connection with two twos, two threes, two fours, are very rare.
+
+Hence, the names, two, three, etc., mean to these children in the main,
+"one something and one something," "one something usually called one,
+and one something usually called one, and another something usually
+called one," and more rarely and imperfectly "two times anything,"
+"three times anything," etc.
+
+With respect to Mr. Phillips' emphasis of the importance of the
+series-idea in children's minds, the matters of importance are: first,
+that the knowledge of a series of number names in order is of very
+little consequence to the teaching of arithmetic and of still less to
+the origin of awareness of number. Second, the habit of applying this
+series of words in counting in such a way that 8 is associated with the
+eighth thing, 9 with the ninth thing, etc., is of consequence because it
+does so much mischief. Third, the really valuable idea of the number
+series, the idea of a series of groups or of magnitudes varying by
+steps, is acquired later, as a result, not a cause, of awareness of
+numbers.
+
+With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers
+should be emphasized in schools, not because it is the main origin of
+the child's awareness of number, but because it is _not_, and because
+the ordinary practical issues of child life do _not_ adequately
+stimulate its action. It also seems both more economical and more
+scientific to introduce it through multiplication, division, and
+fractions rather than to insist that 4 and 5 shall from the start mean 4
+or 5 times anything that is called 1, for instance, that 8 inches shall
+be called 4 two-inches, or 10 cents, 5 two-cents. If I interpret
+Professor Dewey's writings correctly, he would agree that the use of
+inch, foot, yard, pint, quart, ounce, pound, glassful, cupful, handful,
+spoonful, cent, nickel, dime, and dollar gives a sufficient range of
+units for the first two school years. Teaching the meanings of 1/2 of 4,
+1/2 of 6, 1/2 of 8, 1/2 of 10, 1/2 of 20, 1/3 of 6, 1/3 of 9, 1/3 of 30,
+1/4 of 8, two 2s, five 2s, and the like, in early grades, each in
+connection with many different units of measure, provides a sufficient
+assurance that numbers will connect with relationships as well as with
+collections.
+
+
+
+
+CHAPTER XII
+
+INTEREST IN ARITHMETIC
+
+
+CENSUSES OF PUPILS' INTERESTS
+
+Arithmetic, although it makes little or no appeal to collecting,
+muscular manipulation, sensory curiosity, or the potent original
+interests in things and their mechanisms and people and their passions,
+is fairly well liked by children. The censuses of pupils' likes and
+dislikes that have been made are not models of scientific investigation,
+and the resulting percentages should not be used uncritically. They are,
+however, probably not on the average over-favorable to arithmetic in any
+unfair way. Some of their results are summarized below. In general they
+show arithmetic to be surpassed in interest clearly by only the manual
+arts (shopwork and manual training for boys, cooking and sewing for
+girls), drawing, certain forms of gymnastics, and history. It is about
+on a level with reading and science. It clearly surpasses grammar,
+language, spelling, geography, and religion.
+
+Lobsien ['03], who asked one hundred children in each of the first five
+grades (_Stufen_) of the elementary schools of Kiel, "Which part of the
+school work (literally, 'which instruction period') do you like best?"
+found arithmetic led only by drawing and gymnastics in the case of the
+boys, and only by handwork in the case of the girls.
+
+This is an exaggerated picture of the facts, since no count is made of
+those who especially dislike arithmetic. Arithmetic is as unpopular with
+some as it is popular with others. When full allowance is made for this,
+arithmetic still has popularity above the average. Stern ['05] asked,
+"Which subject do you like most?" and "Which subject do you like least?"
+The balance was greatly in favor of gymnastics for boys (28--1),
+handwork for girls (32--1-1/2), and drawing for both (16-1/2--6).
+Writing (6-1/2--4), arithmetic (14-1/2--13), history (9--6-1/2),
+reading (8-1/2--8), and singing (6--7-1/2) come next. Religion,
+nature study, physiology, geography, geometry, chemistry, language,
+and grammar are low.
+
+McKnight ['07] found with boys and girls in grades 7 and 8 of certain
+American cities that arithmetic was liked better than any of the school
+subjects except gymnastics and manual training. The vote as compared
+with history was:--
+
+  Arithmetic    327 liked greatly,     96 disliked greatly.
+  History       164 liked greatly,    113 disliked greatly.
+
+In a later study Lobsien ['09] had 6248 pupils from 9 to 15 years old
+representing all grades of the elementary school report, so far as they
+could, the subject most disliked, the subject most liked, the subject
+next most liked, and the subject next in order. No child was forced to
+report all of these four judgments, or even any of them. Lobsien counts
+the likes and the dislikes for each subject. Gymnastics, handwork, and
+cooking are by far the most popular. History and drawing are next,
+followed by arithmetic and reading. Below these are geography, writing,
+singing, nature study, biblical history, catechism, and three minor
+subjects.
+
+Lewis ['13] secured records from English children in elementary schools
+of the order of preference of all the studies listed below. He reports
+the results in the following table of percents:
+
+  ===================================================================
+                  | TOP THIRD OF | MIDDLE THIRD OF | LOWEST THIRD OF
+                  | STUDIES FOR  | STUDIES FOR     | STUDIES FOR
+                  | INTEREST     | INTEREST        | INTEREST
+  ----------------+--------------+-----------------+-----------------
+  Drawing         |      78      |       20        |        2
+  Manual Subjects |      66      |       26        |        8
+  History         |      64      |       24        |       12
+  Reading         |      53      |       38        |        9
+  Singing         |      32      |       48        |       20
+                  |              |                 |
+  Drill           |      20      |       55        |       25
+  Arithmetic      |      16      |       53        |       31
+  Science         |      23      |       37        |       40
+  Nature Study    |      16      |       36        |       48
+  Dictation       |       4      |       57        |       39
+                  |              |                 |
+  Composition     |      18      |       28        |       54
+  Scripture       |       4      |       38        |       58
+  Recitation      |       9      |       23        |       68
+  Geography       |       4      |       24        |       72
+  Grammar         |      --      |        6        |       94
+  ===================================================================
+
+Brandell ['13] obtained data from 2137 Swedish children in Stockholm
+(327), Norrk�ping (870), and Gothenburg (940).
+
+In general he found, as others have, that handwork, shopwork for boys
+and household work for girls, and drawing were reported as much better
+liked than arithmetic. So also was history, and (in this he differs from
+most students of this matter) so were reading and nature study.
+Gymnastics he finds less liked than arithmetic. Religion, geography,
+language, spelling, and writing are, as in other studies, much less
+popular than arithmetic.
+
+Other studies are by Lilius ['11] in Finland, Walsemann ['07],
+Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern ['13 and '14],
+in Germany. They confirm the general results stated.
+
+The reasons for the good showing that arithmetic makes are probably the
+strength of its appeal to the interest in definite achievement, success,
+doing what one attempts to do; and of its appeal, in grades 5 to 8, to
+the practical interest of getting on in the world, acquiring abilities
+that the world pays for. Of these, the former is in my opinion much the
+more potent interest. Arithmetic satisfies it especially well, because,
+more than any other of the 'intellectual' studies of the elementary
+school, it permits the pupil to see his own progress and determine his
+own success or failure.
+
+The most important applications of the psychology of satisfiers and
+annoyers to arithmetic will therefore be in the direction of utilizing
+still more effectively this interest in achievement. Next in importance
+come the plans to attach to arithmetical learning the satisfyingness of
+bodily action, play, sociability, cheerfulness, and the like, and of
+significance as a means of securing other desired ends than arithmetical
+abilities themselves. Next come plans to relieve arithmetical learning
+from certain discomforts such as the eyestrain of some computations and
+excessive copying of figures. These will be discussed here in the
+inverse order.
+
+
+RELIEVING EYESTRAIN
+
+At present arithmetical work is, hour for hour, probably more of a tax
+upon the eyes than reading. The task of copying numbers from a book to a
+sheet of paper is one of the very hardest tasks that the eyes of a pupil
+in the elementary schools have to perform. A certain amount of such
+work is desirable to teach a child to write numbers, to copy exactly,
+and to organize material in shape for computation. But beyond that,
+there is no more reason for a pupil to copy every number with which he
+is to compute than for him to copy every word he is to read. The
+meaningless drudgery of copying figures should be mitigated by arranging
+much work in the form of exercises like those shown on pages 216, 217,
+and 218, and by having many of the textbook examples in addition,
+subtraction, and multiplication done with a slip of paper laid below the
+numbers, the answers being written on it. There is not only a resulting
+gain in interest, but also a very great saving of time for the pupil
+(very often copying an example more than quadruples the time required to
+get its answer), and a much greater efficiency in supervision.
+Arithmetical errors are not confused with errors of copying,[16] and the
+teacher's task of following a pupil's work on the page is reduced to a
+minimum, each pupil having put the same part of the day's work in just
+the same place. The use of well-printed and well-spaced pages of
+exercises relieves the eyestrain of working with badly made gray
+figures, unevenly and too closely or too widely spaced. I reproduce in
+Fig. 25 specimens taken at random from one hundred random samples of
+arithmetical work by pupils in grade 8. Contrast the task of the eyes in
+working with these and their task in working with pages 216 to 218. The
+customary method of always copying the numbers to be used in computation
+from blackboard or book to a sheet of paper is an utterly unjustifiable
+cruelty and waste.
+
+  [16] Courtis finds in the case of addition that "of all the
+       individuals making mistakes at any given time in a class,
+       at least one third, and usually two thirds, will be making
+       mistakes in carrying or copying."
+
+    [Illustration: FIG. 25_a_.--Specimens taken at random from the
+    computation work of eighth-grade pupils. This computation
+    occurred in a genuine test. In the original gray of the pencil
+    marks the work is still harder to make out.]
+
+    [Illustration: FIG. 25_b_.--Specimens taken at random from the
+    computation work of eighth-grade pupils. This computation
+    occurred in a genuine test. In the original gray of the pencil
+    marks the work is still harder to make out.]
+
+Write the products:--
+
+ A. 3   4s=           B. 5   7s=           C. 9   2s=
+    5   2s=              8   3s=              4   4s=
+    7   2s=              4   2s=              2   7s=
+    1   6 =              4   5s=              6   4s=
+    1   3 =              4   7s=              5   5s=
+    3   7s=              5   9s=              3   6s=
+    4   1s=              7   5s=              3   2s=
+    6   8s=              7   1s=              3   9s=
+    9   8s=              6   3s=              5   1s=
+    4   3s=              4   9s=              8   6s=
+    2   4s=              3   5s=              8   4s=
+    2   2s=              9   6s=              8   5s=
+    8   7s=              2   5s=              7   9s=
+    5   8s=              5   4s=              6   2s=
+    7   6s=              8   2s=              7   4s=
+    7   3s=              8   9s=              9   3s=
+
+ D. 4   20s =     E.    9   60s =       F.    40 � 2 = 80
+    4  200s =           9  600s =             20 � 2 =
+    6   30s =           5   30s =             30 � 2 =
+    6  300s =           5  300s =             40 � 2 =
+    7 �  50 =           8 �  20 =             20 � 3 =
+    7 � 500 =           8 � 200 =             30 � 3 =
+    3 �  40 =           2 �  70 =            300 � 3 = 900
+    3 � 400 =           2 � 700 =            300 � 2 =
+
+Write the missing numbers: (_r_ stands for remainder.)
+
+  25 = .... 3s and .... _r_.
+  25 = .... 4s  "  .... _r_.
+  25 = .... 5s  "  .... _r_.
+  25 = .... 6s  "  .... _r_.
+  25 = .... 7s  "  .... _r_.
+  25 = .... 8s  "  .... _r_.
+  25 = .... 9s  "  .... _r_.
+
+  26 = .... 3s and .... _r_.
+  26 = .... 4s  "  .... _r_.
+  26 = .... 5s  "  .... _r_.
+  26 = .... 6s  "  .... _r_.
+  26 = .... 7s  "  .... _r_.
+  26 = .... 8s  "  .... _r_.
+  26 = .... 9s  "  .... _r_.
+
+  30 = .... 4s and .... _r_.
+  30 = .... 5s  "  .... _r_.
+  30 = .... 6s  "  .... _r_.
+  30 = .... 7s  "  .... _r_.
+  30 = .... 8s  "  .... _r_.
+  30 = .... 9s  "  .... _r_.
+
+  31 = .... 4s and .... _r_.
+  31 = .... 5s  "  .... _r_.
+  31 = .... 6s  "  .... _r_.
+  31 = .... 7s  "  .... _r_.
+  31 = .... 8s  "  .... _r_.
+  31 = .... 9s  "  .... _r_.
+
+Write the whole numbers or mixed numbers which these fractions equal:--
+
+      5      4      9      4      7
+      -      -      -      -      -
+      4      3      5      2      3
+
+      7      5      11     3      8
+      -      -      --     -      -
+      4      3      8      2      8
+
+      8      6      9      9      16
+      -      -      -      -      --
+      4      3      8      4      8
+
+      11     7      13     8      6
+      --     -      --     -      -
+      4      5      8      5      6
+
+Write the missing figures:--
+
+  6         2         8          1          2
+  - = -     - = -     -- = -     - = --     - = -
+  8   4     4   2     10   5     5   10     3   6
+
+Write the missing numerators:--
+
+  1
+  - = --  -  --  -  --  -  --
+  2   12  8  10  4  16  6  14
+
+  1
+  - = --  -  --  -  --  --  --
+  3   12  9  18  6  15  24  21
+
+  1
+  - = --  --  -  --  --  --  --
+  4   12  16  8  24  20  28  32
+
+  1
+  - = --  --  --  --  --  --  --
+  5   10  20  15  25  40  35  30
+
+  2
+  - = --  --  --  -  --  --  -
+  3   12  18  21  6  15  24  9
+
+  3
+  - = -  --  --  --  --  --  --
+  4   8  16  12  20  24  32  28
+
+Find the products. Cancel when you can:--
+
+  5            11           2
+  -- � 4 =     -- � 3 =     - � 5 =
+  16           12           3
+
+  7            8            1
+  -- � 8 =     - � 15 =     - � 8 =
+  12           5            6
+
+
+SIGNIFICANCE FOR RELATED ACTIVITIES
+
+The use of bodily action, social games, and the like was discussed in
+the section on original tendencies. "Significance as a means of securing
+other desired ends than arithmetical learning itself" is therefore our
+next topic. Such significance can be given to arithmetical work by using
+that work as a means to present and future success in problems of
+sports, housekeeping, shopwork, dressmaking, self-management, other
+school studies than arithmetic, and general school life and affairs.
+Significance as a means to future ends alone can also be more clearly
+and extensively attached to it than it now is.
+
+Whatever is done to supply greater strength of motive in studying
+arithmetic must be carefully devised so as not to get a strong but wrong
+motive, so as not to get abundant interest but in something other than
+arithmetic, and so as not to kill the goose that after all lays the
+golden eggs--the interest in intellectual activity and achievement
+itself. It is easy to secure an interest in laying out a baseball
+diamond, measuring ingredients for a cake, making a balloon of a certain
+capacity, or deciding the added cost of an extra trimming of ribbon for
+one's dress. The problem is to _attach_ that interest to arithmetical
+learning. Nor should a teacher be satisfied with attaching the interest
+as a mere tail that steers the kite, so long as it stays on, or as a
+sugar-coating that deceives the pupil into swallowing the pill, or as an
+anodyne whose dose must be increased and increased if it is to retain
+its power. Until the interest permeates the arithmetical activity itself
+our task is only partly done, and perhaps is made harder for the next
+time.
+
+One important means of really interfusing the arithmetical learning
+itself with these derived interests is to lead the pupil to seek the
+help of arithmetic himself--to lead him, in Dewey's phrase, to 'feel the
+need'--to take the 'problem' attitude--and thus appreciate the
+technique which he actively hunts for to satisfy the need. In so far as
+arithmetical learning is organized to satisfy the practical demands of
+the pupil's life at the time, he should, so to speak, come part way to
+get its help.
+
+Even if we do not make the most skillful use possible of these interests
+derived from the quantitative problems of sports, housekeeping,
+shopwork, dressmaking, self-management, other school studies, and school
+life and affairs, the gain will still be considerable. To have them in
+mind will certainly preserve us from giving to children of grades 3 and
+4 problems so devoid of relation to their interests as those shown
+below, all found (in 1910) in thirty successive pages of a book of
+excellent repute:--
+
+  A chair has 4 legs. How many legs have 8 chairs? 5 chairs?
+
+  A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?
+
+              (Eight more of the same sort.)
+
+  In 1890 New York had 1,513,501 inhabitants, Milwaukee had
+  206,308, Boston had 447,720, San Francisco 297,990. How many
+  had these cities together?
+
+              (Five more of the same sort.)
+
+  Milton was born in 1608 and died in 1674. How many years
+  did he live?
+
+             (Several others of the same sort.)
+
+  The population of a certain city was 35,629 in 1880 and 106,670
+  in 1890. Find the increase.
+
+              (Several others of this sort.)
+
+  A number of others about the words in various inaugural addresses
+  and the Psalms in the Bible.
+
+It also seems probable that with enough care other systematic plans of
+textbooks can be much improved in this respect. From every point of
+view, for example, the early work in arithmetic should be adapted to
+some extent to the healthy childish interests in home affairs, the
+behavior of other children, and the activities of material things,
+animals, and plants.
+
+TABLE 9
+
+FREQUENCY OF APPEARANCE OF CERTAIN WORDS ABOUT FAMILY LIFE, PLAY, AND
+ACTION IN EIGHT ELEMENTARY TEXTBOOKS IN ARITHMETIC, pp. 1-50.
+
+  ================================================================
+                  |  A  |  B  |  C  |  D  |  E  |  F  |  G  |  H
+  ----------------+-----+-----+-----+-----+-----+-----+-----+-----
+  baby            |     |     |     |  2  |     |  4  |     |
+  brother         |  2  |     |  6  |  1  |  1  |     |  1  |
+  family          |     |     |  2  |     |  2  |     |  4  |
+  father          |  1  |     |  3  |  5  |     |  2  |  1  |
+  help            |     |     |     |     |     |     |     |
+  home            |  2  |     |  4  |  4  |  2  |  2  |  7  |  1
+  mother          |  4  |  2  |  9  |  5  |     |  5  |  1  |  7
+  sister          |     |     |  1  |  2  |  2  |  9  |  1  |  1
+                  |     |     |     |     |     |     |     |
+  fork            |     |     |     |     |     |     |     |
+  knife           |     |     |     |     |     |     |     |
+  plate           |  4  |  2  |     |  2  |     |  1  |     |
+  spoon           |     |     |     |     |     |     |     |
+                  |     |     |     |     |     |     |     |
+  doll            | 10  |  1  | 10  |  6  |     | 10  |     |  9
+  game            |  1  |     |     |  3  |     |     |  5  |  5
+  jump            |     |     |     |     |     |     |     |  4
+  marbles         | 10  |  4  | 10  |     | 10  |     |  1  |
+  play            |     |     |  1  |     |     |  3  |     |
+  run             |     |     |     |     |     |  1  |     |  3
+  sing            |     |     |     |     |     |     |     |
+  tag             |     |     |     |     |     |     |     |
+  toy             |     |     |     |     |     |     |     |  1
+                  |     |     |     |     |     |     |     |
+  car             |     |     |  2  |  4  |     |  2  |  3  |  1
+  cut             |     |     | 10  |     |  6  |  2  |     |  8
+  dig             |     |     |     |     |     |     |  2  |
+  flower          |  1  |     |     |  4  |  1  |  1  |  2  |
+  grow            |     |     |     |  1  |     |     |     |
+  plant           |     |     |  2  |     |     |     |     |
+  seed            |     |     |     |  3  |     |     |  1  |
+  string          |     |     |     |     |  1  | 10  |  1  |  1
+  wheel           |  5  |     |     |     |     | 10  |     |
+  ================================================================
+
+The words used by textbooks give some indication of how far this aim is
+being realized, or rather of how far short we are of realizing it.
+Consider, for example, the words home, mother, father, brother, sister,
+help, plate, knife, fork, spoon, play, game, toy, tag, marbles, doll,
+run, jump, sing, plant, seed, grow, flower, car, wheel, string, cut,
+dig. The frequency of appearance in the first fifty pages of eight
+beginners' arithmetics was as shown in Table 9. The eight columns refer
+to the eight books (the first fifty pages of each). The numbers refer to
+the number of times the word in question appeared, the number 10 meaning
+10 _or more_ times in the fifty pages. Plurals, past tenses, and the
+like were counted. _Help_, _fork_, _knife_, _spoon_, _jump_, _sing_, and
+_tag_ did not appear at all! _Toy_ and _grow_ appeared each once in the
+400 pages! _Play_, _run_, _dig_, _plant_, and _seed_ appeared once in a
+hundred or more pages. _Baby_ did not appear as often as _buggy_.
+_Family_ appeared no oftener than _fence_ or _Friday_. _Father_ appears
+about a third as often as _farmer_.
+
+Book A shows only 10 of these thirty words in the fifty pages; book B
+only 4; book C only 12; and books D, E, F, G, and H only 13, 8, 14, 13,
+10, respectively. The total number of appearances (counting the 10s as
+only 10 in each case) is 40 for A, 9 for B, 60 for C, 42 for D, 25 for
+E, 62 for F, 30 for G, and 37 for H. The five words--apple, egg, Mary,
+milk, and orange--are used oftener than all these thirty together.
+
+If it appeared that this apparent neglect of childish affairs and
+interests was deliberate to provide for a more systematic treatment of
+pure arithmetic, a better gradation of problems, and a better
+preparation for later genuine use than could be attained if the author
+of the textbook were tied to the child's apron strings, the neglect
+could be defended. It is not at all certain that children in grade 2 get
+much more enjoyment or ability from adding the costs of purchases for
+Christmas or Fourth of July, or multiplying the number of cakes each
+child is to have at a party by the number of children who are to be
+there, than from adding gravestones or multiplying the number of hairs
+of bald-headed men. When, however, there is nothing gained by
+substituting remote facts for those of familiar concern to children, the
+safe policy is surely to favor the latter. In general, the neglect of
+childish data does not seem to be due to provision for some other end,
+but to the same inertia of tradition which has carried over the problems
+of laying walls and digging wells into city schools whose children never
+saw a stone wall or dug well.
+
+       *       *       *       *       *
+
+I shall not go into details concerning the arrangement of courses of
+study, textbooks, and lesson-plans to make desirable connections between
+arithmetical learning and sports, housework, shopwork, and the rest. It
+may be worth while, however, to explain the term _self-management_,
+since this source of genuine problems of real concern to the pupils has
+been overlooked by most writers.
+
+By self-management is meant the pupil's use of his time, his abilities,
+his knowledge, and the like. By the time he reaches grade 5, and to some
+extent before then, a boy should keep some account of himself, of how
+long it takes him to do specified tasks, of how much he gets done in a
+specified time at a certain sort of work and with how many errors, of
+how much improvement he makes month by month, of which things he can do
+best, and the like. Such objective, matter-of-fact, quantitative study
+of one's behavior is not a stimulus to morbid introspection or egotism;
+it is one of the best preventives of these. To treat oneself
+impersonally is one of the essential elements of mental balance and
+health. It need not, and should not, encourage priggishness. On the
+contrary, this matter-of-fact study of what one is and does may well
+replace a certain amount of the exhortations and admonitions concerning
+what one ought to do and be. All this is still truer for a girl.
+
+The demands which such an accounting of one's own activities make of
+arithmetic have the special value of connecting directly with the
+advanced work in computation. They involve the use of large numbers,
+decimals, averaging, percentages, approximations, and other facts and
+processes which the pupil has to learn for later life, but to which his
+childish activities as wage-earner, buyer and seller, or shopworker from
+10 to 14 do not lead. Children have little money, but they have time in
+thousands of units! They do not get discounts or bonuses from commercial
+houses, but they can discount their quantity of examples done for the
+errors made, and credit themselves with bonuses of all sorts for extra
+achievements.
+
+
+INTRINSIC INTEREST IN ARITHMETICAL LEARNING
+
+There remains the most important increase of interest in arithmetical
+learning--an increase in the interest directly bound to achievement and
+success in arithmetic itself. "Arithmetic," says David Eugene Smith, "is
+a game and all boys and girls are players." It should not be a _mere_
+game for them and they should not _merely_ play, but their unpractical
+interest in doing it because they can do it and can see how well they do
+do it is one of the school's most precious assets. Any healthy means to
+give this interest more and better stimulus should therefore be eagerly
+sought and cherished.
+
+Two such means have been suggested in other connections. The first is
+the extension of training in checking and verifying work so that the
+pupil may work to a standard of approximately 100% success, and may
+know how nearly he is attaining it. The second is the use of
+standardized practice material and tests, whereby the pupil may measure
+himself against his own past, and have a clear, vivid, and trustworthy
+idea of just how much better or faster he can do the same tasks than he
+could do a month or a year ago, and of just how much harder things he
+can do now than then.
+
+Another means of stimulating the essential interest in quantitative
+thinking itself is the arrangement of the work so that real arithmetical
+thinking is encouraged more than mere imitation and assiduity. This
+means the avoidance of long series of applied problems all of one type
+to be solved in the same way, the avoidance of miscellaneous series and
+review series which are almost verbatim repetitions of past problems,
+and in general the avoidance of excessive repetition of any one
+problem-situation. Stimulation to real arithmetical thinking is weak
+when a whole day's problem work requires no choice of methods, or when a
+review simply repeats without any step of organization or progress, or
+when a pupil meets a situation (say the 'buy _x_ things at _y_ per
+thing, how much pay' situation) for the five-hundredth time.
+
+Another matter worthy of attention in this connection is the unwise
+tendency to omit or present in diluted form some of the topics that
+appeal most to real intellectual interests, just because they are hard.
+The best illustration, perhaps, is the problem of ratio or "How many
+times as large (long, heavy, expensive, etc.) as _x_ is _y_?" Mastery of
+the 'times as' relation is hard to acquire, but it is well worth
+acquiring, not only because of its strong intellectual appeal, but also
+because of its prime importance in the applications of arithmetic to
+science. In the older arithmetics it was confused by pedantries and
+verbal difficulties and penalized by unreal problems about fractions of
+men doing parts of a job in strange and devious times. Freed from these,
+it should be reinstated, beginning as early as grade 5 with such simple
+exercises as those shown below and progressing to the problems of food
+values, nutritive ratios, gears, speeds, and the like in grade 8.
+
+  John is 4 years old.
+  Fred is 6 years old.
+  Mary is 8 years old.
+  Nell is 10 years old.
+  Alice is 12 years old.
+  Bert is 15 years old.
+
+  Who is twice as old as John?
+  Who is half as old as Alice?
+  Who is three times as old as John?
+  Who is one and one half times as old as Nell?
+  Who is two thirds as old as Fred?
+      etc., etc., etc.
+
+  Alice is .... times as old as John.
+  John is .... as old as Mary.
+  Fred is .... times as old as John.
+  Alice is .... times as old as Fred.
+  Fred is .... as old as Mary.
+      etc., etc., etc.
+
+Finally it should be remembered that all improvements in making
+arithmetic worth learning and helping the pupil to learn it will in the
+long run add to its interest. Pupils like to learn, to achieve, to gain
+mastery. Success is interesting. If the measures recommended in the
+previous chapters are carried out, there will be little need to entice
+pupils to take arithmetic or to sugar-coat it with illegitimate
+attractions.
+
+
+
+
+CHAPTER XIII
+
+THE CONDITIONS OF LEARNING
+
+
+We shall consider in this chapter the influence of time of day, size of
+class, and amount of time devoted to arithmetic in the school program,
+the hygiene of the eyes in arithmetical work, the use of concrete
+objects, and the use of sounds, sights, and thoughts as situations and
+of speech and writing and thought as responses.[17]
+
+  [17] Facts concerning the conditions of learning in general will
+       be found in the author's _Educational Psychology_, Vol. 2,
+       Chapter 8, or in the _Educational Psychology, Briefer Course_,
+       Chapter 15.
+
+
+EXTERNAL CONDITIONS
+
+Computation of one or another sort has been used by several
+investigators as a test of efficiency at different times in the day.
+When freed from the effects of practice on the one hand and lack of
+interest due to repetition on the other, the results uniformly show an
+increase in speed late in the school session with a falling off in
+accuracy that about balances it.[18] There is no wisdom in putting
+arithmetic early in the session because of its _difficulty_. Lively and
+sociable exercises in mental arithmetic with oral answers in fact seem
+to be admirably fitted for use late in the session. Except for the
+general principles (1) of starting the day with work that will set a
+good standard of cheerful, efficient production and (2) of getting the
+least interesting features of the day's work done fairly early in the
+day, psychology permits practical exigencies to rule the program, so far
+as present knowledge extends. Adequate measurements of the effect of
+time of day on _improvement_ have not been made, but there is no reason
+to believe that any one time between 9 A.M. and 4 P.M. is appreciably
+more favorable to arithmetical learning than to learning geography,
+history, spelling, and the like.
+
+  [18] See Thorndike ['00], King ['07], and Heck ['13].
+
+The influence of size of class upon progress in school studies is very
+difficult to measure because (1) within the same city system the average
+of the six (or more) sizes of class that a pupil has experienced will
+tend to approximate closely to the corresponding average for any other
+child; because further (2) there may be a tendency of supervisory
+officers to assign more pupils to the better teachers; and because
+(3) separate systems which differ in respect to size of class probably
+differ in other respects also so that their differences in achievement
+may be referable to totally different differences.
+
+Elliott ['14] has made a beginning by noting size of class during the
+year of test in connection with his own measures of the achievements of
+seventeen hundred pupils, supplemented by records from over four hundred
+other classes. As might be expected from the facts just stated, he finds
+no appreciable difference between classes of different sizes within the
+same school system, the effect of the few months in a small class being
+swamped by the antecedents or concomitants thereof.
+
+The effect of the amount of time devoted to arithmetic in the school
+program has been studied extensively by Rice ['02 and '03] and Stone
+['08].
+
+Dr. Rice ['02] measured the arithmetical ability of some 6000 children
+in 18 different schools in 7 different cities. The results of these
+measurements are summarized in Table 10. This table "gives two averages
+for each grade as well as for each school as a whole. Thus, the school
+at the top shows averages of 80.0 and 83.1, and the one at the bottom,
+25.3 and 31.5. The first represents the percentage of answers which were
+absolutely correct; the second shows what per cent of the problems were
+correct in principle, _i.e._ the average that would have been received
+if no mechanical errors had been made."
+
+The facts of Dr. Rice's table show that there is a positive relation
+between the general standing of a school system in the tests and the
+amount of time devoted to arithmetic by its program. The relation is
+not close, however, being that expressed by a correlation coefficient
+of .36-1/2. Within any one school system there is no relation between
+the standing of a particular school and the amount of time devoted to
+arithmetic in that school's program. It must be kept in mind that the
+amount of time given in the school program may be counterbalanced by
+emphasizing work at home and during study periods, or, on the other
+hand, may be a symptom of correspondingly small or great emphasis on
+arithmetic in work set for the study periods at home.
+
+A still more elaborate investigation of this same topic was made by
+Stone ['08]. I quote somewhat fully from it, since it is an instructive
+sample of the sort of studies that will doubtless soon be made in the
+case of every elementary school subject. He found that school systems
+differed notably in the achievements made by their sixth-grade pupils in
+his tests of computation (the so-called 'fundamentals') and of the
+solution of verbally described problems (the so-called 'reasoning'). The
+facts were as shown in Table 11.
+
+TABLE  10
+
+AVERAGES FOR INDIVIDUAL SCHOOLS IN ARITHMETIC
+
+  KEY    A: CITY
+         B: SCHOOL
+         C: Result
+         D: Principle
+         E: Percent of Mechanical Errors
+         F: Minutes Daily
+
+  ===========================================================
+     |   |6TH YEAR |7TH YEAR |8TH YEAR |SCHOOL AVERAGE |
+     |   |----+----+----+----+----+----+----+----+-----+-----
+   A | B | C  | D  | C  | D  | C  | D  | C  | D  |  E  |  F
+  ---+---+----+----+----+----+----+----+----+----+-----+-----
+  III| 1 |79.3|80.3|81.1|82.3|91.7|93.9|80.0|83.1| 3.7 |  53
+    I| 1 |80.4|81.5|64.2|67.2|80.9|82.8|76.6|80.3| 4.6 |  60
+    I| 2 |80.9|83.4|43.5|50.9|72.7|79.1|69.3|75.1| 7.7 |  25
+    I| 3 |72.2|74.0|63.5|66.2|74.5|76.6|67.8|72.2| 6.1 |  45
+    I| 4 |69.9|72.2|54.6|57.8|66.5|69.1|64.3|70.3| 8.5 |  45
+   II| 1 |71.2|75.3|33.6|35.7|36.8|40.0|60.2|64.8| 7.1 |  60
+  III| 2 |43.7|45.0|53.9|56.7|51.1|53.1|54.5|58.9| 7.4 |  60
+   IV| 1 |58.9|60.4|31.2|34.1|41.6|43.5|55.1|58.4| 5.6 |  60
+   IV| 2 |59.8|63.1| -- | -- |22.5|22.5|53.9|58.8| 8.3 |  --
+   IV| 3 |54.9|58.1|35.2|38.6|43.5|45.0|51.5|57.6|10.5 |  60
+   IV| 4 |42.3|45.1|16.1|19.2|48.7|48.7|42.8|48.2|11.2 |  --
+    V| 1 |44.1|48.7|29.2|32.5|51.1|58.3|45.9|51.3|10.5 |  40
+   VI| 1 |68.3|71.3|33.5|36.6|26.9|30.7|39.0|42.9| 9.0 |  33
+   VI| 2 |46.1|49.5|19.5|24.2|30.2|40.6|36.5|43.6|16.2 |  30
+   VI| 3 |34.5|36.4|30.5|35.1|23.3|24.1|36.0|42.5|15.2 |  48
+  VII| 1 |35.2|37.7|29.1|32.5|25.1|27.2|40.5|45.9|11.7 |  42
+  VII| 2 |35.2|38.7|15.0|16.4|19.6|21.2|36.5|40.6|10.1 |  75
+  VII| 3 |27.6|33.7| 8.9|10.1|11.3|11.3|25.3|31.5|19.6 |  45
+  ===========================================================
+
+High achievement by a system in computation went with high achievement
+in solving the problems, the correlation being about .50; and the
+system that scored high in addition or subtraction or multiplication or
+division usually showed closely similar excellence in the other three,
+the correlations being about .90.
+
+TABLE 11
+
+SCORES MADE BY THE SIXTH-GRADE PUPILS OF EACH OF TWENTY-SIX SCHOOL
+SYSTEMS
+
+  =================================================
+  SYSTEM | SCORE IN TESTS WITH | SCORE IN TESTS IN
+         |      PROBLEMS       |     COMPUTING
+  -------+---------------------+-------------------
+    23   |         356         |       1841
+    24   |         429         |       3513
+    17   |         444         |       3042
+     4   |         464         |       3563
+    25   |         464         |       2167
+    22   |         468         |       2311
+    16   |         469         |       3707
+    20   |         491         |       2168
+    18   |         509         |       3758
+    15   |         532         |       2779
+     3   |         533         |       2845
+     8   |         538         |       2747
+     6   |         550         |       3173
+     1   |         552         |       2935
+    10   |         601         |       2749
+     2   |         615         |       2958
+    21   |         627         |       2951
+    13   |         636         |       3049
+    14   |         661         |       3561
+     9   |         691         |       3404
+     7   |         734         |       3782
+    12   |         736         |       3410
+    11   |         759         |       3261
+    26   |         791         |       3682
+    19   |         848         |       4099
+     5   |         914         |       3569
+  =================================================
+
+Of the conditions under which arithmetical learning took place, the one
+most elaborately studied was the amount of time devoted to arithmetic.
+On the basis of replies by principals of schools to certain questions,
+he gave each of the twenty-six school systems a measure for the
+probable time spent on arithmetic up through grade 6. Leaving home study
+out of account, there seems to be little or no correlation between the
+amount of time a system devotes to arithmetic and its score in
+problem-solving, and not much more between time expenditure and score in
+computation. With home study included there is little relation to the
+achievement of the system in solving problems, but there is a clear
+effect on achievement in computation. The facts as given by Stone are:--
+
+TABLE 12
+
+CORRELATION OF TIME EXPENDITURES WITH ABILITIES
+
+  Without Home Study      { Reasoning and Time Expenditure        -.01
+                          { Fundamentals and Time Expenditure      .09
+
+  Including Home Study    { Reasoning and Time Expenditure         .13
+                          { Fundamentals and Time Expenditure      .49
+
+These correlations, it should be borne in mind, are for school systems,
+not for individual pupils. It might be that, though the system which
+devoted the most time to arithmetic did not show corresponding
+superiority in the product over the system devoting only half as much
+time, the pupils within the system did achieve in exact proportion to
+the time they gave to study. Neither correlation would permit inference
+concerning the effect of different amounts of time spent by the same
+pupil.
+
+Stone considered also the printed announcements of the courses of study
+in arithmetic in these twenty-six systems. Nineteen judges rated these
+announced courses of study for excellence according to the instructions
+quoted below:--
+
+CONCERNING THE RATING OF COURSES OF STUDY
+
+Judges please read before scoring
+
+I. Some Factors Determining Relative Excellence.
+
+(N. B. The following enumeration is meant to be suggestive rather than
+complete or exclusive. And each scorer is urged to rely primarily on his
+own judgment.)
+
+  1. Helpfulness to the teacher in teaching the subject matter outlined.
+
+  2. Social value or concreteness of sources of problems.
+
+  3. The arrangement of subject matter.
+
+  4. The provision made for adequate drill.
+
+  5. A reasonable minimum requirement with suggestions for valuable
+     additional work.
+
+  6. The relative values of any predominating so-called methods--such as
+     Speer, Grube, etc.
+
+  7. The place of oral or so-called mental arithmetic.
+
+  8. The merit of textbook references.
+
+II. Cautions and Directions.
+
+(Judges please follow as implicitly as possible.)
+
+  1. Include references to textbooks as parts of the Course of Study.
+
+     This necessitates judging the parts of the texts referred to.
+
+  2. As far as possible become equally familiar with all courses before
+     scoring any.
+
+  3. When you are ready to begin to score, (1) arrange in serial
+     order according to excellence, (2) starting with the middle one
+     score it 50, then score above and below 50 according as courses
+     are better or poorer, indicating relative differences in
+     excellence by relative differences in scores, _i.e._ in so far
+     as you find that the courses differ by about equal steps, score
+     those better than the middle one 51, 52, etc., and those poorer
+     49, 48, etc., but if you find that the courses differ by
+     unequal steps show these inequalities by omitting numbers.
+
+  4. Write ratings on the slip of paper attached to each course.
+
+The systems whose courses of study were thus rated highest did not
+manifest any greater achievement in Stone's tests than the rest. The
+thirteen with the most approved announcements of courses of study were
+in fact a little inferior in achievement to the other thirteen, and the
+correlation coefficients were slightly negative.
+
+Stone also compared eighteen systems where there was supervision of the
+work by superintendents or supervisors as well as by principals with
+four systems where the principals and teachers had no such help. The
+scores in his tests were very much lower in the four latter cities.
+
+
+THE HYGIENE OF THE EYES IN ARITHMETIC
+
+We have already noted that the task of reading and copying numbers is
+one of the hardest that the eyes have to perform in the elementary
+school, and that it should be alleviated by arranging much of the work
+so that only answers need be written by the pupil. The figures to be
+read and copied should obviously be in type of suitable size and style,
+so arranged and spaced on the page or blackboard as to cause a minimum
+of effort and strain.
+
+    [Illustration: FIG. 26.--Type too large.]
+
+    [Illustration: FIG. 27.--12-point, 11-point, and 10-point type.]
+
+_Size._--Type may be too large as well as too small, though the latter
+is the commoner error. If it is too large, as in Fig. 26, which is a
+duplicate of type actually used in a form of practice pad, the eye has
+to make too many fixations to take in a given content. All things
+considered, 12-point type in grades 3 and 4, 11-point in grades 5 and 6,
+and 10-point in grades 7 and 8 seem the most desirable sizes. These are
+shown in Fig. 27. Too small type occurs oftenest in fractions and in the
+dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30
+are samples from actual school practice. Samples of the desirable size
+are shown in Figs. 31 and 32. The technique of modern typesetting makes
+it very difficult and expensive to make fractions of the horizontal type
+
+  (1  3  5
+   -  -  -
+   4, 8, 6)
+
+large enough without making the whole-number figures with which they
+are mingled too large or giving an uncouth appearance to the total.
+Consequently fractions somewhat smaller than are desirable may have
+to be used occasionally in textbooks.[19] There is no valid excuse,
+however, for the excessively small fractions which often are made in
+blackboard work.
+
+  [19] A special type could be constructed that would use a large
+       type body, say 14 point, with integers in 10 or 12 point and
+       fractions much larger than now.
+
+    [Illustration: FIG. 28.--Type of measurements too small.
+
+        This is a picture of Mary's garden. How many feet is it
+        around the garden?]
+
+    [Illustration: FIG. 29.--Type too small.]
+
+    [Illustration: FIG. 30.--Numbers too small and badly designed.]
+
+    [Illustration: FIG. 31.--Figure 28 with suitable numbers.]
+
+    [Illustration: FIG. 32.--Figure 30 with suitable numbers.]
+
+_Style._--The ordinary type forms often have 3 and 8 so made as to
+require strain to distinguish them. 5 is sometimes easily confused with
+3 and even with 8. 1, 4, and 7 may be less easily distinguishable than
+is desirable. Figure 33 shows a specially good type in which each figure
+is represented by its essential[20] features without any distracting
+shading or knobs or turns. Figure 34 shows some of the types in common
+use. There are no demonstrably great differences amongst these. In
+fractions there is a notable gain from using the slant form (2/3, 3/4)
+for exercises in addition and subtraction, and for almost all mixed
+numbers. This appears clearly to the eye in the comparison of Fig. 35
+below, where the same fractions all in 10-point type are displayed in
+horizontal and in slant form. The figures in the slant form are in
+general larger and the space between them and the fraction-line is
+wider. Also the slant form makes it easier for the eye to examine the
+denominators to see whether reductions are necessary. Except for a few
+cases to show that the operations can be done just as truly with the
+horizontal forms, the book and the blackboard should display mixed
+numbers and fractions to be added or subtracted in the slant form. The
+slant line should be at an angle of approximately 45 degrees. Pupils
+should be taught to use this form in their own work of this sort.
+
+  [20] It will be still better if the 4 is replaced by an open-top 4.
+
+When script figures are presented they should be of simple design,
+showing clearly the essential features of the figure, the line being
+everywhere of equal or nearly equal width (that is, without shading, and
+without ornamentation or eccentricity of any sort). The opening of the 3
+should be wide to prevent confusion with 8; the top of the 3 should be
+curved to aid its differentiation from 5; the down stroke of the 9
+should be almost or quite straight; the 1, 4, 7, and 9 should be clearly
+distinguishable. There are many ways of distinguishing them clearly, the
+best probably being to use the straight line for 1, the open 4 with
+clear angularity, a wide top to the 7, and a clearly closed curve for
+the top of the 9.
+
+    [Illustration: FIG. 33.--Block type; a very desirable type except
+    that it is somewhat too heavy.]
+
+    [Illustration: FIG. 34.--Common styles of printed numbers.]
+
+    [Illustration: FIG. 35.--Diagonal and horizontal fractions
+    compared.]
+
+    [Illustration: FIG. 36.--Good vertical spacing.]
+
+    [Illustration: FIG. 37.--Bad vertical spacing.]
+
+    [Illustration: FIGS. 38 (above) and 39 (below).--Good and bad
+    left-right spacing.]
+
+The pupil's writing of figures should be clear. He will thereby be saved
+eyestrain and errors in his school work as well as given a valuable
+ability for life. Handwriting of figures is used enormously in spite of
+the development of typewriters; illegible figures are commonly more
+harmful than illegible letters or words, since the context far less
+often tells what the figure is intended to be; the habit of making clear
+figures is not so hard to acquire, since they are written unjoined and
+require only the automatic action of ten minor acts of skill. The
+schools have missed a great opportunity in this respect. Whereas the
+hand writing of words is often better than it needs to be for life's
+purposes, the writing of figures is usually much worse. The figures
+presented in books on penmanship are also commonly bad, showing neglect
+or misunderstanding of the matter on the part of leaders in penmanship.
+
+_Spacing._--Spacing up and down the column is rarely too wide, but very
+often too narrow. The specimens shown in Figs. 36 and 37 show good
+practice contrasted with the common fault.
+
+Spacing from right to left is generally fairly satisfactory in books,
+though there is a bad tendency to adopt some one routine throughout and
+so to miss chances to use reductions and increases of spacing so as to
+help the eye and the mind in special cases. Specimens of good and bad
+spacing are shown in Figs. 38 and 39. In the work of the pupils, the
+spacing from right to left is often too narrow. This crowding of
+letters, together with unevenness of spacing, adds notably to the task
+of eye and mind.
+
+_The composition or make-up of the page._--Other things being equal,
+that arrangement of the page is best which helps a child most to keep
+his place on a page and to find it after having looked away to work on
+the paper on which he computes, or for other good reasons. A good page
+and a bad page in this respect are shown in Figs. 40 and 41.
+
+    [Illustration: FIG. 40.--A page well made up to suit the action
+    of the eye.]
+
+    [Illustration: FIG. 41.--The same matter as in Fig. 40, much
+    less well made up.]
+
+_Objective presentations._--Pictures, diagrams, maps, and other
+presentations should not tax the eye unduly,
+
+  (_a_) by requiring too fine distinctions, or
+
+  (_b_) by inconvenient arrangement of the data, preventing easy
+        counting, measuring, comparison, or whatever the task is, or
+
+  (_c_) by putting too many facts in one picture so that the eye
+        and mind, when trying to make out any one, are confused by the
+        others.
+
+Illustrations of bad practices in these respects are shown in Figs. 42
+to 52. A few specimens of work well arranged for the eye are shown in
+Figs. 53 to 56.
+
+Good rules to remember are:--
+
+Other things being equal, make distinctions by the clearest method, fit
+material to the tendency of the eye to see an 'eyeful' at a time
+(roughly 1-1/2 inch by 1/2 inch in a book; 1-1/2 ft. by 1/2 ft. on the
+blackboard), and let one picture teach only one fact or relation, or
+such facts and relations as do not interfere in perception.
+
+The general conditions of seating, illumination, paper, and the like are
+even more important when the eyes are used with numbers than when they
+are used with words.
+
+    [Illustration: FIG. 42.--Try to count the rungs on the ladder,
+    or the shocks in the wagon.]
+
+    [Illustration: FIG. 43.--How many oars do you see? How many
+    birds? How many fish?]
+
+    [Illustration: FIG. 44.--Count the birds in each of the three
+    flocks of birds.]
+
+    [Illustration: FIG. 45.--Note the lack of clear division of the
+    hundreds. Consider the difficulty of counting one of these
+    columns of dots.]
+
+    [Illustration: FIG. 46.--What do you suppose these pictures are
+    intended to show?]
+
+    [Illustration: FIG. 47.--Would a beginner know that after
+    THIRTEEN he was to switch around and begin at the other end?
+    Could you read the SIX of TWENTY-SIX if you did not already know
+    what it ought to be? What meaning would all the brackets have
+    for a little child in grade 2? Does this picture illustrate or
+    obfuscate?]
+
+    [Illustration: FIG. 48.--How long did it take you to find out
+    what these pictures mean?]
+
+    [Illustration: FIG. 49.--Count the figures in the first row,
+    using your eyes alone; have some one make lines of 10, 11, 12,
+    13, and more repetitions of this figure spaced closely as here.
+    Count 20 or 30 such lines, using the eye unaided by fingers,
+    pencil, etc. ]
+
+    [Illustration: FIG. 50.--Can you answer the question without
+    measuring? Could a child of seven or eight?]
+
+    [Illustration: FIG. 51.--What are these drawings intended to
+    show? Why do they show the facts only obscurely and dubiously?]
+
+    [Illustration: FIG. 52.--What are these drawings intended to
+    show? What simple change would make them show the facts much
+    more clearly?]
+
+    [Illustration: FIG. 53.--Arranged in convenient "eye-fulls."]
+
+    [Illustration: FIG. 54.--Clear, simple, and easy of comparison.]
+
+    [Illustration: FIG. 55.--Clear, simple, and well spaced.]
+
+    [Illustration: FIG. 56.--Well arranged, though a little wider
+    spacing between the squares would make it even better.]
+
+
+THE USE OF CONCRETE OBJECTS IN ARITHMETIC
+
+We mean by concrete objects actual things, events, and relations
+presented to sense, in contrast to words and numbers and symbols which
+mean or stand for these objects or for more abstract qualities and
+relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart
+measures, bank books, and checks are such concrete things. A foot rule
+put successively along the three thirds of a yard rule, a bell rung five
+times, and a pound weight balancing sixteen ounce weights are such
+concrete events. A pint beside a quart, an inch beside a foot, an apple
+shown cut in halves display such concrete relations to a pupil who is
+attentive to the issue.
+
+Concrete presentations are obviously useful in arithmetic to teach
+meanings under the general law that a word or number or sign or symbol
+acquires meaning by being connected with actual things, events,
+qualities, and relations. We have also noted their usefulness as means
+to verifying the results of thinking and computing, as when a pupil,
+having solved, "How many badges each 5 inches long can be made from
+3-1/3 yd. of ribbon?" by using 10 � 12/5, draws a line 3-1/3 yd. long
+and divides it into 5-inch lengths.
+
+Concrete experiences are useful whenever the meaning of a number, like 9
+or 7/8 or .004, or of an operation, like multiplying or dividing or
+cubing, or of some term, like rectangle or hypothenuse or discount, or
+some procedure, like voting or insuring property against fire or
+borrowing money from a bank, is absent or incomplete or faulty. Concrete
+work thus is by no means confined to the primary grades but may be
+appropriate at all stages when new facts, relations, and procedures are
+to be taught.
+
+How much concrete material shall be presented will depend upon the fact
+or relation or procedure which is to be made intelligible, and the
+ability and knowledge of the pupil. Thus 'one half' will in general
+require less concrete illustration than 'five sixths'; and five sixths
+will require less in the case of a bright child who already knows 2/3,
+3/4, 3/8, 5/8, 7/8, 2/5, 3/5, and 4/5 than in the case of a dull child
+or one who only knows 2/3 and 3/4. As a general rule the same topic will
+require less concrete material the later it appears in the school
+course. If the meanings of the numbers are taught in grade 2 instead of
+grade 1, there will be less need of blocks, counters, splints, beans,
+and the like. If 1-1/2 + 1/2 = 2 is taught early in grade 3, there will
+be more gain from the use of 1-1/2 inches and 1/2 inch on the foot rule
+than if the same relations were taught in connection with the general
+addition of like fractions late in grade 4. Sometimes the understanding
+can be had either by connecting the idea with the reality directly, or
+by connecting the two indirectly _via_ some other idea. The amount of
+concrete material to be used will depend on its relative advantage per
+unit of time spent. Thus it might be more economical to connect 5/12,
+7/12, and 11/12 with real meanings indirectly by calling up the
+resemblance to the 2/3, 3/4, 3/8, 5/8, 7/8, 2/5, 3/5, 4/5, and 5/6
+already studied, than by showing 5/12 of an apple, 7/12 of a yard, 11/12
+of a foot, and the like.
+
+In general the economical course is to test the understanding of the
+matter from time to time, using more concrete material if it is needed,
+but being careful to encourage pupils to proceed to the abstract ideas
+and general principles as fast as they can. It is wearisome and
+debauching to pupils' intellects for them to be put through elaborate
+concrete experiences to get a meaning which they could have got
+themselves by pure thought. We should also remember that the new idea,
+say of the meaning of decimal fractions, will be improved and clarified
+by using it (see page 183 f.), so that the attainment of a _perfect_
+conception of decimal fractions before doing anything with them is
+unnecessary and probably very wasteful.
+
+A few illustrations may make these principles more instructive.
+
+(_a_) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000,
+need more concrete aids than are commonly given. Guessing contests about
+the value in dollars of the school building and other buildings, the
+area of the schoolroom floor and other surfaces in square inches, the
+number of minutes in a week, and year, and the like, together with
+proper computations and measurements, are very useful to re�nforce the
+concrete presentations and supply genuine problems in multiplication and
+subtraction with large numbers.
+
+(_b_) Numbers very much smaller than one, such as 1/32, 1/64, .04,
+and .002, also need some concrete aids. A diagram like that of
+Fig. 57 is useful.
+
+(_c_) _Majority_ and _plurality_ should be understood by every citizen.
+They can be understood without concrete aid, but an actual vote is well
+worth while for the gain in vividness and surety.
+
+    [Illustration: FIG. 57.--Concrete aid to understanding fractions
+    with large denominators. A = 1/1000 sq. ft.; B = 1/100 sq. ft.;
+    C = 1/50 sq. ft.; D = 1/10 sq. ft.]
+
+(_d_) Insurance against loss by fire can be taught by explanation and
+analogy alone, but it will be economical to have some actual insuring
+and payment of premiums and a genuine loss which is reimbursed.
+
+(_e_) Four play banks in the corners of the room, receiving deposits,
+cashing checks, and later discounting notes will give good educational
+value for the time spent.
+
+(_f_) Trade discount, on the contrary, hardly requires more concrete
+illustration than is found in the very problems to which it is applied.
+
+(_g_) The process of finding the number of square units in a rectangle
+by multiplying with the appropriate numbers representing length and
+width is probably rather hindered than helped by the ordinary objective
+presentation as an introduction. The usual form of objective
+introduction is as follows:--
+
+    [Illustration: FIG. 58.]
+
+    How long is this rectangle? How large is each square? How many
+    square inches are there in the top row? How many rows are
+    there? How many square inches are there in the whole rectangle?
+    Since there are three rows each containing 4 square inches, we
+    have 3 � 4 square inches = 12 square inches.
+
+    Draw a rectangle 7 inches long and 2 inches wide. If you divide
+    it into inch squares how many rows will there be? How many inch
+    squares will there be in each row? How many square inches are
+    there in the rectangle?
+
+    [Illustration: FIG. 59.]
+
+It is better actually to hide the individual square units as in Fig. 59.
+There are four reasons: (1) The concrete rows and columns rather
+distract attention from the essential thing to be learned. This is not
+that "_x_ rows one square wide, _y_ squares in a row will make _xy_
+squares in all," but that "by using proper units and the proper
+operation the area of any rectangle can be found from its length and
+width." (2) Children have little difficulty in learning to multiply
+rather than add, subtract, or divide when computing area. (3) The habit
+so formed holds good for areas like 1-2/3 by 4-1/2, with fractional
+dimensions, in which any effort to count up the areas of rows is very
+troublesome and confusing. (4) The notion that a square inch is an area
+1' by 1' rather than 1/2' by 2' or 1/3 in. by 3 in. or 1-1/2 in. by 2/3
+in. is likely to be formed too emphatically if much time is spent upon
+the sort of concrete presentation shown above. It is then better to use
+concrete counting of rows of small areas as a means of _verification
+after_ the procedure is learned, than as a means of deriving it.
+
+There has been, especially in Germany, much argument concerning what
+sort of number-pictures (that is, arrangement of dots, lines, or the
+like, as shown in Fig. 60) is best for use in connection with the number
+names in the early years of the teaching of arithmetic.
+
+Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and
+others have measured the accuracy of children in estimating the number
+of dots in arrangements of one or more of these different types.[21]
+Many writers interpret a difference in favor of estimating, say, the
+square arrangements of Born or Lay as meaning that such is the best
+arrangement to use in teaching. The inference is, however, unjustified.
+That certain number-pictures are easier to estimate numerically does not
+necessarily mean that they are more instructive in learning. One set may
+be easier to estimate just because they are more familiar, having been
+oftener experienced. Even if the favored set was so after equal
+experience with all sets, accuracy of estimation would be a sign of
+superiority for use in instruction only if all other things were equal
+(or in favor of the arrangement in question). Obviously the way to
+decide which of these is best to use in teaching is by using them in
+teaching and measuring all relevant results, not by merely recording
+which of them are most accurately estimated in certain time exposures.
+
+  [21] For an account in English of their main findings see
+       Howell ['14], pp. 149-251.
+
+It may be noted that the Born, Lay, and Freeman pictures have claims for
+special consideration on grounds of probable instructiveness. Since they
+are also superior in the tests in respect to accuracy of estimate,
+choice should probably be made from these three by any teacher who
+wishes to connect one set of number-pictures systematically with the
+number names, as by drills with the blackboard or with cards.
+
+    [Illustration: FIG. 60.--Various proposed arrangements of dots
+    for use in teaching the meanings of the numbers 1 to 10.]
+
+Such drills are probably useful if undertaken with zeal, and if kept as
+supplementary to more realistic objective work with play money, children
+marching, material to be distributed, garden-plot lengths to be
+measured, and the like, and if so administered that the pupils soon get
+the generalized abstract meaning of the numbers freed from dependence on
+an inner picture of any sort. This freedom is so important that it may
+make the use of many types of number-pictures advisable rather than the
+use of the one which in and of itself is best.
+
+As Meumann says: "Perceptual reckoning can be overdone. It had its chief
+significance for the surety and clearness of the first foundation of
+arithmetical instruction. If, however, it is continued after the first
+operations become familiar to the child, and extended to operations
+which develop from these elementary ones, it necessarily works as a
+retarding force and holds back the natural development of arithmetic.
+This moves on to work with abstract number and with mechanical
+association and reproduction." ['07, Vol. 2, p. 357.]
+
+Such drills are commonly overdone by those who make use of them, being
+given too often, and continued after their instructiveness has waned,
+and used instead of more significant, interesting, and varied work in
+counting and estimating and measuring real things. Consequently, there
+is now rather a prejudice against them in our better schools. They
+should probably be reinstated but to a moderate and judicious use.
+
+
+ORAL, MENTAL, AND WRITTEN ARITHMETIC
+
+There has been much dispute over the relative merits of oral and written
+work in arithmetic--a question which is much confused by the different
+meanings of 'oral' and 'written.' _Oral_ has meant (1) work where the
+situations are presented orally and the pupil's final responses are
+given orally, or (2) work where the situations are presented orally and
+the pupils' final responses are written or partly written and partly
+oral, or (3) work where the situations are presented in writing or print
+and the final responses are oral. _Written_ has meant (1) work where the
+situations are presented in writing or print and the final responses are
+made in writing, or (2) work where also many of the intermediate
+responses are written, or (3) work where the situations are presented
+orally but the final responses and a large percentage of the
+intermediate computational responses are written. There are other
+meanings than these.
+
+It is better to drop these very ambiguous terms and ask clearly what are
+the merits and demerits, in the case of any specified arithmetical work,
+of auditory and of visual presentation of the situations, and of saying
+and of writing each specified step in the response.
+
+The disputes over mental _versus_ written arithmetic are also confused
+by ambiguities in the use of 'mental.' Mental has been used to mean
+"done without pencil and paper" and also "done with few overt
+responses, either written or spoken, between the setting of the task and
+the announcement of the answer." In neither case is the word _mental_
+specially appropriate as a description of the total fact. As before, we
+should ask clearly, "What are the merits and demerits of making certain
+specified intermediate responses in inner speech or imaged sounds or
+visual images or imageless thought--that is, _without_ actual writing or
+overt speech?"
+
+It may be said at the outset that oral, written, and inner presentations
+of initial situations, oral, written, and inner announcements of final
+responses, and oral, written, and inner management of intermediate
+processes have varying degrees of merit according to the particular
+arithmetical exercise, pupil, and context. Devotion to oralness or
+mentalness as such is simply fanatical. Various combinations, such as
+the written presentation of the situation with inner management of the
+intermediate responses and oral announcement of the final response have
+their special merits for particular cases.
+
+These merits the reader can evaluate for himself for any given sort of
+work for a given class by considering: (1) The amount of practice
+received by the class per hour spent; (2) the ease of correction of the
+work; (3) the ease of understanding the tasks; (4) the prevention of
+cheating; (5) the cheerfulness and sociability of the work; (6) the
+freedom from eyestrain, and other less important desiderata.
+
+It should be noted that the stock schemes A, B, C, and D below are only
+a few of the many that are possible and that schemes E, F, G, and H have
+special merits.
+
+    PRESENTATION OF         MANAGEMENT OF           ANNOUNCEMENT OF
+    INITIAL SITUATION       INTERMEDIATE PROCESSES  FINAL RESPONSE
+
+  A. Printed or written      Written                 Written
+
+  B.    "          "         Inner                   Oral by one pupil,
+                                                     inner by the rest
+
+  C. Oral (by teacher)       Written                 Written
+
+  D.   "         "           Inner                   Oral by one pupil,
+                                                     inner by the rest
+
+  E. As in A or C            A mixture, the pupil    As in A or B or H
+                            writing what he needs
+
+  F. The real situation      As in E                 As in A or B or H
+     itself, in part at
+     least
+
+  G. Both read by the pupil  As in E                 As in A or B or H
+     and put orally by the
+     teacher
+
+  H. As in A or C or G       As in E                 Written by all
+                                                     pupils, announced
+                                                     orally by one pupil
+
+The common practice of either having no use made of pencil and paper or
+having all computations and even much verbal analysis written out
+elaborately for examination is unfavorable for learning. The demands
+which life itself will make of arithmetical knowledge and skill will
+range from tasks done with every percentage of written work from zero up
+to the case where every main result obtained by thought is recorded for
+later use by further thought. In school the best way is that which, for
+the pupils in question, has the best total effect upon quality of
+product, speed, and ease of production, re�nforcement of training
+already given, and preparation for training to be given. There is
+nothing intellectually criminal about using a pencil as well as inner
+thought; on the other hand there is no magical value in writing out for
+the teacher's inspection figures that the pupil does not need in order
+to attain, preserve, verify, or correct his result.
+
+The common practice of having the final responses of all _easy_ tasks
+given orally has no sure justification. On the contrary, the great
+advantage of having all pupils really do the work should be secured in
+the easy work more than anywhere else. If the time cost of copying the
+figures is eliminated by the simple plan of having them printed, and if
+the supervision cost of examining the papers is eliminated by having the
+pupils correct each other's work in these easy tasks, written answers
+are often superior to oral except for the elements of sociability and
+'go' and freedom from eyestrain of the oral exercise. Such written work
+provides the gifted pupils with from two to ten times as much practice
+as they would get in an oral drill on the same material, supposing them
+to give inner answers to every exercise done by the class as a whole; it
+makes sure that the dull pupils who would rarely get an inner answer at
+the rate demanded by the oral exercise, do as much as they are able to
+do.
+
+Two arguments often made for the oral statement of problems by the
+teacher are that problems so put are better understood, especially in
+the grades up through the fifth, and that the problems are more likely
+to be genuine and related to the life the pupils know. When these
+statements are true, the first is a still better argument for having the
+pupils read the problems _aided by the teacher's oral statement of
+them_. For the difficulty is largely that the pupils cannot read well
+enough; and it is better to help them to surmount the difficulty rather
+than simply evade it. The second is not an argument for oralness
+_versus_ writtenness, but for good problems _versus_ bad; the teacher
+who makes up such good problems should, in fact, take special care to
+write them down for later use, which may be by voice or by the
+blackboard or by printed sheet, as is best.
+
+
+
+
+CHAPTER XIV
+
+THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE
+
+
+Dewey, and others following him, have emphasized the desirability of
+having pupils do their work as active seekers, conscious of problems
+whose solution satisfies some real need of their own natures. Other
+things being equal, it is unwise, they argue, for pupils to be led along
+blindfold as it were by the teacher and textbook, not knowing where they
+are going or why they are going there. They ought rather to have some
+living purpose, and be zealous for its attainment.
+
+This doctrine is in general sound, as we shall see, but it is often
+misused as a defense of practices which neglect the formation of
+fundamental habits, or as a recommendation to practices which are quite
+unworkable under ordinary classroom conditions. So it seems probable
+that its nature and limitations are not thoroughly known, even to its
+followers, and that a rather detailed treatment of it should be given
+here.
+
+
+ILLUSTRATIVE CASES
+
+Consider first some cases where time spent in making pupils understand
+the end to be attained before attacking the task by which it is
+attained, or care about attaining the end (well or ill understood) is
+well spent.
+
+It is well for a pupil who has learned (1) the meanings of the numbers
+one to ten, (2) how to count a collection of ten or less, and (3) how to
+measure in inches a magnitude of ten, nine, eight inches, etc., to be
+confronted with the problem of true adding without counting or
+measuring, as in 'hidden' addition and measurement by inference. For
+example, the teacher has three pencils counted and put under a book; has
+two more counted and put under the book; and asks, "How many pencils are
+there under the book?" Answers, when obtained, are verified or refuted
+by actual counting and measuring.
+
+The time here is well spent because the children can do the necessary
+thinking if the tasks are well chosen; because they are thereby
+prevented from beginning their study of addition by the bad habit of
+pseudo-adding by looking at the two groups of objects and counting their
+number instead of real adding, that is, thinking of the two numbers and
+inferring their sum; and further, because facing the problem of adding
+as a real problem is in the end more economical for learning arithmetic
+and for intellectual training in general than being enticed into adding
+by objective or other processes which conceal the difficulty while
+helping the pupil to master it.
+
+The manipulation of short multiplication may be introduced by
+confronting the pupils with such problems as, "How to tell how many
+Uneeda biscuit there are in four boxes, by opening only one box."
+Correct solutions by addition should be accepted. Correct solutions by
+multiplication, if any gifted children think of this way, should be
+accepted, even if the children cannot justify their procedure.
+(Inferring the manipulation from the place-values of numbers is beyond
+all save the most gifted and probably beyond them.) Correct solution by
+multiplication by some child who happens to have learned it elsewhere
+should be accepted. Let the main proof of the trustworthiness of the
+manipulation be by measurement and by addition. Proof by the stock
+arguments from the place-values of numbers may also be used. If no child
+hits on the manipulation in question, the problem of finding the length
+_without_ adding may be set. If they still fail, the problem may be made
+easier by being put as "4 times 22 gives the answer. Write down what you
+think 4 times 22 will be." Other reductions of the difficulty of the
+problem may be made, or the teacher may give the answer without very
+great harm being done. The important requirement is that the pupils
+should be aware of the problem and treat the manipulation as a solution
+of it, not as a form of educational ceremonial which they learn to
+satisfy the whims of parents and teachers. In the case of any particular
+class a situation that is more appealing to the pupils' practical
+interests than the situation used here can probably be devised.
+
+The time spent in this way is well spent (1) because all but the very
+dull pupils can solve the problem in some way, (2) because the
+significance of the manipulation as an economy over addition is worth
+bringing out, and (3) because there is no way of beginning training in
+short multiplication that is much better.
+
+In the same fashion multiplication by two-place numbers may be
+introduced by confronting pupils with the problem of the number of
+sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square
+inches in 35 square feet, or the number of days in 32 years, or whatever
+similar problem can be brought up so as to be felt as a problem.
+
+Suppose that it is the 35 square feet. Solutions by (5 � 144) +
+(30 � 144), however arranged, or by (10 � 144) + (10 � 144) +
+(10 � 144) + (5 � 144), or by 3500 + (35 � 40) + (35 � 4), or by
+7 � (5 � 144), however arranged, should all be listed for verification
+or rejection. The pupils need not be required to justify their
+procedures by a verbal statement. Answers like 432,720, or 720,432,
+or 1152, or 4220, or 3220 should be listed for verification or
+rejection. Verification may be by a mixture of short multiplication
+and objective work, or by a mixture of short multiplication and
+addition, or by addition abbreviated by taking ten 144s as 1440, or
+even (for very stupid pupils) by the authority of the teacher. Or the
+manipulation in cases like 53 � 9 or 84 � 7 may be verified by the
+reverse short multiplication. The deductive proof of the correctness
+of the manipulation may be given in whole or in part in connection
+with exercises like
+
+  10 �   2 =          30 �  14 =
+  10 �   3 =           3 �  44 =
+  10 �   4 =          30 �  44 =
+  10 �  14 =           3 � 144 =
+  10 �  44 =          20 � 144 =
+  10 � 144 =          40 � 144 =
+  20 �   2 =          30 � 144 =
+  20 �   3 =           5 � 144 =
+  30 �   3 =          35 = 30 + ....
+  30 �   4 =          30 � 144 added to 5 � 144 =
+   3 �  14 =
+
+Certain wrong answers may be shown to be wrong in many ways; _e.g._,
+432,720 is too big, for 35 times a thousand square inches is only
+35,000; 1152 is too small, for 35 times a hundred square inches would be
+3500, or more than 1152.
+
+The time spent in realizing the problem here is fairly well spent
+because (1) any successful original manipulation in this case
+represents an excellent exercise of thought, because (2) failures show
+that it is useless to juggle the figures at random, and because (3) the
+previous experience with short multiplication makes it possible for the
+pupils to realize the problem in a very few minutes. It may, however, be
+still better to give the pupils the right method just as soon as the
+problem is realized, without having them spend more time in trying to
+solve it. Thus:--
+
+1 square foot has 144 square inches. How many square inches are there in
+35 square feet (marked out in chalk on the floor as a piece 10 ft. � 3
+ft. plus a piece 5 ft. � 1 ft.)?
+
+1 yard = 36 inches. How many inches long is this wall (found by measure
+to be 13 yards)?
+
+Here is a quick way to find the answers:--
+
+   144
+    35
+  ----
+   720
+  432
+  ----
+  5040 sq. inches in 35 sq. ft.
+
+   36
+   13
+  ---
+  108
+  36
+  ---
+  468 inches in 13 yd.
+
+Consider now the following introduction to dividing by a decimal:--
+
+  Dividing by a Decimal
+
+  1. How many minutes will it take a motorcycle, to go 12.675 miles
+     at the rate of .75 mi. per minute?
+
+                                      16.9
+                                      ______
+                                  .75|12.675
+                                       7 5
+                                       ---
+                                       5 17
+                                       4 50
+                                       ----
+                                         675
+                                         675
+                                         ---
+
+  2. Check by multiplying 16.9 by .75.
+
+  3. How do you know that the quotient cannot be as little as 1.69?
+
+  4. How do you know that the quotient cannot be as large as 169?
+
+  5. Find the quotient for 3.75 � 1.5.
+
+  6. Check your result by multiplying the quotient by the divisor.
+
+  7. How do you know that the quotient cannot be .25 or 25?
+                                  ____
+  8. Look at this problem.     .25|7.5
+
+     How do you know that 3.0 is wrong for the quotient?
+
+     How do you know that 300 is wrong for the quotient?
+
+  State which quotient is right for each of these:--
+
+      .021 or .21 or 2.1 or 21 or 210
+         ______
+  9.  1.8|3.78
+
+
+      .021 or .21 or 21 or 210
+         ______
+  10. 1.8|37.8
+
+
+      .03 or .3 or 3 or 30 or 300
+          ______
+  11. 1.25|37.5
+
+
+      .03 or .3 or 3 or 30 or 300
+          ______
+  12. 12.5|37.5
+
+
+      .05 or .5 or 5 or 50 or 500
+          ______
+  13. 1.25|6.25
+
+
+      .05 or .5 or 5 or 50 or 500
+          ______
+  14. 12.5|6.25
+
+
+  15. Is this rule true? If it is true, learn it.
+
+        #In a correct result, the number of decimal places in
+        the divisor and quotient together equals the number
+        of decimal places in the dividend.#
+
+These and similar exercises excite the problem attitude in children _who
+have a general interest in getting right answers_. Such a series
+carefully arranged is a desirable introduction to a statement of the
+rule for placing the decimal point in division with decimals. For it
+attracts attention to the general principle (divisor � quotient should
+equal dividend), which is more important than the rule for convenient
+location of the decimal point, and it gives training in placing the
+point by inspection of the divisor, quotient, and dividend, which
+suffices for nineteen out of twenty cases that the pupil will ever
+encounter outside of school. He is likely to remember this method by
+inspection long after he has forgotten the fixed rule.
+
+It is well for the pupil to be introduced to many arithmetical facts by
+way of problems about their common uses. The clockface, the railroad
+distance table in hundredths of a mile, the cyclometer and speedometer,
+the recipe, and the like offer problems which enlist his interest and
+energy and also connect the resulting arithmetical learning with the
+activities where it is needed. There is no time cost, but a time-saving,
+for the learning as a means to the solution of the problems is quicker
+than the mere learning of the arithmetical facts by themselves alone. A
+few samples of such procedure are shown below:--
+
+  GRADE 3
+
+  To be Done at Home
+
+  Look at a watch. Has it any hands besides the hour hand and the
+  minute hand? Find out all that you can about how a watch tells
+  seconds, how long a second is, and how many seconds make a minute.
+
+
+  GRADE 5
+
+  Measuring Rainfall
+
+         =Rainfall per Week=
+    (=cu. in. per sq. in. of area=)
+    June   1-7      1.056
+           8-14     1.103
+          15-21     1.040
+          22-28      .960
+          29-July 5  .915
+    July   6-12      .782
+          13-19      .790
+          20-26      .670
+          27-Aug. 2  .503
+    Aug.   3-9       .512
+          10-16      .240
+          17-23      .215
+          24-30      .811
+
+  1. In which weeks was the rainfall 1 or more?
+
+  2. Which week of August had the largest rainfall for that month?
+
+  3. Which was the driest week of the summer? (Driest means with
+     the least rainfall.)
+
+  4. Which week was the next to the driest?
+
+  5. In which weeks was the rainfall between .800 and 1.000?
+
+  6. Look down the table and estimate whether the average rainfall for
+     one week was about .5, or about .6, or about .7, or about .8, or
+     about .9.
+
+  Dairy Records
+
+    =Record of Star Elsie=
+
+          Pounds of Milk     Butter-Fat per Pound of Milk
+      Jan.    1742                  .0461
+      Feb.    1690                  .0485
+      Mar.    1574                  .0504
+      Apr.    1226                  .0490
+      May     1202                  .0466
+      June    1251                  .0481
+
+  Read this record of the milk given by the cow Star Elsie. The first
+  column tells the number of pounds of milk given by Star Elsie each
+  month. The second column tells what fraction of a pound of butter-fat
+  each pound of milk contained.
+
+  1. Read the first line, saying, "In January this cow gave 1742 pounds
+      of milk. There were 461 ten thousandths of a pound of butter-fat
+      per pound of milk." Read the other lines in the same way.
+
+  2. How many pounds of butter-fat did the cow produce in Jan.?
+      3. In Feb.? 4. In Mar.? 5. In Apr.? 6. In May? 7. In June?
+
+
+  GRADE 5 OR LATER
+
+  Using Recipes to Make Larger or Smaller Quantities
+
+  I. State how much you would use of each material in the following
+  recipes: (_a_) To make double the quantity. (_b_) To make half the
+  quantity. (_c_) To make 1-1/2 times the quantity. You may use pencil
+  and paper when you cannot find the right amount mentally.
+
+    1. PEANUT PENUCHE
+
+      1 tablespoon butter
+      2 cups brown sugar
+      1/3 cup milk or cream
+      3/4 cup chopped peanuts
+      1/3 teaspoon salt
+
+    2. MOLASSES CANDY
+
+      1/2 cup butter
+      2 cups sugar
+      1 cup molasses
+      1-1/2 cups boiling water
+
+    3. RAISIN OPERA CARAMELS
+
+      2 cups light brown sugar
+      7/8 cup thin cream
+      1/2 cup raisins
+
+    4. WALNUT MOLASSES SQUARES
+
+      2 tablespoons butter
+      1 cup molasses
+      1/3 cup sugar
+      1/2 cup walnut meats
+
+    5. RECEPTION ROLLS
+
+      1 cup scalded milk
+      1-1/2 tablespoons sugar
+      1 teaspoon salt
+      1/4 cup lard
+      1 yeast cake
+      1/4 cup lukewarm water
+      White of 1 egg
+      3-1/2 cups flour
+
+    6. GRAHAM RAISED LOAF
+
+      2 cups milk
+      6 tablespoons molasses
+      1-1/2 teaspoons salt
+      1/3 yeast cake
+      1/4 cup lukewarm water
+      2 cups sifted Graham flour
+      1/2 cup Graham bran
+      3/4 cup flour (to knead)
+
+  II. How much would you use of each material in the following recipes:
+  (_a_) To make 2/3 as large a quantity? (_b_) To make 1-1/3 times as
+  much? (_c_) To make 2-1/2 times as much?
+
+    1. ENGLISH DUMPLINGS
+
+      1/2 pound beef suet
+      1-1/4 cups flour
+      3 teaspoons baking powder
+      1 teaspoon salt
+      1/2 teaspoon pepper
+      1 teaspoon minced parsley
+      1/2 cup cold water
+
+    2. WHITE MOUNTAIN ANGEL CAKE
+
+      1-1/2 cups egg whites
+      1-1/2 cups sugar
+      1 teaspoon cream of tartar
+      1 cup bread flour
+      1/4 teaspoon salt
+      1 teaspoon vanilla
+
+In many cases arithmetical facts and principles can be well taught in
+connection with some problem or project which is not arithmetical, but
+which has special potency to arouse an intellectual activity in the
+pupil which by some ingenuity can be directed to arithmetical learning.
+Playing store is the most fundamental case. Planning for a party, seeing
+who wins a game of bean bag, understanding the calendar for a month,
+selecting Christmas presents, planning a picnic, arranging a garden, the
+clock, the watch with second hand, and drawing very simple maps are
+situations suggesting problems which may bring a living purpose into
+arithmetical learning in grade 2. These are all available under ordinary
+conditions of class instruction. A sample of such problems for a higher
+grade (6) is shown below.
+
+  Estimating Areas
+
+  The children in the geography class had a contest in estimating
+  the areas of different surfaces. Each child wrote his estimates
+  for each of these maps, A, B, C, D, and E. (Only C and D are
+  shown here.) In the arithmetic class they learned how to find
+  the exact areas. Then they compared their estimates with the
+  exact areas to find who came nearest.
+
+    [Illustration]
+
+  Write your estimates for A, B, C, D, and E. Then study the
+  next 6 pages and learn how to find the exact areas.
+
+  (The next 6 pages comprise training in the mensuration of
+  parallelograms and triangles.)
+
+In some cases the affairs of individual pupils include problems which
+may be used to guide the individual in question to a zealous study of
+arithmetic as a means of achieving his purpose--of making a canoe,
+surveying an island, keeping the accounts of a Girls' Canning Club, or
+the like. It requires much time and very great skill to direct the work
+of thirty or more pupils each busy with a special type of his own, so as
+to make the work instructive for each, but in some cases the expense of
+time and skill is justified.
+
+
+GENERAL PRINCIPLES
+
+In general what should be meant when one says that it is desirable to
+have pupils in the problem-attitude when they are studying arithmetic is
+substantially as follows:--
+
+_First._--Information that comes as an answer to questions is better
+attended to, understood, and remembered than information that just
+comes.
+
+_Second._--Similarly, movements that come as a step toward achieving an
+end that the pupil has in view are better connected with their
+appropriate situations, and such connections are longer retained, than
+is the case with movements that just happen.
+
+_Third._--The more the pupil is set toward getting the question answered
+or getting the end achieved, the greater is the satisfyingness attached
+to the bonds of knowledge or skill which mean progress thereto.
+
+_Fourth._--It is bad policy to rely exclusively on the purely
+intellectualistic problems of "How can I do this?" "How can I get the
+right answer?" "What is the reason for this?" "Is there a better way to
+do that?" and the like. It is bad policy to supplement these
+intellectualistic problems by only the remote problems of "How can I be
+fitted to earn a higher wage?" "How can I make sure of graduating?" "How
+can I please my parents?" and the like. The purely intellectualistic
+problems have too weak an appeal for many pupils; the remote problems
+are weak so long as they are remote and, what is worse, may be deprived
+of the strength that they would have in due time if we attempt to use
+them too soon. It is the extreme of bad policy to neglect those personal
+and practical problems furnished by life outside the class in arithmetic
+the solution of which can really be furthered by arithmetic then and
+there. It is good policy to spend time in establishing certain mental
+sets--stimulating, or even creating, certain needs--setting up problems
+themselves--when the time so spent brings a sufficient improvement in
+the quality and quantity of the pupils' interest in arithmetical
+learning.
+
+_Fifth._--It would be still worse policy to rely exclusively on
+problems arising outside arithmetic. To learn arithmetic is itself a
+series of problems of intrinsic interest and worth to healthy-minded
+children. The need for ability to multiply with United States money or
+to add fractions or to compute percents may be as truly vital and
+engaging as the need for skill to make a party dress or for money to buy
+it or for time to play baseball. The intellectualistic needs and
+problems should be considered along with all others, and given whatever
+weight their educational value deserves.
+
+
+DIFFICULTY AND SUCCESS AS STIMULI
+
+There are certain misconceptions of the doctrine of the
+problem-attitude. The most noteworthy is that difficulty--temporary
+failure--an inadequacy of already existing bonds--is the essential and
+necessary stimulus to thinking and learning. Dewey himself does not, as
+I understand him, mean this, but he has been interpreted as meaning it
+by some of his followers.[22]
+
+  [22] In his _How We Think_.
+
+Difficulty--temporary failure, inadequacy of existing bonds--on the
+contrary does nothing whatsoever in and of itself; and what is done by
+the annoying lack of success which sometimes accompanies difficulty
+sometimes hinders thinking and learning.
+
+Mere difficulty, mere failure, mere inadequacy of existing bonds, does
+nothing. It is hard for me to add three eight-place numbers at a glance;
+I have failed to find as effective illustrations for pages 276 to 277 as
+I wished; my existing sensori-motor connections are inadequate to
+playing a golf course in 65. But these events and conditions have done
+nothing to stimulate me in respect to the behavior in question. In the
+first of the three there is no annoying lack and no dynamic influence at
+all; in the second there was to some degree an annoying lack--a slight
+irritation at not getting just what I wanted,--and this might have
+impelled me to further thinking (though it did not, and getting one
+tiptop illustration would as a rule stimulate me to hunt for others more
+than failing to get such). In the third case the lack of the 65 does not
+annoy me or have any noteworthy dynamic effect. The lack of 90 instead
+of 95-100 is annoying and is at times a stimulus to further learning,
+though not nearly so strong a stimulus as the attainment of the 90 would
+be! At other times this annoying lack is distinctly inhibitory--a
+stimulus to ceasing to learn. In the intellectual life the inhibitory
+effect seems far the commoner of the two. Not getting answers seems as a
+rule to make us stop trying to get them. The annoying lack of success
+with a theoretical problem most often makes us desert it for problems to
+whose solution the existing bonds promise to be more adequate.
+
+The real issue in all this concerns the relative strength, in the
+pupil's intellectual life, of the "negative reaction" of behavior in
+general. An animal whose life processes are interfered with so that an
+annoying state of affairs is set up, changes his behavior, making one
+after another responses as his instincts and learned tendencies
+prescribe, until the annoying state of affairs is terminated, or the
+animal dies, or suffers the annoyance as less than the alternatives
+which his responses have produced. When the annoying state of affairs is
+characterized by the failure of things as they are to minister to a
+craving--as in cases of hunger, loneliness, sex-pursuit, and the
+like,--we have stimulus to action by an annoying lack or need, with
+relief from action by the satisfaction of the need.
+
+Such is in some measure true of man's intellectual life. In recalling a
+forgotten name, in solving certain puzzles, or in simplifying an
+algebraic complex, there is an annoying lack of the name, solution, or
+factor, a trial of one after another response, until the annoyance is
+relieved by success or made less potent by fatigue or distraction. Even
+here the _difficulty_ does not do anything--but only the annoying
+interference with our intellectual peace by the problem. Further,
+although for the particular problem, the annoying lack stimulates, and
+the successful attainment stops thinking, the later and more important
+general effect on thinking is the reverse. Successful attainment stops
+our thinking _on that problem_ but makes us more predisposed later to
+thinking _in general_.
+
+Overt negative reaction, however, plays a relatively small part in man's
+intellectual life. Filling intellectual voids or relieving intellectual
+strains in this way is much less frequent than being stimulated
+positively by things seen, words read, and past connections acting under
+modified circumstances. The notion of thinking as coming to a lack,
+filling it, meeting an obstacle, dodging it, being held up by a
+difficulty and overcoming it, is so one-sided as to verge on phantasy.
+The overt lacks, strains, and difficulties come perhaps once in five
+hours of smooth straightforward use and adaptation of existing
+connections, and they might as truly be called hindrances to
+thought--barriers which past successes help the thinker to surmount.
+Problems themselves come more often as cherished issues which new facts
+reveal, and whose contemplation the thinker enjoys, than as strains or
+lacks or 'problems which I need to solve.' It is just as true that the
+thinker gets many of his problems as results from, or bonuses along
+with, his information, as that he gets much of his information as
+results of his efforts to solve problems.
+
+As between difficulty and success, success is in the long run more
+productive of thinking. Necessity is not the mother of invention.
+Knowledge of previous inventions is the mother; original ability is the
+father. The solutions of previous problems are more potent in producing
+both new problems and their solutions than is the mere awareness of
+problems and desire to have them solved.
+
+In the case of arithmetic, learning to cancel instead of getting the
+product of the dividends and the product of the divisors and dividing
+the former by the latter, is a clear case of very valuable learning,
+with ease emphasized rather than difficulty, with the adequacy of
+existing bonds (when slightly redirected) as the prime feature of the
+process rather than their inadequacy, and with no sense of failure or
+lack or conflict. It would be absurd to spend time in arousing in the
+pupil, before beginning cancellation, a sense of a difficulty--viz.,
+that the full multiplying and dividing takes longer than one would like.
+A pupil in grade 4 or 5 might well contemplate that difficulty for years
+to no advantage. He should at once begin to cancel and prove by checking
+that errorless cancellation always gives the right answer. To emphasize
+before teaching cancellation the inadequacy of the old full multiplying
+and dividing would, moreover, not only be uneconomical as a means to
+teaching cancellation; it would amount to casting needless slurs on
+valuable past acquisitions, and it would, scientifically, be false.
+For, until a pupil has learned to cancel, the old full multiplying is
+not inadequate; it is admirable in every respect. The issue of its
+inadequacy does not truly appear until the new method is found. It is
+the best way until the better way is mastered.
+
+In the same way it is unwise to spend time in making pupils aware of the
+annoying lacks to be supplied by the multiplication tables, the division
+tables, the casting out of nines, or the use of the product of the
+length and breadth of a rectangle as its area, the unit being changed to
+the square erected on the linear unit as base. The annoying lack will
+be unproductive, while the learning takes place readily as a
+modification of existing habits, and is sufficiently appreciated as soon
+as it does take place. The multiplication tables come when instead of
+merely counting by 7s from 0 up saying "7, 14, 21," etc., the pupil
+counts by 7s from 0 up saying "Two sevens make 14, three sevens make 21,
+four sevens make 28," etc. The division tables come as easy selections
+from the known multiplications; the casting out of nines comes as an
+easy device. The computation of the area of a rectangle is best
+facilitated, not by awareness of the lack of a process for doing it, but
+by awareness of the success of the process as verified objectively.
+
+In all these cases, too, the pupil would be misled if we aroused first a
+sense of the inadequacy of counting, adding, and objective division, an
+awareness of the difficulties which the multiplication and division
+tables and nines device and area theorem relieve. The displaced
+processes are admirable and no unnecessary fault should be found with
+them, and they are _not_ inadequate until the shorter ways have been
+learned.
+
+
+FALSE INFERENCES
+
+One false inference about the problem-attitude is that the pupil should
+always understand the aim or issue before beginning to form the bonds
+which give the method or process that provides the solution. On the
+contrary, he will often get the process more easily and value it more
+highly if he is taught what it is _for_ gradually while he is learning
+it. The system of decimal notation, for example, may better be taken
+first as a mere fact, just as we teach a child to talk without trying
+first to have him understand the value of verbal intercourse, or to keep
+clean without trying first to have him understand the bacteriological
+consequences of filth.
+
+A second inference--that the pupil should always be taught to care about
+an issue and crave a process for managing it before beginning to learn
+the process--is equally false. On the contrary, the best way to become
+interested in certain issues and the ways of handling them is to learn
+the process--even to learn it by sheer habituation--and then note what
+it does for us. Such is the case with ".1666-2/3 � = divide by 6,"
+".333-1/3 � = divide by 3," "multiply by .875 = divide the number by 8
+and subtract the quotient from the number."
+
+A third unwise tendency is to degrade the mere giving of information--to
+belittle the value of facts acquired in any other way than in the course
+of deliberate effort by the pupil to relieve a problem or conflict or
+difficulty. As a protest against merely verbal knowledge, and merely
+memoriter knowledge, and neglect of the active, questioning search for
+knowledge, this tendency to belittle mere facts has been healthy, but as
+a general doctrine it is itself equally one-sided. Mere facts not got by
+the pupil's thinking are often of enormous value. They may stimulate to
+active thinking just as truly as that may stimulate to the reception of
+facts. In arithmetic, for example, the names of the numbers, the use of
+the fractional form to signify that the upper number is divided by the
+lower number, the early use of the decimal point in U. S. money to
+distinguish dollars from cents, and the meanings of "each," "whole,"
+"part," "together," "in all," "sum," "difference," "product,"
+"quotient," and the like are self-justifying facts.
+
+A fourth false inference is that whatever teaching makes the pupil face
+a question and think out its answer is thereby justified. This is not
+necessarily so unless the question is a worthy one and the answer that
+is thought out an intrinsically valuable one and the process of thinking
+used one that is appropriate for that pupil for that question. Merely
+to think may be of little value. To rely much on formal discipline is
+just as pernicious here as elsewhere. The tendency to emphasize the
+methods of learning arithmetic at the expense of what is learned is
+likely to lead to abuses different in nature but as bad in effect as
+that to which the emphasis on disciplinary rather than content value has
+led in the study of languages and grammar, or in the old puzzle problems
+of arithmetic.
+
+The last false inference that I shall discuss here is the inference that
+most of the problems by which arithmetical learning is stimulated had
+better be external to arithmetic itself--problems about Noah's Ark or
+Easter Flowers or the Merry Go Round or A Trip down the Rhine.
+
+Outside interests should be kept in mind, as has been abundantly
+illustrated in this volume, but it is folly to neglect the power, even
+for very young or for very stupid children, of the problem "How can I
+get the right answer?" Children do have intellectual interests. They do
+like dominoes, checkers, anagrams, and riddles as truly as playing tag,
+picking flowers, and baking cake. With carefully graded work that is
+within their powers they like to learn to add, subtract, multiply, and
+divide with integers, fractions, and decimals, and to work out
+quantitative relations.
+
+In some measure, learning arithmetic is like learning to typewrite. The
+learner of the latter has little desire to present attractive-looking
+excuses for being late, or to save expense for paper. He has no desire
+to hoard copies of such and such literary gems. He may gain zeal from
+the fact that a school party is to be given and invitations are to be
+sent out, but the problem "To typewrite better" is after all his main
+problem. Learning arithmetic is in some measure a game whose moves are
+motivated by the general set of the mind toward victory--winning right
+answers. As a ball-player learns to throw the ball accurately to
+first-base, not primarily because of any particular problem concerning
+getting rid of the ball, or having the man at first-base possess it, or
+putting out an opponent against whom he has a grudge, but because that
+skill is required by the game as a whole, so the pupil, in some measure,
+learns the technique of arithmetic, not because of particular concrete
+problems whose solutions it furnishes, but because that technique is
+required by the game of arithmetic--a game that has intrinsic worth and
+many general recommendations.
+
+
+
+
+CHAPTER XV
+
+INDIVIDUAL DIFFERENCES
+
+
+The general facts concerning individual variations in abilities--that
+the variations are large, that they are continuous, and that for
+children of the same age they usually cluster around one typical or
+modal ability, becoming less and less frequent as we pass to very high
+or very low degrees of the ability--are all well illustrated by
+arithmetical abilities.
+
+
+NATURE AND AMOUNT
+
+The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In
+these diagrams each space along the baseline represents a certain score
+or degree of ability, and the height of the surface above it represents
+the number of individuals obtaining that score. Thus in Fig. 61, 63 out
+of 1000 soldiers had no correct answer, 36 out of 1000 had one correct
+answer, 49 had two, 55 had three, 67 had four, and so on, in a test with
+problems (stated in words).
+
+Figure 61 shows that these adults varied from no problems solved
+correctly to eighteen, around eight as a central tendency. Figure 62
+shows that children of the same year-age (they were also from the same
+neighborhood and in the same school) varied from under 40 to over 200
+figures correct. Figure 63 shows that even among children who have all
+reached the same school grade and so had rather similar educational
+opportunities in arithmetic, the variation is still very great. It
+requires a range from 15 to over 30 examples right to include even nine
+tenths of them.
+
+    [Illustration: FIG. 61.--The scores of 1000 soldiers in the
+    National Army born in English-speaking countries, in Test 2 of the
+    Army Alpha. The score is the number of correct answers obtained in
+    five minutes. Probably 10 to 15 percent of these men were unable to
+    read or able to read only very easy sentences at a very slow rate.
+    Data furnished by the Division of Psychology in the office of the
+    Surgeon General.]
+
+It should, however, be noted that if each individual had been scored by
+the average of his work on eight or ten different days instead of by his
+work in just one test, the variability would have been somewhat less
+than appears in Figs. 61, 62, and 63.
+
+    [Illustration: FIG. 62.--The scores of 100 11-year-old pupils
+    in a test of computation. Estimated from the data given by Burt
+    ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the
+    number of correct figures.]
+
+It is also the case that if each individual had been scored, not in
+problem-solving alone or division alone, but in an elaborate examination
+on the whole field of arithmetic, the variability would have been
+somewhat less than appears in Figs. 61, 62, and 63. On the other hand,
+if the officers and the soldiers rejected for feeblemindedness had been
+included in Fig. 61, if the 11-year-olds in special classes for the very
+dull had been included in Fig. 62, and if all children who had been to
+school six years had been included in Fig. 63, no matter what grade they
+had reached, the effect would have been to _increase_ the variability.
+
+    [Illustration: FIG. 63.--The scores of pupils in grade 6 in city
+    schools in the Woody Division Test A. The score is the number of
+    correct answers obtained in 20 minutes. From Woody ['16, p. 61].]
+
+In spite of the effort by school officers to collect in any one school
+grade those somewhat equal in ability or in achievement or in a mixture
+of the two, the population of the same grades in the same school system
+shows a very wide range in any arithmetical ability. This is partly
+because promotion is on a more general basis than arithmetical ability
+so that some very able arithmeticians are deliberately held back on
+account of other deficiencies, and some very incompetent arithmeticians
+are advanced on account of other excellencies. It is partly because of
+general inaccuracy in classifying and promoting pupils.
+
+In a composite score made up of the sum of the scores in Woody
+tests,--Add. A, Subt. A, Mult. A, and Div. A, and two tests in
+problem-solving (ten and six graded problems, with maximum attainable
+credits of 30 and 18), Kruse ['18] found facts from which I compute
+those of Table 13, and Figs. 64 to 66, for pupils all having the
+training of the same city system, one which sought to grade its pupils
+very carefully.
+
+    [Illustration: FIGS. 64, 65, and 66.--The scores of pupils in
+    grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a
+    composite of tests in computation and problem-solving. The time
+    was about 120 minutes. The maximum score attainable was 196.]
+
+The overlapping of grade upon grade should be noted. Of the pupils in
+grade 6 about 18 percent do better than the average pupil in grade 7,
+and about 7 percent do better than the average pupil in grade 8. Of the
+pupils in grade 8 about 33 percent do worse than the average pupil in
+grade 7 and about 12 percent do worse than the average pupil in grade 6.
+
+TABLE 13
+
+RELATIVE FREQUENCIES OF SCORES IN AN EXTENSIVE TEAM OF ARITHMETICAL
+TESTS.[23] IN PERCENTS
+
+  ==============================================
+     SCORE    |  GRADE 6  |  GRADE 7  |  GRADE 8
+  ------------+-----------+-----------+---------
+    70 to  79 |    1.3    |     .9    |     .4
+    80  "  89 |    5.5    |    2.3    |     .4
+    90  "  99 |   10.6    |    4.3    |    2.9
+   100  " 109 |   19.4    |    5.2    |    4.4
+   110  " 119 |   19.8    |   18.5    |    5.8
+   120  " 129 |   23.5    |   16.2    |   16.8
+   130  " 139 |   12.6    |   17.5    |   16.8
+   140  " 149 |    4.6    |   13.9    |   22.9
+   150  " 159 |    1.7    |   13.6    |   17.1
+   160  " 169 |    1.2    |    4.8    |    9.4
+   170  " 179 |           |    2.5    |    3.3
+  ==============================================
+
+  [23] Compiled from data on p. 89 of Kruse ['18].
+
+
+DIFFERENCES WITHIN ONE CLASS
+
+The variation within a single class for which a single teacher has to
+provide is great. Even when teaching is departmental and promotion is by
+subjects, and when also the school is a large one and classification
+within a grade is by ability--there may be a wide range for any given
+special component ability. Under ordinary circumstances the range is so
+great as to be one of the chief limiting conditions for the teaching of
+arithmetic. Many methods appropriate to the top quarter of the class
+will be almost useless for the bottom quarter, and _vice versa_.
+
+    [Illustration: FIGS. 67 and 68.--The scores of ten 6 B classes in
+    a 12-minute test in computation with integers (the Courtis Test 7).
+    The score is the number of units done. Certain long tasks are
+    counted as two units.]
+
+Figures 67 and 68 show the scores of ten classes taken at random from
+ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of
+computation done in 12 minutes. Observe the very wide variation present
+in the case of every class. The variation within a class would be
+somewhat reduced if each pupil were measured by his average in eight or
+ten such tests given on different days. If a rather generous allowance
+is made for this we still have a variation in speed as great as that
+shown in Fig. 69, as the fact to be expected for a class of thirty-two 6
+B pupils.
+
+    [Illustration: FIG. 69.--A conservative estimate of the amount of
+    variation to be expected within a single class of 32 pupils in
+    grade 6, in the number of units done in Courtis Test 7 when all
+    chance variations are eliminated.]
+
+The variations within a class in respect to what processes are
+understood so as to be done with only occasional errors may be
+illustrated further as follows:--A teacher in grade 4 at or near the
+middle of the year in a city doing the customary work in arithmetic will
+probably find some pupil in her class who cannot do column addition even
+without carrying, or the easiest written subtraction
+
+  (8   9      78)
+  (5   3  or  37)
+  (-   -      --),
+
+who does not know his multiplication tables or how to derive them, or
+understand the meanings of + - � and �, or have any useful ideas
+whatever about division.
+
+There will probably be some child in the class who can do such work as
+that shown below, and with very few errors.
+
+  Add       3/8 + 5/8 + 7/8 + 1/8       2-1/2       1/6 + 3/8
+                                        6-3/8
+                                        3-3/4
+                                        -----
+
+  Subtract     10.00                4 yd.    1 ft.   6 in.
+                3.49                2 yd.    2 ft.   3 in.
+               -----                ----------------------
+
+  Multiply    1-1/4 � 8           16              145
+                                   2-5/8          206
+                                  ------          ---
+                   _______               _____
+  Divide          2)13.50              25)9750
+
+The invention of means of teaching thirty so different children at once
+with the maximum help and minimum hindrance from their different
+capacities and acquisitions is one of the great opportunities for
+applied science.
+
+Courtis, emphasizing the social demand for a certain moderate
+arithmetical attainment in the case of nearly all elementary school
+children of, say, grade 6, has urged that definite special means be
+taken to bring the deficient children up to certain standards, without
+causing undesirable 'overlearning' by the more gifted children. Certain
+experimental work to this end has been carried out by him and others,
+but probably much more must be done before an authoritative program for
+securing certain minimum standards for all or nearly all pupils can be
+arranged.
+
+
+THE CAUSES OF INDIVIDUAL DIFFERENCES
+
+The differences found among children of the same grade in the same city
+are due in large measure to inborn differences in their original
+natures. If, by a miracle, the children studied by Courtis, or by Woody,
+or by Kruse had all received exactly the same nurture from birth to
+date, they would still have varied greatly in arithmetical ability,
+perhaps almost as much as they now do vary.
+
+The evidence for this is the general evidence that variation in original
+nature is responsible for much of the eventual variation found in
+intellectual and moral traits, plus certain special evidence in the case
+of arithmetical abilities themselves.
+
+Thorndike found ['05] that in tests with addition and multiplication
+twins were very much more alike than siblings[24] two or three years
+apart in age, though the resemblance in home and school training in
+arithmetic should be nearly as great for the latter as for the former.
+Also the young twins (9-11) showed as close a resemblance in addition
+and multiplication as the older twins (12-15), although the similarities
+of training in arithmetic have had twice as long to operate in the
+latter case.
+
+  [24] Siblings is used for children of the same parents.
+
+If the differences found, say among children in grade 6 in addition,
+were due to differences in the quantity and quality of training in
+addition which they have had, then by giving each of them 200 minutes of
+additional identical training the differences should be reduced. For the
+200 minutes of identical training is a step toward equalizing training.
+It has been found in many investigations of the matter that when we make
+training in arithmetic more nearly equal for any group the variation
+within the group is not reduced.
+
+On the contrary, equalizing training seems rather to increase
+differences. The superior individual seems to have attained his
+superiority by his own superiority of nature rather than by superior
+past training, for, during a period of equal training for all, he
+increases his lead. For example, compare the gains of different
+individuals due to about 300 minutes of practice in mental
+multiplication of a three-place number by a three-place number shown
+in Table 14 below, from data obtained by the author ['08].[25]
+
+  [25] Similar results have been obtained in the case of arithmetical
+       and other abilities by Thorndike ['08, '10, '15, '16], Whitley
+       ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and
+       Thorndike ['13], Hahn and Thorndike ['14], and on a very
+       large scale by Race in a study as yet unpublished.
+
+TABLE 14
+
+THE EFFECT OF EQUAL AMOUNTS OF PRACTICE UPON INDIVIDUAL DIFFERENCE IN
+THE MULTIPLICATION OF THREE-PLACE NUMBERS
+
+  ====================================================================
+                                     |     AMOUNT     | PERCENTAGE OF
+                                     |                |CORRECT FIGURES
+                                     |----------------+---------------
+                                     | Initial |      | Initial |
+                                     |  Score  | Gain |  Score  | Gain
+  -----------------------------------+---------+------+---------+-----
+  Initially highest five individuals |    85   |  61  |    70   |  18
+            next    five     "       |    56   |  51  |    68   |  10
+            next    six      "       |    46   |  22  |    74   |   8
+            next    six      "       |    38   |   8  |    58   |  12
+            next    six      "       |    29   |  24  |    56   |  14
+  ====================================================================
+
+
+THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES
+
+Achievement in arithmetic depends upon a number of different abilities.
+For example, accuracy in copying numbers depends upon eyesight, ability
+to perceive visual details, and short-term memory for these. Long
+column addition depends chiefly upon great strength of the addition
+combinations especially in higher decades, 'carrying,' and keeping one's
+place in the column. The solution of problems framed in words requires
+understanding of language, the analysis of the situation described into
+its elements, the selection of the right elements for use at each step
+and their use in the right relations.
+
+Since the abilities which together constitute arithmetical ability are
+thus specialized, the individual who is the best of a thousand of his
+age or grade in respect to, say, adding integers, may occupy different
+stations, perhaps from 1st to 600th, in multiplying with integers,
+placing the decimal point in division with decimals, solving novel
+problems, copying figures, etc., etc. Such specialization is in part due
+to his having had, relatively to the others in the thousand, more or
+better training in certain of these abilities than in others, and to
+various circumstances of life which have caused him to have, relatively
+to the others in the thousand, greater interest in certain of these
+achievements than in others. The specialization is not wholly due
+thereto, however. Certain inborn characteristics of an individual
+predispose him to different degrees of superiority or inferiority to
+other men in different features of arithmetic.
+
+We measure the extent to which ability of one sort goes with or fails to
+go with ability of some other sort by the coefficient of correlation
+between the two. If every individual keeps the same rank in the second
+ability--if the individual who is the best of the thousand in one is the
+best of the group in the other, and so on down the list--the correlation
+is 1.00. In proportion as the ranks of individuals vary in the two
+abilities the coefficient drops from 1.00, a coefficient of 0 meaning
+that the best individual in ability A is no more likely to be in first
+place in ability B than to be in any other rank.
+
+The meanings of coefficients of correlation of .90, .70, .50, and 0 are
+shown by Tables 15, 16, 17 and 18.[26]
+
+  [26] Unless he has a thorough understanding of the underlying
+       theory, the student should be very cautious in making
+       inferences from coefficients of correlation.
+
+TABLE 15
+
+ DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .90
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |     |     |     |     |  .1 |  .4 | 1.8 | 6.6 |22.4 |68.7
+  2d tenth  |     |     |  .1 |  .4 | 1.4 | 4.7 |11.5 |23.5 |36.0 |22.4
+  3d tenth  |     |  .1 |  .5 | 2.1 | 5.8 |12.8 |21.1 |27.4 |23.5 | 6.6
+  4th tenth |     |  .4 | 2.1 | 6.4 |12.8 |20.1 |23.8 |21.2 |11.5 | 1.8
+  5th tenth |  .1 | 1.4 | 5.8 |12.8 |19.3 |22.6 |20.1 |12.8 | 4.7 |  .4
+  6th tenth |  .4 | 4.7 |12.8 |20.1 |22.6 |19.3 |12.8 | 5.8 | 1.4 |  .1
+  7th tenth | 1.8 |11.5 |21.2 |23.8 |20.1 |12.8 | 6.4 | 2.1 |  .4 |
+  8th tenth | 6.6 |23.5 |27.4 |21.1 |12.8 | 5.8 | 2.1 |  .5 |  .1 |
+  9th tenth |22.4 |36.0 |23.5 |11.5 | 4.7 | 1.4 |  .4 |  .1 |     |
+  10th tenth|68.7 |22.4 | 6.6 | 1.8 |  .4 |  .1 |     |     |     |
+  ======================================================================
+
+TABLE 16
+
+DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .70
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |     |  .2 |  .7 | 1.5 | 2.8 | 4.8 | 8.0 |13.0 |22.3 |46.7
+  2d tenth  |  .2 | 1.2 | 2.6 | 4.5 | 7.0 | 9.8 |13.4 |17.3 |21.7 |22.3
+  3d tenth  |  .7 | 2.6 | 5.0 | 7.3 |10.0 |12.5 |14.9 |16.7 |17.3 |13.0
+  4th tenth | 1.5 | 4.5 | 7.3 | 9.8 |12.0 |13.7 |14.8 |14.9 |13.4 | 8.0
+  5th tenth | 2.8 | 7.0 |10.0 |12.0 |13.4 |14.0 |13.7 |12.5 | 9.8 | 4.8
+  6th tenth | 4.8 | 9.8 |12.5 |13.7 |14.0 |13.4 |12.0 |10.0 | 7.0 | 2.8
+  7th tenth | 8.0 |13.4 |14.9 |14.8 |13.7 |12.0 | 9.8 | 7.3 | 4.5 | 1.5
+  8th tenth |13.0 |17.3 |16.7 |14.9 |12.5 |10.0 | 7.3 | 5.0 | 2.6 |  .7
+  9th tenth |22.3 |21.7 |17.3 |13.4 | 9.8 | 7.0 | 4.5 | 2.6 | 1.2 |  .2
+  10th tenth|46.7 |22.3 |13.0 | 8.0 | 4.8 | 2.8 | 1.5 |  .7 |  .2 |
+  ======================================================================
+
+TABLE 17
+
+DISTRIBUTION OF ARRAYS OF SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .50
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |  .8 | 2.0 | 3.2 | 4.6 | 6.2 | 8.1 |10.5 |13.9 |18.0 |31.8
+  2d tenth  | 2.0 | 4.1 | 5.7 | 7.3 | 8.8 |10.5 |12.2 |14.1 |16.4 |18.9
+  3d tenth  | 3.2 | 5.7 | 7.4 | 8.9 |10.0 |11.2 |12.3 |13.3 |14.1 |13.9
+  4th tenth | 4.6 | 7.3 | 8.8 | 9.9 |10.8 |11.6 |12.0 |12.3 |12.2 |10.5
+  5th tenth | 6.2 | 8.8 |10.0 |10.8 |11.3 |11.5 |11.6 |11.2 |10.5 | 8.1
+  6th tenth | 8.1 |10.5 |11.2 |11.6 |11.5 |11.3 |10.8 |10.0 | 8.8 | 6.2
+  7th tenth |10.5 |12.2 |12.3 |12.0 |11.6 |10.8 | 9.9 | 8.8 | 7.5 | 4.6
+  8th tenth |13.9 |14.1 |13.3 |12.3 |11.2 |10.0 | 8.8 | 7.4 | 5.7 | 3.2
+  9th tenth |18.9 |16.4 |14.1 |12.2 |10.5 | 8.8 | 7.3 | 5.7 | 4.1 | 2.0
+  10th tenth|31.8 |18.9 |13.9 |10.5 | 8.1 | 6.2 | 4.6 | 3.2 | 2.0 |  .8
+  ======================================================================
+
+TABLE 18
+
+DISTRIBUTION OF ARRAYS, IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .0
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  2d tenth  |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  3d tenth  |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  4th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  5th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  6th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  7th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  8th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  9th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  10th tenth|10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  ======================================================================
+
+The significance of any coefficient of correlation depends upon the
+group of individuals for which it is determined. A correlation of .40
+between computation and problem-solving in eighth-grade pupils of 14
+years would mean a much closer real relation than a correlation of .40
+in all 14-year-olds, and a very, very much closer relation than a
+correlation of .40 for all children 8 to 15.
+
+Unless the individuals concerned are very elaborately tested on several
+days, the correlations obtained are "attenuated" toward 0 by the
+"accidental" errors in the original measurements. This effect was not
+known until 1904; consequently the correlations in the earlier studies
+of arithmetic are all too low.
+
+In general, the correlation between ability in any one important feature
+of computation and ability in any other important feature of computation
+is high. If we make enough tests to measure each individual exactly
+in:--
+
+  (_A_) Subtraction with integers and decimals,
+  (_B_) Multiplication with integers and decimals,
+  (_C_) Division with integers and decimals,
+  (_D_) Multiplication and division with common fractions, and
+  (_E_) Computing with percents,
+
+we shall probably find the intercorrelations for a thousand 14-year-olds
+to be near .90. Addition of integers (_F_) will, however, correlate less
+closely with any of the above, being apparently dependent on simpler and
+more isolated abilities.
+
+The correlation between problem-solving (_G_) and computation will be
+very much less, probably not over .60.
+
+It should be noted that even when the correlation is as high as .90,
+there will be some individuals very high in one ability and very low in
+the other. Such disparities are to some extent, as Courtis ['13, pp.
+67-75] and Cobb ['17] have argued, due to inborn characteristics of the
+individual in question which predispose him to very special sorts of
+strength and weakness. They are often due, however, to defects in his
+learning whereby he has acquired more ability than he needs in one line
+of work or has failed to acquire some needed ability which was well
+within his capacity.
+
+In general, all correlations between an individual's divergence from the
+common type or average of his age for one arithmetical function, and his
+divergences from the average for any other arithmetical function, are
+positive. The correlation due to original capacity more than
+counterbalances the effects that robbing Peter to pay Paul may have.
+
+Speed and accuracy are thus positively correlated. The individuals who
+do the most work in ten minutes will be above the average in a test of
+accuracy. The common notion that speed is opposed to accuracy is correct
+when it means that the same person will tend to make more errors if he
+works at too rapid a rate; but it is entirely wrong when it means that
+the kind of person who works more rapidly than the average person is
+likely to be less accurate than the average person.
+
+Interest in arithmetic and ability at arithmetic are probably correlated
+positively in the sense that the pupil who has more interest than other
+pupils of his age tends in the long run to have more ability than they.
+They are certainly correlated in the sense that the pupil who 'likes'
+arithmetic better than geography or history tends to have relatively
+more ability in arithmetic, or, in other words, that the pupil who is
+more gifted at arithmetic than at drawing or English tends also to like
+it better than he likes these. These correlations are high.
+
+It is correct then to think of mathematical ability as, in a sense, a
+unitary ability of which any one individual may have much or little,
+most individuals possessing a moderate amount of it. This is
+consistent, however, with the occasional appearance of individuals
+possessed of very great talents for this or that particular feature of
+mathematical ability and equally notable deficiencies in other features.
+
+Finally it may be noted that ability in arithmetic, though occasionally
+found in men otherwise very stupid, is usually associated with superior
+intelligence in dealing with ideas and symbols of all sorts, and is one
+of the best early indications thereof.
+
+
+
+
+BIBLIOGRAPHY OF REFERENCES MADE IN THE TEXT
+
+
+  Ames, A. F., and McLellan, J. F.; '00
+    Public School Arithmetic.
+
+  Ballou, F. W.; '16
+    Determining the Achievements of Pupils in the Addition of Fractions.
+    School Document No. 3, 1916, Boston Public Schools.
+
+  Brandell, G.; '13
+    Skolbarns intressen.  Translated ['15] by W. Stern as, Das Interesse
+    der Schulkinder an den Unterrichtsf�chern.
+
+  Brandford, B.; '08
+    A Study of Mathematical Education.
+
+  Brown, J. C.; '11, '12
+    An Investigation on the Value of Drill Work in the Fundamental
+    Operations in Arithmetic. Journal of Educational Psychology, vol. 2,
+    pp. 81-88, vol. 3, pp. 485-492 and 561-570.
+
+  Brown, J. C. and Coffman, L. D.; '14
+    How to Teach Arithmetic.
+
+  Burgerstein, L.; '91
+    Die Arbeitscurve einer Schulstunde. Zeitschrift f�r
+    Schulgesundheitspflege, vol. 4, pp. 543-562 and 607-627.
+
+  Burnett, C. J.; '06
+    The Estimation of Number. Harvard Psychological Studies, vol. 2,
+    pp. 349-404.
+
+  Burt, C.; '17
+    The Distribution and Relations of Educational Abilities. Report of
+    The London County Council, No. 1868.
+
+
+  Chapman, J. C.; '14
+    Individual Differences in Ability and Improvement and Their
+    Correlations. Teachers College Contributions to Education, No. 63.
+
+  Chapman, J. C.; '17
+    The Scientific Measurement of Classroom Products. (With G. P. Rush.)
+
+  Cobb, M. V.; '17
+    A Preliminary Study of the Inheritance of Arithmetical Abilities.
+    Jour. of Educational Psychology, vol. 8, pp. 1-20. Jan., 1917.
+
+  Coffman, L. D., and Brown, J. C.; '14
+    How to Teach Arithmetic.
+
+  Coffman, L. D., and Jessup, W. A.; '16
+    The Supervision of Arithmetic.
+
+  Courtis, S. A.; '09, '10, '11
+    Measurement of Growth and Efficiency in Arithmetic. Elementary
+    School Teacher, vol. 10, pp. 58-74 and 177-199, vol. 11,
+    pp. 171-185, 360-370, and 528-539.
+
+  Courtis, S. A.; '11, '12
+    Report on Educational Aspects of the Public School System of the
+    City of New York. Part II, Subdivision 1, Section D. Report on the
+    Courtis Tests in Arithmetic.
+
+  Courtis, S. A.; '13
+    Courtis Standard Tests. Second Annual Accounting.
+
+  Courtis, S. A.; '14
+    Manual of Instructions for Giving and Scoring the Courtis Standard
+    Tests in the Three R's. Department of Comparative Research, 82
+    Elliot St., Detroit, Mich., 1914.
+
+  Decroly, M., and Degand, J.; '12
+    L'evolution des notions de quantit�s continues et discontinues chez
+    l'enfant. Archives de psychologie, vol. 12, pp. 81-121.
+
+  Degand, J.  _See_ Decroly.
+
+  De Voss, J. C. _See_ Monroe, De Voss, and Kelly.
+
+  Dewey, J.; '10
+    How We Think.
+
+  Dewey, J., and McLellan, J. A.; '95
+    Psychology of Number and Its Applications to Methods of Teaching
+    Arithmetic.
+
+  Donovan, M. E., and Thorndike, E. L.; '13
+    Improvement in a Practice Experiment under School Conditions.
+    American Journal of Psychology, vol. 24, pp. 426-428.
+
+  Elliott, C. H.; '14
+    Variation in the Achievements of Pupils. Teachers College, Columbia
+    University, Contributions to Education, No. 72.
+
+  Flynn, F. J.; '12
+    Mathematical Games--Adaptations from Games Old and New. Teachers
+    College Record, vol. 13, pp. 399-412.
+
+  Freeman, F. N.; '10
+    Untersuchungen �ber den Aufmerksamkeitsumfang und die
+    Zahlauffassung. P�dagogische-Psychologische Arbeiten, I, 88-168.
+
+  Friedrich, J.; '97
+    Untersuchungen �ber die Einfl�sse der Arbeitsdauer und die
+    Arbeitspausen auf die geistige Leistungsf�higkeit der Schulkinder.
+    Zeitschrift f�r Psychologie, vol. 13, pp. 1-53.
+
+  Gilbert, J. A.; '94
+    Researches on the Mental and Physical Development of School
+    Children. Studies from the Yale Psychological Laboratory, vol. 2,
+    pp. 40-100.
+
+  Greenleaf, B.; '73
+    Practical Arithmetic.
+
+  Hahn, H. H., and Thorndike, E. L.; '14
+    Some Results of Practice in Addition under School Conditions.
+    Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.
+
+  Hall, G. S.; '83
+    The Contents of Children's Minds on Entering School. Princeton
+    Review, vol. II, pp. 249-272, May, 1883. Reprinted in Aspects of
+    Child Life and Education, 1907.
+
+  Hartmann, B.; '90
+    Die Analyze des Kindlichen Gedanken-Kreises als die Naturgem�ssedes
+    Ersten Schulunterrichts, 1890.
+
+  Heck, W. H.; '13
+    A Study of Mental Fatigue.
+
+  Heck, W. H.; '13
+    A Second Study in Mental Fatigue in the Daily School Program.
+    Psychological Clinic, vol. 7, pp. 29-34.
+
+  Hoffmann, P.; '11
+    Das Interesse der Sch�ler an den Unterrichtsf�chern. Zeitschrift
+    f�r p�dagogische Psychologie, XII, 458-470.
+
+  Hoke, K. J., and Wilson, G. M.; '20
+    How to Measure.
+
+  Holmes, M. E.; '95
+    The Fatigue of a School Hour. Pedagogical Seminary, vol. 3,
+    pp. 213-234.
+
+  Howell, H. B.; '14
+    A Foundation Study in the Pedagogy of Arithmetic.
+
+  Hunt, C. W.; '12
+    Play and Recreation in Arithmetic. Teachers College Record, vol. 13,
+    pp. 388-398.
+
+  Jessup, W. A., and Coffman, L. D.; '16
+    The Supervision of Arithmetic.
+
+  Kelly, F. J. _See_ Monroe, De Voss and Kelly.
+
+  King, A. C.; '07
+    The Daily Program in Elementary Schools. MSS.
+
+  Kirby, T. J.; '13
+    Practice in the Case of School Children. Teachers College
+    Contributions to Education, No. 58.
+
+  Klapper, P.; '16
+    The Teaching of Arithmetic.
+
+  Kruse, P. J.; '18
+    The Overlapping of Attainments in Certain Sixth, Seventh, and Eighth
+    Grades. Teachers College, Columbia University, Contributions to
+    Education, No. 92.
+
+  Laser, H.; '94
+    Ueber geistige Erm�dung beim Schulunterricht. Zeitschrift f�r
+    Schulgesundheitspflege, vol. 7, pp. 2-22.
+
+  Lay, W. A.; '98
+    F�hrer durch den ersten Rechenunterricht.
+
+  Lay, W. A.; '07
+    F�hrer durch den Rechenunterricht der Unterstufe.
+
+  Lewis, E. O.; '13
+    Popular and Unpopular School-Subjects. The Journal of Experimental
+    Pedagogy, vol. 2, pp. 89-98.
+
+  Lobsien, M.; '03
+    Kinderideale.  Zeitschrift f�r p�dagogische Psychologie, V, 323-344
+    and 457-494.
+
+  Lobsien, M.; '09
+    Beliebtheit und Unbeliebtheit der Unterrichtsf�cher. P�dagogisches
+    Magazin, Heft 361.
+
+  McCall, W. A.; '21
+    How to Measure in Education.
+
+  McDougle, E. C.; '14
+    A Contribution to the Pedagogy of Arithmetic. Pedagogical Seminary,
+    vol. 21, pp. 161-218.
+
+  McKnight, J.A.; '07
+    Differentiation of the Curriculum in the Upper Grammar Grades.
+    MSS. in the library of Teachers College, Columbia University.
+
+  McLellan, J.A., and Dewey, J.; '95
+    Psychology of Number and Its Applications to Methods of Teaching.
+
+  McLellan, J.A., and Ames, A.F.; '00
+    Public School Arithmetic.
+
+  Messenger, J.F.; '03
+    The Perception of Number. Psychological Review, Monograph Supplement
+    No. 22.
+
+  Meumann, E.; '07
+    Vorlesungen zur Einf�hrung in die experimentelle P�dagogik.
+
+  Mitchell, H.E.; '20
+    Unpublished studies of the uses of arithmetic in factories, shops,
+    farms, and the like.
+
+  Monroe, W.S., De Voss, J.C., and Kelly, F.J.; '17
+    Educational Tests and Measurements.
+
+  Nanu, H.A.; '04
+    Zur Psychologie der Zahl Auffassung.
+
+  National Intelligence Tests; '20
+    Scale A, Form 1, Edition 1.
+
+  Phillips, D.E.; '97
+    Number and Its Application Psychologically Considered. Pedagogical
+    Seminary, vol. 5, pp. 221-281.
+
+  Pommer, O.; '14
+    Die Erforschung der Beliebtheit der Unterrichtsf�cher.
+    Ihre psychologischen Grundlagen und ihre p�dagog. Bedeutung. VII.
+    Jahresber. des k.k. Ssaatsgymn. im XVIII Bez. v. Wien.
+
+  Rice, J.M.; '02
+    Test in Arithmetic. Forum, vol. 34, pp. 281-297.
+
+  Rice, J.M.; '03
+    Causes of Success and Failure in Arithmetic.
+    Forum, vol. 34, pp. 437-452.
+
+  Rush, G.P.; '17
+    The Scientific Measurement of Classroom Products. (With J. C.
+    Chapman.)
+
+  Seekel, E.; '14
+    Ueber die Beziehung zwischen der Beliebtheit und der Schwierigkeit
+    der Schulf�cher. Ergebnisse einer Erhebung. Zeitschrift f�r
+    Angewandte Psychologie 9. S. 268-277.
+
+  Selkin, F. B.; '12
+    Number Games Bordering on Arithmetic and Algebra.
+    Teachers College Record, vol. 13, pp. 452-493.
+
+  Smith, D. E.; '01
+    The Teaching of Elementary Mathematics.
+
+  Smith, D. E.; '11
+    The Teaching of Arithmetic.
+
+  Speer, W. W.; '97
+    Arithmetic: Elementary for Pupils.
+
+  Starch, D.; '11
+    Transfer of Training in Arithmetical Operations.
+    Journal of Educational Psychology, vol. 2, pp. 306-310.
+
+  Starch, D.; '16
+    Educational Measurements.
+
+  Stern, W.; '05
+    Ueber Beliebtheit und Unbeliebtheit der Schulf�cher.
+    Zeitschrift f�r p�dagogische Psychologie, VII, 267-296.
+
+  Stern, C., and Stern, W.; '13
+    Beliebtheit und Schwierigkeit der Schulf�cher.
+    (Freie Schulgemeinde Wickersdorf.)
+    Auf Grund der von Herrn Luserke beschafften Materialien bearbeitet.
+    In: "Die Ausstellung zur vergleichenden Jungendkunde
+    der Geschlechter in Breslau."
+    Arbeit 7 des Bundes f�r Schulreform. S. 24-26.
+
+  Stern, W.; '14
+    Zur vergleichenden Jugendkunde der Geschlechter.
+    Vortrag. III. Deutsch. Kongr. f. Jugendkunde
+    usw. Arbeiten 8 des Bundes f�r Schulreform. S. 17-38.
+
+  Stone, C.W.; '08
+    Arithmetical Abilities and Some Factors Determining Them.
+    Teachers College Contributions to Education, No. 19.
+
+  Suzzallo, H.; '11
+    The Teaching of Primary Arithmetic.
+
+  Thorndike, E.L.; '00
+    Mental Fatigue. Psychological Review, vol. 7, pp. 466-482 and
+    547-579.
+
+  Thorndike, E.L.; '08
+    The Effect of Practice in the Case of a Purely Intellectual
+    Function. American Journal of Psychology, vol. 10, pp. 374-384.
+
+  Thorndike, E.L.; '10
+    Practice in the case of Addition.
+    American Journal of Psychology, vol. 21, pp. 483-486.
+
+  Thorndike, E.L., and Donovan, M.E.; '13
+    Improvement in a Practice Experiment under School Conditions.
+    American Journal of Psychology, vol. 24, pp. 426-428.
+
+  Thorndike, E.L., and Donovan, M.E., and Hahn, H.H.; '14
+    Some Results of Practice in Addition under School Conditions.
+    Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.
+
+  Thorndike, E.L.; '15
+    The Relation between Initial Ability and Improvement in a
+    Substitution Test. School and Society, vol. 12, p. 429.
+
+  Thorndike, E.L.; '16
+    Notes on Practice, Improvability and the Curve of Work.
+    American Journal of Psychology, vol 27, pp. 550-565.
+
+  Walsh, J.H.; '06
+    Grammar School Arithmetic.
+
+  Wells, F.L.; '12
+    The Relation of Practice to Individual Differences.
+    American Journal of Psychology, vol. 23, pp. 75-88.
+
+  White, E. E.; '83
+    A New Elementary Arithmetic.
+
+  Whitley, M. T.; '11
+    An Empirical Study of Certain Tests for Individual Differences.
+    Archives of Psychology, No. 19.
+
+  Wiederkehr, G.; '07
+    Statistiche Untersuchungen �ber die
+    Art und den Grad des Interesses bei Kindern der Volksschule.
+    Neue Bahnen, vol. 19, pp. 241-251, 289-299.
+
+  Wilson, G. M.; '19
+    A Survey of the Social and Business Usage of Arithmetic. Teachers
+    College Contributions to Education, No. 100.
+
+  Wilson, G. M., and Hoke, K. J.; '20
+    How to Measure.
+
+  Woody, C.; '10
+    Measurements of Some Achievements in Arithmetic. Teachers College
+    Contributions to Education, No. 80.
+
+
+
+
+INDEX
+
+
+  Abilities, arithmetical, nature of, 1 ff.;
+    measurement of, 27 ff.;
+    constitution of, 51 ff.;
+    organization of, 137 ff.
+
+  Abstract numbers, 85 ff.
+
+  Abstraction, 169 ff.
+
+  Accuracy, in relation to speed, 31;
+    in fundamental operations, 102 ff.
+
+  Addition, measurement of, 27 ff., 34;
+    constitution of, 52 f.;
+    habit in relation to, 71 f.;
+    in the higher decades, 75 f.;
+    accuracy in, 108 f.;
+    amount of practice in, 122 ff.;
+    interest in 196 f.
+
+  Aims of the teaching of arithmetic, 23 f.
+
+  AMES, A. F., 89
+
+  Analysis, learning by, 169 ff.;
+    systematic and opportunistic stimuli to, 178 f.;
+    gradual progress in, 180 ff.
+
+  Area, 257 f., 275
+
+  Arithmetic, sociology of, 24 ff.
+
+  Arithmetical abilities. _See_ Abilities.
+
+  Arithmetical language, 8 f., 19, 89 ff., 94 ff.
+
+  Arithmetical learning, before school, 199 ff.;
+    conditions of, 227 ff.;
+    in relation to time of day, 227 ff.;
+    in relation to time devoted to arithmetic, 228 ff.
+
+  Arithmetical reasoning. _See_ Reasoning.
+
+  Arithmetical terms, 8, 19
+
+  Averages, 40 f.; 135 f.
+
+
+  BALLOU, F. W., 34, 38
+
+  Banking, 256 f.
+
+  BINET, A., 201
+
+  Bonds, selection of, 70 ff.;
+    strength of, 102 ff.;
+    for temporary service, 111 ff.;
+    order of formation of, 141 ff.
+    _See also_ Habits.
+
+  BRANDELL, G., 211
+
+  BRANDFORD, B., 198 f.
+
+  BROWN, J. C., xvi, 103
+
+  BURGERSTEIN, L., 103
+
+  BURNETT, C. J., 202
+
+  BURT, C., 286
+
+
+  Cardinal and ordinal numbers confused, 206
+
+  Catch problems, 21 ff.
+
+  CHAPMAN, J. C., 49
+
+  Class, size of, in relation to arithmetical learning, 228;
+    variation within a, 289 ff.
+
+  COBB, M. V., 299
+
+  COFFMAN, L. D., xvi
+
+  Collection meaning of numbers, 3 ff.
+
+  Computation, measurements of, 33 ff.;
+    explanations of the processes in, 60 ff.;
+    accuracy in, 102 ff.
+    _See also_ Addition, Subtraction, Multiplication, Division,
+          Fractions, Decimal numbers, Percents.
+
+  Concomitants, law of varying, 172 ff.;
+    law of contrasting, 173 ff.
+
+  Concrete numbers, 85 ff.
+
+  Concrete objects, use of, 253 ff.
+
+  Conditions of arithmetical learning, 227 ff.
+
+  Constitution of arithmetical abilities, 51 ff.
+
+  Copying of numbers, eyestrain due to, 212 f.
+
+  Correlations of arithmetical abilities, 295 ff.
+
+  Courses of study, 232 f.
+
+  COURTIS, S. A., 28 ff., 43 ff., 49, 103, 291, 293, 299
+
+  Crutches, 112 f.
+
+  Culture-epoch theory, 198 f.
+
+
+  Dairy records, 273
+
+  Decimal numbers, uses of, 24 f.;
+    measurement of ability with, 36 ff.;
+    learning, 181 ff.;
+    division by, 270 f.
+
+  DE CROLY, M., 205
+
+  Deductive reasoning, 60 ff., 185 ff.
+
+  DEGAND, J., 205
+
+  Denominate numbers, 141 f., 147 f.
+
+  Described problems, 10 ff.
+
+  Development of knowledge of number, 205 ff.
+
+  DE VOSS, J. C., 49
+
+  DEWEY, J., 3, 83, 150, 205, 207, 208, 219, 266, 277
+
+  Differences in arithmetical ability, 285 ff.;
+    within a class, 289 ff.
+
+  Difficulty as a stimulus, 277 ff.
+
+  Drill, 102 ff.
+
+  Discipline, mental, 20
+
+  Distribution of practice, 156 ff.
+
+  Division, measurement of, 35 f., 37;
+    constitution of, 57 ff.;
+    deductive explanations of, 63, 64 f.;
+    inductive explanations of, 63 f., 65 f.;
+    habit in relation to, 72;
+    with remainders, 76;
+    with fractions, 78 ff.;
+    amount of practice in, 122 ff.;
+    distribution of practice in, 167;
+    use of the problem attitude in teaching, 270 f.
+
+  DONOVAN, M. E., 295
+
+
+  Elements, responses to, 169 ff.
+
+  Eleven, multiples of, 85
+
+  ELLIOTT, C. H., 228
+
+  Equation form, importance of, 77 f.
+
+  Explanations of the processes of computation, 60 ff.;
+    memory of, 115 f.;
+    time for giving, 154 ff.
+
+  Eyestrain in arithmetical work, 212 ff.
+
+
+  Facilitation, 143 ff.
+
+  Figures, printing of, 235 ff.;
+    writing of, 214 f., 241
+
+  FLYNN, F. J., 196
+
+  Fractions, uses of, 24 f.;
+    measurement of ability with, 36 ff.;
+    knowledge of the meaning of, 54 ff.
+
+  FREEMAN, F. N., 259, 261
+
+  FRIEDRICH, J., 103
+
+
+  Generalization, 169 ff.
+
+  GILBERT, J. A., 203
+
+  Graded tests, 28 ff., 36 ff.
+
+  Greatest common divisor, 88 f.
+
+
+  Habits, importance of, in arithmetical learning, 70 ff.;
+    now neglected, 75 ff.;
+    harmful or wasteful, 83 ff.; 91 ff.;
+    prop�deutic, 117 ff.;
+    organization of, 137 ff.;
+    arrangement of, 141 ff.
+
+  HAHN, H. H., 295
+
+  HALL, G. S., 200 f.
+
+  HARTMANN, B., 200 f.
+
+  HECK, W. H., 227
+
+  Heredity in arithmetical abilities, 293 ff.
+
+  Highest common factor, 88 f.
+
+  HOKE, K. J., 49
+
+  HOLMES, M. E., 103
+
+  HOWELL, H. B., 259
+
+  HUNT, C. W., 196
+
+  Hygiene of arithmetic, 212 ff., 234 ff.
+
+
+  Individual differences, 285 ff.
+
+  Inductive reasoning, 60 ff., 169 ff.
+
+  Insurance, 256
+
+  Interest as a principle determining the order of topics, 150 ff.
+
+  Interests, instinctive 195 ff.;
+    censuses of, 209 ff.;
+    neglect of childish, 226 ff.;
+    in self-management, 223 f.;
+    intrinsic, 224 ff.
+
+  Interference, 143 ff.
+
+  Inventories of arithmetical knowledge and skill, 199 ff.
+
+
+  JESSUP, W. A., xvi
+
+
+  KELLY, F. J., 49
+
+  KING, A. C., 103, 227
+
+  KIRBY, T. J., 76 f., 104, 295
+
+  KLAPPER, P., xvi
+
+  KRUSE, P. J., 289, 293
+
+
+  Ladder tests, 28 ff., 36 ff.
+
+  Language in arithmetic, 8 f., 19, 89 ff., 94 ff.
+
+  LASER, H., 103
+
+  LAY, W. A., 259, 261
+
+  Learning, nature of arithmetical, 1 ff.
+
+  Least common multiple, 88 f.
+
+  LEWIS, E. O., 210 f.
+
+  LOBSIEN, M., 209 f.
+
+
+  MCCALL, W. A., 49
+
+  MCDOUGLE, E. C., 85 ff.
+
+  MCKNIGHT, J. A., 210
+
+  MCLELLAN, J. A., 3, 83, 89, 205, 207
+
+  Manipulation of numbers, 60 ff.
+
+  Meaning, of numbers, 2 ff., 171;
+    of a fraction, 54 ff.;
+    of decimals, 181 f.
+
+  Measurement of arithmetical abilities, 27 ff.
+
+  Mental arithmetic, 262 ff.
+
+  MESSENGER, J. F., 202
+
+  Metric system, 147
+
+  MEUMANN, E., 261
+
+  MITCHELL, H. E., 24
+
+  MONROE, W. S., 49
+
+  Multiplication, measurement of, 35, 36;
+    constitution of, 51;
+    deductive explanations of, 61;
+    inductive explanations of, 61 f.;
+    with fractions, 78 ff.;
+    by eleven, 85;
+    amount of practice in, 122 ff.;
+    order of learning the elementary facts of, 144 f.;
+    distribution of practice in, 158 ff.;
+    use of the problem attitude in teaching, 267 ff.
+
+
+  NANU, H. A., 202
+
+  National Intelligence Tests, 49 f.
+
+  Negative reaction in intellectual life, 278 f.
+
+  Number pictures, 259 ff.
+
+  Numbers, meaning of, 2;
+    as measures of continuous quantities, 75;
+    abstract and concrete, 85 ff.;
+    denominate, 141 f., 147 f.;
+    use of large, 145 f.;
+    perception of, 205 ff.;
+    early awareness of, 205 ff.;
+    confusion of cardinal and ordinal, 206.
+    _See also_ Decimal numbers _and_ Fractions.
+
+
+  Objective aids, used for verification, 154;
+    in general, 243 ff.
+
+  Oral arithmetic, 262 ff.
+
+  Order of topics, 141 ff.
+
+  Ordinal numbers, confused with cardinal, 206
+
+  Original tendencies and arithmetic, 195 ff.
+
+  Overlearning, 134 ff.
+
+
+  Percents, 80 f.
+
+  Perception of number, 202 ff.
+
+  PHILLIPS, D. E., 3, 4, 205, 207
+
+  Pictures, hygiene of, 246 ff.;
+    number, 259 ff.
+
+  POMMER, O., 212
+
+  Practice, amount of, 122 ff.;
+    distribution of, 156 ff.
+
+  Precision in fundamental operations, 102 ff.
+
+  Problem attitude, 266 ff.
+
+  Problems, 9 ff.;
+    "catch," 21 ff.;
+    measurement of ability with, 42 ff.;
+    whose answer must be known in order to frame them, 93 f.;
+    verbal form of, 111 f.;
+    interest in, 220 ff.;
+    as introductions to arithmetical learning, 266 ff.
+
+  Prop�deutic bonds, 117 ff.
+
+  Purposive thinking, 193 ff.
+
+
+  Quantity, number and, 85 ff.;
+    perception of, 202 ff.
+
+
+  RACE, H., 295
+
+  Rainfall, 272
+
+  Ratio, 225 f.;
+    meaning of numbers, 3 ff.
+
+  Reaction, negative, 278 f.
+
+  Reality, in problems, 9 ff.
+
+  Reasoning, arithmetical, nature of, 19 ff.;
+    measurement of ability in, 42 ff.;
+    derivation of tables by, 58 f.;
+    about the rationale of computations, 60 ff.;
+    habit in relation to, 73 f., 190 ff.;
+    problems which provoke false, 100 f.;
+    the essentials of arithmetical, 185 ff.;
+    selection in, 187 ff.;
+    as the co�peration of organized habits, 190 ff.
+
+  Recapitulation theory, 198 f.
+
+  Recipes, 273 f.
+
+  Rectangle, area of, 257 f.
+
+  RICE, J. M., 228 ff.
+
+  RUSH, G. P., 49
+
+
+  SEEKEL, E., 212
+
+  SELKIN, F. B., 196 f.
+
+  Sequence of topics, 141 ff.
+
+  Series meaning of numbers, 2 ff.
+
+  Size of class in relation to arithmetical learning, 228
+
+  SMITH, D. E., xvi, 224
+
+  Social instincts, use of, 195 f.
+
+  Sociology of arithmetic, 24 ff.
+
+  Speed in relation to accuracy, 31, 108
+
+  SPEER, W. W., 3, 5, 83
+
+  Spiral order, 141, 145
+
+  STARCH, D., 49, 295
+
+  STERN, W., 210, 212
+
+  STONE, C. W., 27 ff., 42 ff., 228 ff.
+
+  Subtraction, measurement of, 34 f.;
+    constitution of, 57 f.;
+    amount of practice in, 122 ff.
+
+  Supervision, 233 f.
+
+  SUZZALLO, H., xvi
+
+
+  Temporary bonds, 111 ff.
+
+  Terms, 113 f.
+
+  Tests of arithmetical abilities, 27 ff.
+
+  THORNDIKE, E. L., 34, 38 ff., 227, 294
+
+  Time, devoted to arithmetic, 228 ff.;
+    of day, in relation to arithmetical learning, 227 f.
+
+  Type, hygiene of, 235 ff.
+
+
+  Underlearning, 134 ff.
+
+  United States money, 148 ff.
+
+  Units of measure, arbitrary, 5, 83 f.
+
+
+  Variation, among individuals, 285 ff.
+
+  Variety, in teaching, 153
+
+  Verification, 81 f.;
+    aided by greater strength of the fundamental bonds, 107 ff.
+
+
+  WALSH, J. H., 11
+
+  WELLS, F. L., 295
+
+  WHITE, E. E., 5
+
+  WHITLEY, M. T., 295
+
+  WIEDERKEHR, G., 212
+
+  WILSON, G. M., 24, 49
+
+  WOODY, C., 29 ff., 52, 287, 293
+
+  Words. _See_ Language _and_ Terms.
+
+  Written arithmetic, 262 ff.
+
+
+  Zero in multiplication, 179 f.
+
+
+
+
+TRANSCRIBER'S NOTES:
+
+
+1. Passages in italics are surrounded by _underscores_.
+
+2. Passages in bold are indicated by #bold#.
+
+3. Mixed fractions are represented using forward slash and hyphen in
+this text version. For example, 5-1/2 represents five and a half.
+
+4. Images and footnotes have been moved from the middle of a paragraph
+to the closest paragraph break.
+
+5. Obvious errors in spelling and punctuation have been silently closed.
+
+
+
+
+
+
+End of the Project Gutenberg EBook of The Psychology of Arithmetic, by 
+Edward L. Thorndike
+
+*** END OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC ***
+
+***** This file should be named 39300-8.txt or 39300-8.zip *****
+This and all associated files of various formats will be found in:
+        http://www.gutenberg.org/3/9/3/0/39300/
+
+Produced by Jonathan Ingram and the Online Distributed
+Proofreading Team at http://www.pgdp.net
+
+
+Updated editions will replace the previous one--the old editions
+will be renamed.
+
+Creating the works from public domain print editions means that no
+one owns a United States copyright in these works, so the Foundation
+(and you!) can copy and distribute it in the United States without
+permission and without paying copyright royalties.  Special rules,
+set forth in the General Terms of Use part of this license, apply to
+copying and distributing Project Gutenberg-tm electronic works to
+protect the PROJECT GUTENBERG-tm concept and trademark.  Project
+Gutenberg is a registered trademark, and may not be used if you
+charge for the eBooks, unless you receive specific permission.  If you
+do not charge anything for copies of this eBook, complying with the
+rules is very easy.  You may use this eBook for nearly any purpose
+such as creation of derivative works, reports, performances and
+research.  They may be modified and printed and given away--you may do
+practically ANYTHING with public domain eBooks.  Redistribution is
+subject to the trademark license, especially commercial
+redistribution.
+
+
+
+*** START: FULL LICENSE ***
+
+THE FULL PROJECT GUTENBERG LICENSE
+PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
+
+To protect the Project Gutenberg-tm mission of promoting the free
+distribution of electronic works, by using or distributing this work
+(or any other work associated in any way with the phrase "Project
+Gutenberg"), you agree to comply with all the terms of the Full Project
+Gutenberg-tm License (available with this file or online at
+http://gutenberg.org/license).
+
+
+Section 1.  General Terms of Use and Redistributing Project Gutenberg-tm
+electronic works
+
+1.A.  By reading or using any part of this Project Gutenberg-tm
+electronic work, you indicate that you have read, understand, agree to
+and accept all the terms of this license and intellectual property
+(trademark/copyright) agreement.  If you do not agree to abide by all
+the terms of this agreement, you must cease using and return or destroy
+all copies of Project Gutenberg-tm electronic works in your possession.
+If you paid a fee for obtaining a copy of or access to a Project
+Gutenberg-tm electronic work and you do not agree to be bound by the
+terms of this agreement, you may obtain a refund from the person or
+entity to whom you paid the fee as set forth in paragraph 1.E.8.
+
+1.B.  "Project Gutenberg" is a registered trademark.  It may only be
+used on or associated in any way with an electronic work by people who
+agree to be bound by the terms of this agreement.  There are a few
+things that you can do with most Project Gutenberg-tm electronic works
+even without complying with the full terms of this agreement.  See
+paragraph 1.C below.  There are a lot of things you can do with Project
+Gutenberg-tm electronic works if you follow the terms of this agreement
+and help preserve free future access to Project Gutenberg-tm electronic
+works.  See paragraph 1.E below.
+
+1.C.  The Project Gutenberg Literary Archive Foundation ("the Foundation"
+or PGLAF), owns a compilation copyright in the collection of Project
+Gutenberg-tm electronic works.  Nearly all the individual works in the
+collection are in the public domain in the United States.  If an
+individual work is in the public domain in the United States and you are
+located in the United States, we do not claim a right to prevent you from
+copying, distributing, performing, displaying or creating derivative
+works based on the work as long as all references to Project Gutenberg
+are removed.  Of course, we hope that you will support the Project
+Gutenberg-tm mission of promoting free access to electronic works by
+freely sharing Project Gutenberg-tm works in compliance with the terms of
+this agreement for keeping the Project Gutenberg-tm name associated with
+the work.  You can easily comply with the terms of this agreement by
+keeping this work in the same format with its attached full Project
+Gutenberg-tm License when you share it without charge with others.
+
+1.D.  The copyright laws of the place where you are located also govern
+what you can do with this work.  Copyright laws in most countries are in
+a constant state of change.  If you are outside the United States, check
+the laws of your country in addition to the terms of this agreement
+before downloading, copying, displaying, performing, distributing or
+creating derivative works based on this work or any other Project
+Gutenberg-tm work.  The Foundation makes no representations concerning
+the copyright status of any work in any country outside the United
+States.
+
+1.E.  Unless you have removed all references to Project Gutenberg:
+
+1.E.1.  The following sentence, with active links to, or other immediate
+access to, the full Project Gutenberg-tm License must appear prominently
+whenever any copy of a Project Gutenberg-tm work (any work on which the
+phrase "Project Gutenberg" appears, or with which the phrase "Project
+Gutenberg" is associated) is accessed, displayed, performed, viewed,
+copied or distributed:
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+1.E.2.  If an individual Project Gutenberg-tm electronic work is derived
+from the public domain (does not contain a notice indicating that it is
+posted with permission of the copyright holder), the work can be copied
+and distributed to anyone in the United States without paying any fees
+or charges.  If you are redistributing or providing access to a work
+with the phrase "Project Gutenberg" associated with or appearing on the
+work, you must comply either with the requirements of paragraphs 1.E.1
+through 1.E.7 or obtain permission for the use of the work and the
+Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or
+1.E.9.
+
+1.E.3.  If an individual Project Gutenberg-tm electronic work is posted
+with the permission of the copyright holder, your use and distribution
+must comply with both paragraphs 1.E.1 through 1.E.7 and any additional
+terms imposed by the copyright holder.  Additional terms will be linked
+to the Project Gutenberg-tm License for all works posted with the
+permission of the copyright holder found at the beginning of this work.
+
+1.E.4.  Do not unlink or detach or remove the full Project Gutenberg-tm
+License terms from this work, or any files containing a part of this
+work or any other work associated with Project Gutenberg-tm.
+
+1.E.5.  Do not copy, display, perform, distribute or redistribute this
+electronic work, or any part of this electronic work, without
+prominently displaying the sentence set forth in paragraph 1.E.1 with
+active links or immediate access to the full terms of the Project
+Gutenberg-tm License.
+
+1.E.6.  You may convert to and distribute this work in any binary,
+compressed, marked up, nonproprietary or proprietary form, including any
+word processing or hypertext form.  However, if you provide access to or
+distribute copies of a Project Gutenberg-tm work in a format other than
+"Plain Vanilla ASCII" or other format used in the official version
+posted on the official Project Gutenberg-tm web site (www.gutenberg.org),
+you must, at no additional cost, fee or expense to the user, provide a
+copy, a means of exporting a copy, or a means of obtaining a copy upon
+request, of the work in its original "Plain Vanilla ASCII" or other
+form.  Any alternate format must include the full Project Gutenberg-tm
+License as specified in paragraph 1.E.1.
+
+1.E.7.  Do not charge a fee for access to, viewing, displaying,
+performing, copying or distributing any Project Gutenberg-tm works
+unless you comply with paragraph 1.E.8 or 1.E.9.
+
+1.E.8.  You may charge a reasonable fee for copies of or providing
+access to or distributing Project Gutenberg-tm electronic works provided
+that
+
+- You pay a royalty fee of 20% of the gross profits you derive from
+     the use of Project Gutenberg-tm works calculated using the method
+     you already use to calculate your applicable taxes.  The fee is
+     owed to the owner of the Project Gutenberg-tm trademark, but he
+     has agreed to donate royalties under this paragraph to the
+     Project Gutenberg Literary Archive Foundation.  Royalty payments
+     must be paid within 60 days following each date on which you
+     prepare (or are legally required to prepare) your periodic tax
+     returns.  Royalty payments should be clearly marked as such and
+     sent to the Project Gutenberg Literary Archive Foundation at the
+     address specified in Section 4, "Information about donations to
+     the Project Gutenberg Literary Archive Foundation."
+
+- You provide a full refund of any money paid by a user who notifies
+     you in writing (or by e-mail) within 30 days of receipt that s/he
+     does not agree to the terms of the full Project Gutenberg-tm
+     License.  You must require such a user to return or
+     destroy all copies of the works possessed in a physical medium
+     and discontinue all use of and all access to other copies of
+     Project Gutenberg-tm works.
+
+- You provide, in accordance with paragraph 1.F.3, a full refund of any
+     money paid for a work or a replacement copy, if a defect in the
+     electronic work is discovered and reported to you within 90 days
+     of receipt of the work.
+
+- You comply with all other terms of this agreement for free
+     distribution of Project Gutenberg-tm works.
+
+1.E.9.  If you wish to charge a fee or distribute a Project Gutenberg-tm
+electronic work or group of works on different terms than are set
+forth in this agreement, you must obtain permission in writing from
+both the Project Gutenberg Literary Archive Foundation and Michael
+Hart, the owner of the Project Gutenberg-tm trademark.  Contact the
+Foundation as set forth in Section 3 below.
+
+1.F.
+
+1.F.1.  Project Gutenberg volunteers and employees expend considerable
+effort to identify, do copyright research on, transcribe and proofread
+public domain works in creating the Project Gutenberg-tm
+collection.  Despite these efforts, Project Gutenberg-tm electronic
+works, and the medium on which they may be stored, may contain
+"Defects," such as, but not limited to, incomplete, inaccurate or
+corrupt data, transcription errors, a copyright or other intellectual
+property infringement, a defective or damaged disk or other medium, a
+computer virus, or computer codes that damage or cannot be read by
+your equipment.
+
+1.F.2.  LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
+of Replacement or Refund" described in paragraph 1.F.3, the Project
+Gutenberg Literary Archive Foundation, the owner of the Project
+Gutenberg-tm trademark, and any other party distributing a Project
+Gutenberg-tm electronic work under this agreement, disclaim all
+liability to you for damages, costs and expenses, including legal
+fees.  YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
+LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
+PROVIDED IN PARAGRAPH 1.F.3.  YOU AGREE THAT THE FOUNDATION, THE
+TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
+LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
+INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
+DAMAGE.
+
+1.F.3.  LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
+defect in this electronic work within 90 days of receiving it, you can
+receive a refund of the money (if any) you paid for it by sending a
+written explanation to the person you received the work from.  If you
+received the work on a physical medium, you must return the medium with
+your written explanation.  The person or entity that provided you with
+the defective work may elect to provide a replacement copy in lieu of a
+refund.  If you received the work electronically, the person or entity
+providing it to you may choose to give you a second opportunity to
+receive the work electronically in lieu of a refund.  If the second copy
+is also defective, you may demand a refund in writing without further
+opportunities to fix the problem.
+
+1.F.4.  Except for the limited right of replacement or refund set forth
+in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER
+WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
+WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.
+
+1.F.5.  Some states do not allow disclaimers of certain implied
+warranties or the exclusion or limitation of certain types of damages.
+If any disclaimer or limitation set forth in this agreement violates the
+law of the state applicable to this agreement, the agreement shall be
+interpreted to make the maximum disclaimer or limitation permitted by
+the applicable state law.  The invalidity or unenforceability of any
+provision of this agreement shall not void the remaining provisions.
+
+1.F.6.  INDEMNITY - You agree to indemnify and hold the Foundation, the
+trademark owner, any agent or employee of the Foundation, anyone
+providing copies of Project Gutenberg-tm electronic works in accordance
+with this agreement, and any volunteers associated with the production,
+promotion and distribution of Project Gutenberg-tm electronic works,
+harmless from all liability, costs and expenses, including legal fees,
+that arise directly or indirectly from any of the following which you do
+or cause to occur: (a) distribution of this or any Project Gutenberg-tm
+work, (b) alteration, modification, or additions or deletions to any
+Project Gutenberg-tm work, and (c) any Defect you cause.
+
+
+Section  2.  Information about the Mission of Project Gutenberg-tm
+
+Project Gutenberg-tm is synonymous with the free distribution of
+electronic works in formats readable by the widest variety of computers
+including obsolete, old, middle-aged and new computers.  It exists
+because of the efforts of hundreds of volunteers and donations from
+people in all walks of life.
+
+Volunteers and financial support to provide volunteers with the
+assistance they need, are critical to reaching Project Gutenberg-tm's
+goals and ensuring that the Project Gutenberg-tm collection will
+remain freely available for generations to come.  In 2001, the Project
+Gutenberg Literary Archive Foundation was created to provide a secure
+and permanent future for Project Gutenberg-tm and future generations.
+To learn more about the Project Gutenberg Literary Archive Foundation
+and how your efforts and donations can help, see Sections 3 and 4
+and the Foundation web page at http://www.pglaf.org.
+
+
+Section 3.  Information about the Project Gutenberg Literary Archive
+Foundation
+
+The Project Gutenberg Literary Archive Foundation is a non profit
+501(c)(3) educational corporation organized under the laws of the
+state of Mississippi and granted tax exempt status by the Internal
+Revenue Service.  The Foundation's EIN or federal tax identification
+number is 64-6221541.  Its 501(c)(3) letter is posted at
+http://pglaf.org/fundraising.  Contributions to the Project Gutenberg
+Literary Archive Foundation are tax deductible to the full extent
+permitted by U.S. federal laws and your state's laws.
+
+The Foundation's principal office is located at 4557 Melan Dr. S.
+Fairbanks, AK, 99712., but its volunteers and employees are scattered
+throughout numerous locations.  Its business office is located at
+809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email
+business@pglaf.org.  Email contact links and up to date contact
+information can be found at the Foundation's web site and official
+page at http://pglaf.org
+
+For additional contact information:
+     Dr. Gregory B. Newby
+     Chief Executive and Director
+     gbnewby@pglaf.org
+
+
+Section 4.  Information about Donations to the Project Gutenberg
+Literary Archive Foundation
+
+Project Gutenberg-tm depends upon and cannot survive without wide
+spread public support and donations to carry out its mission of
+increasing the number of public domain and licensed works that can be
+freely distributed in machine readable form accessible by the widest
+array of equipment including outdated equipment.  Many small donations
+($1 to $5,000) are particularly important to maintaining tax exempt
+status with the IRS.
+
+The Foundation is committed to complying with the laws regulating
+charities and charitable donations in all 50 states of the United
+States.  Compliance requirements are not uniform and it takes a
+considerable effort, much paperwork and many fees to meet and keep up
+with these requirements.  We do not solicit donations in locations
+where we have not received written confirmation of compliance.  To
+SEND DONATIONS or determine the status of compliance for any
+particular state visit http://pglaf.org
+
+While we cannot and do not solicit contributions from states where we
+have not met the solicitation requirements, we know of no prohibition
+against accepting unsolicited donations from donors in such states who
+approach us with offers to donate.
+
+International donations are gratefully accepted, but we cannot make
+any statements concerning tax treatment of donations received from
+outside the United States.  U.S. laws alone swamp our small staff.
+
+Please check the Project Gutenberg Web pages for current donation
+methods and addresses.  Donations are accepted in a number of other
+ways including checks, online payments and credit card donations.
+To donate, please visit: http://pglaf.org/donate
+
+
+Section 5.  General Information About Project Gutenberg-tm electronic
+works.
+
+Professor Michael S. Hart is the originator of the Project Gutenberg-tm
+concept of a library of electronic works that could be freely shared
+with anyone.  For thirty years, he produced and distributed Project
+Gutenberg-tm eBooks with only a loose network of volunteer support.
+
+
+Project Gutenberg-tm eBooks are often created from several printed
+editions, all of which are confirmed as Public Domain in the U.S.
+unless a copyright notice is included.  Thus, we do not necessarily
+keep eBooks in compliance with any particular paper edition.
+
+
+Most people start at our Web site which has the main PG search facility:
+
+     http://www.gutenberg.org
+
+This Web site includes information about Project Gutenberg-tm,
+including how to make donations to the Project Gutenberg Literary
+Archive Foundation, how to help produce our new eBooks, and how to
+subscribe to our email newsletter to hear about new eBooks.
diff --git a/39300-8.zip b/39300-8.zip
new file mode 100644
index 0000000..79f1c61
--- /dev/null
+++ b/39300-8.zip
diff --git a/39300-h.zip b/39300-h.zip
new file mode 100644
index 0000000..2efa0a3
--- /dev/null
+++ b/39300-h.zip
diff --git a/39300-h/39300-h.htm b/39300-h/39300-h.htm
new file mode 100644
index 0000000..8fe1cbf
--- /dev/null
+++ b/39300-h/39300-h.htm
@@ -0,0 +1,13736 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+    "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+  <head>
+    <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1" />
+    <title>
+      The Project Gutenberg eBook of The Psychology of Arithmetic, by Edward L. Thorndike.
+    </title>
+    <style type="text/css">
+
+    p {  margin-top: .75em;
+         text-indent: 1em;
+         text-align: justify;
+         margin-bottom: .75em;
+         }
+    h1,h2,h3,h4,h5,h6 {
+         text-align: center; /* all headings centered */
+	 margin-top: 2em;
+	 margin-bottom: 1em;
+         width: 80%;
+         margin-left: auto;
+         margin-right: auto;
+         clear: both;
+         }
+    hr { width: 33%;
+	 margin-top: 2em;
+	 margin-bottom: 2em;
+         margin-left: auto;
+         margin-right: auto;
+         clear: both;
+       }
+
+    table {margin-left: auto; margin-right: auto; border-collapse: collapse;}
+
+    body{margin-left: 10%;
+         margin-right: 10%;
+        }
+
+  /*   sup { vertical-align: 0.8ex; font-size:95%; }  */
+/*     sub { vertical-align: -0.6ex; font-size:95%; } */
+    span.inline_eqn { white-space: nowrap; }
+    .pagenum  { /* uncomment the next line for invisible page numbers */
+            /*  visibility: hidden;  */
+                position: absolute;
+                left: 92%;
+                font-size: smaller;
+                text-align: right;
+              } /* page numbers */
+
+    .dblockquot{font-size: 1.1em; font-weight: bold;}
+    .iblockquot{font-style: italic;}
+    .blockquot{margin-left: 10%; margin-right: 10%;}
+    .pblockquot{margin-top: 2em; margin-bottom: 2em; font-size: 0.9em;}
+    .sblockquot{font-size: 0.9em; margin-left: 2em;}
+    .nblockquot{text-indent: -3em;  margin-left: 3em;}
+    .nnblockquot{text-indent: -1.5em;  margin-left: 2.5em;}
+
+    .spacer {padding: 1em; text-align: left;}
+    td {padding-right: 1em; padding-left: 1em; }
+    th {padding-right: 2em; padding-left: 2em; }
+    .bb       {border-bottom: solid 1px;}
+    .bl       {border-left: solid 1px;}
+    .bt       {border-top: solid 1px;}
+    .br       {border-right: solid 1px;}
+    .bbt      {border-bottom: solid 1px; border-top: solid 1px;}
+    .bbox     {border: solid 1px;}
+    .bbox1     {border: solid 1px; font-size: 85%; padding-right: 1em; padding-left: 1em;}
+
+    .center   {text-align: center;  text-indent: 0em;}
+    .smcap    {font-variant: small-caps;}
+    .u        {text-decoration: underline;}
+    .overline {text-decoration: overline;}
+    .sz30     {font-size: 3em;}
+    .ft05     {font-size: 60%;}
+    .noidt     {text-indent: 0em;}
+    .alignleft {float: left;}
+    .alignright {float: right;}
+
+    .caption  {font-weight: bold; text-align: center;}
+    .tabcap   {font-weight: bold; text-align: center;  margin-top: 1.5em;}
+
+    .figcenter   {margin: auto; text-align: center;}
+
+    .figleft     {float: left; clear: left; margin-left: 0; margin-bottom: 1em; margin-top:
+                 0em; margin-right: 1em; padding: 0; text-align: center;}
+
+    .figright    {float: right; clear: right; margin-left: 1em; margin-bottom: 1em;
+                 margin-top: 1em; margin-right: 0; padding: 0; text-align: center;}
+
+    .footnotes        {border: dashed 1px;}
+    .footnote         {margin-left: 10%; margin-right: 10%; font-size: 0.9em; margin-top: 1.5em;}
+    .footnote .label  {position: absolute; right: 82%; text-align: right;}
+    .fnanchor         {vertical-align: super; font-size: .8em; text-decoration: none;}
+
+    .poem             {margin-left:10%; margin-right:10%; text-align: left;}
+    .poem br          {display: none;}
+    .poem .stanza     {margin: 1em 0em 1em 0em;}
+    .poem span.i0     {display: block; margin-left: 0em; padding-left: 3em; text-indent: -3em;}
+    .poem span.i2     {display: block; margin-left: 2em; padding-left: 3em; text-indent: -3em;}
+    .poem span.i4     {display: block; margin-left: 4em; padding-left: 3em; text-indent: -3em;}
+
+    .notebox    {border: solid 2px; padding: 1em; margin-left: 15%; margin-right: 15%; background: #CCCCB2;}
+    </style>
+  </head>
+<body>
+
+
+<pre>
+
+Project Gutenberg's The Psychology of Arithmetic, by Edward L. Thorndike
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: The Psychology of Arithmetic
+
+Author: Edward L. Thorndike
+
+Release Date: March 29, 2012 [EBook #39300]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC ***
+
+
+
+
+Produced by Jonathan Ingram and the Online Distributed
+Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+</pre>
+
+
+
+
+<div class="figcenter" style="width: 216px;">
+<img src="images/publogo.png" width="216" height="90" alt="Logo" title="Logo" />
+</div>
+
+<p class="center">
+THE MACMILLAN COMPANY<br />
+
+<small>NEW YORK &nbsp; &middot; &nbsp; BOSTON &nbsp; &middot; &nbsp; CHICAGO &nbsp; &middot; &nbsp; DALLAS<br />
+ATLANTA &nbsp; &middot; &nbsp; SAN FRANCISCO</small><br /><br />
+
+MACMILLAN &amp; CO., LIMITED<br />
+
+<small>LONDON &nbsp; &middot; &nbsp; BOMBAY &nbsp; &middot; &nbsp; CALCUTTA<br />
+MELBOURNE</small><br /><br />
+
+THE MACMILLAN COMPANY<br />
+OF CANADA, <span class="smcap">Limited</span><br />
+<small>TORONTO</small></p>
+
+
+<hr style="width: 65%;" />
+<h1>THE PSYCHOLOGY OF<br />
+ARITHMETIC</h1>
+
+
+
+<h3><small>BY<br /><br /></small>
+<big>EDWARD L. THORNDIKE</big></h3>
+
+<p class="center">TEACHERS COLLEGE, COLUMBIA<br />
+UNIVERSITY</p>
+
+<p>&nbsp;</p>
+
+<p class="center"><big>New York<br />
+THE MACMILLAN COMPANY<br />
+1929</big><br />
+
+<i>All rights reserved</i></p>
+
+
+<hr style="width: 65%;" />
+<p class="center"><span class="smcap">Copyright</span>, 1922,<br />
+<span class="smcap">By</span> THE MACMILLAN COMPANY.</p>
+<hr style="width: 10%;" />
+<p class="center">Set up and electrotyped. Published January, 1922.<br />
+Reprinted October, 1924; May, 1926; August, 1927; October, 1929.</p>
+
+
+<p>&nbsp;</p>
+<p class="center"><b>&middot; &nbsp; PRINTED IN THE UNITED STATES OF AMERICA  &nbsp; &middot;</b></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'>[Pg v]</span></p>
+<h2>PREFACE</h2>
+
+
+<p>Within recent years there have been three lines of advance
+in psychology which are of notable significance for teaching.
+The first is the new point of view concerning the
+general process of learning. We now understand that
+learning is essentially the formation of connections or
+bonds between situations and responses, that the satisfyingness
+of the result is the chief force that forms them, and
+that habit rules in the realm of thought as truly and as
+fully as in the realm of action.</p>
+
+<p>The second is the great increase in knowledge of the
+amount, rate, and conditions of improvement in those
+organized groups or hierarchies of habits which we call
+abilities, such as ability to add or ability to read. Practice
+and improvement are no longer vague generalities, but
+concern changes which are definable and measurable by
+standard tests and scales.</p>
+
+<p>The third is the better understanding of the so-called
+"higher processes" of analysis, abstraction, the formation
+of general notions, and reasoning. The older view of a
+mental chemistry whereby sensations were compounded
+into percepts, percepts were duplicated by images, percepts
+and images were amalgamated into abstractions and concepts,
+and these were manipulated by reasoning, has given
+way to the understanding of the laws of response to elements
+or aspects of situations and to many situations or elements
+thereof in combination. James' view of reasoning as
+"selection of essentials" and "thinking things together"
+<span class='pagenum'>[Pg vi]</span>
+in a revised and clarified form has important applications
+in the teaching of all the school subjects.</p>
+
+<p>This book presents the applications of this newer dynamic
+psychology to the teaching of arithmetic. Its contents are
+substantially what have been included in a course of lectures
+on the psychology of the elementary school subjects given
+by the author for some years to students of elementary
+education at Teachers College. Many of these former
+students, now in supervisory charge of elementary schools,
+have urged that these lectures be made available to teachers
+in general. So they are now published in spite of the
+author's desire to clarify and reinforce certain matters by
+further researches.</p>
+
+<p>A word of explanation is necessary concerning the exercises
+and problems cited to illustrate various matters, especially
+erroneous pedagogy. These are all genuine, having
+their source in actual textbooks, courses of study, state
+examinations, and the like. To avoid any possibility of
+invidious comparisons they are not quotations, but equivalent
+problems such as represent accurately the spirit and
+intent of the originals.</p>
+
+<p>I take pleasure in acknowledging the courtesy of Mr. S. A.
+Courtis, Ginn and Company, D. C. Heath and Company,
+The Macmillan Company, The Oxford University Press,
+Rand, McNally and Company, Dr. C. W. Stone, The
+Teachers College Bureau of Publications, and The World
+Book Company, in permitting various quotations.</p>
+
+<p style='text-align:right'>
+<span class="smcap">Edward L. Thorndike.</span><br />
+</p>
+
+<p>
+ &nbsp;<span class="smcap">Teachers College</span><br />
+<span class="smcap">Columbia University</span><br />
+&nbsp; &nbsp; &nbsp; &nbsp; April 1, 1920</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'>[Pg vii]</span></p>
+<h2>CONTENTS</h2>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="Contents">
+<tr><td align='right'><small>CHAPTER</small></td><td>&nbsp;</td><td align='right'><small>PAGE</small></td></tr>
+<tr><td align='right'>&nbsp;</td><td align='left'><a href="#GENERAL_INTRODUCTION"><span class="smcap">Introduction: The Psychology of the Elementary School Subjects</span></a></td><td align='right'><a href="#Page_xi">xi</a></td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_I">I.</a></td><td align='left'><span class="smcap">The Nature of Arithmetical Abilities</span></td><td align='right'><a href="#Page_1">1</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Knowledge of the Meanings of Numbers</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Arithmetical Language</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Problem Solving</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Arithmetical Reasoning</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Summary</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Sociology of Arithmetic</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_II">II.</a></td><td align='left'><span class="smcap">The Measurement of Arithmetical Abilities</span></td><td align='right'><a href="#Page_27">27</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;A Sample Measurement of an Arithmetical Ability</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Ability to Add Integers</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Measurements of Ability in Computation</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Measurements of Ability in Applied Arithmetic: the Solution of Problems</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_III">III.</a></td><td align='left'><span class="smcap">The Constitution of Arithmetical Abilities</span></td><td align='right'><a href="#Page_51">51</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Elementary Functions of Arithmetical Learning</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Knowledge of the Meaning of a Fraction</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Learning the Processes of Computation</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_IV">IV.</a></td><td align='left'><span class="smcap">The Constitution of Arithmetical Abilities</span> (<i>continued</i>)</td><td align='right'><a href="#Page_70">70</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Selection of the Bonds to Be Formed</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Importance of Habit Formation</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Desirable Bonds Now Often Neglected</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Wasteful and Harmful Bonds</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Guiding Principles</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_V">V.</a></td><td align='left'><span class="smcap">The Psychology of Drill in Arithmetic: The Strength of Bonds</span><span class='pagenum'>[Pg viii]</span></td><td align='right'><a href="#Page_102">102</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Need of Stronger Elementary Bonds</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Early Mastery</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Strength of Bonds for Temporary Service</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Strength of Bonds with Technical Facts and Terms</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Strength of Bonds Concerning the Reasons for Arithmetical Processes</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Prop�deutic Bonds</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_VI">VI.</a></td><td align='left'><span class="smcap">The Psychology of Drill in Arithmetic: The Amount of Practice and the Organization of Abilities</span></td><td align='right'><a href="#Page_122">122</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Amount of Practice</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Under-learning and Over-learning</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Organization of Abilities</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_VII">VII.</a></td><td align='left'><span class="smcap">The Sequence of Topics: The Order of Formation of Bonds</span></td><td align='right'><a href="#Page_141">141</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Conventional <i>versus</i> Effective Orders</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Decreasing Interference and Increasing Facilitation</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Interest</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;General Principles</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_VIII">VIII.</a></td><td align='left'><span class="smcap">The Distribution of Practice</span></td><td align='right'><a href="#Page_156">156</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Problem</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Sample Distributions</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Possible Improvements</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_IX">IX.</a></td><td align='left'><span class="smcap">The Psychology of Thinking: Abstract Ideas and General Notions in Arithmetic</span></td><td align='right'><a href="#Page_169">169</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Responses to Elements and Classes</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Facilitating the Analysis of Elements</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Systematic and Opportunistic Stimuli to Analysis</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Adaptations to Elementary-school Pupils</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_X">X.</a></td><td align='left'><span class="smcap">The Psychology of Thinking: Reasoning in Arithmetic</span></td><td align='right'><a href="#Page_185">185</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Essentials of Arithmetical Reasoning</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Reasoning as the Co�peration of Organized Habits</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_XI">XI.</a></td><td align='left'><span class="smcap">Original Tendencies and Acquisitions before School</span><span class='pagenum'>[Pg ix]</span></td><td align='right'><a href="#Page_195">195</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Utilization of Instinctive Interests</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Order of Development of Original Tendencies</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Inventories of Arithmetical Knowledge and Skill</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Perception of Number and Quantity</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Early Awareness of Number</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_XII">XII.</a></td><td align='left'><span class="smcap">Interest in Arithmetic</span></td><td align='right'><a href="#Page_209">209</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Censuses of Pupils' Interests</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Relieving Eye Strain</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Significance for Related Activities</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Intrinsic Interest in Arithmetical Learning</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_XIII">XIII.</a></td><td align='left'><span class="smcap">The Conditions of Learning</span></td><td align='right'><a href="#Page_227">227</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;External Conditions</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Hygiene of the Eyes in Arithmetic</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Use of Concrete Objects in Arithmetic</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Oral, Mental, and Written Arithmetic</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_XIV">XIV.</a></td><td align='left'><span class="smcap">The Conditions of Learning: the Problem Attitude</span></td><td align='right'><a href="#Page_266">266</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Illustrative Cases</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;General Principles</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Difficulty and Success as Stimuli</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;False Inferences</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'><a href="#CHAPTER_XV">XV.</a></td><td align='left'><span class="smcap">Individual Differences</span></td><td align='right'><a href="#Page_285">285</a></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Nature and Amount</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;Differences within One Class</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Causes of Individual Differences</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left' colspan='2'>&nbsp; &nbsp;The Interrelations of Individual Differences</td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'>&nbsp;</td><td align='left'><a href="#BIBLIOGRAPHY"><span class="smcap">Bibliography of References</span></a></td><td align='right'><a href="#Page_302">302</a></td></tr>
+<tr><td align='left' colspan='3'>&nbsp;</td></tr>
+<tr><td align='right'>&nbsp;</td><td align='left'><a href="#INDEX"><span class="smcap">Index</span></a></td><td align='right'><a href="#Page_311">311</a></td></tr>
+</table></div>
+
+
+
+<p><span class='pagenum'><a name="Page_x" id="Page_x">[Pg x]</a></span></p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_xi" id="Page_xi">[Pg xi]</a></span></p>
+<h2><a name="GENERAL_INTRODUCTION" id="GENERAL_INTRODUCTION"></a>GENERAL INTRODUCTION</h2>
+
+<h2>THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL
+SUBJECTS</h2>
+
+
+<p>The psychology of the elementary school subjects is
+concerned with the connections whereby a child is able to
+respond to the sight of printed words by thoughts of their
+meanings, to the thought of "six and eight" by thinking
+"fourteen," to certain sorts of stories, poems, songs, and
+pictures by appreciation thereof, to certain situations by
+acts of skill, to certain others by acts of courtesy and justice,
+and so on and on through the series of situations and responses
+which are provided by the systematic training of
+the school subjects and the less systematic training of
+school life during their study. The aims of elementary
+education, when fully defined, will be found to be the production
+of changes in human nature represented by an almost
+countless list of connections or bonds whereby the pupil
+thinks or feels or acts in certain ways in response to the
+situations the school has organized and is influenced to think
+and feel and act similarly to similar situations when life
+outside of school confronts him with them.</p>
+
+<p>We are not at present able to define the work of the elementary
+school in detail as the formation of such and such
+bonds between certain detached situations and certain
+specified responses. As elsewhere in human learning, we
+are at present forced to think somewhat vaguely in terms
+of mental functions, like "ability to read the vernacular,"
+"ability to spell common words," "ability to add, subtract,
+<span class='pagenum'><a name="Page_xii" id="Page_xii">[Pg xii]</a></span>
+multiply, and divide with integers," "knowledge of
+the history of the United States," "honesty in examinations,"
+and "appreciation of good music," defined by some
+general results obtained rather than by the elementary
+bonds which constitute them.</p>
+
+<p>The psychology of the school subjects begins where our
+common sense knowledge of these functions leaves off and
+tries to define the knowledge, interest, power, skill, or ideal
+in question more adequately, to measure improvement in
+it, to analyze it into its constituent bonds, to decide what
+bonds need to be formed and in what order as means to the
+most economical attainment of the desired improvement,
+to survey the original tendencies and the tendencies already
+acquired before entrance to school which help or hinder
+progress in the elementary school subjects, to examine
+the motives that are or may be used to make the desired
+connections satisfying, to examine any other special conditions
+of improvement, and to note any facts concerning
+individual differences that are of special importance to the
+conduct of elementary school work.</p>
+
+<p>Put in terms of problems, the task of the psychology of
+the elementary school subjects is, in each case:&mdash;</p>
+
+<p>(1) <i>What is the function?</i> For example, just what is
+"ability to read"? Just what does "the understanding of
+decimal notation" mean? Just what are "the moral effects
+to be sought from the teaching of literature"?</p>
+
+<p>(2) <i>How are degrees of ability or attainment, and degrees of
+progress or improvement in the function or a part of the function
+measured?</i> For example, how can we determine how well
+a pupil should write, or how hard words we expect him
+to spell, or what good taste we expect him to show?
+How can we define to ourselves what knowledge of the
+meaning of a fraction we shall try to secure in grade
+4?<span class='pagenum'><a name="Page_xiii" id="Page_xiii">[Pg xiii]</a></span></p>
+
+<p>(3) <i>What can be done toward reducing the function to terms
+of particular situation-response connections, whose formation
+can be more surely and easily controlled?</i> For example, how
+far does ability to spell involve the formation one by one
+of bonds between the thought of almost every word in the
+language and the thought of that word's letters in their
+correct order; and how far does, say, the bond leading from
+the situation of the sound of <i>ceive</i> in <i>receive</i> and <i>deceive</i> to
+their correct spelling insure the correct spelling of that part
+of <i>perceive</i>? Does "ability to add" involve special bonds
+leading from "27 and 4" to "31," from "27 and 5" to "32,"
+and "27 and 6" to "33"; or will the bonds leading from "7
+and 4" to "11," "7 and 5" to "12" and "7 and 6" to
+"13" (each plus a simple inference) serve as well? What
+are the situations and responses that represent in actual
+behavior the quality that we call school patriotism?</p>
+
+<p>(4) <i>In almost every case a certain desired change of knowledge
+or skill or power can be attained by any one of several
+sets of bonds. Which of them is the best? What are the
+advantages of each?</i> For example, learning to add may include
+the bonds "0 and 0 are 0," "0 and 1 are 1," "0 and
+2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these
+may be all left unformed, the pupil being taught the habits
+of entering 0 as the sum of a column that is composed of
+zeros and otherwise neglecting 0 in addition. Are the rules
+of usage worth teaching as a means toward correct speech,
+or is the time better spent in detailed practice in correct
+speech itself?</p>
+
+<p>(5) <i>A bond to be formed may be formed in any one of many
+degrees of strength. Which of these is, at any given stage of
+learning the subject, the most desirable, all things considered?</i>
+For example, shall the dates of all the early settlements
+of North America be learned so that the exact year will be
+<span class='pagenum'><a name="Page_xiv" id="Page_xiv">[Pg xiv]</a></span>
+remembered for ten years, or so that the exact date will be
+remembered for ten minutes and the date with an error plus
+or minus of ten years will be remembered for a year or two?
+Shall the tables of inches, feet, and yards, and pints, quarts,
+and gallons be learned at their first appearance so as to be
+remembered for a year, or shall they be learned only well
+enough to be usable in the work of that week, which in
+turn fixes them to last for a month or so? Should a pupil
+in the first year of study of French have such perfect
+connections between the sounds of French words and their
+meanings that he can understand simple sentences containing
+them spoken at an ordinary rate of speaking? Or is
+slow speech permissible, and even imperative, on the part of
+the teacher, with gradual increase of rate?</p>
+
+<p>(6) <i>In almost every case, any set of bonds may produce the
+desired change when presented in any one of several orders.
+Which is the best order? What are the advantages of each?</i>
+Certain systems for teaching handwriting perfect the elementary
+movements one at a time and then teach their
+combination in words and sentences. Others begin and
+continue with the complex movement-series that actual
+words require. What do the latter lose and gain? The
+bonds constituting knowledge of the metric system are
+now formed late in the pupil's course. Would it be better
+if they were formed early as a means of facilitating knowledge
+of decimal fractions?</p>
+
+<p>(7) <i>What are the original tendencies and pre-school acquisitions
+upon which the connection-forming of the elementary
+school may be based or which it has to counteract?</i> For
+example, if a pupil knows the meaning of a heard word,
+he may read it understandingly from getting its sound,
+as by phonic reconstruction. What words does the average
+beginner so know? What are the individual differences in
+<span class='pagenum'><a name="Page_xv" id="Page_xv">[Pg xv]</a></span>
+this respect? What do the instincts of gregariousness,
+attention-getting, approval, and helpfulness recommend
+concerning group-work <i>versus</i> individual-work, and concerning
+the size of a group that is most desirable? The original
+tendency of the eyes is certainly not to move along a line
+from left to right of a page, then back in one sweep and along
+the next line. What is their original tendency when confronted
+with the printed page, and what must we do with it
+in teaching reading?</p>
+
+<p>(8) <i>What armament of satisfiers and annoyers, of positive
+and negative interests and motives, stands ready for use in the
+formation of the intrinsically uninteresting connections between
+black marks and meanings, numerical exercises and their
+answers, words and their spelling, and the like?</i> School
+practice has tried, more or less at random, incentives and
+deterrents from quasi-physical pain to the most sentimental
+fondling, from sheer cajolery to philosophical argument,
+from appeals to assumed savage and primitive traits to
+appeals to the interest in automobiles, flying-machines, and
+wireless telegraphy. Can not psychology give some rules
+for guidance, or at least limit experimentation to its more
+hopeful fields?</p>
+
+<p>(9) <i>The general conditions of efficient learning are described
+in manuals of educational psychology. How do these
+apply in the case of each task of the elementary school?</i> For
+example, the arrangement of school drills in addition and
+in short division in the form of practice experiments has
+been found very effective in producing interest in the work
+and in improvement at it. In what other arithmetical
+functions may we expect the same?</p>
+
+<p>(10) <i>Beside the general principles concerning the nature and
+causation of individual differences, there must obviously be,
+in existence or obtainable as a possible result of proper investigation,</i>
+<span class='pagenum'><a name="Page_xvi" id="Page_xvi">[Pg xvi]</a></span>
+<i>a great fund of knowledge of special differences relevant
+to the learning of reading, spelling, geography, arithmetic, and
+the like. What are the facts as far as known? What are the
+means of learning more of them?</i> Courtis finds that a child
+may be specially strong in addition and yet be specially
+weak in subtraction in comparison with others of his age
+and grade. It even seems that such subtle and intricate
+tendencies are inherited. How far is such specialization
+the rule? Is it, for example, the case that a child may have
+a special gift for spelling certain sorts of words, for drawing
+faces rather than flowers, for learning ancient history rather
+than modern?</p>
+
+<p>&nbsp;</p>
+
+<p>Such are our problems: this volume discusses them in the
+case of arithmetic. The student who wishes to relate the
+discussion to the general pedagogy of arithmetic may
+profitably read, in connection with this volume: The
+Teaching of Elementary Mathematics, by D. E. Smith
+['01], The Teaching of Primary Arithmetic, by H. Suzzallo
+['11], How to Teach Arithmetic, by J. C. Brown and L. D.
+Coffman ['14], The Teaching of Arithmetic, by Paul Klapper
+['16], and The New Methods in Arithmetic, by the author
+['21].</p>
+
+
+<hr style="width: 100%;" />
+<h2>THE PSYCHOLOGY OF ARITHMETIC</h2>
+
+
+<hr style="width: 100%;" />
+<h1>THE PSYCHOLOGY OF<br />
+ARITHMETIC</h1>
+
+<hr style="width: 15%;" />
+
+
+<p><span class='pagenum'><a name="Page_1" id="Page_1">[Pg 1]</a></span></p>
+<h2><a name="CHAPTER_I" id="CHAPTER_I"></a>CHAPTER I</h2>
+
+<h2>THE NATURE OF ARITHMETICAL ABILITIES</h2>
+
+
+<p>According to common sense, the task of the elementary
+school is to teach:&mdash;(1) the meanings of numbers, (2) the
+nature of our system of decimal notation, (3) the meanings
+of addition, subtraction, multiplication, and division, and
+(4) the nature and relations of certain common measures;
+to secure (5) the ability to add, subtract, multiply, and
+divide with integers, common and decimal fractions, and
+denominate numbers, (6) the ability to apply the knowledge
+and power represented by (1) to (5) in solving problems,
+and (7) certain specific abilities to solve problems concerning
+percentage, interest, and other common occurrences in
+business life.</p>
+
+<p>This statement of the functions to be developed and improved
+is sound and useful so far as it goes, but it does not
+go far enough to make the task entirely clear. If teachers
+had nothing but the statement above as a guide to what
+changes they were to make in their pupils, they would often
+leave out important features of arithmetical training, and<span class='pagenum'><a name="Page_2" id="Page_2">[Pg 2]</a></span>
+put in forms of training that a wise educational plan would
+not tolerate. It is also the case that different leaders in
+arithmetical teaching, though they might all subscribe to
+the general statement of the previous paragraph, certainly
+do not in practice have identical notions of what arithmetic
+should be for the elementary school pupil.</p>
+
+<p>The ordinary view of the nature of arithmetical learning
+is obscure or inadequate in four respects. It does not define
+what 'knowledge of the meanings of numbers' is; it does
+not take account of the very large amount of teaching of
+<i>language</i> which is done and should be done as a part of the
+teaching of arithmetic; it does not distinguish between the
+ability to meet certain quantitative problems as life offers
+them and the ability to meet the problems provided by
+textbooks and courses of study; it leaves 'the ability to
+apply arithmetical knowledge and power' as a rather
+mystical general faculty to be improved by some educational
+magic. The four necessary amendments may be discussed
+briefly.</p>
+
+
+<h4>KNOWLEDGE OF THE MEANINGS OF NUMBERS</h4>
+
+<p>Knowledge of the meanings of the numbers from one to
+ten may mean knowledge that 'one' means a single thing
+of the sort named, that two means one more than one, that
+three means one more than two, and so on. This we may
+call the <i>series</i> meaning. To know the meaning of 'six' in
+this sense is to know that it is one more than five and one
+less than seven&mdash;that it is between five and seven in the
+number series. Or we may mean by knowledge of the
+meanings of numbers, knowledge that two fits a collection of
+two units, that three fits a collection of three units, and so
+on, each number being a name for a certain sized collection
+of discrete things, such as apples, pennies, boys, balls, fingers,<span class='pagenum'><a name="Page_3" id="Page_3">[Pg 3]</a></span>
+and the other customary objects of enumeration in the
+primary school. This we may call the <i>collection</i> meaning.
+To know the meaning of six in this sense is to be able to name
+correctly any collection of six separate, easily distinguishable
+individual objects. In the third place, knowledge of the
+numbers from one to ten may mean knowledge that two is
+twice whatever is called one, that three is three times whatever
+is one, and so on. This is, of course, the <i>ratio</i> meaning.
+To know the meaning of six in this sense is to know that
+if _________ is one, a line half a foot long is six, that if
+[ __ ] is one, [ ____________ ] is about six, while
+if [ _ ] is one, [ ______ ] is about six, and the like. In
+the fourth place, the meaning of a number may be a smaller
+or larger fraction of its <i>implications</i>&mdash;its numerical relations,
+facts about it. To know six in this sense is to know
+that it is more than five or four, less than seven or eight,
+twice three, three times two, the sum of five and one, or of
+four and two, or of three and three, two less than eight&mdash;that
+with four it makes ten, that it is half of twelve, and the
+like. This we may call the '<i>nucleus of facts</i>' or <i>relational</i>
+meaning of a number.</p>
+
+<p>Ordinary school practice has commonly accepted the
+second meaning as that which it is the task of the school to
+teach beginners, but each of the other meanings has been
+alleged to be the essential one&mdash;the series idea by Phillips
+['97], the ratio idea by McLellan and Dewey ['95] and Speer
+['97], and the relational idea by Grube and his followers.</p>
+
+<p>This diversity of views concerning what the function is
+that is to be improved in the case of learning the meanings
+of the numbers one to ten is not a trifling matter of definition,
+but produces very great differences in school practice.
+Consider, for example, the predominant value assigned to
+counting by Phillips in the passage quoted below, and the
+<span class='pagenum'><a name="Page_4" id="Page_4">[Pg 4]</a></span>
+samples of the sort of work at which children were kept
+employed for months by too ardent followers of Speer and
+Grube.</p>
+
+
+<h4>THE SERIES IDEA OVEREMPHASIZED</h4>
+
+<div class="pblockquot">
+<p>"This is essentially the counting period, and any words that
+can be arranged into a series furnish all that is necessary. Counting
+is fundamental, and counting that is spontaneous, free from
+sensible observation, and from the strain of reason. A study of
+these original methods shows that multiplication was developed
+out of counting, and not from addition as nearly all textbooks
+treat it. Multiplication is counting. When children count by
+4's, etc., they accent the same as counting gymnastics or music.
+When a child now counts on its fingers it simply reproduces a
+stage in the growth of the civilization of all nations.</p>
+
+<p>I would emphasize again that during the counting period there
+is a somewhat spontaneous development of the number series-idea
+which Preyer has discussed in his Arithmogenesis; that an
+immense momentum is given by a systematic series of names;
+and that these names are generally first learned and applied to
+objects later. A lady teacher told me that the Superintendent
+did not wish the teachers to allow the children to count on their
+fingers, but she failed to see why counting with horse-chestnuts
+was any better. Her children could hardly avoid using their
+fingers in counting other objects yet they followed the series to
+100 without hesitation or reference to their fingers. This spontaneous
+counting period, or naming and following the series, should
+precede its application to objects." [D.E. Phillips, '97, p. 238.]</p>
+</div>
+
+<h4>THE RATIO IDEA OVEREMPHASIZED</h4>
+
+<div class="figcenter" style="width: 448px;">
+<img src="images/fig1.jpg" width="448" height="195" alt="Fig. 1." title="Fig. 1." />
+<span class="caption"><span class="smcap">Fig. 1.</span></span>
+</div>
+
+<p><span class='pagenum'><a name="Page_5" id="Page_5">[Pg 5]</a></span></p>
+
+<div class="pblockquot"><p>"Ratios.&mdash;<b>1.</b> Select solids having the relation, or ratio, of
+<i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>o</i>, <i>e</i>.</p>
+
+<p><b>2.</b> Name the solids, <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>o</i>, <i>e</i>.</p>
+
+<p>The means of expressing must be as freely supplied as the means
+of discovery. The pupil is not expected to invent terms.</p>
+
+<p><b>3.</b> Tell all you can about the relation of these units.</p>
+
+<p><b>4.</b> Unite units and tell what the sum equals.</p>
+
+<p><b>5.</b> Make statements like this: <i>o</i> less <i>e</i> equals <i>b</i>.</p>
+
+<p><b>6.</b> <i>c</i> can be separated into how many <i>d</i>'s?  into how
+many <i>b</i>'s?</p>
+
+<p><b>7.</b> <i>c</i> can be separated into how many <i>b</i>'s? What is the name
+of the largest unit that can be found in both <i>c</i> and <i>d</i> an exact
+number of times?</p>
+
+<p><b>8.</b> Each of the other units equals what part of <i>c</i>?</p>
+
+<p><b>9.</b> If <i>b</i> is 1, what is each of the other units?</p>
+
+<p><b>10.</b> If <i>a</i> is 1, what is each of the other units?</p>
+
+<p><b>11.</b> If <i>b</i> is 1, how many 1's are there in each of the other units?</p>
+
+<p><b>12.</b> If <i>d</i> is 1, how many 1's and parts of 1 in each of the other units?</p>
+
+<p><b>13.</b> 2 is the relation of what units?</p>
+
+<p><b>14.</b> 3 is the relation of what units?</p>
+
+<p><b>15.</b> <sup>1</sup>&frasl;<sub>2</sub> is the relation of what units?</p>
+
+<p><b>16.</b> <sup>2</sup>&frasl;<sub>3</sub> is the relation of what units?</p>
+
+<p><b>17.</b> Which units have the relation <sup>3</sup>&frasl;<sub>2</sub>?</p>
+
+<p><b>18.</b> Which unit is 3 times as large as <sup>1</sup>&frasl;<sub>2</sub> of <i>b</i>?</p>
+
+<p><b>19.</b> <i>c</i> equals 6 times <sup>1</sup>&frasl;<sub>3</sub> of what unit?</p>
+
+<p><b>20.</b> <sup>1</sup>&frasl;<sub>3</sub> of what unit equals <sup>1</sup>&frasl;<sub>6</sub> of <i>c</i>?</p>
+
+<p><b>21.</b> What equals <sup>1</sup>&frasl;<sub>2</sub> of <i>c</i>? <i>d</i> equals how many sixths of <i>c</i>?</p>
+
+<p><b>22.</b> <i>o</i> equals 5 times <sup>1</sup>&frasl;<sub>3</sub> of what unit?</p>
+
+<p><b>23.</b> <sup>1</sup>&frasl;<sub>3</sub> of what unit equals <sup>1</sup>&frasl;<sub>5</sub> of <i>o</i>?</p>
+
+<p><b>24.</b> <sup>2</sup>&frasl;<sub>3</sub> of <i>d</i> equals what unit? <i>b</i> equals how many thirds of <i>d</i>?</p>
+
+<p><b>25.</b> 2 is the ratio of <i>d</i> to <sup>1</sup>&frasl;<sub>3</sub> of what unit? 3 is the ratio of <i>d</i> to <sup>1</sup>&frasl;<sub>2</sub>
+of what unit?</p>
+
+<p><b>26.</b> <i>d</i> equals <sup>3</sup>&frasl;<sub>4</sub> of what unit? <sup>3</sup>&frasl;<sub>4</sub> is the ratio of what units?"
+[Speer, '97, p. 9f.]</p></div>
+
+
+<h4>THE RELATIONAL IDEA OVEREMPHASIZED</h4>
+
+<div class="pblockquot">
+<p>An inspection of books of the eighties which followed the
+"Grube method" (for example, the <i>New Elementary Arithmetic</i>
+by E.E. White ['83]) will show undue emphasis on the relational
+ideas. There will be over a hundred and fifty successive tasks all,
+or nearly all, on &#43; 7 and &minus; 7. There will be much written work
+of the sort shown below:</p></div>
+<p><span class='pagenum'><a name="Page_6" id="Page_6">[Pg 6]</a></span></p>
+
+<div class="pblockquot"><p><i>Add:</i></p></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="60%">
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr>
+<tr><td align='center'>4</td><td align='center'>1</td><td align='center'>2</td></tr>
+<tr><td align='center'>&mdash;&mdash;</td><td align='center'>&mdash;&mdash;</td><td align='center'>&mdash;&mdash;</td></tr>
+</table></div>
+
+<div class="pblockquot">
+<p class="noidt">which must have sorely tried the eyes of all concerned. Pupils
+are taught to "give the analysis and synthesis of each of the nine
+digits." Yet the author states that he does not carry the principle
+of the Grube method "to the extreme of useless repetition
+and mechanism."</p></div>
+
+<p>It should be obvious that all four meanings have claims
+upon the attention of the elementary school. Four is the
+thing between three and five in the number series; it is the
+name for a certain sized collection of discrete objects; it is
+also the name for a continuous magnitude equal to four
+units&mdash;for four quarts of milk in a gallon pail as truly as
+for four separate quart-pails of milk; it is also, if we know it
+well, the thing got by adding one to three or subtracting
+six from ten or taking two two's or half of eight. To know
+the meaning of a number means to know somewhat about
+it in all of these respects. The difficulty has been the
+narrow vision of the extremists. A child must not be left
+interminably counting; in fact the one-more-ness of the
+number series can almost be had as a by-product. A child
+must not be restricted to exercises with collections objectified
+<span class='pagenum'><a name="Page_7" id="Page_7">[Pg 7]</a></span>
+as in Fig. 2 or stated in words as so many apples, oranges,
+hats, pens, etc., when work with measurement of continuous
+quantities with varying units&mdash;inches, feet, yards, glassfuls,
+<span class='pagenum'><a name="Page_8" id="Page_8">[Pg 8]</a></span>
+pints, quarts, seconds, minutes, hours, and the like&mdash;is so
+easy and so significant. On the other hand, the elaboration
+of artificial problems with fictitious units of measure just to
+have relative magnitudes as in the exercises on page 5 is
+a wasteful sacrifice. Similarly, special drills emphasizing
+the fact that eighteen is eleven and seven, twelve and six,
+three less than twenty-one, and the like, are simply idolatrous;
+these facts about eighteen, so far as they are needed,
+are better learned in the course of actual column-addition
+and -subtraction.</p>
+
+
+<div class="figcenter" style="width: 548px;">
+<img src="images/fig2.jpg" width="548" height="740" alt="Fig. 2." title="Fig. 2." />
+<span class="caption"><span class="smcap">Fig. 2.</span></span>
+</div>
+
+
+<h4>ARITHMETICAL LANGUAGE</h4>
+
+<p>The second improvement to be made in the ordinary
+notion of what the functions to be improved are in the case
+of arithmetic is to include among these functions the knowledge
+of certain words. The understanding of such words
+as <i>both</i>, <i>all</i>, <i>in all</i>, <i>together</i>, <i>less</i>, <i>difference</i>, <i>sum</i>, <i>whole</i>, <i>part</i>,
+<i>equal</i>, <i>buy</i>, <i>sell</i>, <i>have left</i>, <i>measure</i>, <i>is contained in</i>, and the
+like, is necessary in arithmetic as truly as is the understanding
+of numbers themselves. It must be provided
+for by the school; for pre-school and extra-school training
+does not furnish it, or furnishes it too late. It can be
+provided for much better in connection with the teaching
+of arithmetic than in connection with the teaching of
+English.</p>
+
+<p>It has not been provided for. An examination of the first
+fifty pages of eight recent textbooks for beginners in arithmetic
+reveals very slight attention to this matter at the
+best and no attention at all in some cases. Three of the
+books do not even use the word <i>sum</i>, and one uses it only
+once in the fifty pages. In all the four hundred pages
+the word <i>difference</i> occurs only twenty times. When the
+words are used, no great ingenuity or care appears in
+<span class='pagenum'><a name="Page_9" id="Page_9">[Pg 9]</a></span>
+the means of making sure that their meanings are understood.</p>
+
+<p>The chief reason why it has not been provided for is precisely
+that the common notion of what the functions are that
+arithmetic is to develop has left out of account this function
+of intelligent response to quantitative terms, other than the
+names of the numbers and processes.</p>
+
+<p>Knowledge of language over a much wider range is a
+necessary element in arithmetical ability in so far as the
+latter includes ability to solve verbally stated problems.
+As arithmetic is now taught, it does include that ability, and
+a large part of the time of wise teaching is given to improving
+the function 'knowing what a problem states and what it asks
+for.' Since, however, this understanding of verbally stated
+problems may not be an absolutely necessary element of
+arithmetic, it is best to defer its consideration until we have
+seen what the general function of problem-solving is.</p>
+
+
+<h4>PROBLEM-SOLVING</h4>
+
+<p>The third respect in which the function, 'ability in arithmetic,'
+needs clearer definition, is this 'problem-solving.'
+The aim of the elementary school is to provide for correct
+and economical response to genuine problems, such as
+knowing the total due for certain real quantities at certain
+real prices, knowing the correct change to give or get,
+keeping household accounts, calculating wages due, computing
+areas, percentages, and discounts, estimating quantities
+needed of certain materials to make certain household
+or shop products, and the like. Life brings these problems
+usually either with a real situation (as when one buys and
+counts the cost and his change), or with a situation that one
+imagines or describes to himself (as when one figures out
+how much money he must save per week to be able to buy
+<span class='pagenum'><a name="Page_10" id="Page_10">[Pg 10]</a></span>
+a forty-dollar bicycle before a certain date). Sometimes,
+however, the problem is described in words to the person
+who must solve it by another person (as when a life insurance
+agent says, 'You pay only 25 cents a week from now till&mdash;and
+you get $250 then'; or when an employer says, 'Your
+wages would be 9 dollars a week, with luncheon furnished
+and bonuses of such and such amounts'). Sometimes also
+the problem is described in printed or written words to the
+person who must solve it (as in an advertisement or in the
+letter of a customer asking for an estimate on this or that).
+The problem may be in part real, in part imagined or described
+to oneself, and in part described to one orally or in
+printed or written words (as when the proposed articles
+for purchase lie before one, the amount of money one
+has in the bank is imagined, the shopkeeper offers 10
+percent discount, and the printed price list is there to be
+read).</p>
+
+<p>To fit pupils to solve these real, personally imagined, or
+self-described problems, and 'described-by-another' problems,
+schools have relied almost exclusively on training with
+problems of the last sort only. The following page taken
+almost at random from one of the best recent textbooks
+could be paralleled by thousands of others; and the oral
+problems put by teachers have, as a rule, no real situation
+supporting them.</p>
+
+<div class="pblockquot"><p><b>1.</b> At 70 cents per 100 pounds, what will be the amount of
+duty on an invoice of 3622 steel rails, each rail being 27 feet long
+and weighing 60 pounds to the yard?</p>
+
+<p><b>2.</b> A man had property valued at $6500. What will be his
+taxes at the rate of $10.80 per $1000?</p>
+
+<p><b>3.</b> Multiply seventy thousand fourteen hundred-thousandths
+by one hundred nine millionths, and divide the product by five
+hundred forty-five.</p>
+
+<p><b>4.</b> What number multiplied by 43&frac34; will produce 265<sup>5</sup>&frasl;<sub>8</sub>?</p>
+
+<p><span class='pagenum'><a name="Page_11" id="Page_11">[Pg 11]</a></span><b>5.</b>
+What decimal of a bushel is 3 quarts?</p>
+
+<p><b>6.</b> A man sells <sup>5</sup>&frasl;<sub>8</sub> of an acre of land for $93.75. What would
+be the value of his farm of 150&frac34; acres at the same rate?</p>
+
+<p><b>7.</b> A coal dealer buys 375 tons coal at $4.25 per ton of 2240
+pounds. He sells it at $4.50 per ton of 2000 pounds. What is his
+profit?</p>
+
+<p><b>8.</b> Bought 60 yards of cloth at the rate of 2 yards for $5, and
+80 yards more at the rate of 4 yards for $9. I immediately sold
+the whole of it at the rate of 5 yards for $12. How much did I
+gain?</p>
+
+<p><b>9.</b> A man purchased 40 bushels of apples at $1.50 per bushel.
+Twenty-five hundredths of them were damaged, and he sold them
+at 20 cents per peck. He sold the remainder at 50 cents per peck.
+How much did he gain or lose?</p>
+
+<p><b>10.</b> If oranges are 37&frac12; cents per dozen, how many boxes, each
+containing 480, can be bought for $60?</p>
+
+<p><b>11.</b> A man can do a piece of work in 18&frac34; days. What part of
+it can he do in 6<sup>2</sup>&frasl;<sub>3</sub> days?</p>
+
+<p><b>12.</b> How old to-day is a boy that was born Oct. 29, 1896?
+[Walsh, '06, Part I, p. 165.]</p></div>
+
+<p>As a result, teachers and textbook writers have come to
+think of the functions of solving arithmetical problems as
+identical with the function of solving the described problems
+which they give in school in books, examination papers, and
+the like. If they do not think explicitly that this is so, they
+still act in training and in testing pupils as if it were so.</p>
+
+<p>It is not. Problems should be solved in school to the
+end that pupils may solve the problems which life offers.
+To know what change one should receive after a given real
+purchase, to keep one's accounts accurately, to adapt a
+recipe for six so as to make enough of the article for four
+persons, to estimate the amount of seed required for a plot
+of a given size from the statement of the amount required
+per acre, to make with surety the applications that the
+household, small stores, and ordinary trades require&mdash;such
+is the ability that the elementary school should develop.
+<span class='pagenum'><a name="Page_12" id="Page_12">[Pg 12]</a></span>
+Other things being equal, the school should set problems in
+arithmetic which life then and later will set, should favor
+the situations which life itself offers and the responses which
+life itself demands.</p>
+
+<p>Other things are not always equal. The same amount of
+time and effort will often be more productive toward the
+final end if directed during school to 'made-up' problems.
+The keeping of personal financial accounts as a school
+exercise is usually impracticable, partly because some of the
+children have no earnings or allowance&mdash;no accounts to
+keep, and partly because the task of supervising work when
+each child has a different problem is too great for the teacher.
+The use of real household and shop problems will be easy
+only when the school program includes the household arts
+and industrial education, and when these subjects themselves
+are taught so as to improve the functions used by
+real life. Very often the most efficient course is to make
+sure that arithmetical procedures are applied to the real
+and personally initiated problems which they fit, by having
+a certain number of such problems arise and be solved;
+then to make sure that the similarity between these real
+problems and certain described problems of the textbook
+or teacher's giving is appreciated; and then to give the
+needed drill work with described problems. In many cases
+the school practice is fairly well justified in assuming that
+solving described problems will prepare the pupil to solve
+the corresponding real problems actually much better than
+the same amount of time spent on the real problems themselves.</p>
+
+<p>All this is true, yet the general principle remains that,
+other things being equal, the school should favor real situations,
+should present issues as life will present them.</p>
+
+<p>Where other things make the use of verbally described
+<span class='pagenum'><a name="Page_13" id="Page_13">[Pg 13]</a></span>
+problems of the ordinary type desirable, these should be
+chosen so as to give a maximum of preparation for the real
+applications of arithmetic in life. We should not, for
+example, carelessly use any problem that comes to mind
+in applying a certain principle, but should stop to consider
+just what the situations of life really require
+and show clearly the application of that principle.
+For example, contrast these two problems applying cancellation:&mdash;</p>
+
+<div class="pblockquot"><p>A. A man sold 24 lambs at $18 apiece on each of six days,
+and bought 8 pounds of metal with the proceeds. How much did
+he pay per ounce for the metal?</p>
+
+<p>B. How tall must a rectangular tank 16" long by 8" wide be
+to hold as much as a rectangular tank 24" by 18" by 6"?</p></div>
+
+<p class="noidt">The first problem not only presents a situation that
+would rarely or never occur, but also takes a way to find
+the answer that would not, in that situation, be taken
+since the price set by another would determine the
+amount.</p>
+
+<p>Much thought and ingenuity should in the future be
+expended in eliminating problems whose solution does not
+improve the real function to be improved by applied arithmetic,
+or improves it at too great cost, and in devising
+problems which prepare directly for life's demands and still
+can fit into a curriculum that can be administered by one
+teacher in charge of thirty or forty pupils, under the limitations
+of school life.</p>
+
+<p>The following illustrations will to some extent show concretely
+what the ability to apply the knowledge and power
+represented by abstract or pure arithmetic&mdash;the so-called
+fundamentals&mdash;in solving problems should mean and what
+it should not mean.</p>
+
+<p><span class='pagenum'><a name="Page_14" id="Page_14">[Pg 14]</a></span></p>
+
+<h4><i>Samples of Desirable Applications of Arithmetic in Problems
+where the Situation is Actually Present to
+Sense in Whole or in Part</i></h4>
+
+<p>Keeping the scores and deciding which side beat and by
+how much in appropriate classroom games, spelling matches,
+and the like.</p>
+
+<p>Computing costs, making and inspecting change, taking
+inventories, and the like with a real or play store.</p>
+
+<p>Mapping the school garden, dividing it into allotments,
+planning for the purchase of seeds, and the like.</p>
+
+<p>Measuring one's own achievement and progress in tests
+of word-knowledge, spelling, addition, subtraction, speed of
+writing, and the like. Measuring the rate of improvement
+per hour of practice or per week of school life, and the like.</p>
+
+<p>Estimating costs of food cooked in the school kitchen,
+articles made in the school shops, and the like.</p>
+
+<p>Computing the cost of telegrams, postage, expressage,
+for a real message or package, from the published tariffs.</p>
+
+<p>Computing costs from mail order catalogues and the like.</p>
+
+
+<h4><i>Samples of Desirable Applications of Arithmetic where the
+Situation is Not Present to Sense</i></h4>
+
+<p>The samples given here all concern the subtraction of
+fractions. Samples concerning any other arithmetical principle
+may be found in the appropriate pages of any text
+which contains problem-material selected with consideration
+of life's needs.</p>
+
+<h4>A</h4>
+
+<div class="pblockquot">
+<p><b>1.</b> Dora is making jelly. The recipe calls for 24 cups of sugar
+and she has only 21&frac12;. She has no time to go to the store
+so she has to borrow the sugar from a neighbor. How
+much must she get?</p>
+
+<p><i>Subtract</i></p>
+</div>
+<p><span class='pagenum'><a name="Page_15" id="Page_15">[Pg 15]</a></span></p>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>24</td><td align='left'><i>Think "&frac12; and &frac12; = 1." Write &frac12;.</i></td></tr>
+<tr><td align='left'>21&frac12;</td><td align='left'><i>Think "2 and 2 = 4." Write the 2.</i></td></tr>
+<tr><td align='left'>&mdash;&mdash;&mdash;</td><td>&nbsp;</td></tr>
+<tr><td align='left'>&nbsp; 2&frac12;</td><td>&nbsp;</td></tr>
+</table></div>
+
+<div class="pblockquot">
+<p><b>2.</b> A box full of soap weighs 29&frac12; lb. The empty box weighs
+3&frac12; lb. How much does the soap alone weigh?</p>
+
+<p><b>3.</b> On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt.
+On July 15 there were just 11&frac12; lb. left. How much had he
+used in the two weeks?</p>
+
+<p><b>4.</b> Grace promised to pick 30 qt. blueberries for her mother.
+So far she has picked 18&frac12; qt. How many more quarts must
+she pick?</p>
+</div>
+
+<h4>B</h4>
+
+<div class="pblockquot"><p>This table of numbers tells
+what Nell's baby sister Mary
+weighed every two months from
+the time she was born till she
+was a year old.</p></div>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="30%">
+<tr><th colspan='2'>Weight of Mary Adams</th></tr>
+<tr><td align='left'><b> When born</b></td><td align='right'>7<sup>3</sup>&frasl;<sub>8</sub> lb.</td></tr>
+<tr><td align='left'><b> 2 months old</b></td><td align='right'>11<sup>1</sup>&frasl;<sub>4</sub> lb.</td></tr>
+<tr><td align='left'><b> 4 months old</b></td><td align='right'>14<sup>1</sup>&frasl;<sub>8</sub> lb.</td></tr>
+<tr><td align='left'><b> 6 months old</b></td><td align='right'>15<sup>3</sup>&frasl;<sub>4</sub> lb.</td></tr>
+<tr><td align='left'><b> 8 months old</b></td><td align='right'>17<sup>5</sup>&frasl;<sub>8</sub> lb.</td></tr>
+<tr><td align='left'><b>10 months old</b></td><td align='right'>19<sup>1</sup>&frasl;<sub>2</sub> lb.</td></tr>
+<tr><td align='left'><b>12 months old</b></td><td align='right'>21<sup>3</sup>&frasl;<sub>8</sub> lb.</td></tr>
+</table></div>
+
+<div class="pblockquot">
+<p><b>1.</b> How much did the Adams baby gain in the first two months?</p>
+
+<p><b>2.</b> How much did the Adams baby gain in the second two months?</p>
+
+<p><b>3.</b> In the third two months?</p>
+<p><b>4.</b> In the fourth two months?</p>
+
+<p><b>5.</b> From the time it was 8 months old till it was 10 months old?</p>
+
+<p><b>6.</b> In the last two months?</p>
+
+<p><b>7.</b> From the time it was born till it was 6 months old?</p>
+</div>
+
+
+<h4>C</h4>
+
+<div class="pblockquot"><p><b>1.</b> Helen's exact average for December was 87<sup>1</sup>&frasl;<sub>3</sub>. Kate's was
+84<sup>1</sup>&frasl;<sub>2</sub>. How much higher was Helen's than Kate's?</p></div>
+
+<p class="center"><br />
+87<sup>1</sup>&frasl;<sub>3</sub><br />
+84<sup>1</sup>&frasl;<sub>2</sub><br />
+&mdash;&mdash;&mdash;
+</p>
+
+<div class="pblockquot"><p>How do you think of <sup>1</sup>&frasl;<sub>2</sub> and <sup>1</sup>&frasl;<sub>3</sub>?</p>
+<p>How do you think of 1<sup>2</sup>&frasl;<sub>6</sub>?</p>
+<p>How do you change the 4?</p>
+
+<p><b>2.</b> Find the exact average for each girl in the following list.
+Write the answers clearly so that you can see them easily. You
+will use them in solving problems 3, 4, 5, 6, 7, and 8.</p></div>
+
+
+<p><span class='pagenum'><a name="Page_16" id="Page_16">[Pg 16]</a></span></p>
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><th>&nbsp;</th><th>Alice</th><th>Dora</th><th>Emma</th><th>Grace</th><th>Louise</th><th>Mary</th><th>Nell</th><th>Rebecca</th></tr>
+<tr><td align='left'><b>Reading</b></td><td align='center'>91</td><td align='center'>87</td><td align='center'>83</td><td align='center'>81</td><td align='center'>79</td><td align='center'>77</td><td align='center'>76</td><td align='center'>73</td></tr>
+<tr><td align='left'><b>Language</b></td><td align='center'>88</td><td align='center'>78</td><td align='center'>82</td><td align='center'>79</td><td align='center'>73</td><td align='center'>78</td><td align='center'>73</td><td align='center'>75</td></tr>
+<tr><td align='left'><b>Arithmetic</b></td><td align='center'>89</td><td align='center'>85</td><td align='center'>79</td><td align='center'>75</td><td align='center'>84</td><td align='center'>87</td><td align='center'>89</td><td align='center'>80</td></tr>
+<tr><td align='left'><b>Spelling</b></td><td align='center'>90</td><td align='center'>79</td><td align='center'>75</td><td align='center'>80</td><td align='center'>82</td><td align='center'>91</td><td align='center'>68</td><td align='center'>81</td></tr>
+<tr><td align='left'><b>Geography</b></td><td align='center'>91</td><td align='center'>87</td><td align='center'>83</td><td align='center'>75</td><td align='center'>78</td><td align='center'>85</td><td align='center'>73</td><td align='center'>79</td></tr>
+<tr><td align='left'><b>Writing</b></td><td align='center'>90</td><td align='center'>88</td><td align='center'>75</td><td align='center'>72</td><td align='center'>93</td><td align='center'>92</td><td align='center'>95</td><td align='center'>78</td></tr>
+</table></div>
+
+<div class="pblockquot">
+<p><b>3.</b> Which girl had the highest average?</p>
+
+<p><b>4.</b> How much higher was her average than the next highest?</p>
+
+<p><b>5.</b> How much difference was there between the highest and the
+lowest girl?</p>
+
+<p><b>6.</b> Was Emma's average higher or lower than Louise's? How
+much?</p>
+
+<p><b>7.</b> How much difference was there between Alice's average and
+Dora's?</p>
+
+<p><b>8.</b> How much difference was there between Mary's average and
+Nell's?</p>
+
+<p><b>9.</b> Write five other problems about these averages, and solve
+each of them.</p>
+</div>
+
+<h4><i>Samples of Undesirable Applications of Arithmetic</i><a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a></h4>
+
+<div class="pblockquot">
+<p>Will has XXI marbles, XII jackstones, XXXVI pieces of
+string. How many things had he?</p>
+
+<p>George's kite rose CDXXXV feet and Tom's went LXIII feet
+higher. How high did Tom's kite rise?</p>
+
+<p>If from DCIV we take CCIV the result will be a number IV
+times as large as the number of dollars Mr. Dane paid for his horse.
+How much did he pay for his horse?</p>
+
+<p>Hannah has <sup>5</sup>&frasl;<sub>8</sub> of a dollar, Susie <sup>7</sup>&frasl;<sub>25</sub>, Nellie <sup>3</sup>&frasl;<sub>4</sub>, Norah <sup>13</sup>&frasl;<sub>16</sub>. How
+much money have they all together?</p>
+
+<p>A man saves 3<sup>17</sup>&frasl;<sub>80</sub> dollars a week. How much does he save in
+a year?</p>
+
+<p>A tree fell and was broken into 4 pieces, 13<sup>1</sup>&frasl;<sub>6</sub> feet, 10<sup>3</sup>&frasl;<sub>7</sub> feet, 8<sup>1</sup>&frasl;<sub>2</sub>
+feet, and 7<sup>16</sup>&frasl;<sub>21</sub> feet long. How tall was the tree?<span class='pagenum'><a name="Page_17" id="Page_17">[Pg 17]</a></span></p>
+
+<p>Annie's father gave her 20 apples to divide among her friends.
+She gave each one 2<sup>2</sup>&frasl;<sub>9</sub> apples apiece. How many playmates had she?</p>
+
+<p>John had 17<sup>2</sup>&frasl;<sub>5</sub> apples. He divided his whole apples into fifths.
+How many pieces had he in all?</p>
+
+<p>A landlady has 3<sup>3</sup>&frasl;<sub>7</sub> pies to be divided among her 8 boarders.
+How much will each boarder receive?</p>
+
+<p>There are twenty columns of spelling words in Mary's lesson
+and 16 words in each column. How many words are in her
+lesson?</p>
+
+<p>There are 9 nuts in a pint. How many pints in a pile of
+5,888,673 nuts?</p>
+
+<p>The Adams school contains eight rooms; each room contains
+48 pupils; if each pupil has eight cents, how much have they
+together?</p>
+
+<p>A pile of wood in the form of a cube contains 15&frac12; cords. What
+are the dimensions to the nearest inch?</p>
+
+<p>A man 6 ft. high weighs 175 lb. How tall is his wife who is of
+similar build, and weighs 125 lb.?</p>
+
+<p>A stick of timber is in the shape of the frustum of a square
+pyramid, the lower base being 22 in. square and the upper 14 in.
+square. How many cubic feet in the log, if it is 22 ft. long?</p>
+
+<p>Mr. Ames, being asked his age, replied: "If you cube one half
+of my age and add 41,472 to the result, the sum will be one half the
+cube of my age. How old am I?"</p></div>
+
+<p>These samples, just given, of the kind of problem-solving
+that should not be emphasized in school training refer in
+some cases to books of forty years back, but the following
+represent the results of a collection made in 1910 from books
+then in excellent repute. It required only about an hour to
+collect them; and I am confident that a thousand such
+problems describing situations that the pupil will never
+encounter in real life, or putting questions that he will never
+be asked in real life, could easily be found in any ten textbooks
+of the decade from 1900 to 1910.</p>
+
+<div class="pblockquot">
+<p>If there are 250 kernels of corn on one ear, how many are there
+on 24 ears of corn the same size?</p>
+
+<p>Maud is four times as old as her sister, who is 4 years old. What
+is the sum of their ages?</p>
+<p><span class='pagenum'><a name="Page_18" id="Page_18">[Pg 18]</a></span></p>
+<p>If the first century began with the year 1, with what year does
+it end?</p>
+
+<p>Every spider has 8 compound eyes. How many eyes have 21
+spiders?</p>
+
+<p>A nail 4 inches long is driven through a board so that it projects
+1.695 inches on one side and 1.428 on the other. How thick is the
+board?</p>
+
+<p>Find the perimeter of an envelope 5 in. by 3&frac14; in.</p>
+
+<p>How many minutes in <sup>5</sup>&frasl;<sub>9</sub> of <sup>9</sup>&frasl;<sub>4</sub> of an hour?</p>
+
+<p>Mrs. Knox is <sup>3</sup>&frasl;<sub>4</sub> as old as Mr. Knox, who is 48 years old. Their
+son Edward is <sup>4</sup>&frasl;<sub>9</sub> as old as his mother. How old is Edward?</p>
+
+<p>Suppose a pie to be exactly round and 10&frac12; miles in diameter.
+If it were cut into 6 equal pieces, how long would the curved edge
+of each piece be?</p>
+
+<p>8<sup>1</sup>&frasl;<sub>3</sub>% of a class of 36 boys were absent on a rainy day. 33<sup>1</sup>&frasl;<sub>3</sub>%
+of those present went out of the room to the school yard. How
+many were left in the room?</p>
+
+<p>Just after a ton of hay was weighed in market, a horse ate one
+pound of it. What was the ratio of what he ate to what was left?</p>
+
+<p>If a fan having 15 rays opens out so that the outer rays form a
+straight line, how many degrees are there between any two adjacent
+rays?</p>
+
+<p>One half of the distance between St. Louis and New Orleans
+is 280 miles more than <sup>1</sup>&frasl;<sub>10</sub> of the distance; what is the distance
+between these places?</p>
+
+<p>If the pressure of the atmosphere is 14.7 lb. per square inch what
+is the pressure on the top of a table 1&frac14; yd. long and <sup>2</sup>&frasl;<sub>3</sub> yd. wide?</p>
+
+<p><sup>13</sup>&frasl;<sub>28</sub> of the total acreage of barley in 1900 was 100,000 acres;
+what was the total acreage?</p>
+
+<p>What is the least number of bananas that a mother can exactly
+divide between her 2 sons, or among her 4 daughters, or among all
+her children?</p>
+
+<p>If Alice were two years older than four times her actual age she
+would be as old as her aunt, who is 38 years old. How old is
+Alice?</p>
+
+<p>Three men walk around a circular island, the circumference of
+which is 360 miles. A walks 15 miles a day, B 18 miles a day, and
+C 24 miles a day. If they start together and walk in the same direction,
+how many days will elapse before they will be together again?</p></div>
+
+<p>With only thirty or forty dollars a year to spend on a
+pupil's education, of which perhaps eight dollars are spent<span class='pagenum'><a name="Page_19" id="Page_19">[Pg 19]</a></span>
+on improving his arithmetical abilities, the immediate
+guidance of his responses to real situations and personally
+initiated problems has to be supplemented largely by
+guidance of his responses to problems described in words,
+diagrams, pictures, and the like. Of these latter, words will
+be used most often. As a consequence the understanding
+of the words used in these descriptions becomes a part
+of the ability required in arithmetic. Such word knowledge
+is also required in so far as the problems to be solved in real
+life are at times described, as in advertisements, business
+letters, and the like.</p>
+
+<p>This is recognized by everybody in the case of words like
+<i>remainder</i>, <i>profit</i>, <i>loss</i>, <i>gain</i>, <i>interest</i>, <i>cubic capacity</i>, <i>gross</i>,
+<i>net</i>, and <i>discount</i>, but holds equally of <i>let</i>, <i>suppose</i>, <i>balance</i>,
+<i>average</i>, <i>total</i>, <i>borrowed</i>, <i>retained</i>, and many such semi-technical
+words, and may hold also of hundreds of other
+words unless the textbook and teacher are careful to use
+only words and sentence structures which daily life and the
+class work in English have made well known to the pupils.
+To apply arithmetic to a problem a pupil must understand
+what the problem is; problem-solving depends on problem-reading.
+In actual school practice training in problem-reading
+will be less and less necessary as we get rid of problems
+to be solved simply for the sake of solving them,
+unnecessarily unreal problems, and clumsy descriptions,
+but it will remain to some extent as an important joint task
+for the 'arithmetic' and 'reading' of the elementary school.</p>
+
+
+<h4>ARITHMETICAL REASONING</h4>
+
+<p>The last respect in which the nature of arithmetical
+abilities requires definition concerns arithmetical reasoning.
+An adequate treatment of the reasoning that may be
+expected of pupils in the elementary school and of the most
+<span class='pagenum'><a name="Page_20" id="Page_20">[Pg 20]</a></span>
+efficient ways to encourage and improve it cannot be given
+until we have studied the formation of habits. For reasoning
+is essentially the organization and control of habits of
+thought. Certain matters may, however, be decided here.
+The first concerns the use of computation and problems
+merely for discipline,&mdash;that is, the emphasis on training
+in reasoning regardless of whether the problem is otherwise
+worth reasoning about. It used to be thought that the mind
+was a set of faculties or abilities or powers which grew strong
+and competent by being exercised in a certain way, no
+matter on what they were exercised. Problems that could
+not occur in life, and were entirely devoid of any worthy
+interest, save the intellectual interest in solving them, were
+supposed to be nearly or quite as useful in training the mind
+to reason as the genuine problems of the home, shop, or
+trade. Anything that gave the mind a chance to reason
+would do; and pupils labored to find when the minute hand
+and hour hand would be together, or how many sheep a
+shepherd had if half of what he had plus ten was one third
+of twice what he had!</p>
+
+<p>We now know that the training depends largely on the
+particular data used, so that efficient discipline in reasoning
+requires that the pupil reason about matters of real importance.
+There is no magic essence or faculty of reasoning that
+works in general and irrespective of the particular facts and relations
+reasoned about. So we should try to find problems which
+not only stimulate the pupil to reason, but also direct his
+reasoning in useful channels and reward it by results that are
+of real significance. We should replace the purely disciplinary
+problems by problems that are also valuable as special
+training for important particular situations of life. Reasoning
+sought for reasoning's sake alone is too wasteful an expenditure
+of time and is also likely to be inferior as reasoning.</p>
+
+<p><span class='pagenum'><a name="Page_21" id="Page_21">[Pg 21]</a></span></p>
+<p>The second matter concerns the relative merits of 'catch'
+problems, where the pupil has to go against some customary
+habit of thinking, and what we may call 'routine' problems,
+where the regular ways of thinking that have served him in
+the past will, except for some blunder, guide him rightly.</p>
+
+<p>Consider, for example, these four problems:</p>
+
+<div class="pblockquot"><p><b>1.</b> "A man bought ten dozen eggs for $2.50 and sold them for
+30 cents a dozen. How many cents did he lose?"</p>
+
+<p><b>2.</b> "I went into Smith's store at 9 <span class="smcap">A.M.</span> and remained until
+10 <span class="smcap">A.M.</span> I bought six yards of gingham at 40 cents a yard and
+three yards of muslin at 20 cents a yard and gave a $5.00 bill.
+How long was I in the store?"</p>
+
+<p><b>3.</b> "What must you divide 48 by to get half of twice 6?"</p>
+
+<p><b>4.</b> "What must you add to 19 to get 30?"</p></div>
+
+<p>The 'catch' problem is now in disrepute, the wise teacher
+feeling by a sort of intuition that to willfully require a pupil
+to reason to a result sharply contrary to that to which previous
+habits lead him is risky. The four illustrations just
+given show, however, that mere 'catchiness' or 'contra-previous-habit-ness'
+in a problem is not enough to condemn
+it. The fourth problem is a catch problem, but so useful a
+one that it has been adopted in many modern books as a
+routine drill! The first problem, on the contrary, all, save
+those who demand no higher criterion for a problem than
+that it make the pupil 'think,' would reject. It demands the
+reversal of fixed habits <i>to no valid purpose</i>; for in life the
+question in such case would never (or almost never) be
+'How many cents did he lose?' but 'What was the result?'
+or simply 'What of it?' This problem weakens without excuse
+the child's confidence in the training he has had. Problems
+like (2) are given by teachers of excellent reputation,
+but probably do more harm than good. If a pupil should
+interrupt his teacher during the recitation in arithmetic by
+<span class='pagenum'><a name="Page_22" id="Page_22">[Pg 22]</a></span>
+saying, "I got up at 7 o'clock to multiply 9 by 2&frac34; and got 24&frac34;
+for my answer; was that the right time to get up?" the
+teacher would not thank fortune for the stimulus to thought
+but would think the child a fool. Such catch questions may
+be fairly useful as an object lesson on the value of search
+for the essential element in a situation if a great variety of
+them are given one after another with routine problems
+intermixed and with warning of the general nature of the
+exercise at the beginning. Even so, it should be remembered
+that reasoning should be chiefly a force organizing habits,
+not opposing them; and also that there are enough bad
+habits to be opposed to give all necessary training. Fabricated
+puzzle situations wherein a peculiar hidden element
+of the situation makes the good habits called up by the
+situation misleading are useful therefore rather as a relief
+and amusing variation in arithmetical work than as stimuli
+to thought.</p>
+
+<p>Problems like the third quoted above we might call puzzling
+rather than 'catch' problems. They have value as
+drills in analysis of a situation into its elements that will
+amuse the gifted children, and as tests of certain abilities.
+They also require that of many confusing habits, the right one
+be chosen, rather than that ordinary habits be set aside by
+some hidden element in the situation. Not enough is known
+about their effect to enable us to decide whether or not the
+elementary school should include special facility with them
+as one of the arithmetical functions that it specially trains.</p>
+
+<p>The fourth 'catch' quoted above, which all would admit
+is a good problem, is good because it opposes a good habit
+for the sake of another good habit, forces the analysis of an
+element whose analysis life very much requires, and does it
+with no obvious waste. It is not safe to leave a child with
+the one habit of responding to 'add, 19, 30' by 49, for in<span class='pagenum'><a name="Page_23" id="Page_23">[Pg 23]</a></span>
+life the 'have 19, must get .... to have 30' situation is very
+frequent and important.</p>
+
+<p>On the whole, the ordinary problems which ordinary life
+proffers seem to be the sort that should be reasoned out,
+though the elementary school may include the less noxious
+forms of pure mental gymnastics for those pupils who like
+them.</p>
+
+
+<h4>SUMMARY</h4>
+
+<p>These discussions of the meanings of numbers, the linguistic
+demands of arithmetic, the distinction between
+scholastic and real applications of arithmetic, and the possible
+restrictions of training in reasoning,&mdash;may serve as
+illustrations of the significance of the question, "What are
+the functions that the elementary school tries to improve in
+its teaching of arithmetic?" Other matters might well be
+considered in this connection, but the main outline of the
+work of the elementary school is now fairly clear. The
+arithmetical functions or abilities which it seeks to improve
+are, we may say:&mdash;</p>
+
+<p>(1) Working knowledge of the meanings of numbers as
+names for certain sized collections, for certain relative magnitudes,
+the magnitude of unity being known, and for certain
+centers or nuclei of relations to other numbers.</p>
+
+<p>(2) Working knowledge of the system of decimal notation.</p>
+
+<p>(3) Working knowledge of the meanings of addition,
+subtraction, multiplication, and division.</p>
+
+<p>(4) Working knowledge of the nature and relations of
+certain common measures.</p>
+
+<p>(5) Working ability to add, subtract, multiply, and divide
+with integers, common and decimal fractions, and denominate
+numbers, all being real positive numbers.</p>
+
+<p>(6) Working knowledge of words, symbols, diagrams, and<span class='pagenum'><a name="Page_24" id="Page_24">[Pg 24]</a></span>
+the like as required by life's simpler arithmetical demands
+or by economical preparation therefor.</p>
+
+<p>(7) The ability to apply all the above as required by life's
+simpler arithmetical demands or by economical preparation
+therefor, including (7 <i>a</i>) certain specific abilities to solve
+problems concerning areas of rectangles, volumes of rectangular
+solids, percents, interest, and certain other common
+occurrences in household, factory, and business life.</p>
+
+
+<h4>THE SOCIOLOGY OF ARITHMETIC</h4>
+
+<div class="pblockquot">
+<p>The phrase 'life's simpler arithmetical demands' is necessarily
+left vague. Just what use is being made of arithmetic in this
+country in 1920 by each person therein, we know only very roughly.
+What may be called a 'sociology' of arithmetic is very much needed
+to investigate this matter. For rare or difficult demands the elementary
+school should not prepare; there are too many other
+desirable abilities that it should improve.</p>
+
+<p>A most interesting beginning at such an inventory of the actual
+uses of arithmetic has been made by Wilson ['19] and Mitchell.<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a>
+Although their studies need to be much extended and checked by
+other methods of inquiry, two main facts seem fairly certain.</p>
+
+<p>First, the great majority of people in the great majority of their
+doings use only very elementary arithmetical processes. In
+1737 cases of addition reported by Wilson, seven eighths were of
+five numbers or less. Over half of the multipliers reported were
+one-figure numbers. Over 95 per cent of the fractions operated
+with were included in this list: <sup>1</sup>&frasl;<sub>2</sub> <sup>1</sup>&frasl;<sub>4</sub> <sup>3</sup>&frasl;<sub>4</sub>
+<sup>1</sup>&frasl;<sub>3</sub> <sup>2</sup>&frasl;<sub>3</sub> <sup>1</sup>&frasl;<sub>8</sub> <sup>3</sup>&frasl;<sub>8</sub>
+<sup>1</sup>&frasl;<sub>5</sub> <sup>2</sup>&frasl;<sub>5</sub> <sup>4</sup>&frasl;<sub>5</sub>. Three fourths
+of all the cases reported were simple one-step computations with
+integers or United States money.</p>
+
+<p>Second, they often use these very elementary processes, not
+because such are the quickest and most convenient, but because
+they have lost, or maybe never had, mastery of the more advanced
+processes which would do the work better. The 5 and 10 cent
+stores, the counter with "Anything on this counter for 25&cent;," and
+the arrangements for payments on the installment plan are familiar
+instances of human avoidance of arithmetic. Wilson found very
+slight use of decimals; and Mitchell found men computing with
+<span class='pagenum'><a name="Page_25" id="Page_25">[Pg 25]</a></span>
+49ths as common fractions when the use of decimals would have
+been more efficient. If given 120 seconds to do a test like that
+shown below, leading lawyers, physicians, manufacturers, and
+business men and their wives will, according to my experience, get
+only about half the work right. Many women, finding on their
+meat bill "7<sup>3</sup>&frasl;<sub>8</sub> lb. roast beef $2.36," will spend time and money to
+telephone the butcher asking how much roast beef was per pound,
+because they have no sure power in dividing by a mixed number.</p>
+</div>
+
+<div class="pblockquot">
+
+<p class="tabcap">Test</p>
+
+<p>Perform the operations indicated. Express all fractions in
+answers in lowest terms.</p>
+
+<div class="blockquot"><p>
+<i>Add:</i></p>
+
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'><sup>3</sup>&frasl;<sub>4</sub> &#43; <sup>1</sup>&frasl;<sub>6</sub> &#43; .25 &nbsp; &nbsp; &nbsp;</td>
+<td align='left' class='bb'>
+                      4 yr. 6 mo.<br />
+                      1 yr. 2 mo.<br />
+                      6 yr. 9 mo.<br />
+                      3 yr. 6 mo.<br />
+                      4 yr. 5 mo.
+</td></tr>
+</table>
+
+
+<p><i>Subtract:</i></p>
+
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>8.6 &minus; 6.05007  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
+<sup>7</sup>&frasl;<sub>8</sub> &minus; <sup>2</sup>&frasl;<sub>3</sub> =   &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
+5<sup>7</sup>&frasl;<sub>16</sub> &minus; 2<sup>3</sup>&frasl;<sub>16</sub> =</td></tr>
+</table>
+
+
+<p><i>Multiply:</i></p>
+
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>7 &times; 8 &times; 4&frac12; = &nbsp; &nbsp; &nbsp;</td>
+<td align='left' valign='top' class='bb'>29 ft.</td><td align='left' class='bb'>6 in.<br />8</td></tr>
+</table>
+
+<p><i>Divide:</i></p>
+<p class="center">4&frac12; &divide; 7 =</p>
+</div>
+
+<p>It seems probable that the school training in arithmetic of the
+past has not given enough attention to perfecting the more elementary
+abilities. And we shall later find further evidence of this.
+On the other hand, the fact that people in general do not at present
+use a process may not mean that they ought not to use it.</p>
+
+<p>Life's simpler arithmetical demands certainly do not include
+matters like the rules for finding cube root or true discount, which
+no sensible person uses. They should not include matters like
+computing the lateral surface or volume of pyramids and cones,
+or knowing the customs of plasterers and paper hangers, which
+are used only by highly specialized trades. They should not include
+matters like interest on call loans, usury, exact interest, and<span class='pagenum'><a name="Page_26" id="Page_26">[Pg 26]</a></span>
+the rediscounting of notes, which concern only brokers, bank clerks,
+and rich men. They should not include the technique of customs
+which are vanishing from efficient practice, such as simple interest
+on amount for times longer than a year, days of grace, or extremes
+and means in proportions. They should not include any elaborate
+practice with very large numbers, or decimals beyond thousandths,
+or the addition and subtraction of fractions which not one person
+in a hundred has to add or subtract oftener than once a year.</p>
+
+<p>When we have an adequate sociology of arithmetic, stating
+accurately just who should use each arithmetical ability and how
+often, we shall be able to define the task of the elementary school
+in this respect. For the present, we may proceed by common
+sense, guided by two limiting rules. The first is,&mdash;"It is no more
+desirable for the elementary school to teach all the facts of arithmetic
+than to teach all the words in the English language, or all the
+topography of the globe, or all the details of human physiology."
+The second is,&mdash;"It is not desirable to eliminate any element
+of arithmetical training until you have something better to put in
+its place."</p>
+</div>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_27" id="Page_27">[Pg 27]</a></span></p>
+<h2><a name="CHAPTER_II" id="CHAPTER_II"></a>CHAPTER II</h2>
+
+<h2>THE MEASUREMENT OF ARITHMETICAL ABILITIES</h2>
+
+
+<p>One of the best ways to clear up notions of what the
+functions are which schools should develop and improve
+is to get measures of them. If any given knowledge or
+skill or power or ideal exists, it exists in some amount. A
+series of amounts of it, varying from less to more, defines
+the ability itself in a way that no general verbal description
+can do. Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb.,
+etc., helps to tell us what we mean by weight. By finding a
+series of words like <i>only</i>, <i>smoke</i>, <i>another</i>, <i>pretty</i>, <i>answer</i>, <i>tailor</i>,
+<i>circus</i>, <i>telephone</i>, <i>saucy</i>, and <i>beginning</i>, which are spelled
+correctly by known and decreasing percentages of children
+of the same age, or of the same school grade, we know better
+what we mean by 'spelling-difficulty.' Indeed, until we
+can measure the efficiency and improvement of a function,
+we are likely to be vague and loose in our ideas of what the
+function is.</p>
+
+
+<h4>A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY:
+THE ABILITY TO ADD INTEGERS</h4>
+
+<p>Consider first, as a sample, the measurement of ability
+to add integers.</p>
+
+<p>The following were the examples used in the measurements
+made by Stone ['08]:<span class='pagenum'><a name="Page_28" id="Page_28">[Pg 28]</a></span></p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="40%">
+<tr><td align='right'>&nbsp;</td><td align='right'>596</td><td align='right'>4695</td></tr>
+<tr><td align='right'>&nbsp;</td><td align='right'>428</td><td align='right'>872</td></tr>
+<tr><td align='right'>2375</td><td align='right'>94</td><td align='right'>7948</td></tr>
+<tr><td align='right'>4052</td><td align='right'>75</td><td align='right'>6786</td></tr>
+<tr><td align='right'>6354</td><td align='right'>304</td><td align='right'>567</td></tr>
+<tr><td align='right'>260</td><td align='right'>645</td><td align='right'>858</td></tr>
+<tr><td align='right'>5041</td><td align='right'>984</td><td align='right'>9447</td></tr>
+<tr><td align='right'>1543</td><td align='right'>897</td><td align='right'>7499</td></tr>
+<tr><td align='right'>&mdash;&mdash;&mdash;</td><td align='right'>&mdash;&mdash;&mdash;</td><td align='right'>&mdash;&mdash;&mdash;</td></tr>
+</table></div>
+
+<p>The scoring was as follows: Credit of 1 for each column
+added correctly. Stone combined measures of other abilities
+with this in a total score for amount done correctly in 12
+minutes. Stone also scored the correctness of the additions
+in certain work in multiplication.</p>
+
+<p>Courtis uses a sheet of twenty-four tasks or 'examples,'
+each consisting of the addition of nine three-place numbers
+as shown below. Eight minutes is allowed. He scores the
+amount done by the number of examples, and also scores
+the number of examples done correctly, but does not suggest
+any combination of these two into a general-efficiency
+score.</p>
+
+<p class="center"><br />
+927<br />
+379<br />
+756<br />
+837<br />
+924<br />
+110<br />
+854<br />
+965<br />
+344<br />
+&mdash;&mdash;&mdash;<br />
+</p>
+
+<p>The author long ago proposed that pupils be measured
+also with series like <i>a</i> to <i>g</i> shown below, in which the difficulty
+increases step by step.</p>
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>a.</i></td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>1</td><td align='right'>2</td></tr>
+<tr><td align='right'></td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>4</td><td align='right'>5</td><td align='right'>5</td><td align='right'>1</td></tr>
+<tr><td align='right'></td><td align='right'>4</td><td align='right'>2</td><td align='right'>3</td><td align='right'>3</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>2</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+<p><span class='pagenum'><a name="Page_29" id="Page_29">[Pg 29]</a></span></p>
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>b.</i></td><td align='right'>21</td><td align='right'>32</td><td align='right'>12</td><td align='right'>24</td><td align='right'>34</td><td align='right'>34</td><td align='right'>22</td><td align='right'>12</td></tr>
+<tr><td align='right'></td><td align='right'>23</td><td align='right'>12</td><td align='right'>52</td><td align='right'>31</td><td align='right'>33</td><td align='right'>12</td><td align='right'>23</td><td align='right'>13</td></tr>
+<tr><td align='right'></td><td align='right'>24</td><td align='right'>25</td><td align='right'>15</td><td align='right'>14</td><td align='right'>32</td><td align='right'>23</td><td align='right'>43</td><td align='right'>61</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>c.</i></td><td align='right'>22</td><td align='right'>3</td><td align='right'>4</td><td align='right'>35</td><td align='right'>32</td><td align='right'>83</td><td align='right'>22</td><td align='right'>&nbsp; 3</td></tr>
+<tr><td align='right'></td><td align='right'>&nbsp; 3</td><td align='right'>31</td><td align='right'>3</td><td align='right'>2</td><td align='right'>33</td><td align='right'>11</td><td align='right'>3</td><td align='right'>21</td></tr>
+<tr><td align='right'></td><td align='right'>38</td><td align='right'>45</td><td align='right'>52</td><td align='right'>52</td><td align='right'>2</td><td align='right'>4</td><td align='right'>33</td><td align='right'>64</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>d.</i></td><td align='right'>30</td><td align='right'>20</td><td align='right'>10</td><td align='right'>22</td><td align='right'>10</td><td align='right'>20</td><td align='right'>52</td><td align='right'>12</td></tr>
+<tr><td align='right'></td><td align='right'>20</td><td align='right'>50</td><td align='right'>40</td><td align='right'>43</td><td align='right'>30</td><td align='right'>4</td><td align='right'>6</td><td align='right'>22</td></tr>
+<tr><td align='right'></td><td align='right'>40</td><td align='right'>17</td><td align='right'>24</td><td align='right'>13</td><td align='right'>40</td><td align='right'>23</td><td align='right'>30</td><td align='right'>44</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>e.</i></td><td align='right'></td><td align='right'>4</td><td align='right'>5</td><td align='right'>20</td><td align='right'>12</td><td align='right'>12</td><td align='right'>20</td><td align='right'>10</td></tr>
+<tr><td align='right'></td><td align='right'>20</td><td align='right'>30</td><td align='right'>3</td><td align='right'>40</td><td align='right'>4</td><td align='right'>11</td><td align='right'>20</td><td align='right'>20</td></tr>
+<tr><td align='right'></td><td align='right'>10</td><td align='right'>30</td><td align='right'>20</td><td align='right'>4</td><td align='right'>1</td><td align='right'>23</td><td align='right'>7</td><td align='right'>2</td></tr>
+<tr><td align='right'></td><td align='right'>20</td><td align='right'>2</td><td align='right'>40</td><td align='right'>23</td><td align='right'>40</td><td align='right'>11</td><td align='right'>10</td><td align='right'>30</td></tr>
+<tr><td align='right'></td><td align='right'>20</td><td align='right'>20</td><td align='right'>10</td><td align='right'>11</td><td align='right'>20</td><td align='right'>22</td><td align='right'>30</td><td align='right'>25</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>f.</i></td><td align='right'></td><td align='right'></td><td align='right'>19</td><td align='right'>9</td><td align='right'></td><td align='right'></td><td align='right'>9</td></tr>
+<tr><td align='right'></td><td align='right'>14</td><td align='right'>2</td><td align='right'>19</td><td align='right'>24</td><td align='right'>9</td><td align='right'>4</td><td align='right'>13</td></tr>
+<tr><td align='right'></td><td align='right'>9</td><td align='right'>14</td><td align='right'>13</td><td align='right'>12</td><td align='right'>13</td><td align='right'>13</td><td align='right'>9</td><td align='right'>14</td></tr>
+<tr><td align='right'></td><td align='right'>17</td><td align='right'>23</td><td align='right'>13</td><td align='right'>15</td><td align='right'>15</td><td align='right'>34</td><td align='right'>12</td><td align='right'>25</td></tr>
+<tr><td align='right'></td><td align='right'>26</td><td align='right'>29</td><td align='right'>18</td><td align='right'>19</td><td align='right'>25</td><td align='right'>28</td><td align='right'>18</td><td align='right'>39</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+
+
+<p>&nbsp;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><td align='right'><i>g.</i></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'>13</td></tr>
+<tr><td align='right'></td><td align='right'></td><td align='right'>13</td><td align='right'></td><td align='right'>9</td><td align='right'>14</td><td align='right'>12</td><td align='right'>9</td></tr>
+<tr><td align='right'></td><td align='right'></td><td align='right'>9</td><td align='right'></td><td align='right'>13</td><td align='right'>12</td><td align='right'>9</td><td align='right'>14</td><td align='right'>24</td></tr>
+<tr><td align='right'></td><td align='right'>23</td><td align='right'>19</td><td align='right'>19</td><td align='right'>29</td><td align='right'>9</td><td align='right'>9</td><td align='right'>13</td><td align='right'>21</td></tr>
+<tr><td align='right'></td><td align='right'>28</td><td align='right'>26</td><td align='right'>26</td><td align='right'>14</td><td align='right'>8</td><td align='right'>8</td><td align='right'>29</td><td align='right'>23</td></tr>
+<tr><td align='right'></td><td align='right'>29</td><td align='right'>16</td><td align='right'>15</td><td align='right'>19</td><td align='right'>17</td><td align='right'>19</td><td align='right'>19</td><td align='right'>22</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td><td align='right'>&mdash;</td></tr>
+</table></div>
+
+<p>Woody ['16] has constructed his well-known tests on this
+principle, though he uses only one example at each step of
+difficulty instead of eight or ten as suggested above. His
+test, so far as addition of integers goes, is:&mdash;</p>
+<p><span class='pagenum'><a name="Page_30" id="Page_30">[Pg 30]</a></span></p>
+
+<p class="tabcap">SERIES A. &nbsp; &nbsp; ADDITION SCALE (in part)</p>
+<p class="tabcap">By Clifford Woody</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><th align='right'>(1)</th><th align='right'>(2)</th><th align='right'>(3)</th><th align='right'>(4)</th><th align='right'>(5)</th><th align='right'>(6)</th><th align='center'>(7)</th><th align='center'>(8)</th><th align='right'>(9)</th></tr>
+<tr>
+  <td align='right'>2<br />3<br />&mdash;</td>
+  <td align='right'>2<br />4<br />3<br />&mdash;</td>
+  <td align='right'>17<br />2<br />&mdash;</td>
+  <td align='right'>53<br />45<br />&mdash;</td>
+  <td align='right'>72<br />26<br />&mdash;</td>
+  <td align='right'>60<br />37<br />&mdash;</td>
+  <td align='center'>3 &#43; 1 =</td>
+  <td align='center'>2 &#43; 5 &#43; 1 =</td>
+  <td align='right'>20<br />10<br />2<br />30<br />25<br />&mdash;</td>
+</tr>
+<tr><th align='right'>(10)</th><th align='right'>(11)</th><th align='right'>(12)</th><th align='right'>(13)</th><th align='center'>(14)</th><th align='right'>(15)</th><th align='right'>(16)</th><th align='right'>(17)</th><th align='right'>(18)</th></tr>
+<tr>
+  <td align='right'>21<br />33<br />35<br />&mdash;</td>
+  <td align='right'>32<br />59<br />17<br />&mdash;</td>
+  <td align='right'>43<br />1<br />2<br />13<br />&mdash;</td>
+  <td align='right'>23<br />25<br />16<br />&mdash;</td>
+  <td align='center'>25 &#43; 42 =</td>
+  <td align='right'>100<br />33<br />45<br />201<br />46<br />&mdash;</td>
+  <td align='right'>9<br />24<br />12<br />15<br />19<br />&mdash;</td>
+  <td align='right'>199<br />194<br />295<br />156<br />&mdash;&mdash;</td>
+  <td align='right'>2563<br />1387<br />4954<br />2065<br />&mdash;&mdash;</td>
+</tr>
+<tr><th align='right'>(19)</th><th align='right'>(20)</th><th align='right'>(21)</th><th align='right'>(22)</th></tr>
+<tr>
+  <td align='right'>$ .75<br />1.25<br />.49<br />&mdash;&mdash;</td>
+  <td align='right'>$12.50<br />16.75<br />15.75<br />&mdash;&mdash;</td>
+  <td align='right'>$8.00<br />5.75<br />2.33<br />4.16<br />.94<br />6.32<br />&mdash;&mdash;</td>
+  <td align='right'>547<br />197<br />685<br />678<br />456<br />393<br />525<br />240<br />152<br />&mdash;&mdash;</td>
+</tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_31" id="Page_31">[Pg 31]</a></span></p>
+
+<p>In his original report, Woody gives no scheme for scoring
+an individual, wisely assuming that, with so few samples at
+each degree of difficulty, a pupil's score would be too unreliable
+for individual diagnosis. The test is reliable for a
+class; and for a class Woody used the degree of difficulty
+such that a stated fraction of the class can do the work
+correctly, if twenty minutes is allowed for the thirty-eight
+examples of the entire test.</p>
+
+<p>The measurement of even so simple a matter as the efficiency
+of a pupil's responses to these tests in adding integers
+is really rather complex. There is first of all the problem
+of combining speed and accuracy into some single estimate.
+Stone gives no credit for a column unless it is correctly
+added. Courtis evades the difficulty by reporting both
+number done and number correct. The author's scheme,
+which gives specified weights to speed and accuracy at each
+step of the series, involves a rather intricate computation.</p>
+
+<p>This difficulty of equating speed and accuracy in adding
+means precisely that we have inadequate notions of what the
+ability is that the elementary school should improve. Until,
+for example, we have decided whether, for a given group
+of pupils, fifteen Courtis attempts with ten right, is or is
+not a better achievement than ten Courtis attempts with
+nine right, we have not decided just what the business of
+the teacher of addition is, in the case of that group of pupils.</p>
+
+<p>There is also the difficulty of comparing results when
+short and long columns are used. Correctness with a short
+column, say of five figures, testifies to knowledge of the process
+and to the power to do four successive single additions
+without error. Correctness with a long column, say of ten
+digits, testifies to knowledge of the process and to the
+power to do nine successive single additions without error.
+Now if a pupil's precision was such that on the average he<span class='pagenum'><a name="Page_32" id="Page_32">[Pg 32]</a></span>
+made one mistake in eight single additions, he would get
+about half of his five-digit columns right and almost none of
+his ten-digit columns right. (He would do this, that is,
+if he added in the customary way. If he were taught to
+check results by repeated addition, by adding in half-columns
+and the like, his percentages of accurate answers might be
+greatly increased in both cases and be made approximately
+equal.) Length of column in a test of addition under
+ordinary conditions thus automatically overweights precision
+in the single additions as compared with knowledge of
+the process, and ability at carrying.</p>
+
+<p>Further, in the case of a column of whatever size, the
+result as ordinarily scored does not distinguish between one,
+two, three, or more (up to the limit) errors in the single
+additions. Yet, obviously, a pupil who, adding with ten-digit
+columns, has half of his answer-figures wrong, probably
+often makes two or more errors within a column, whereas a
+pupil who has only one column-answer in ten wrong, probably
+almost never makes more than one error within a
+column. A short-column test is then advisable as a means
+of interpreting the results of a long-column test.</p>
+
+<p>Finally, the choice of a short-column or of a long-column
+test is indicative of the measurer's notion of the kind of
+efficiency the world properly demands of the school. Twenty
+years ago the author would have been readier to accept a
+long-column test than he now is. In the world at large,
+long-column addition is being more and more done by
+machine, though it persists still in great frequency in the
+bookkeeping of weekly and monthly accounts in local
+groceries, butcher shops, and the like.</p>
+
+<p>The search for a measure of ability to add thus puts the
+problem of speed <i>versus</i> precision, and of short-column
+<i>versus</i> long-column additions clearly before us. The latter<span class='pagenum'><a name="Page_33" id="Page_33">[Pg 33]</a></span>
+problem has hardly been realized at all by the ordinary
+definitions of ability to add.</p>
+
+<p>It may be said further that the measurement of ability to
+add gives the scientific student a shock by the lack of precision
+found everywhere in schools. Of what value is it to a
+graduate of the elementary school to be able to add with
+examples like those of the Courtis test, getting only eight
+out of ten right? Nobody would pay a computer for that
+ability. The pupil could not keep his own accounts with it.
+The supposed disciplinary value of habits of precision
+runs the risk of turning negative in such a case. It appears,
+at least to the author, imperative that checking should be
+taught and required until a pupil can add single columns of
+ten digits with not over one wrong answer in twenty columns.
+Speed is useful, especially indirectly as an indication of
+control of the separate higher-decade additions, but the
+social demand for addition below a certain standard of
+precision is <i>nil</i>, and its disciplinary value is <i>nil</i> or negative.
+This will be made a matter of further study later.</p>
+
+
+<h4>MEASUREMENTS OF ABILITIES IN COMPUTATION</h4>
+
+<p>Measurements of these abilities may be of two sorts&mdash;(1)
+of the speed and accuracy shown in doing one same sort
+of task, as illustrated by the Courtis test for addition shown
+on page 28; and (2) of how hard a task can be done perfectly
+(or with some specified precision) within a certain
+assigned time or less, as illustrated by the author's rough
+test for addition shown on pages 28 and 29, and by the
+Woody tests, when extended to include alternative forms.</p>
+
+<p>The Courtis tests, originated as an improvement on the
+Stone tests and since elaborated by the persistent devotion
+of their author, are a standard instrument of the first
+sort for measuring the so-called 'fundamental' arithmetical<span class='pagenum'><a name="Page_34" id="Page_34">[Pg 34]</a></span>
+abilities with integers. They are shown on this and the
+following page.</p>
+
+<p>Tests of the second sort are the Woody tests, which
+include operations with integers, common and decimal
+fractions, and denominate numbers, the Ballou test for
+common fractions ['16], and the "Ladder" exercises of the
+Thorndike arithmetics. Some of these are shown on pages
+36 to 41.</p>
+
+
+<p class="tabcap">Courtis Test</p>
+
+<p class="tabcap">Arithmetic. &nbsp; &nbsp; Test No. 1. &nbsp; &nbsp; Addition</p>
+
+<p class="center">Series B</p>
+
+<div class="pblockquot">
+<p>You will be given eight minutes to find the answers to as many
+of these addition examples as possible. Write the answers on this
+paper directly underneath the examples. You are not expected
+to be able to do them all. You will be marked for both speed and
+accuracy, but it is more important to have your answers right than
+to try a great many examples.</p></div>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="80%">
+<tr><td align='right'>927</td><td align='right'>297</td><td align='right'>136</td><td align='right'>486</td><td align='right'>384</td><td align='right'>176</td><td align='right'>277</td><td align='right'>837</td></tr>
+<tr><td align='right'>379</td><td align='right'>925</td><td align='right'>340</td><td align='right'>765</td><td align='right'>477</td><td align='right'>783</td><td align='right'>445</td><td align='right'>882</td></tr>
+<tr><td align='right'>756</td><td align='right'>473</td><td align='right'>988</td><td align='right'>524</td><td align='right'>881</td><td align='right'>697</td><td align='right'>682</td><td align='right'>959</td></tr>
+<tr><td align='right'>837</td><td align='right'>983</td><td align='right'>386</td><td align='right'>140</td><td align='right'>266</td><td align='right'>200</td><td align='right'>594</td><td align='right'>603</td></tr>
+<tr><td align='right'>924</td><td align='right'>315</td><td align='right'>353</td><td align='right'>812</td><td align='right'>679</td><td align='right'>366</td><td align='right'>481</td><td align='right'>118</td></tr>
+<tr><td align='right'>110</td><td align='right'>661</td><td align='right'>904</td><td align='right'>466</td><td align='right'>241</td><td align='right'>851</td><td align='right'>778</td><td align='right'>781</td></tr>
+<tr><td align='right'>854</td><td align='right'>794</td><td align='right'>547</td><td align='right'>355</td><td align='right'>796</td><td align='right'>535</td><td align='right'>849</td><td align='right'>756</td></tr>
+<tr><td align='right'>965</td><td align='right'>177</td><td align='right'>192</td><td align='right'>834</td><td align='right'>850</td><td align='right'>323</td><td align='right'>157</td><td align='right'>222</td></tr>
+<tr><td align='right'>344</td><td align='right'>124</td><td align='right'>439</td><td align='right'>567</td><td align='right'>733</td><td align='right'>229</td><td align='right'>953</td><td align='right'>525</td></tr>
+<tr><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td></tr>
+</table></div>
+
+<div class="pblockquot">
+<p class="noidt">and sixteen more addition examples of nine three-place numbers.</p>
+</div>
+
+<p class="tabcap">Courtis Test</p>
+
+<p class="tabcap">Arithmetic. &nbsp; &nbsp; Test No. 2. &nbsp; &nbsp; Subtraction</p>
+
+<p class="center">Series B</p>
+
+<div class="pblockquot">
+<p>You will be given four minutes to find the answers to as many
+of these subtraction examples as possible. Write the answers
+on this paper directly underneath the examples. You are not<span class='pagenum'><a name="Page_35" id="Page_35">[Pg 35]</a></span>
+expected to be able to do them all. You will be marked for both
+speed and accuracy, but it is more important to have your answers
+right than to try a great many examples.</p></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="80%">
+<tr><td align='right'>107795491</td><td align='right'>75088824</td><td align='right'>91500053</td><td align='right'>87939983</td></tr>
+<tr><td align='right'>77197029</td><td align='right'>57406394</td><td align='right'>19901563</td><td align='right'>72207316</td></tr>
+<tr><td align='right'>&mdash;&mdash;&mdash;&mdash;&mdash;</td><td align='right'>&mdash;&mdash;&mdash;&mdash;&mdash;</td><td align='right'>&mdash;&mdash;&mdash;&mdash;&mdash;</td><td align='right'>&mdash;&mdash;&mdash;&mdash;&mdash;</td></tr>
+</table></div>
+<div class="pblockquot"><p class="noidt">and twenty more tasks of the same sort.</p></div>
+
+
+<p class="tabcap">Courtis Test</p>
+
+<p class="tabcap">Arithmetic. &nbsp; &nbsp; Test No. 3. &nbsp; &nbsp; Multiplication</p>
+
+<p class="center">Series B</p>
+
+<div class="pblockquot">
+<p>You will be given six minutes to work as many of these multiplication
+examples as possible. You are not expected to be able
+to do them all. Do your work directly on this paper; use no other.
+You will be marked for both speed and accuracy, but it is more
+important to get correct answers than to try a large number of
+examples.</p></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="80%">
+<tr><td align='right'>8246</td><td align='right'>7843</td><td align='right'>4837</td><td align='right'>3478</td><td align='right'>6482</td></tr>
+<tr><td align='right'>29</td><td align='right'>702</td><td align='right'>83</td><td align='right'>15</td><td align='right'>46</td></tr>
+<tr><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td></tr>
+</table></div>
+
+<div class="pblockquot">
+<p class="noidt">and twenty more multiplication examples of the same sort.</p>
+</div>
+
+
+<p class="tabcap">Courtis Test</p>
+
+<p class="tabcap">Arithmetic. &nbsp; &nbsp; Test No. 4. &nbsp; &nbsp; Division</p>
+
+<p class="center">Series B</p>
+
+<div class="pblockquot">
+<p>You will be given eight minutes to work as many of these division
+examples as possible. You are not expected to be able to
+do them all. Do your work directly on this paper; use no other.
+You will be marked for both speed and accuracy, but it is more
+important to get correct answers than to try a large number of
+examples.</p></div>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='right'>25 <span class='overline'>) 6775</span></td><td align='right'>94 <span class='overline'>) 85352</span></td><td align='right'>37 <span class='overline'>) 9990</span></td><td align='right'>86 <span class='overline'>) 80066</span></td></tr>
+</table></div>
+
+<div class="pblockquot"><p class="noidt">and twenty more division examples of the same sort.</p></div>
+
+<p><span class='pagenum'><a name="Page_36" id="Page_36">[Pg 36]</a></span></p>
+
+<p class="tabcap">SERIES B. &nbsp; &nbsp;    MULTIPLICATION  SCALE</p>
+<p class="tabcap">By Clifford Woody</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><th align='right'>(1)</th><th align='right'>(3)</th><th align='right'>(4)</th><th align='right'>(5)</th></tr>
+<tr>
+  <td align='right'>3 &times; 7 =</td>
+  <td align='right'>2 &times; 3 =</td>
+  <td align='right'>4 &times; 8 =</td>
+  <td align='right'>23<br />3<br />&mdash;</td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='right'>(8)</th><th align='right'>(9)</th><th align='right'>(11)</th><th align='right'>(12)</th></tr>
+<tr>
+  <td align='right'>50<br />3<br />&mdash;</td>
+  <td align='right'>254<br />6<br />&mdash;</td>
+  <td align='right'>1036<br />8<br />&mdash;</td>
+  <td align='right'>5096<br />6<br />&mdash;</td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='right'>(13)</th><th align='right'>(16)</th><th align='right'>(18)</th><th align='right'>(20)</th></tr>
+<tr>
+  <td align='right'>8754<br />8<br />&mdash;&mdash;</td>
+  <td align='right'>7898<br />9<br />&mdash;&mdash;</td>
+  <td align='right'>24<br />234<br />&mdash;&mdash;</td>
+  <td align='right'>287<br />.05<br />&mdash;&mdash;</td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='right'>(24)</th><th align='right'>(26)</th><th align='right'>(27)</th><th align='right'>(29)</th></tr>
+<tr>
+  <td align='right'>16 &nbsp; &nbsp;<br />2<sup>5</sup>&frasl;<sub>8</sub><br />&mdash;&mdash;</td>
+  <td align='right'>9742<br />59<br />&mdash;&mdash;</td>
+  <td align='right'>6.25<br />3.2<br />&mdash;&mdash;</td>
+  <td align='right'><sup>1</sup>&frasl;<sub>8</sub> &times; 2 =</td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='right'>(33)</th><th align='right'>(35)</th><th align='right'>(37)</th><th align='right'>(38)</th></tr>
+<tr>
+  <td align='right'>2&frac12; &times; 3&frac12; =</td>
+  <td align='right'>987&frac34;<br />25 &nbsp; <br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>2&frac14; &times; 4&frac12; &times; 1&frac12; =</td>
+  <td align='right'>.0963<sup>1</sup>&frasl;<sub>8</sub><br />.084 &nbsp; &nbsp;<br />&mdash;&mdash;</td>
+</tr>
+</table></div>
+
+
+
+
+<p><span class='pagenum'><a name="Page_37" id="Page_37">[Pg 37]</a></span></p>
+
+<p class="tabcap">SERIES B. DIVISION  SCALE<br />
+By Clifford Woody</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><th align='center'>(1)</th><th align='center'>(2)</th><th align='center'>(7)</th><th align='center'>(8)</th></tr>
+<tr>
+  <td align='center'>3 )<span class="overline"> 6 &nbsp;&nbsp;</span></td>
+  <td align='center'>9 )<span class="overline"> 27 &nbsp;</span></td>
+  <td align='center'>4 &divide; 2 =</td>
+  <td align='center'>9 )<span class="overline"> 0 &nbsp;&nbsp;</span></td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='center'>(11)</th><th align='center'>(14)</th><th align='center'>(15)</th><th align='center'>(17)</th></tr>
+<tr>
+  <td align='center'>2 )<span class="overline"> 13 &nbsp;&nbsp;</span></td>
+  <td align='center'>8 )<span class="overline"> 5856 &nbsp;</span></td>
+  <td align='center'>&frac14; of 128 =</td>
+  <td align='center'>50 &divide; 7 =</td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='center'>(19)</th><th align='center'>(23)</th><th align='center'>(27)</th><th align='center'>(28)</th></tr>
+<tr>
+  <td align='center'>248 &divide; 7 =</td>
+  <td align='center'>23 )<span class="overline"> 469 &nbsp;&nbsp;</span></td>
+  <td align='center'><sup>7</sup>&frasl;<sub>8</sub> of 624 =</td>
+  <td align='center'>.003 )<span class="overline"> .0936 &nbsp;</span></td>
+</tr>
+<tr><td colspan='4'>&nbsp;</td></tr>
+<tr><th align='center'>(30)</th><th align='center'>(34)</th><th align='center'>(36)</th></tr>
+<tr>
+  <td align='center'><sup>3</sup>&frasl;<sub>4</sub> &divide; 5 =</td>
+  <td align='center'>62.50 &divide; 1&frac14; =</td>
+  <td align='center'>9 )<span class="overline"> 69 lbs. 9 oz. </span></td>
+</tr>
+</table></div>
+
+
+<p><span class='pagenum'><a name="Page_38" id="Page_38">[Pg 38]</a></span></p>
+
+<p class="tabcap">Ballou Test</p>
+<p class="tabcap">Addition of Fractions</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%">
+<tr><th></th><th><i>Test 1</i></th><th></th><th><i>Test 2</i></th></tr>
+<tr>
+  <td align='right'><b>(1)</b> &nbsp; &frac14;<br />&frac14;<br />&mdash;</td>
+  <td align='right'><b>(2)</b> &nbsp; <sup>3</sup>&frasl;<sub>14</sub><br /><sup>1</sup>&frasl;<sub>14</sub><br />&mdash;</td>
+  <td align='right'><b>(1)</b> &nbsp; <sup>1</sup>&frasl;<sub>3</sub><br /><sup>1</sup>&frasl;<sub>6</sub><br />&mdash;</td>
+  <td align='right'><b>(2)</b> &nbsp; <sup>2</sup>&frasl;<sub>7</sub><br /><sup>3</sup>&frasl;<sub>14</sub><br />&mdash;</td>
+</tr>
+
+<tr><th></th><th><i>Test 3</i></th><th></th><th><i>Test 4</i></th></tr>
+<tr>
+  <td align='right'><b>(1)</b> &nbsp; <sup>3</sup>&frasl;<sub>5</sub><br /><sup>11</sup>&frasl;<sub>15</sub><br />&mdash;</td>
+  <td align='right'><b>(2)</b> &nbsp; <sup>5</sup>&frasl;<sub>6</sub><br /><sup>1</sup>&frasl;<sub>2</sub><br />&mdash;</td>
+  <td align='right'><b>(1)</b> &nbsp; <sup>1</sup>&frasl;<sub>7</sub><br /><sup>9</sup>&frasl;<sub>10</sub><br />&mdash;</td>
+  <td align='right'><b>(2)</b> &nbsp; <sup>7</sup>&frasl;<sub>9</sub><br /><sup>1</sup>&frasl;<sub>4</sub><br />&mdash;</td>
+</tr>
+
+<tr><th></th><th><i>Test 5</i></th><th></th><th><i>Test 6</i></th></tr>
+<tr>
+  <td align='right'><b>(1)</b> &nbsp; <sup>1</sup>&frasl;<sub>10</sub><br /><sup>1</sup>&frasl;<sub>6</sub><br />&mdash;</td>
+  <td align='right'><b>(2)</b> &nbsp; <sup>4</sup>&frasl;<sub>9</sub><br /><sup>5</sup>&frasl;<sub>12</sub><br />&mdash;</td>
+  <td align='right'><b>(1)</b> &nbsp; <sup>1</sup>&frasl;<sub>6</sub><br /><sup>9</sup>&frasl;<sub>10</sub><br />&mdash;</td>
+  <td align='right'><b>(2)</b> &nbsp; <sup>5</sup>&frasl;<sub>6</sub><br /><sup>3</sup>&frasl;<sub>8</sub><br />&mdash;</td>
+</tr>
+
+</table></div>
+
+
+<p class="tabcap">An Addition Ladder [Thorndike, '17, III, 5]</p>
+
+<p>Begin at the bottom of the ladder. See if you can climb to the
+top without making a mistake. Be sure to copy the numbers
+correctly.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='left'><b>Step 6.</b></td><td align='left'><i>a.</i> Add 1<sup>1</sup>&frasl;<sub>3</sub> yd., <sup>7</sup>&frasl;<sub>8</sub> yd., 1&frac14; yd., <sup>3</sup>&frasl;<sub>4</sub> yd., <sup>7</sup>&frasl;<sub>8</sub> yd., and 1&frac12; yd.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> Add 62&frac12;&cent;, 66<sup>2</sup>&frasl;<sub>3</sub>&cent;, 56&frac14;&cent;, 60&cent;, and 62&frac12;&cent;.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> Add 1<sup>5</sup>&frasl;<sub>16</sub>, 1<sup>9</sup>&frasl;<sub>32</sub>, 1<sup>3</sup>&frasl;<sub>8</sub>, 1<sup>11</sup>&frasl;<sub>32</sub>, and 1<sup>7</sup>&frasl;<sub>16</sub>.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> Add 1<sup>1</sup>&frasl;<sub>3</sub> yd., 1&frac14; yd., 1&frac12; yd., 2 yd., <sup>3</sup>&frasl;<sub>4</sub> yd., and <sup>2</sup>&frasl;<sub>3</sub> yd.</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 5.</b></td><td align='left'><i>a.</i> Add 4 ft. 6&frac12; in., 53&frac14; in., 5 ft. &frac12; in., 56&frac34; in., and 5 ft.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> Add 7 lb., 6 lb. 11 oz., 7&frac12; lb., 6 lb. 4&frac12; oz., and 8&frac12; lb.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min., and 55 min.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes, and 19 nickels.</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 4.</b></td><td align='left'><i>a.</i> Add .05&frac12;, .06, .04&frac34;, .02&frac34;, and .05&frac14;.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> Add .33<sup>1</sup>&frasl;<sub>3</sub>, .12&frac12;, .18, .16<sup>2</sup>&frasl;<sub>3</sub>, .08<sup>1</sup>&frasl;<sub>3</sub> and .15.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> Add .08<sup>1</sup>&frasl;<sub>3</sub>, .06&frac14;, .21, .03&frac34;, and .16<sup>2</sup>&frasl;<sub>3</sub>.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> Add .62, .64&frac12;, .66<sup>2</sup>&frasl;<sub>3</sub>, .10&frac14;, and .68.</td></tr>
+<tr><td align='left'>&nbsp;<span class='pagenum'><a name="Page_39" id="Page_39">[Pg 39]</a></span></td></tr>
+<tr><td align='left'><b>Step 3.</b></td><td align='left'><i>a.</i> Add 7&frac14;, 6&frac12;, 8<sup>3</sup>&frasl;<sub>8</sub>, 5&frac34;, 9<sup>5</sup>&frasl;<sub>8</sub> and 3<sup>7</sup>&frasl;<sub>8</sub>.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> Add 4<sup>5</sup>&frasl;<sub>8</sub>, 12, 7&frac12;, 8&frac34;, 6 and 5&frac14;.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> Add 9&frac34;, 5<sup>7</sup>&frasl;<sub>8</sub>, 4<sup>1</sup>&frasl;<sub>8</sub>, 6&frac12;, 7, 3<sup>5</sup>&frasl;<sub>8</sub>.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> Add 12, 8&frac12;, 7<sup>1</sup>&frasl;<sub>3</sub>, 5, 6<sup>2</sup>&frasl;<sub>3</sub>, and 9&frac12;.</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 2.</b></td><td align='left'><i>a.</i> Add 12.04, .96, 4.7, 9.625, 3.25, and 20.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> Add .58, 6.03, .079, 4.206, 2.75, and 10.4.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> Add 52, 29.8, 41.07, 1.913, 2.6, and 110.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05.</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 1.</b></td><td align='left'><i>a.</i> Add 10<sup>3</sup>&frasl;<sub>5</sub>, 11<sup>1</sup>&frasl;<sub>5</sub>, 10<sup>4</sup>&frasl;<sub>5</sub>, 11, 11<sup>2</sup>&frasl;<sub>5</sub>, 10<sup>3</sup>&frasl;<sub>5</sub>, and 11.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> Add 7<sup>3</sup>&frasl;<sub>8</sub>, 6<sup>5</sup>&frasl;<sub>8</sub>, 8, 9<sup>1</sup>&frasl;<sub>8</sub>, 7<sup>7</sup>&frasl;<sub>8</sub>, 5<sup>3</sup>&frasl;<sub>8</sub>, and 8<sup>1</sup>&frasl;<sub>8</sub>.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> Add 21&frac12;, 18&frac34;, 31&frac12;, 19&frac14;, 17&frac14;, 22, and 16&frac12;.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> Add 14<sup>5</sup>&frasl;<sub>12</sub>, 12<sup>7</sup>&frasl;<sub>12</sub>, 9<sup>11</sup>&frasl;<sub>12</sub>, 6<sup>1</sup>&frasl;<sub>12</sub>, and 5.</td></tr>
+</table></div>
+
+
+
+<p class="tabcap">A Subtraction Ladder [Thorndike, '17, III, 11]</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><th><b>Step 9.</b></th></tr>
+<tr><td></td>
+  <td align='left' colspan='2'><i>a.</i> 2.16 mi. &minus; 1&frac34; mi.<br />
+     <i>c.</i> 2 min. 10&frac12; sec. &minus; 93.4 sec.<br />
+     <i>e.</i> 10 gal. 2&frac12; qt. &minus; 4.623 gal.</td>
+  <td align='left' colspan='2'><i>b.</i> 5.72 ft. &minus; 5 ft. 3 in.<br />
+     <i>d.</i> 30.28 A. &minus; 10<sup>1</sup>&frasl;<sub>5</sub> A.<br /> &nbsp;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 8.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>25<sup>7</sup>&frasl;<sub>12</sub><br />12<sup>3</sup>&frasl;<sub>4</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>10<sup>1</sup>&frasl;<sub>4</sub><br />7<sup>1</sup>&frasl;<sub>3</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>9<sup>5</sup>&frasl;<sub>16</sub><br />6<sup>3</sup>&frasl;<sub>8</sub>&nbsp;<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>5<sup>7</sup>&frasl;<sub>16</sub><br />2<sup>3</sup>&frasl;<sub>4</sub>&nbsp;<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>4<sup>2</sup>&frasl;<sub>3</sub><br />1<sup>3</sup>&frasl;<sub>4</sub><br />&mdash;&mdash;&mdash;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 7.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>28<sup>3</sup>&frasl;<sub>4</sub><br />16<sup>1</sup>&frasl;<sub>8</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>40<sup>1</sup>&frasl;<sub>2</sub><br />14<sup>3</sup>&frasl;<sub>8</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>10<sup>1</sup>&frasl;<sub>4</sub><br />6<sup>1</sup>&frasl;<sub>2</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>24<sup>1</sup>&frasl;<sub>3</sub><br />11<sup>1</sup>&frasl;<sub>2</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>37<sup>1</sup>&frasl;<sub>2</sub><br />14<sup>3</sup>&frasl;<sub>4</sub><br />&mdash;&mdash;&mdash;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 6.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>10<sup>1</sup>&frasl;<sub>3</sub><br />4<sup>2</sup>&frasl;<sub>3</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>7<sup>1</sup>&frasl;<sub>4</sub><br />2<sup>3</sup>&frasl;<sub>4</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>15<sup>1</sup>&frasl;<sub>8</sub><br />6<sup>3</sup>&frasl;<sub>8</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>12<sup>1</sup>&frasl;<sub>5</sub><br />11<sup>4</sup>&frasl;<sub>5</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>4<sup>1</sup>&frasl;<sub>16</sub><br />2<sup>7</sup>&frasl;<sub>16</sub><br />&mdash;&mdash;&mdash;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 5.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>58<sup>4</sup>&frasl;<sub>5</sub><br />52<sup>1</sup>&frasl;<sub>5</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>66<sup>2</sup>&frasl;<sub>3</sub><br />33<sup>1</sup>&frasl;<sub>3</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>28<sup>7</sup>&frasl;<sub>8</sub><br />7<sup>5</sup>&frasl;<sub>8</sub><br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>62&frac12;<br />37&frac12;<br />&mdash;&mdash;</td>
+  <td align='right'>9<sup>7</sup>&frasl;<sub>12</sub><br />4<sup>5</sup>&frasl;<sub>12</sub><br />&mdash;&mdash;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 4.</b></th></tr>
+<tr><td></td>
+  <td align='left' colspan='2'><i>a.</i> 4 hr. &minus; 2 hr. 17 min.<br />
+     <i>c.</i> 1 lb. 5 oz. &minus; 13 oz.<br />
+     <i>e.</i> 1 bu. &minus; 1 pk.</td>
+  <td align='left' colspan='2'><i>b.</i> 4 lb. 7 oz. &minus; 2 lb. 11 oz.<br />
+     <i>d.</i> 7 ft. &minus; 2 ft. 8 in.<br /> &nbsp;</td>
+</tr>
+<tr><td align='left'>&nbsp;<span class='pagenum'><a name="Page_40" id="Page_40">[Pg 40]</a></span></td></tr>
+<tr><th><b>Step 3.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>92 &nbsp;&nbsp; &nbsp; mi.<br />84.15 mi.<br />&mdash;&mdash;&mdash;&mdash;</td>
+  <td align='right'>6735 mi.<br />6689 mi.<br />&mdash;&mdash;&mdash;&mdash;</td>
+  <td align='right'>$3 &minus; 89&cent;<br /><br />&mdash;&mdash;&mdash;&mdash;</td>
+  <td align='right'>28.4 &nbsp; mi.<br />18.04 mi.<br />&mdash;&mdash;&mdash;&mdash;</td>
+  <td align='right'>$508.40<br />208.62<br />&mdash;&mdash;&mdash;&mdash;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 2.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>$25.00<br />9.36<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>$100.00<br />71.28<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>$750.00<br />736.50<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>6124 sq. mi.<br />2494 sq. mi.<br />&mdash;&mdash;&mdash;&mdash;&mdash;</td>
+  <td align='right'>7846 sq. mi.<br />2789 sq. mi.<br />&mdash;&mdash;&mdash;&mdash;&mdash;</td>
+</tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><th><b>Step 1.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr>
+<tr><td></td>
+  <td align='right'>$18.64<br />7.40<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>$25.39<br />13.37<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>$56.70<br />45.60<br />&mdash;&mdash;&mdash;</td>
+  <td align='right'>819.4 mi.<br />209.2 mi.<br />&mdash;&mdash;&mdash;&mdash;</td>
+  <td align='right'>67.55 mi.<br />36.14 mi.<br />&mdash;&mdash;&mdash;&mdash;</td>
+</tr>
+</table></div>
+
+<p class="tabcap">An Average Ladder [Thorndike, '17, III, 132]</p>
+
+<p>Find the average of the quantities on each line. Begin with
+<b>Step 1</b>. Climb to the top without making a mistake. Be sure
+to copy the numbers correctly. Extend the division to two
+decimal places if necessary.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='left'><b>Step 6.</b></td><td align='left'><i>a</i>. 2<sup>2</sup>&frasl;<sub>3</sub>, 1<sup>7</sup>&frasl;<sub>8</sub>, 2&frac34;, 4&frac14;, 3<sup>5</sup>&frasl;<sub>8</sub>, 3&frac12;</td></tr>
+<tr><td align='left'></td><td align='left'><i>b</i>. 62&frac12;&cent;, 66<sup>2</sup>&frasl;<sub>3</sub>&cent;, 40&cent;, 83<sup>1</sup>&frasl;<sub>3</sub>&cent;, $1.75, $2.25</td></tr>
+<tr><td align='left'></td><td align='left'><i>c</i>. 3<sup>11</sup>&frasl;<sub>16</sub>, 3<sup>9</sup>&frasl;<sub>32</sub>, 3<sup>3</sup>&frasl;<sub>8</sub>, 3<sup>17</sup>&frasl;<sub>32</sub>, 3<sup>7</sup>&frasl;<sub>16</sub></td></tr>
+<tr><td align='left'></td><td align='left'><i>d</i>. .17, 19, .16<sup>2</sup>&frasl;<sub>3</sub>, .15&frac12;, .23&frac14;, .18</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 5.</b></td><td align='left'><i>a</i>. 5 ft. 3&frac12; in., 61&frac14; in., 58&frac34; in., 4 ft. 11 in.</td></tr>
+<tr><td align='left'></td><td align='left'><i>b</i>. 6 lb. 9 oz., 6 lb. 11 oz., 7&frac14; lb., 7<sup>3</sup>&frasl;<sub>8</sub> lb.</td></tr>
+<tr><td align='left'></td><td align='left'><i>c</i>. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1&frac14; hr.</td></tr>
+<tr><td align='left'></td><td align='left'><i>d</i>. 2.8 miles, 3&frac12; miles, 2.72 miles</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 4.</b></td><td align='left'><i>a</i>. .03&frac12;, .06, .04&frac34;, .05&frac12;, .05&frac14;</td></tr>
+<tr><td align='left'></td><td align='left'><i>b</i>. .043, .045, .049, .047, .046, .045</td></tr>
+<tr><td align='left'></td><td align='left'><i>c</i>. 2.20, .87&frac12;, 1.18, .93&frac34;, 1.2925, .80</td></tr>
+<tr><td align='left'></td><td align='left'><i>d</i>. .14&frac12;, .12&frac12;, .33<sup>1</sup>&frasl;<sub>3</sub>, .16<sup>2</sup>&frasl;<sub>3</sub>, .15, .17</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 3.</b></td><td align='left'><i>a</i>. 5&frac14;, 4&frac12;, 8<sup>3</sup>&frasl;<sub>8</sub>, 7&frac34;, 6<sup>5</sup>&frasl;<sub>8</sub>, 9<sup>3</sup>&frasl;<sub>8</sub></td></tr>
+<tr><td align='left'></td><td align='left'><i>b</i>. 9<sup>5</sup>&frasl;<sub>8</sub>, 12, 8&frac12;, 8&frac34;, 6, 5&frac14;, 9</td></tr>
+<tr><td align='left'></td><td align='left'><i>c</i>. 9<sup>3</sup>&frasl;<sub>8</sub>, 5&frac34;, 4<sup>1</sup>&frasl;<sub>8</sub>, 7&frac12;, 6</td></tr>
+<tr><td align='left'></td><td align='left'><i>d</i>. 11, 9&frac12;, 10<sup>1</sup>&frasl;<sub>3</sub>, 13, 16<sup>2</sup>&frasl;<sub>3</sub>, 9&frac12;</td></tr>
+<tr><td align='left'>&nbsp;<span class='pagenum'><a name="Page_41" id="Page_41">[Pg 41]</a></span></td></tr>
+<tr><td align='left'><b>Step 2.</b></td><td align='left'><i>a.</i> 13.05, .97, 4.8, 10.625, 3.37</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> 68, 71.4, 59.8, 112, 96.1, 79.8</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> 2.079, 3.908, 4.165, 2.74</td></tr>
+<tr><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>Step 1.</b></td><td align='left'><i>a.</i> 4, 9&frac12;, 6, 5, 7&frac12;, 8, 10, 9</td></tr>
+<tr><td align='left'></td><td align='left'><i>b.</i> 6, 5, 3.9, 7.1, 8</td></tr>
+<tr><td align='left'></td><td align='left'><i>c.</i> 1086, 1141, 1059, 1302, 1284</td></tr>
+<tr><td align='left'></td><td align='left'><i>d.</i> $100.82, $206.49, $317.25, $244.73</td></tr>
+</table></div>
+
+<p>As such tests are widened to cover the whole task of the
+elementary school in respect to arithmetic, and accepted by
+competent authorities as adequate measures of achievement
+in computing, they will give, as has been said, a working
+definition of the task. The reader will observe, for example,
+that work such as the following, though still found in many
+textbooks and classrooms, does not, in general, appear in
+the modern tests and scales.</p>
+
+<p>Reduce the following improper fractions to mixed numbers:&mdash;</p>
+
+<p class="center">
+<sup>19</sup>&frasl;<sub>13</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>43</sup>&frasl;<sub>21</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>176</sup>&frasl;<sub>25</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>198</sup>&frasl;<sub>14</sub>
+</p>
+
+
+<p>Reduce to integral or mixed numbers:&mdash;</p>
+
+<p class="center">
+<sup>61381</sup>&frasl;<sub>37</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>2134</sup>&frasl;<sub>67</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>413</sup>&frasl;<sub>413</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>697</sup>&frasl;<sub>225</sub>
+</p>
+
+
+<p>Simplify:&mdash;</p>
+
+<p class="center">
+<sup>3</sup>&frasl;<sub>4</sub> &nbsp;  of &nbsp;
+<sup>8</sup>&frasl;<sub>9</sub> &nbsp;  of &nbsp;
+<sup>3</sup>&frasl;<sub>5</sub>  &nbsp; of &nbsp;
+<sup>15</sup>&frasl;<sub>22</sub>
+</p>
+
+
+<p>Reduce to lowest terms:&mdash;</p>
+
+<p class="center">
+<sup>357</sup>&frasl;<sub>527</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>264</sup>&frasl;<sub>312</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>492</sup>&frasl;<sub>779</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>418</sup>&frasl;<sub>874</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>854</sup>&frasl;<sub>1769</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>30</sup>&frasl;<sub>735</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>44</sup>&frasl;<sub>242</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>77</sup>&frasl;<sub>847</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>18</sup>&frasl;<sub>243</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>96</sup>&frasl;<sub>224</sub>
+</p>
+
+<p>Find differences:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr>
+<td align='left'>6<sup>2</sup>&frasl;<sub>7</sub><br />3<sup>1</sup>&frasl;<sub>14</sub><br />&mdash;&mdash;</td>
+<td align='left'>8<sup>5</sup>&frasl;<sub>11</sub><br />5<sup>1</sup>&frasl;<sub>7</sub><br />&mdash;&mdash;</td>
+<td align='left'>8<sup>4</sup>&frasl;<sub>13</sub><br />3<sup>7</sup>&frasl;<sub>13</sub><br />&mdash;&mdash;</td>
+<td align='left'>5<sup>1</sup>&frasl;<sub>4</sub><br />2<sup>11</sup>&frasl;<sub>14</sub><br />&mdash;&mdash;</td>
+<td align='left'>7<sup>1</sup>&frasl;<sub>8</sub><br />2<sup>1</sup>&frasl;<sub>7</sub><br />&mdash;&mdash;</td>
+</tr>
+</table></div>
+
+
+<p>Square:&mdash;</p>
+
+
+<p class="center">
+<sup>2</sup>&frasl;<sub>3</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>4</sup>&frasl;<sub>5</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>5</sup>&frasl;<sub>7</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>6</sup>&frasl;<sub>9</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>10</sup>&frasl;<sub>11</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>12</sup>&frasl;<sub>13</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>2</sup>&frasl;<sub>7</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>15</sup>&frasl;<sub>16</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>19</sup>&frasl;<sub>20</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>17</sup>&frasl;<sub>18</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>25</sup>&frasl;<sub>30</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+<sup>41</sup>&frasl;<sub>53</sub>
+</p>
+
+<p>Multiply:&mdash;</p>
+
+<p class="center">
+<sup>2</sup>&frasl;<sub>11</sub> &times; 33
+&nbsp; &nbsp; &nbsp; &nbsp;
+ 32 &times; <sup>3</sup>&frasl;<sub>14</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+ 39 &times; <sup>2</sup>&frasl;<sub>13</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+ 60 &times; <sup>11</sup>&frasl;<sub>28</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+ 77 &times; <sup>4</sup>&frasl;<sub>11</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+ 63 &times; <sup>2</sup>&frasl;<sub>27</sub>
+
+<br />
+
+54 &times; <sup>8</sup>&frasl;<sub>45</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+65 &times; <sup>3</sup>&frasl;<sub>13</sub>
+&nbsp; &nbsp; &nbsp; &nbsp;
+344<sup>16</sup>&frasl;<sub>21</sub> &nbsp;&nbsp; 432<sup>2</sup>&frasl;<sub>7</sub>
+</p>
+
+<p><span class='pagenum'><a name="Page_42" id="Page_42">[Pg 42]</a></span></p>
+
+<h4>MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE
+SOLUTION OF PROBLEMS</h4>
+
+<p>Stone ['08] measured achievement with the following
+problems, fifteen minutes being the time allowed.</p>
+
+<p>"Solve as many of the following problems as you have
+time for; work them in order as numbered:</p>
+
+<div class="pblockquot"><p>1. If you buy 2 tablets at 7 cents each and a book for 65 cents,
+how much change should you receive from a two-dollar bill?</p>
+
+<p>2. John sold 4 Saturday Evening Posts at 5 cents each. He
+kept <sup>1</sup>&frasl;<sub>2</sub> the money and with the other <sup>1</sup>&frasl;<sub>2</sub> he bought Sunday papers
+at 2 cents each. How many did he buy?</p>
+
+<p>3. If James had 4 times as much money as George, he would
+have $16. How much money has George?</p>
+
+<p>4. How many pencils can you buy for 50 cents at the rate of
+2 for 5 cents?                                                                                        '</p>
+
+<p>5. The uniforms for a baseball nine cost $2.50 each. The
+shoes cost $2 a pair. What was the total cost of uniforms and
+shoes for the nine?</p>
+
+<p>6. In the schools of a certain city there are 2200 pupils; <sup>1</sup>&frasl;<sub>2</sub>
+are in the primary grades, <sup>1</sup>&frasl;<sub>4</sub> in the grammar grades, <sup>1</sup>&frasl;<sub>8</sub> in the high
+school, and the rest in the night school. How many pupils are
+there in the night school?</p>
+
+<p>7. If 3&frac12; tons of coal cost $21, what will 5&frac12; tons cost?</p>
+
+<p>8. A news dealer bought some magazines for $1. He sold
+them for $1.20, gaining 5 cents on each magazine. How many
+magazines were there?</p>
+
+<p>9. A girl spent <sup>1</sup>&frasl;<sub>8</sub> of her money for car fare, and three times
+as much for clothes. Half of what she had left was 80 cents.
+How much money did she have at first?</p>
+
+<p>10. Two girls receive $2.10 for making buttonholes. One
+makes 42, the other 28. How shall they divide the money?</p>
+
+<p>11. Mr. Brown paid one third of the cost of a building; Mr.
+Johnson paid <sup>1</sup>&frasl;<sub>2</sub> the cost. Mr. Johnson received $500 more annual
+rent than Mr. Brown. How much did each receive?</p>
+
+<p>12. A freight train left Albany for New York at 6 o'clock. An
+express left on the same track at 8 o'clock. It went at the rate
+of 40 miles an hour. At what time of day will it overtake the freight
+train if the freight train stops after it has gone 56 miles?"</p></div>
+<p><span class='pagenum'><a name="Page_43" id="Page_43">[Pg 43]</a></span></p>
+
+<p>The criteria he had in mind in selecting the problems
+were as follows:&mdash;</p>
+
+<p>"The main purpose of the reasoning test is the determination
+of the ability of VI A children to reason in arithmetic.
+To this end, the problems, as selected and arranged,
+are meant to embody the following conditions:&mdash;</p>
+
+<div class="pblockquot">
+<p>1. Situations equally concrete to all VI A children.</p>
+
+<p>2. Graduated difficulties.<br />
+<span style="margin-left: 2.5em;"><i>a.</i> As to arithmetical thinking.</span><br />
+<span style="margin-left: 2.5em;"><i>b.</i> As to familiarity with the situation presented.</span></p>
+
+<p>3. The omission of<br />
+<span style="margin-left: 2.5em;"><i>a.</i> Large numbers.</span><br />
+<span style="margin-left: 2.5em;"><i>b.</i> Particular memory requirements.</span><br />
+<span style="margin-left: 2.5em;"><i>c.</i> Catch problems.</span><br />
+<span style="margin-left: 2.5em;"><i>d.</i> All subject matter except whole numbers, fractions, and United States money.</span></p>
+</div>
+
+<p>The test is purposely so long that only very rarely did any
+pupil fully complete it in the fifteen minute limit."</p>
+
+<p>Credits were given of 1, for each of the first five problems,
+1.4, 1.2, and 1.6 respectively for problems 6, 7, and 8, and of
+2 for each of the others.</p>
+
+<p>Courtis sought to improve the Stone test of problem-solving,
+replacing it by the two tests reproduced below.</p>
+
+<p class="tabcap">ARITHMETIC&mdash;Test No. 6. &nbsp; &nbsp; &nbsp; Speed Test&mdash;Reasoning</p>
+
+<div class="pblockquot">
+
+<p><b>Do not work</b> the following examples. Read each example through, make
+up your mind what operation you would use if you were going to work it,
+then write the name of the operation selected in the blank space after the
+example. Use the following abbreviations:&mdash;"Add." for addition, "Sub."
+for subtraction, "Mul." for multiplication, and "Div." for division.</p>
+
+<div class='center'>
+<table cellpadding="4" cellspacing="0" summary="">
+<tr><td align='justify'>&nbsp;</td><td class='bbox'><span class="smcap">Operation</span></td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>1. A girl brought a collection of 37 colored postal cards
+to school one day, and gave away 19 cards to her friends.
+How many cards did she have left to take home?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>2. Five boys played marbles. When the game was
+over, each boy had the same number of marbles. If there
+were 45 marbles altogether, how many did each boy have?</td>
+<td class='bbox'>&nbsp;<span class='pagenum'><a name="Page_44" id="Page_44">[Pg 44]</a></span></td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>3. A girl, watching from a window, saw 27 automobiles
+pass the school the first hour, and 33 the second. How
+many autos passed by the school in the two hours?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>4. In a certain school there were eight rooms and each
+room had seats for 50 children. When all the places were
+taken, how many children were there in the school?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>5. A club of boys sent their treasurer to buy baseballs.
+They gave him $3.15 to spend. How many balls did they
+expect him to buy, if the balls cost 45&cent;. apiece?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>6. A teacher weighed all the girls in a certain grade. If
+one girl weighed 79 pounds and another 110 pounds, how
+many pounds heavier was one girl than the other?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>7. A girl wanted to buy a 5-pound box of candy to give
+as a present to a friend. She decided to get the kind worth
+35&cent;. a pound. What did she pay for the present?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>8. One day in vacation a boy went on a fishing trip and
+caught 12 fish in the morning, and 7 in the afternoon. How
+many fish did he catch altogether?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>9. A boy lived 15 blocks east of a school; his chum lived
+on the same street, but 11 blocks west of the school. How
+many blocks apart were the two boys' houses?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>10. A girl was 5 times as strong as her small sister. If
+the little girl could lift a weight of 20 pounds, how large a
+weight could the older girl lift?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>11. The children of a school gave a sleigh-ride party.
+There were 270 children to go on the ride and each sleigh
+held 30 children. How many sleighs were needed?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>12. In September there were 43 children in the eighth
+grade of a certain school; by June there were 59. How
+many children entered the grade during the year?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>13. A girl who lived 17 blocks away walked to school and
+back twice a day. What was the total number of blocks the
+girl walked each day in going to and from school?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>14. A boy who made 67&cent;. a day carrying papers, was
+hired to run on a long errand for which he received 50&cent;.
+What was the total amount the boy earned that day?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='right'>Total Right</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+</table></div>
+
+
+<p>(Two more similar problems follow.)</p>
+
+<p>Test 6 and Test 8 are from the Courtis Standard Test. Used by permission
+of S. A. Courtis.</p></div>
+<p><span class='pagenum'><a name="Page_45" id="Page_45">[Pg 45]</a></span></p>
+
+
+<p class="tabcap">ARITHMETIC&mdash;Test No. 8. &nbsp; &nbsp; &nbsp; Reasoning</p>
+
+<div class="pblockquot">
+<p>In the blank space below, work as many of the following examples as possible
+in the time allowed. Work them in order as numbered, entering each answer
+in the "answer" column before commencing a new example. Do not work
+on any other paper.</p>
+
+<div class='center'>
+<table cellpadding="4" cellspacing="0" summary="">
+<tr><td align='justify'>&nbsp;</td><td class='bbox'><span class="smcap">Answer</span></td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>1. The children in a certain school gave a Christmas
+party. One of the presents was a box of candy. In filling
+the boxes, one grade used 16 pounds of candy, another 17
+pounds, a third 12 pounds, and a fourth 13 pounds. What
+did the candy cost at 26&cent;. a pound?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>2. A school in a certain city used 2516 pieces of chalk
+in 37 school days. Three new rooms were opened, each
+room holding 50 children, and the school was then found
+to use 84 sticks of chalk per day. How many more sticks
+of chalk were used per day than at first?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>3. Several boys went on a bicycle trip of 1500 miles.
+The first week they rode 374 miles, the second week 264
+miles, the third 423 miles, the fourth 401 miles. They
+finished the trip the next week. How many miles did they
+ride the last week?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>4. Forty-five boys were hired to pick apples from 15 trees
+in an apple orchard. In 50 minutes each boy had picked
+48 choice apples. If all the apples picked were packed away
+carefully in 8 boxes of equal size, how many apples were put
+in each box?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>5. In a certain school 216 children gave a sleigh-ride
+party. They rented 7 sleighs at a cost of $30.00 and paid
+$24.00 for the refreshments. The party travelled 15 miles
+in 2&frac12; hours and had a very pleasant time. What was each
+child's share of the expense?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>6. A girl found, by careful counting, that there were
+2400 letters on one page of her history, and only 2295 letters
+on a page of her reader. How many more letters had she
+read in one book than in the other if she had read 47 pages
+in each of the books?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>7. Each of 59 rooms in the schools of a certain city contributed
+25 presents to a Christmas entertainment for poor
+children. The stores of the city gave 1986 other articles for
+presents. What was the total number of presents given
+away at the entertainment?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='justify'>8. Forty-eight children from a certain school paid 10&cent;.
+apiece to ride 7 miles on the cars to a woods. There in a
+few hours they gathered 2765 nuts. 605 of these were bad,
+but the rest were shared equally among the children. How
+many good nuts did each one get?</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+
+<tr><td align='right'>Total</td>
+<td class='bbox'>&nbsp;</td><td class='bbox'>&nbsp;&nbsp;&nbsp;</td></tr>
+</table></div>
+
+</div>
+
+<p><span class='pagenum'><a name="Page_46" id="Page_46">[Pg 46]</a></span></p>
+
+<p>These proposed measures of ability to apply arithmetic
+illustrate very nicely the differences of opinion concerning
+what applied arithmetic and arithmetical reasoning should be.
+The thinker who emphasizes the fact that in life out of school
+the situation demanding quantitative treatment is usually
+real rather than described, will condemn a test all of whose
+constituents are <i>described</i> problems. Unless we are excessively
+hopeful concerning the transfer of ideas of method
+and procedure from one mental function to another we shall
+protest against the artificiality of No. 3 of the Stone series,
+and of the entire Courtis Test 8 except No. 4. The Courtis
+speed-reasoning test (No. 6) is a striking example of the mixture
+of ability to understand quantitative relations with
+the ability to understand words. Consider these five, for
+example, in comparison with the revised versions attached.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></p>
+
+<div class="pblockquot"><p>1. The children of a school gave a sleigh-ride party. There
+were 9 sleighs, and each sleigh held 30 children. How many
+children were there in the party?</p>
+
+<p><span class="smcap">Revision.</span> <i>If one sleigh holds 30 children, 9 sleighs hold ....
+children.</i></p>
+
+<p>2. Two school-girls played a number-game. The score of the
+girl that lost was 57 points and she was beaten by 16 points.
+What was the score of the girl that won?</p>
+
+<p><span class="smcap">Revision.</span> <i>Mary and Nell played a game. Mary had a score
+of 57. Nell beat Mary by 16. Nell had a score of</i> ....</p>
+
+<p>3. A girl counted the automobiles that passed a school. The
+total was 60 in two hours. If the girl saw 27 pass the first hour
+how many did she see the second?</p>
+
+<p><span class="smcap">Revision.</span> <i>In two hours a girl saw 60 automobiles. She saw 27
+the first hour. She saw .... the second hour.</i></p>
+
+<p>4. On a playground there were five equal groups of children
+each playing a different game. If there were 75 children all together,
+how many were there in each group?</p>
+
+<p><span class="smcap">Revision.</span> <i>75 pounds of salt just filled five boxes. The boxes
+were exactly alike. There were .... pounds in a box.</i></p>
+
+<p><span class='pagenum'><a name="Page_47" id="Page_47">[Pg 47]</a></span></p>
+
+<p>5. A teacher weighed all the children in a certain grade. One
+girl weighed 70 pounds. Her older sister was 49 pounds heavier.
+How many pounds did the sister weigh?</p>
+
+<p><span class="smcap">Revision.</span> <i>Mary weighs 70 lb. Jane weighs 49 pounds more
+than Mary. Jane weighs .... pounds.</i></p></div>
+
+<p>The distinction between a problem described as clearly
+and simply as possible and the same problem put awkwardly
+or in ill-known words or willfully obscured should be regarded;
+and as a rule measurements of ability to apply arithmetic
+should eschew all needless obscurity or purely linguistic
+difficulty. For example,</p>
+
+<div class="blockquot"><p><i>A boy bought a two-cent stamp. He gave the man in the store 10
+cents. The right change was .... cents.</i></p></div>
+
+<p class="noidt">is better as a test than</p>
+
+<div class="blockquot"><p><i>If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in
+payment, what change should he be expected to receive in return?</i></p></div>
+
+<p>The distinction between the description of a <i>bona fide</i>
+problem that a human being might be called on to solve out
+of school and the description of imaginary possibilities or
+puzzles should also be considered. Nos. 3 and 9 of Stone
+are bad because to frame the problems one must first know
+the answers, so that in reality there could never be any
+point in solving them. It is probably safe to say that
+nobody in the world ever did or ever will or ever should
+find the number of apples in a box by the task of No. 4
+of the Courtis Test 8.</p>
+
+<p>This attaches no blame to Dr. Stone or to Mr. Courtis.
+Until very recently we were all so used to the artificial
+problems of the traditional sort that we did not expect
+anything better; and so blind to the language demands of
+described problems that we did not see their very great
+influence. Courtis himself has been active in reform and
+has pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.<span class='pagenum'><a name="Page_48" id="Page_48">[Pg 48]</a></span></p>
+
+<p>"Tests Nos. 6 and 8, the so-called reasoning tests, have
+proved the least satisfactory of the series. The judgments
+of various teachers and superintendents as to the inequalities
+of the units in any one test, and of the differences between
+the different editions of the same test, have proved the need
+of investigating these questions. Tests of adults in many
+lines of commercial work have yielded in many cases lower
+scores than those of the average eighth grade children. At
+the same time the scores of certain individuals of marked
+ability have been high, and there appears to be a general
+relation between ability in these tests and accuracy in the
+abstract work. The most significant facts, however, have
+been the difficulties experienced by teachers in attempting
+to remedy the defects in reasoning. It is certain that the
+tests measure abilities of value but the abilities are probably
+not what they seem to be. In an attempt to measure the
+value of different units, for instance, as many problems as possible
+were constructed based upon a single situation. Twenty-one
+varieties were secured by varying the relative form of
+the question and the relative position of the different phrases.
+One of these proved nineteen times as hard as another as measured
+by the number of mistakes made by the children; yet
+the cause of the difference was merely the changes in the
+phrasing. This and other facts of the same kind seem to show
+that Tests 6 and 8 measure mainly the ability to read."</p>
+
+<p>The scientific measurement of the abilities and achievements
+concerned with applied arithmetic or problem-solving
+is thus a matter for the future. In the case of described
+problems a beginning has been made in the series which
+form a part of the National Intelligence Tests ['20], one of
+which is shown on page 49 f. In the case of problems with
+real situations, nothing in systematic form is yet available.</p>
+
+<p>Systematic tests and scales, besides defining the abil<span class='pagenum'><a name="Page_49" id="Page_49">[Pg 49]</a></span>ities
+we are to establish and improve, are of very great
+service in measuring the status and improvement of individuals
+and of classes, and the effects of various methods
+of instruction and of study. They are thus helpful to
+pupils, teachers, supervisors, and scientific investigators;
+and are being more and more widely used every year.
+Information concerning the merits of the different tests,
+the procedure to follow in giving and scoring them, the age
+and grade standards to be used in interpreting results, and
+the like, is available in the manuals of Educational Measurement,
+such as Courtis, <i>Manual of Instructions for Giving and
+Scoring the Courtis Standard Tests in the Three R's</i> ['14];
+Starch, <i>Educational Measurements</i> ['16]; Chapman and
+Rush, <i>Scientific Measurement of Classroom Products</i> ['17];
+Monroe, DeVoss, and Kelly, <i>Educational Tests and Measurements</i>
+['17]; Wilson and Hoke, <i>How to Measure</i> ['20];
+and McCall, <i>How to Measure in Education</i> ['21].</p>
+
+<p class="noidt"><small>National Intelligence Tests.<br />
+Scale A. Form 1, Edition 1</small></p>
+
+<h4>TEST 1</h4>
+
+<p class="center">Find the answers as quickly as you can.<br />
+Write the answers on the dotted lines.<br />
+Use the side of the page to figure on.</p>
+
+<p class="noidt"><b>Begin here</b></p>
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><td align='right'><b>1</b></td><td align='left'>Five cents make 1 nickel. How many nickels make a
+dime?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>2</b></td><td align='left'>John paid 5 dollars for a watch and 3 dollars for a chain.
+How many dollars did he pay for the watch and chain?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>3</b></td><td align='left'>Nell is 13 years old. Mary is 9 years old. How much
+younger is Mary than Nell?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>4</b></td><td align='left'>One quart of ice cream is enough for 5 persons. How
+many quarts of ice cream are needed for 25 persons?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>5</b></td><td align='left'>John's grandmother is 86 years old. If she lives, in
+how many years will she be 100 years old?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>6</b></td><td align='left'>If a man gets $2.50 a day, what will he be paid for six
+days' work?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>7</b></td><td align='left'>How many inches are there in a foot and a half?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>8</b></td><td align='left'>What is the cost of 12 cakes at 6 for 5 cents?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>9</b></td><td align='left'>The uniforms for a baseball team of nine boys cost $2.50
+each. The shoes cost $2 a pair. What was the total
+cost of uniforms and shoes for the nine?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>10</b></td><td align='left'>A train that usually arrives at half-past ten was 17
+minutes late. When did it arrive?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>11</b></td><td align='left'>At 10&cent; a yard, what is the cost of a piece 10&frac12; ft. long?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>12</b></td><td align='left'>A man earns $6 a day half the time, $4.50 a day one
+fourth of the time, and nothing on the remaining days
+for a total period of 40 days. What did he earn in all
+in the 40 days?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>13</b></td><td align='left'>What per cent of $800 is 4% of $1000?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>14</b></td><td align='left'>If 60 men need 1500 lb. flour per month, what is the
+requirement per man per day counting a month as 30
+days?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>15</b></td><td align='left'>A car goes at the rate of a mile a minute. A truck goes 20 miles an hour. How many times as far will the car go as the truck in 10 seconds?</td><td align='right'><i>Answer</i>......</td></tr>
+<tr><td align='right'><b>16</b></td><td align='left'>The area of the base (inside measure) of a cylindrical tank is 90 square feet. How tall must it be to hold 100 cubic yards?</td><td align='right'><i>Answer</i>......</td></tr>
+</table></div>
+
+<p class="center"><small>From National Intelligence Tests by National Research Council.<br />
+Copyright, 1920, by The World Book Company, Yonkers-on-Hudson, New York.<br />
+Used by permission of the publishers.</small></p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_51" id="Page_51">[Pg 51]</a></span></p>
+<h2><a name="CHAPTER_III" id="CHAPTER_III"></a>CHAPTER III</h2>
+
+<h3>THE CONSTITUTION OF ARITHMETICAL ABILITIES</h3>
+
+
+<h4>THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING</h4>
+
+<p>It would be a useful work for some one to try to analyze
+arithmetical learning into the unitary abilities which compose
+it, showing just what, in detail, the mind has to do in
+order to be prepared to pass a thorough test on the whole of
+arithmetic. These unitary abilities would make a very
+long list. Examination of a well-planned textbook will show
+that such an ability as multiplication is treated as a composite
+of the following: knowledge of the multiplications up
+to 9 &times; 9; ability to multiply two (or more)-place numbers
+by 2, 3, and 4 when 'carrying' is not required and no zeros
+occur in the multiplicand; ability to multiply by 2, 3, ... 9,
+with carrying; the ability to handle zeros in the multiplicand;
+the ability to multiply with two-place numbers not ending
+in zero; the ability to handle zero in the multiplier as
+last number; the ability to multiply with three (or more)-place
+numbers not including a zero; the ability to multiply
+with three- and four-place numbers with zero in second
+or third, or second and third, as well as in last place; the
+ability to save time by annexing zeros; and so on and on
+through a long list of further abilities required to multiply
+with United States money, decimal fractions, common
+fractions, mixed numbers, and denominate numbers.<span class='pagenum'><a name="Page_52" id="Page_52">[Pg 52]</a></span></p>
+
+<p>The units or 'steps' thus recognized by careful teaching
+would make a long list, but it is probable that a still more
+careful study of arithmetical ability as a hierarchy of mental
+habits or connections would greatly increase the list. Consider,
+for example, ordinary column addition. The majority
+of teachers probably treat this as a simple application of the
+knowledge of the additions to 9 &#43; 9, plus understanding of
+'carrying.' On the contrary there are at least seven processes
+or minor functions involved in two-place column addition,
+each of which is psychologically distinct and requires
+distinct educational treatment.</p>
+
+<p>These are:&mdash;</p>
+
+<p class="nblockquot">A. Learning to keep one's place in the column as one adds.</p>
+
+<p class="nblockquot">B. Learning to keep in mind the result of each addition until
+the next number is added to it.</p>
+
+<p class="nblockquot">C. Learning to add a seen to a thought-of number.</p>
+
+<p class="nblockquot">D. Learning to neglect an empty space in the columns.</p>
+
+<p class="nblockquot">E. Learning to neglect 0s in the columns.</p>
+
+<p class="nblockquot">F. Learning the application of the combinations to higher
+decades may for the less gifted pupils involve as much
+time and labor as learning all the original addition
+tables. And even for the most gifted child the
+formation of the connection '8 and 7 = 15' probably
+never quite insures the presence of the connections
+'38 and 7 = 45' and '18 &#43; 7 = 25.'</p>
+
+<p class="nblockquot">G. Learning to write the figure signifying units rather than
+the total sum of a column. In particular, learning
+to write 0 in the cases where the sum of the column
+is 10, 20, etc. Learning to 'carry' also involves in
+itself at least two distinct processes, by whatever way
+it is taught.</p>
+
+<p>We find evidence of such specialization of functions in
+the results with such tests as Woody's. For example,
+<span class='pagenum'><a name="Page_53" id="Page_53">[Pg 53]</a></span>
+2 &#43; 5 &#43; 1 = .... surely involves abilities in part different from</p>
+
+<p class="center">2<br />4<br />3<br />&mdash;</p>
+
+<p class="noidt">because only 77 percent of children in grade 3
+do the former correctly, whereas 95 percent of children in
+that grade do the latter correctly. In grade 2 the difference
+is even more marked. In the case of subtraction</p>
+
+<p class="center">4<br />4<br />&mdash;</p>
+
+<p class="noidt">involves abilities different from those involved in</p>
+
+<p class="center">9<br />3<br />&mdash;,</p>
+
+<p class="noidt">being much less often solved correctly in grades 2 and 4.</p>
+
+<p class="center">6<br />0<br />&mdash;</p>
+
+<p class="noidt">is much harder than either of the above.</p>
+
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>43<br /> &nbsp; 1<br /> &nbsp; 2<br />13</td><td align='left' valign='bottom'>is much harder than</td><td align='left' valign='bottom'>21<br />33<br />35.</td></tr>
+<tr><td align='left'>&mdash;</td><td></td><td align='left'>&mdash;</td></tr>
+</table>
+
+
+<p class="noidt">It may be said that these differences in difficulty are due to
+different amounts of practice. This is probably not true,
+but if it were, it would not change the argument; if the two
+abilities were identical, the practice of one would improve
+the other equally.</p>
+
+<p>I shall not undertake here this task of listing and describing
+the elementary functions which constitute arithmetical
+learning, partly because what they are is not fully known,
+partly because in many cases a final ability may be constituted
+in several different ways whose descriptions become
+necessarily tedious, and partly because an adequate statement
+of what is known would far outrun the space limits
+of this chapter. Instead, I shall illustrate the results by
+some samples.</p>
+<p><span class='pagenum'><a name="Page_54" id="Page_54">[Pg 54]</a></span></p>
+
+<h4>KNOWLEDGE OF THE MEANING OF A FRACTION</h4>
+
+<p>As a first sample, consider knowledge of the meaning of
+a fraction. Is the ability in question simply to understand
+that a fraction is a statement of the number of parts, each
+of a certain size, the upper number or numerator telling how
+many parts are taken and the lower number or denominator
+telling what fraction of unity each part is? And is the
+educational treatment required simply to describe and
+illustrate such a statement and have the pupils apply it to
+the recognition of fractions and the interpretation of each
+of them? And is the learning process (1) the formation
+of the notions of part, size of part, number of part, (2) relating
+the last two to the numbers in a fraction, and, as a
+necessary consequence, (3) applying these notions adequately
+whenever one encounters a fraction in operation?</p>
+
+<p>Precisely this was the notion a few generations ago. The
+nature of fractions was taught as one principle, in one step,
+and the habits of dealing with fractions were supposed to be
+deduced from the general law of a fraction's nature. As a
+result the subject of fractions had to be long delayed, was
+studied at great cost of time and effort, and, even so, remained
+a mystery to all save gifted pupils. These gifted
+pupils probably of their own accord built up the ability
+piecemeal out of constituent insights and habits.</p>
+
+<p>At all events, scientific teaching now does build up the
+total ability as a fusion or organization of lesser abilities.
+What these are will be seen best by examining the means
+taken to get them. (1) First comes the association of &frac12;
+of a pie, &frac12; of a cake, &frac12; of an apple, and such like with their
+concrete meanings so that a pupil can properly name a
+clearly designated half of an obvious unit like an orange,
+pear, or piece of chalk. The same degree of understanding<span class='pagenum'><a name="Page_55" id="Page_55">[Pg 55]</a></span>
+of <sup>1</sup>&frasl;<sub>4</sub>, <sup>1</sup>&frasl;<sub>8</sub>, <sup>1</sup>&frasl;<sub>3</sub>, <sup>1</sup>&frasl;<sub>6</sub>, and <sup>1</sup>&frasl;<sub>5</sub> is secured. The pupil is taught that
+1 pie = 2 <sup>1</sup>&frasl;<sub>2</sub>s, 3 <sup>1</sup>&frasl;<sub>3</sub>s,
+4 <sup>1</sup>&frasl;<sub>4</sub>s, 5 <sup>1</sup>&frasl;<sub>5</sub>s, 6 <sup>1</sup>&frasl;<sub>6</sub>s, and 8 <sup>1</sup>&frasl;<sub>8</sub>s; similarly for 1
+cake, 1 apple, and the like.</p>
+
+<p>So far he understands <sup>1</sup>&frasl;<sub><i>x</i></sub> of <i>y</i> in the sense of certain simple
+parts of obviously unitary <i>y</i>s.</p>
+
+<p>(2) Next comes the association with &frac12; of an inch, &frac12; of a
+foot, &frac12; of a glassful and other cases where <i>y</i> is not so obviously
+a unitary object whose pieces still show their derivation
+from it. Similarly for <sup>1</sup>&frasl;<sub>4</sub>, <sup>1</sup>&frasl;<sub>3</sub>, etc.</p>
+
+<p>(3) Next comes the association with <sup>1</sup>&frasl;<sub>2</sub> of a collection of
+eight pieces of candy, <sup>1</sup>&frasl;<sub>3</sub> of a dozen eggs, <sup>1</sup>&frasl;<sub>5</sub> of a squad of ten
+soldiers, etc., until <sup>1</sup>&frasl;<sub>2</sub>, <sup>1</sup>&frasl;<sub>3</sub>, <sup>1</sup>&frasl;<sub>4</sub>, <sup>1</sup>&frasl;<sub>5</sub>, <sup>1</sup>&frasl;<sub>6</sub>, and <sup>1</sup>&frasl;<sub>8</sub> are understood as
+names of certain parts of a collection of objects.</p>
+
+<p>(4) Next comes the similar association when the nature
+of the collection is left undefined, the pupil responding to<br />
+<sup>1</sup>&frasl;<sub>2</sub> of 6 is ..., <sup>1</sup>&frasl;<sub>4</sub> of 8 is ..., 2 is <sup>1</sup>&frasl;<sub>5</sub> of ...,<br />
+<sup>1</sup>&frasl;<sub>3</sub> of 6 is ..., <sup>1</sup>&frasl;<sub>3</sub> of 9 is ..., 2 is <sup>1</sup>&frasl;<sub>3</sub> of ..., and the like.</p>
+
+<p>Each of these abilities is justified in teaching by its intrinsic
+merits, irrespective of its later service in helping to
+constitute the general understanding of the meaning of a
+fraction. The habits thus formed in grades 3 or 4 are of
+constant service then and thereafter in and out of school.</p>
+
+<p>(5) With these comes the use of <sup>1</sup>&frasl;<sub>5</sub> of 10, 15, 20, etc., <sup>1</sup>&frasl;<sub>6</sub> of
+12, 18, 42, etc., as a useful variety of drill on the division
+tables, valuable in itself, and a means of making the notion
+of a unit fraction more general by adding <sup>1</sup>&frasl;<sub>7</sub> and <sup>1</sup>&frasl;<sub>9</sub> to the
+scheme.</p>
+
+<p>(6) Next comes the connection of <sup>3</sup>&frasl;<sub>4</sub>, <sup>2</sup>&frasl;<sub>5</sub>, <sup>3</sup>&frasl;<sub>5</sub>, <sup>4</sup>&frasl;<sub>5</sub>, <sup>2</sup>&frasl;<sub>3</sub>, <sup>1</sup>&frasl;<sub>6</sub>, <sup>5</sup>&frasl;<sub>6</sub>,
+<sup>3</sup>&frasl;<sub>8</sub>, <sup>5</sup>&frasl;<sub>8</sub>,
+<sup>7</sup>&frasl;<sub>8</sub>, <sup>3</sup>&frasl;<sub>10</sub>, <sup>7</sup>&frasl;<sub>10</sub>,
+and <sup>9</sup>&frasl;<sub>10</sub>, each with its meaning as a certain part of
+some conveniently divisible unit, and, (7) and (8), connections
+between these fractions and their meanings as parts
+of certain magnitudes (7) and collections (8) of convenient<span class='pagenum'><a name="Page_56" id="Page_56">[Pg 56]</a></span>
+size, and (9) connections between these fractions and their
+meanings when the nature of the magnitude or collection is
+unstated, as in <sup>4</sup>&frasl;<sub>5</sub> of 15 = ..., <sup>5</sup>&frasl;<sub>8</sub> of 32 = ....</p>
+
+<p>(10) That the relation is general is shown by using it with
+numbers requiring written division and multiplication, such
+as <sup>7</sup>&frasl;<sub>8</sub> of 1736 = ..., and with United States money.</p>
+
+<p>Elements (6) to (10) again are useful even if the pupil never
+goes farther in arithmetic. One of the commonest uses of
+fractions is in calculating the cost of fractions of yards of
+cloth, and fractions of pounds of meat, cheese, etc.</p>
+
+<p>The next step (11) is to understand to some extent the
+principle that the value of any of these fractions is unaltered
+by multiplying or dividing the numerator and denominator
+by the same number. The drills in expressing fractions in
+lower and higher terms which accomplish this are paralleled
+by (12) and (13) simple exercises in adding and subtracting
+fractions to show that fractions are quantities that can be
+operated on like any quantities, and by (14) simple work
+with mixed numbers (addition and subtraction and reductions),
+and (15) improper fractions. All that is done with
+improper fractions is (<i>a</i>) to have the pupil use a few of them
+as he would any fractions and (<i>b</i>) to note their equivalent
+mixed numbers. In (12), (13), and (14) only fractions of
+the same denominators are added or subtracted, and in (12)
+(13), (14), and (15) only fractions with 2, 3, 4, 5, 6, 8, or 10
+in the denominator are used. As hitherto, the work of (11)
+to (15) is useful in and of itself. (16) Definitions are given of
+the following type:&mdash;</p>
+
+<p>Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called
+whole numbers.</p>
+
+<p>Numbers like <sup>7</sup>&frasl;<sub>8</sub>, <sup>1</sup>&frasl;<sub>5</sub>, <sup>2</sup>&frasl;<sub>3</sub>,
+<sup>3</sup>&frasl;<sub>4</sub>, <sup>11</sup>&frasl;<sub>8</sub>,
+<sup>7</sup>&frasl;<sub>6</sub>, <sup>1</sup>&frasl;<sub>3</sub>, <sup>4</sup>&frasl;<sub>3</sub>, <sup>1</sup>&frasl;<sub>8</sub>,
+<sup>1</sup>&frasl;<sub>6</sub> are called fractions.</p>
+
+<p>Numbers like 5&frac14;, 7<small><sup>3</sup>&frasl;<sub>8</sub></small>, 9&frac12;,
+16<small><sup>4</sup>&frasl;<sub>5</sub></small>,
+315<small><sup>7</sup>&frasl;<sub>8</sub></small>,
+1<small><sup>1</sup>&frasl;<sub>3</sub></small>,
+1<small><sup>2</sup>&frasl;<sub>3</sub></small> are called mixed
+numbers.<span class='pagenum'><a name="Page_57" id="Page_57">[Pg 57]</a></span></p>
+
+<p>(17) The terms numerator and denominator are connected
+with the upper and lower numbers composing a fraction.</p>
+
+<p>Building this somewhat elaborate series of minor abilities
+seems to be a very roundabout way of getting knowledge
+of the meaning of a fraction, and is, if we take no account
+of what is got along with this knowledge. Taking account
+of the intrinsically useful habits that are built up, one might
+retort that the pupil gets his knowledge of the meaning of a
+fraction at zero cost.</p>
+
+
+<h4>KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES</h4>
+
+<p>Consider next the knowledge of the subtraction and division
+'Tables.' The usual treatment presupposes that learning
+them consists of forming independently the bonds:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='center'>3 &minus; 1 = 2</td><td align='center'>4 &divide; 2 = 2</td></tr>
+<tr><td align='center'>3 &minus; 2 = 1</td><td align='center'>6 &divide; 2 = 3</td></tr>
+<tr><td align='center'>4 &minus; 1 = 3</td><td align='center'>6 &divide; 3 = 2</td></tr>
+<tr>
+<td align='center'>
+&nbsp;.<br />
+&nbsp;.<br />
+&nbsp;.</td>
+<td align='center'>
+&nbsp;.<br />
+&nbsp;.<br />
+&nbsp;.</td>
+</tr>
+<tr><td align='center'>18 &minus; 9 = 9</td><td align='center'>81 &divide; 9 = 9</td></tr>
+</table></div>
+
+<p>In fact, however, these 126 bonds are not formed independently.
+Except perhaps in the case of the dullest
+twentieth of pupils, they are somewhat facilitated by the
+already learned additions and multiplications. And by
+proper arrangement of the learning they may be enormously
+facilitated thereby. Indeed, we may replace the independent
+memorizing of these facts by a set of instructive
+<span class='pagenum'><a name="Page_58" id="Page_58">[Pg 58]</a></span>exercises wherein the pupil derives the subtractions from
+the corresponding additions by simple acts of reasoning or
+selective thinking. As soon as the additions giving sums of
+9 or less are learned, let the pupil attack an exercise like
+the following:&mdash;</p>
+
+<p>Write the missing numbers:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='center'>A</td><td align='center'>B</td><td align='center'>C</td><td align='center'>D</td></tr>
+<tr><td align='left'>3 and ... are 5.</td><td align='left'>5 and ... are 8.</td><td align='left'>4 and ... are 5.</td><td align='left'>4 and ... are 8.</td></tr>
+<tr><td align='left'>3 and ... are 9.</td><td align='left'>3 and ... are 6.</td><td align='left'>5 and ... are 6.</td><td align='left'>1 and ... are 7.</td></tr>
+<tr><td align='left'>4 and ... are 7.</td><td align='left'>4 and ... are 9.</td><td align='left'>6 and ... are 9.</td><td align='left'>6 and ... are 7.</td></tr>
+<tr><td align='left'>5 and ... are 7.</td><td align='left'>2 and ... &nbsp;=&nbsp; 6.</td><td align='left'>1 and ... are 8.</td><td align='left'>8 and ... are 9.</td></tr>
+<tr><td align='left'>6 and ... are 8.</td><td align='left'>5 and ... &nbsp;=&nbsp; 9.</td><td align='left'>3 and ... are 7.</td><td align='left'>3&nbsp; &#43; &nbsp; ... are 4.</td></tr>
+<tr><td align='left'>4 and ... are 6.</td><td align='left'>2 and ... &nbsp;=&nbsp; 7.</td><td align='left'>1&nbsp; &#43; &nbsp; ... are 3.</td><td align='left'>7&nbsp; &#43; &nbsp; ... are 8.</td></tr>
+<tr><td align='left'>2 and ... are 5.</td><td align='left'>3 and ... &nbsp;=&nbsp; 8.</td><td align='left'>1&nbsp; &#43; &nbsp; ... are 5.</td><td align='left'>4&nbsp; &#43; &nbsp; ... are 9.</td></tr>
+<tr><td align='left'>2 and ... &nbsp;=&nbsp; 8.</td><td align='left'>1 and ... &nbsp;=&nbsp; 4.</td><td align='left'>4&nbsp; &#43; &nbsp; ... are 8.</td><td align='left'>2&nbsp; &#43; &nbsp; ... are 3.</td></tr>
+<tr><td align='left'>3 and ... &nbsp;=&nbsp; 6.</td><td align='left'>2 and ... &nbsp;=&nbsp; 4.</td><td align='left'>7&nbsp; &#43; &nbsp; ... are 9.</td><td align='left'>1&nbsp; &#43; &nbsp; ... are 9.</td></tr>
+<tr><td align='left'>6 and ... &nbsp;=&nbsp; 9.</td><td align='left'>3 and ... &nbsp;=&nbsp; 8.</td><td align='left'>2&nbsp; &#43; &nbsp; ... &nbsp;=&nbsp; 4.</td><td align='left'>3&nbsp; &#43; &nbsp; ... &nbsp;=&nbsp; 6.</td></tr>
+<tr><td align='left'>4 and ... &nbsp;=&nbsp; 6.</td><td align='left'>6 and ... &nbsp;=&nbsp; 7.</td><td align='left'>3&nbsp; &#43; &nbsp; ... &nbsp;=&nbsp; 8.</td><td align='left'>5&nbsp; &#43; &nbsp; ... &nbsp;=&nbsp; 9.</td></tr>
+<tr><td align='left'>4 and ... &nbsp;=&nbsp; 7.</td><td align='left'>2 and ... &nbsp;=&nbsp; 5.</td><td align='left'>4&nbsp; &#43; &nbsp; ... &nbsp;=&nbsp; 5.</td><td align='left'>1&nbsp; &#43; &nbsp; ... &nbsp;=&nbsp; 3.</td></tr>
+</table></div>
+
+<p>The task for reasoning is only to try, one after another,
+numbers that seem promising and to select the right one
+when found. With a little stimulus and direction children
+can thus derive the subtractions up to those with 9 as the
+larger number. Let them then be taught to do the same
+with the printed forms:&mdash;</p>
+
+<p class="center">Subtract</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='center'>9</td><td align='center'>7</td><td align='center'>8</td><td align='center'>5</td><td align='center'>8</td><td align='center'>6</td></tr>
+<tr><td align='center'>3</td><td align='center'>5</td><td align='center'>6</td><td align='center'>2</td><td align='center'>2</td><td align='center'>4</td><td align='center'>etc.</td></tr>
+<tr><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td></tr>
+</table></div>
+
+<p class="noidt">and 9 &minus; 7 = ..., 9 &minus; 5 = ..., 7 &minus; 5 = ..., etc.</p>
+
+<p>In the case of the divisions, suppose that the pupil has
+learned his first table and gained surety in such exercises
+as:<span class='pagenum'><a name="Page_59" id="Page_59">[Pg 59]</a></span>&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>4 5s = ....</td><td align='left'>6 &times; 5 = ....</td><td align='left'>9 nickels = .... cents.</td></tr>
+<tr><td align='left'>8 5s = ....</td><td align='left'>4 &times; 5 = ....</td><td align='left'>6 &nbsp;  &nbsp; " &nbsp; &nbsp;&nbsp; = .... &nbsp;  &nbsp; "</td></tr>
+<tr><td align='left'>3 5s = ....</td><td align='left'>2 &times; 5 = ....</td><td align='left'>5 &nbsp;  &nbsp; " &nbsp; &nbsp;&nbsp; = .... &nbsp;  &nbsp; "</td></tr>
+<tr><td align='left'>7 5s = ....</td><td align='left'>9 &times; 5 = ....</td><td align='left'>7 &nbsp;  &nbsp; " &nbsp; &nbsp;&nbsp; = .... &nbsp;  &nbsp; "</td></tr>
+</table></div>
+
+<p class="noidt">If one ball costs 5 cents,<br />
+<span style="margin-left: 6em;">two balls cost .... cents,</span><br />
+<span style="margin-left: 6em;">three balls cost .... cents, etc.</span><br />
+</p>
+
+<p class="noidt">He may then be set at once to work at the answers to exercises
+like the following:&mdash;</p>
+
+<p>Write the answers and the missing numbers:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='center'>A</td><td align='center'>B</td><td align='center'>C</td><td align='center'>D</td></tr>
+<tr><td align='left'>.... 5s = 15</td><td align='left'>40 = .... 5s</td><td align='left'>.... &times; 5 = 25</td><td align='left'>20 cents = .... nickels.</td></tr>
+<tr><td align='left'>.... 5s = 20</td><td align='left'>20 = .... 5s</td><td align='left'>.... &times; 5 = 50</td><td align='left'>30 cents = .... nickels.</td></tr>
+<tr><td align='left'>.... 5s = 40</td><td align='left'>15 = .... 5s</td><td align='left'>.... &times; 5 = 35</td><td align='left'>15 cents = .... nickels.</td></tr>
+<tr><td align='left'>.... 5s = 25</td><td align='left'>45 = .... 5s</td><td align='left'>.... &times; 5 = 10</td><td align='left'>40 cents = .... nickels.</td></tr>
+<tr><td align='left'>.... 5s = 30</td><td align='left'>50 = .... 5s</td><td align='left'>.... &times; 5 = 40</td></tr>
+<tr><td align='left'>.... 5s = 35</td><td align='left'>25 = .... 5s</td><td align='left'>.... &times; 5 = 45</td></tr>
+</table></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td class="center">E</td></tr>
+<tr><td align='left'>For 5 cents you can buy 1 small loaf of bread.<br />
+For 10 cents you can buy 2 small loaves of bread.<br />
+For 25 cents you can buy .... small loaves of bread.<br />
+For 45 cents you can buy .... small loaves of bread.<br />
+For 35 cents you can buy .... small loaves of bread.
+</td></tr>
+</table></div>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td class="center">F</td></tr>
+<tr><td align='left'> &nbsp; 5 cents pays 1 car fare.<br />
+15 cents pays .... car fares.<br />
+10 cents pays .... car fares.<br />
+20 cents pays .... car fares.
+</td></tr>
+</table></div>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td class="center">G</td></tr>
+<tr><td align='left'>How many 5 cent balls can you buy with 30 cents? ....<br />
+How many 5 cent balls can you buy with 35 cents? ....<br />
+How many 5 cent balls can you buy with 25 cents? ....<br />
+How many 5 cent balls can you buy with 15 cents? ....</td></tr>
+</table></div>
+
+
+<p>In the case of the meaning of a fraction, the ability, and
+so the learning, is much more elaborate than common
+<span class='pagenum'><a name="Page_60" id="Page_60">[Pg 60]</a></span>
+practice has assumed; in the case of the subtraction and
+division tables the learning is much less so. In neither case
+is the learning either mere memorizing of facts or the mere
+understanding of a principle <i>in abstracto</i> followed by its
+application to concrete cases. It is (and this we shall find
+true of almost all efficient learning in arithmetic) the formation
+of connections and their use in such an order that each
+helps the others to the maximum degree, and so that each
+will do the maximum amount for arithmetical abilities
+other than the one specially concerned, and for the general
+competence of the learner.</p>
+
+
+<h4>LEARNING THE PROCESSES OF COMPUTATION</h4>
+
+<p>As another instructive topic in the constitution of arithmetical
+abilities, we may take the case of the reasoning involved
+in understanding the manipulations of figures in two
+(or more)-place addition and subtraction, multiplication and
+division involving a two (or more)-place number, and the
+manipulations of decimals in all four operations. The
+psychology of these is of special interest and importance.
+For there are two opposite explanations possible here,
+leading to two opposite theories of teaching.</p>
+
+<p>The common explanation is that these methods of manipulation,
+if understood at all, are understood as deductions from
+the properties of our system of decimal notation. The other
+is that they are understood partly as inductions from the
+experience that they always give the right answer. The
+first explanation leads to the common preliminary deductive
+explanations of the textbooks. The other leads to explanations
+by verification; <i>e.g.</i>, of addition by counting, of subtraction
+by addition, of multiplication by addition, of division
+by multiplication. Samples of these two sorts of
+explanation are given below.</p>
+<p><span class='pagenum'><a name="Page_61" id="Page_61">[Pg 61]</a></span></p>
+
+
+<div class="pblockquot">
+<p class="center"><b>SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION</b></p>
+
+<p><span class="smcap">Multiplication</span> is the process of taking one number as many
+times as there are units in another number.</p>
+
+<p>The <span class="smcap">Product</span> is the result of the multiplication.</p>
+
+<p>The <span class="smcap">Multiplicand</span> is the number to be taken.</p>
+
+<p>The <span class="smcap">Multiplier</span> is the number denoting how many times the
+multiplicand is to be taken.</p>
+
+<p>The multiplier and multiplicand are the <span class="smcap">Factors</span>.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='center' colspan='2'>Multiply 623 by 3<br /><br />OPERATION<br />&nbsp;</td></tr>
+<tr><td align='left'><i>Multiplicand</i> &nbsp; &nbsp; &nbsp;</td><td align='right'>623</td></tr>
+<tr><td align='left'><i>Multiplier</i></td><td align='right'>3</td></tr>
+<tr><td align='left'><i>Product</i></td><td align='right'><span class="overline">1869</span></td></tr>
+</table></div>
+
+<p><small><span class="smcap">Explanation.</span>&mdash;For convenience we write the multiplier under the multiplicand,
+and begin with units to multiply. 3 times 3 units are 9 units. We write the nine
+units in units' place in the product. 3 times 2 tens are 6 tens. We write the 6 tens
+in tens' place in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and
+8 hundreds. The 1 thousand we write in thousands' place and the 8 hundreds in
+hundreds' place in the product. Therefore, the product is 1 thousand 8 hundreds,
+6 tens and 9 units, or 1869.</small></p>
+
+
+<p class="center"><b>SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE
+EXPLANATION</b></p>
+
+<p><b>1.</b> The children of the third grade are to have a picnic. 32 are
+going. How many sandwiches will they need if each of
+the 32 children has four sandwiches?</p>
+
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr><td align='right'>&nbsp;<br /><i>32</i><br /><span class="u">&nbsp; <i>4</i></span></td>
+<td align='left'><i>Here is a quick way to find out</i>:&mdash;<br />
+<i>Think "4 &times; 2," write 8 under the 2 in the ones column.</i><br />
+<i>Think "4 &times; 3," write 12 under the 3 in the tens column.</i>
+</td></tr>
+</table>
+
+<p><b>2.</b> How many bananas will they need if each of the 32 children
+has two bananas? 32 &times; 2 or 2 &times; 32 will give the answer.</p>
+
+<p><b>3.</b> How many little cakes will they need if each child has three
+cakes? 32 &times; 3 or 3 &times; 32 will give the answer.</p>
+
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr><td align='right'><i>32</i><br /><span class="u">&nbsp; <i>3</i></span></td>
+<td align='left'><i>3 &times; 2 = ....  Where do you write the 6?</i><br />
+<i>3 &times; 3 = .... Where do you write the 9?</i>
+</td></tr>
+</table>
+<p><span class='pagenum'><a name="Page_62" id="Page_62">[Pg 62]</a></span></p>
+
+<p><b>4.</b>  Prove that 128, 64, and 96 are right by adding four 32s,
+two 32s, and three 32s.</p>
+
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr><td align='right' valign='bottom'>32<br />32<br />32<br /><span class="u">32</span></td>
+<td align='right' valign='bottom'>32<br />32<br /><span class="u">32</span></td>
+<td align='right' valign='bottom'>32<br /><span class="u">32</span></td></tr>
+</table>
+
+
+<p class="center"><b>Multiplication</b></p>
+
+<p>You <b>multiply</b> when you find the answers to questions like</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>How many are 9 &times; 3?<br />
+How many are 3 &times; 32?<br />
+How many are 8 &times; 5?<br />
+How many are 4 &times; 42?</td></tr>
+</table></div>
+
+<p>
+<b>1.</b> Read these lines. Say the right numbers where the dots are:<br />
+<span style="margin-left: 4.5em;">If you <b>add</b> 3 to 32, you have .... 35 is the <b>sum</b>.</span><br />
+<span style="margin-left: 4.5em;">If you <b>subtract</b> 3 from 32, the result is .... 29 is the <b>difference</b> or <b>remainder</b>.</span><br />
+<span style="margin-left: 4.5em;">If you <b>multiply</b> 3 by 32 or 32 by 3, you have .... 96 is the <b>product</b>.</span><br />
+</p>
+
+<p>Find the products. Check your answers to the first line by
+adding.</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%">
+<tr><td align='right'><b>2. </b></td><td align='right'><b>3. </b></td><td align='right'><b>4. </b></td><td align='right'><b>5. </b></td><td align='right'><b>6. </b></td><td align='right'><b>7. </b></td><td align='right'><b>8. </b></td><td align='right'><b>9. </b></td></tr>
+<tr><td align='right'>41</td><td align='right'>33</td><td align='right'>42</td><td align='right'>44</td><td align='right'>53</td><td align='right'>43</td><td align='right'>34</td><td align='right'>24</td></tr>
+<tr><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 4</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 2</span></td></tr>
+<tr><td colspan='8'>&nbsp;</td></tr>
+<tr><td align='right'><b>10. </b></td><td align='right'><b>11. </b></td><td align='right'><b>12. </b></td><td align='right'><b>13. </b></td><td align='right'><b>14. </b></td><td align='right'><b>15. </b></td><td align='right'><b>16. </b></td></tr>
+<tr><td align='right'>43</td><td align='right'>52</td><td align='right'>32</td><td align='right'>23</td><td align='right'>41</td><td align='right'>51</td><td align='right'>14</td></tr>
+<tr><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 4</span></td><td align='right'><span class="u"> &nbsp; 2</span></td></tr>
+</table></div>
+
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'><br /><b>17.</b></td></tr>
+<tr><td align='right'><i>213</i><br /><span class="u">&nbsp; &nbsp; <i>3</i></span><br />&nbsp;</td>
+<td align='left'><i>Write the 9 in the ones column.</i><br />
+<i>Write the 6 in the hundreds column.</i> &nbsp; &nbsp;<br />
+<i>Write the 3 in the tens column.</i></td>
+<td align='left'><i>Check your  answer by adding.</i></td>
+<td align='left'>Add<br /><i>213</i><br /><i>213</i><br /><span class="u"><i>213</i></span><br />&nbsp;</td>
+</tr>
+</table>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%">
+<tr><td align='right'><b>18. </b></td><td align='right'><b>19. </b></td><td align='right'><b>20. </b></td><td align='right'><b>21. </b></td><td align='right'><b>22. </b></td><td align='right'><b>23. </b></td><td align='right'><b>24. </b></td></tr>
+<tr><td align='right'>214</td><td align='right'>312</td><td align='right'>432</td><td align='right'>231</td><td align='right'>132</td><td align='right'>314</td><td align='right'>243</td></tr>
+<tr><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 3</span></td><td align='right'><span class="u"> &nbsp; 2</span></td><td align='right'><span class="u"> &nbsp; 2</span></td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_63" id="Page_63">[Pg 63]</a></span></p>
+
+<p class="center"><b>SHORT DIVISION: DEDUCTIVE EXPLANATION</b></p>
+
+<p class="center">Divide 1825 by 4</p>
+
+<p>Divisor &nbsp; 4 |<span class="u">&nbsp; 1825 </span> &nbsp; Dividend<br />
+<span style="margin-left: 6.5em;">456&frac14;</span><br />
+<span style="margin-left: 6em;">Quotient</span></p>
+
+<p><small><span class="smcap">Explanation.</span>&mdash;For convenience
+we write the divisor at the left of
+the dividend, and the quotient below
+it, and begin at the left to divide.
+<b>4</b> is not contained in 1 thousand
+any thousand times, therefore
+the quotient contains no unit of
+any order higher than hundreds.
+Consequently we find how many times 4 is contained in the hundreds of the dividend.
+1 thousand and 8 hundreds are 18 hundreds. 4 is contained in 18 hundreds
+4 hundred times and 2 hundreds remaining. We write the 4 hundreds in the quotient.
+The 2 hundreds we consider as united with the 2 tens, making 22 tens.
+4 is contained in 22 tens 5 tens times, and 2 tens remaining. We write the 5 tens
+in the quotient, and the remaining 2 tens we consider as united with the 5 units,
+making 25 units. 4 is contained in 25 units 6 units times and 1 unit remaining.
+We write the 6 units in the quotient and indicate the division of the remainder, 1
+unit, by the divisor 4.</small></p>
+
+<p><small>Therefore the quotient of 1825 divided by 4 is 456&frac14;, or 456 and 1 remainder.</small></p>
+
+
+<p class="center"><b>SHORT DIVISION: INDUCTIVE EXPLANATION</b></p>
+
+
+<p class="center"><b>Dividing Large Numbers</b></p>
+
+<p><b>1.</b> Tom, Dick, Will, and Fred put in 2 cents each to buy an
+eight-cent bag of marbles. There are 128 marbles in it.
+How many should each boy have, if they divide the
+marbles equally among the four boys?</p>
+
+<p>4 |<span class="overline"> 128</span></p>
+
+<p><i>Think "12 = three 4s." Write the 3 over the 2 in the tens column.</i></p>
+
+<p><i>Think "8 = two 4s." Write the 2 over the 8 in the ones column.</i></p>
+
+<p><i>32 is right, because 4 &times; 32 = 128.</i></p>
+
+
+<p><b>2.</b> Mary, Nell, and Alice are going to buy a book as a present
+for their Sunday-school teacher. The present costs 69
+cents. How much should each girl pay, if they divide
+the cost equally among the three girls?</p>
+
+<p>3 |<span class="overline"> 69</span></p>
+
+<p><i>Think "6 = .... 3s." Write the 2 over the 6 in the tens column</i>.<span class='pagenum'><a name="Page_64" id="Page_64">[Pg 64]</a></span></p>
+
+<p><i>Think "9 = .... 3s." Write the 3 over the 9 in the ones column.</i></p>
+
+<p><i>23 is right, for 3 &times; 23 = 69.</i></p>
+
+<p><b>3.</b> Divide the cost of a 96-cent present equally among three
+girls. How much should each girl pay?
+girls. How much should each girl pay? &nbsp; &nbsp; 3 |<span class="overline"> 96</span></p>
+
+<p><b>4.</b> Divide the cost of an 84-cent present equally among 4 girls.
+How much should each girl pay?</p>
+
+<p><b>5.</b> Learn this: (Read &divide; as "<i>divided by</i>.")</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'><b>12 &#43; 4 = 16.</b> &nbsp; &nbsp; </td><td align='left'><b>16 is the sum.</b></td></tr>
+<tr><td align='left'><b>12 &minus; 4 = &nbsp; 8.</b></td><td align='left'><b> &nbsp; 8 is the difference or remainder.</b></td></tr>
+<tr><td align='left'><b>12 &times; 4 = 48.</b></td><td align='left'><b>48 is the product.</b></td></tr>
+<tr><td align='left'><b>12 &divide; 4 = &nbsp; 3.</b></td><td align='left'><b> &nbsp; 3 is the quotient.</b></td></tr>
+</table></div>
+<p><b>6.</b> Find the quotients. Check your answers by multiplying.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%">
+<tr><td align='left'>3 |<span class="overline"> 99</span></td><td align='left'>2 |<span class="overline"> 86</span></td><td align='left'>5 |<span class="overline"> 155</span></td>
+<td align='left'>6 |<span class="overline"> 246</span></td><td align='left'>4 |<span class="overline"> 168</span></td><td align='left'>3 |<span class="overline"> 219</span></td></tr>
+</table></div>
+<p>[Uneven division is taught by the same general plan, extended.]</p>
+
+
+<p class="center"><b>LONG DIVISION: DEDUCTIVE EXPLANATION</b></p>
+
+
+<p class="center"><b>To Divide by Long Division</b></p>
+
+<p><b>1</b>. Let it be required to divide 34531 by 15.</p>
+
+<p class="center"><i>Operation</i></p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr>
+<td align='right' valign='top'>&nbsp;<br />Divisor &nbsp; </td>
+<td align='right' valign='top'>Divided<br />
+15 ) 34531 (<br />
+30 &nbsp; &nbsp; &nbsp; &nbsp;<br />
+<span class="overline">&nbsp;&nbsp;45</span>&nbsp; &nbsp;&nbsp; &nbsp;<br />
+<span class="u">&nbsp;&nbsp;45</span>&nbsp; &nbsp;&nbsp; &nbsp;<br />
+31 &nbsp;&nbsp;<br />
+<span class="u">30</span> &nbsp;&nbsp;<br />
+1 &nbsp;&nbsp;<br />
+</td>
+<td align='left' valign='top'>&nbsp; <br />2302<sup>1</sup>&frasl;<sub>15</sub> &nbsp; &nbsp; Quotient</td>
+</tr>
+<tr><td align='left'></td><td align='right'>Remainder</td><td align='left'></td></tr>
+</table></div>
+
+
+<p>For convenience we write
+the divisor at the left and the
+quotient at the right of the
+dividend, and begin to divide
+as in Short Division.</p>
+
+<p>15 is contained in 3 ten-thousands
+0 ten-thousands
+times; therefore, there will
+be 0 ten-thousands in the
+quotient. Take 34 thousands;
+15 is contained in
+34 thousands 2 thousands
+times; we write the 2 thousands in the quotient. 15 &times; 2 thousands
+= 30 thousands, which, subtracted from 34 thousands,
+leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we
+have 45 hundreds.</p>
+
+<p>15 in 45 hundreds 3 hundreds times; we write the 3 hundreds
+in the quotient. 15 &times; 3 hundreds = 45 hundreds, which subtracted
+from 45 hundreds, leaves nothing. Adding the 3 tens, we have
+3 tens.<span class='pagenum'><a name="Page_65" id="Page_65">[Pg 65]</a></span></p>
+
+<p>15 in 3 tens 0 tens times; we write 0 tens in the quotient.
+Adding to the three tens, which equal 30 units, the 1 unit, we
+have 31 units.</p>
+
+<p>15 in 31 units 2 units times; we write the 2 units in the quotient.
+15 &times; 2 units = 30 units, which, subtracted from 31 units, leaves 1
+unit as a remainder. Indicating the division of the 1 unit, we
+annex the fractional expression, <sup>1</sup>&frasl;<sub>15</sub> unit, to the integral part of the
+quotient.</p>
+
+<p>Therefore, 34531 divided by 15 is equal to 2302<sup>1</sup>&frasl;<sub>15</sub>.</p>
+
+<p class="noidt">[B. Greenleaf, <i>Practical Arithmetic</i>, '73, p. 49.]</p>
+
+
+<p class="center"><b>LONG DIVISION: INDUCTIVE EXPLANATION</b></p>
+
+<p class="center"><b>Dividing by Large Numbers</b></p>
+
+<p><b>1</b>. Just before Christmas Frank's father sent 360 oranges to
+be divided among the children in Frank's class. There
+are 29 children. How many oranges should each child
+receive? How many oranges will be left over?</p>
+
+<p><i>Here is the best way to find out:</i></p>
+
+<div class='center'>
+<table border="0" cellpadding="5" cellspacing="0" summary="">
+<tr>
+<td align='right' valign='top'><i>
+12<br />
+29|<span class="overline">360</span><br />
+29&nbsp;&nbsp;<br />
+<span class="overline"> &nbsp;&nbsp; 70</span><br />
+58<br />
+<span class="overline">12</span>
+</i></td>
+<td align='right' valign='top'><i><small>and 12 remainder</small></i></td>
+<td align='left'>
+<i>Think how many 29s there are in 36. 1 is right.<br />
+Write 1 over the 6 of 36. Multiply 29 by 1.<br />
+Write the 29 under the 36. Subtract 29 from 36.<br />
+Write the 0 of 360 after the 7.<br />
+Think how many 29s there are in 70. 2 is right.<br />
+Write 2 over the 0 of 360. Multiply 29 by 2.<br />
+Write the 58 under 70. Subtract 58 from 70.<br />
+There is 12 remainder.</i><br />
+<br />
+<i>Each child gets 12 oranges, and there are 12 left
+over. This is right, for 12 multiplied by 29 = 348,
+and 348 &#43; 12 = 360</i>.
+</td></tr>
+</table></div>
+
+
+<hr style='width: 45%;' />
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%">
+<tr>
+<td align='left'><b> &nbsp; 8.</b><br />31 |<span class="overline"> 99,587</span></td>
+<td align='left'><i>In No. 8, keep on dividing by 31 until you
+have used the 5, the 8, and the 7, and have
+four figures in the quotient.</i></td>
+</tr>
+</table></div>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%">
+<tr>
+<td align='left'><b> &nbsp; 9.</b><br />22 |<span class="overline"> 253</span></td>
+<td align='left'><b> &nbsp; 10.</b><br />22 |<span class="overline"> 2895</span></td>
+<td align='left'><b> &nbsp; 11.</b><br />21 |<span class="overline"> 8891</span></td>
+<td align='left'><b> &nbsp; 12.</b><br />22 |<span class="overline"> 290</span></td>
+<td align='left'><b> &nbsp; 13.</b><br />32 |<span class="overline"> 16,368</span></td>
+</tr>
+</table></div>
+
+<p>Check your results for 9, 10, 11, 12, and 13.</p>
+
+<p>&nbsp;</p>
+<p><span class='pagenum'><a name="Page_66" id="Page_66">[Pg 66]</a></span></p>
+
+<p><b>1</b>. The boys and girls of the Welfare Club plan to earn money
+to buy a victrola. There are 23 boys and girls. They
+can get a good second-hand victrola for $5.75. How
+much must each earn if they divide the cost equally?</p>
+
+<p><i>Here is the best way to find out</i>:</p>
+
+<div class='center'>
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr>
+<td align='right'><i>
+$.25<br />
+23|<span class="overline">$5.75</span><br />
+46&nbsp;&nbsp;&nbsp;<br />
+<span class="overline">&nbsp;115</span><br />
+<span class="u">115</span>
+</i></td>
+<td align='left'>
+<i>Think how many 23s there are in 57. 2 is right.<br />
+Write 2 over the 7 of 57. Multiply 23 by 2.<br />
+Write 46 under 57 and subtract. Write the 5 of 575 after the 11.<br />
+Think how many 23s there are in 115. 5 is right.<br />
+Write 5 over the 5 of 575. Multiply 23 by 5.<br />
+Write the 115 under the 115 that is there and subtract.<br />
+There is no remainder.<br />
+Put $ and the decimal point where they belong.<br />
+Each child must earn 25 cents. This is right, for $.25 multiplied by 23 = $5.75</i>.
+</td></tr>
+</table></div>
+
+<p><b>2</b>. Divide $71.76 equally among 23 persons. How much is
+each person's share?</p>
+
+<p><b>3</b>. Check your result for No. 2 by multiplying the quotient by
+the divisor.</p>
+
+<p>Find the quotients. Check each quotient by multiplying it by
+the divisor.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%">
+<tr>
+<td align='left'><b> &nbsp; 4.</b><br />23 |<span class="overline"> $99.13</span></td>
+<td align='left'><b> &nbsp; 5.</b><br />25 |<span class="overline"> $18.50</span></td>
+<td align='left'><b> &nbsp; 6.</b><br />21 |<span class="overline"> $129.15</span></td>
+<td align='left'><b> &nbsp; 7.</b><br />13 |<span class="overline"> $29.25</span></td>
+<td align='left'><b> &nbsp; 8.</b><br />32 |<span class="overline"> $73.92</span></td>
+</tr>
+</table></div>
+
+<p class="center"><b>1 bushel = 32 qt</b>.</p>
+
+<p><b>9</b>. How many bushels are there in 288 qt.? &nbsp; &nbsp; <b>10</b>. In 192 qt.? &nbsp; &nbsp; <b>11</b>. In 416 qt.?</p>
+</div>
+
+<p>Crucial experiments are lacking, but there are several
+lines of well-attested evidence. First of all, there can be
+no doubt that the great majority of pupils learn these manipulations
+at the start from the placing of units under units,
+tens under tens, etc., in adding, to the placing of the decimal
+point in division with decimals, by imitation and blind<span class='pagenum'><a name="Page_67" id="Page_67">[Pg 67]</a></span>
+following of specific instructions, and that a very large proportion
+of the pupils do not to the end, that is to the fifth
+school-year, understand them as necessary deductions from
+decimal notation. It also seems probable that this proportion
+would not be much reduced no matter how ingeniously
+and carefully the deductions were explained by textbooks
+and teachers. Evidence of this fact will appear abundantly
+to any one who will observe schoolroom life. It also appears
+in the fact that after the properties of the decimal notation
+have been thus used again and again; <i>e.g.</i>, for deducing
+'carrying' in addition, 'borrowing' in subtraction, 'carrying'
+in multiplication, the value of the digits in the partial product,
+the value of each remainder in short division, the
+value of the quotient figures in division, the addition, subtraction,
+multiplication, and division of United States money,
+and the placing of the decimal point in multiplication, no
+competent teacher dares to rely upon the pupil, even though
+he now has four or more years' experience with decimal notation,
+to deduce the placing of the decimal point in division
+with decimals. It may be an illusion, but one seems to
+sense in the better textbooks a recognition of the futility
+of the attempt to secure deductive derivations of those
+manipulations. I refer to the brevity of the explanations
+and their insertion in such a form that they will influence
+the pupils' thinking as little as possible. At any rate the
+fact is sure that most pupils do not learn the manipulations
+by deductive reasoning, or understand them as necessary
+consequences of abstract principles.</p>
+
+<p>It is a common opinion that the only alternative is knowing
+them by rote. This, of course, is one common alternative,
+but the other explanation suggests that understanding the
+manipulations by inductive reasoning from their results
+is another and an important alternative. The manipula<span class='pagenum'><a name="Page_68" id="Page_68">[Pg 68]</a></span>tions
+of 'long' multiplication, for instance, learned by imitation
+or mechanical drill, are found to give for 25 &times; <i>A</i> a
+result about twice as large as for 13 &times; <i>A</i>, for 38 or 39 &times; <i>A</i>
+a result about three times as large; for 115 &times; <i>A</i> a result
+about ten times as large as for 11 &times; <i>A</i>. With even the very
+dull pupils the procedure is verified at least to the extent
+that it gives a result which the scientific expert in the case&mdash;the
+teacher&mdash;calls right. With even the very bright
+pupils, who can appreciate the relation of the procedure to
+decimal notation, this relation may be used not as the sole
+deduction of the procedure beforehand, but as one partial
+means of verifying it afterward. Or there may be the condition
+of half-appreciation of the relation in which the pupil
+uses knowledge of the decimal notation to convince himself
+that the procedure <i>does</i>, but not that it <i>must</i> give the right
+answer, the answer being 'right' because the teacher, the
+answer-list, and collateral evidence assure him of it.</p>
+
+<p>I have taken the manipulation of the partial products as
+an illustration because it is one of the least favored cases
+for the explanation I am presenting. If we take the first
+case where a manipulation may be deduced from decimal
+notation, known merely by rote, or verified inductively,
+namely, the addition of two-place numbers, it seems sure
+that the mental processes just described are almost the
+universal rule.</p>
+
+<p>Surely in our schools at present children add the 3 of 23
+to the 3 of 53 and the 2 of 23 to the 5 of 53 at the start, in
+nine cases out of ten because they see the teacher do so and
+are told to do so. They are protected from adding
+3 &#43; 3 &#43; 2 &#43; 5 not by any deduction of any sort but because
+they do not know how to add 8 and 5, because they have
+been taught the habit of adding figures that stand one above
+the other, or with a  &#43;  between them; and because they are<span class='pagenum'><a name="Page_69" id="Page_69">[Pg 69]</a></span>
+shown or told what they are to do. They are protected from
+adding 3 &#43; 5 and 2 &#43; 3, again, by no deductive reasoning but
+for the second and third reasons just given. In nine cases
+out of ten they do not even think of the possibility of adding
+in any other way than the '3 &#43; 3, 2 &#43; 5' way, much less do
+they select that way on account of the facts that 53 = 50 &#43; 3
+and 23 = 20  &#43; 3, that 50 &#43; 20 = 70, that 3 &#43; 3 = 6, and that
+(<i>a</i> &#43; <i>b</i>) &#43; (<i>c</i> &#43; <i>d</i>) = (<i>a</i> &#43; <i>c</i>) &#43; (<i>b</i> &#43; <i>d</i>)!</p>
+
+<p>Just as surely all but the very dullest twentieth or so of
+children come in the end to something more than rote
+knowledge,&mdash;to <i>understand</i>, to <i>know</i> that the procedure
+in question is right.</p>
+
+<p>Whether they know <i>why</i> 76 is right depends upon what is
+meant by <i>why</i>. If it means that 76 is the result which
+competent people agree upon, they do. If it means that 76
+is the result which would come from accurate counting they
+perhaps know why as well as they would have, had they been
+given full explanations of the relation of the procedure in
+two-place addition to decimal notation. If <i>why</i> means
+because 53 = 50 &#43; 3, 23 = 20 &#43; 3, 50 &#43; 20 = 70, and (<i>a</i> &#43; <i>b</i>) &#43; (<i>c</i> &#43; <i>d</i>) = (<i>a</i> &#43; <i>c</i>) &#43; (<i>b</i> &#43; <i>d</i>),
+they do not. Nor, I am tempted
+to add, would most of them by any sort of teaching whatever.</p>
+
+<p>I conclude, therefore, that school children may and do
+reason about and understand the manipulations of numbers
+in this inductive, verifying way without being able to, or at
+least without, under present conditions, finding it profitable
+to derive them deductively. I believe, in fact, that pure
+arithmetic <i>as it is learned and known</i> is largely an <i>inductive
+science</i>. At one extreme is a minority to whom it is a series
+of deductions from principles; at the other extreme is a
+minority to whom it is a series of blind habits; between
+the two is the great majority, representing every gradation
+but centering about the type of the inductive thinker.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_70" id="Page_70">[Pg 70]</a></span></p>
+<h2><a name="CHAPTER_IV" id="CHAPTER_IV"></a>CHAPTER IV</h2>
+
+<h3>THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED):
+THE SELECTION OF THE BONDS TO BE
+FORMED</h3>
+
+
+<p>When the analysis of the mental functions involved in
+arithmetical learning is made thorough it turns into the
+question, 'What are the elementary bonds or connections
+that constitute these functions?' and when the problem of
+teaching arithmetic is regarded, as it should be in the light
+of present psychology, as a problem in the development of a
+hierarchy of intellectual habits, it becomes in large measure
+a problem of the choice of the bonds to be formed and of the
+discovery of the best order in which to form them and the
+best means of forming each in that order.</p>
+
+
+<h4>THE IMPORTANCE OF HABIT-FORMATION</h4>
+
+<p>The importance of habit-formation or connection-making
+has been grossly underestimated by the majority of teachers
+and writers of textbooks. For, in the first place, mastery
+by deductive reasoning of such matters as 'carrying' in
+addition, 'borrowing' in subtraction, the value of the digits
+in the partial products in multiplication, the manipulation
+of the figures in division, the placing of the decimal point
+after multiplication or division with decimals, or the manipulation
+of the figures in the multiplication and division of
+<span class='pagenum'><a name="Page_71" id="Page_71">[Pg 71]</a></span>
+fractions, is impossible or extremely unlikely in the case of
+children of the ages and experience in question. They do
+not as a rule deduce the method of manipulation from their
+knowledge of decimal notation. Rather they learn about
+decimal notation by carrying, borrowing, writing the last
+figure of each partial product under the multiplier which
+gives that product, etc. They learn the method of manipulating
+numbers by seeing them employed, and by more or
+less blindly acquiring them as associative habits.</p>
+
+<p>In the second place, we, who have already formed and
+long used the right habits and are thereby protected against
+the casual misleadings of unfortunate mental connections,
+can hardly realize the force of mere association. When a
+child writes sixteen as 61, or finds 428 as the sum of</p>
+
+<p class="center">
+15<br />
+19<br />
+16<br />
+<span class="u">18</span>
+</p>
+
+<p class="noidt">or gives 642 as an answer to 27 &times; 36, or says that 4 divided
+by &frac14; = 1, we are tempted to consider him mentally perverse,
+forgetting or perhaps never having understood that he
+goes wrong for exactly the same general reason that we go
+right; namely, the general law of habit-formation. If we
+study the cases of 61 for 16, we shall find them occurring
+in the work of pupils who after having been drilled in writing
+26, 36, 46, 62, 63, and so on, in which the order of the six
+in writing is the same as it is in speech, return to writing the
+'teen numbers. If our language said onety-one for eleven
+and onety-six for sixteen, we should probably never find
+such errors except as 'lapses' or as the results of misperception
+or lack of memory. They would then be more
+frequent <i>before</i> the 20s, 30s, etc., were learned.</p>
+
+<p>If pupils are given much drill on written single column
+addition involving the higher decades (each time writing
+<span class='pagenum'><a name="Page_72" id="Page_72">[Pg 72]</a></span>
+the two-figure sum), they are forming a habit of writing 28
+after the sum of 8, 6, 9, and 5 is reached; and it should not
+surprise us if the pupil still occasionally writes the two-figure
+sum for the first column though a second column is
+to be added also. On the contrary, unless some counter
+force influences him, he is absolutely sure to make this
+mistake.</p>
+
+<p>The last mistake quoted (4 &divide; &frac14; = 1) is interesting because
+here we have possibly one of the cases where deduction from
+psychology alone can give constructive aid to teaching.
+Multiplication and division by fractions have been notorious
+for their difficulty. The former is now alleviated by using <i>of</i>
+instead of &times; until the new habit is fixed. The latter is still
+approached with elaborate caution and with various means
+of showing why one must 'invert and multiply' or 'multiply
+by the reciprocal.'</p>
+
+<p>But in the author's opinion it seems clear that the difficulty
+in multiplying and dividing by a fraction was not that
+children felt any logical objections to canceling or inverting.
+I fancy that the majority of them would cheerfully invert
+any fraction three times over or cancel numbers at random
+in a column if they were shown how to do so. But if you
+are a youngster inexperienced in numerical abstractions
+and if you have had <i>divide</i> connected with 'make smaller'
+three thousand times and never once connected with 'make
+bigger,' you are sure to be somewhat impelled to make the
+number smaller the three thousand and first time you are
+asked to divide it. Some of my readers will probably confess
+that even now they feel a slight irritation or doubt in saying
+or writing that <sup>16</sup>&frasl;<sub>1</sub> &divide; <sup>1</sup>&frasl;<sub>8</sub> = 128.</p>
+
+<p>The habits that have been confirmed by every multiplication
+and division by integers are, in this particular of '<i>the
+ratio of result to number operated upon</i>,' directly opposed to<span class='pagenum'><a name="Page_73" id="Page_73">[Pg 73]</a></span>
+the formation of the habits required with fractions. And
+that is, I believe, the main cause of the difficulty. Its
+treatment then becomes easy, as will be shown later.</p>
+
+<p>These illustrations could be added to almost indefinitely,
+especially in the case of the responses made to the so-called
+'catch' problems. The fact is that the learner rarely can,
+and almost never does, survey and analyze an arithmetical
+situation and justify what he is going to do by articulate
+deductions from principles. He usually feels the situation
+more or less vaguely and responds to it as he has responded
+to it or some situation like it in the past. Arithmetic is
+to him not a logical doctrine which he applies to various
+special instances, but a set of rather specialized habits of
+behavior toward certain sorts of quantities and relations.
+And in so far as he does come to know the doctrine it is
+chiefly by doing the will of the master. This is true even
+with the clearest expositions, the wisest use of objective
+aids, and full encouragement of originality on the pupil's
+part.</p>
+
+<p>Lest the last few paragraphs be misunderstood, I hasten to
+add that the psychologists of to-day do not wish to make
+the learning of arithmetic a mere matter of acquiring thousands
+of disconnected habits, nor to decrease by one jot the
+pupil's genuine comprehension of its general truths. They
+wish him to reason not less than he has in the past, but more.
+They find, however, that you do not secure reasoning in a
+pupil by demanding it, and that his learning of a general
+truth without the proper development of organized
+habits back of it is likely to be, not a rational learning
+of that general truth, but only a mechanical memorizing
+of a verbal statement of it. They have come to
+know that reasoning is not a magic force working in
+independence of ordinary habits of thought, but an or<span class='pagenum'><a name="Page_74" id="Page_74">[Pg 74]</a></span>ganization
+and co&ouml;peration of those very habits on a higher
+level.</p>
+
+<p>The older pedagogy of arithmetic stated a general law or
+truth or principle, ordered the pupil to learn it, and gave
+him tasks to do which he could not do profitably unless he
+understood the principle. It left him to build up himself
+the particular habits needed to give him understanding and
+mastery of the principle. The newer pedagogy is careful
+to help him build up these connections or bonds ahead of
+and along with the general truth or principle, so that he
+can understand it better. The older pedagogy commanded
+the pupil to reason and let him suffer the penalty of small
+profit from the work if he did not. The newer provides
+instructive experiences with numbers which will stimulate
+the pupil to reason so far as he has the capacity, but will
+still be profitable to him in concrete knowledge and skill,
+even if he lacks the ability to develop the experiences into
+a general understanding of the principles of numbers. The
+newer pedagogy secures more reasoning in reality by not
+pretending to secure so much.</p>
+
+<p>The newer pedagogy of arithmetic, then, scrutinizes
+every element of knowledge, every connection made in the
+mind of the learner, so as to choose those which provide the
+most instructive experiences, those which will grow together
+into an orderly, rational system of thinking about numbers
+and quantitative facts. It is not enough for a problem
+to be a test of understanding of a principle; it must also
+be helpful in and of itself. It is not enough for an example
+to be a case of some rule; it must help review and
+consolidate habits already acquired or lead up to and
+facilitate habits to be acquired. Every detail of the pupil's
+work must do the maximum service in arithmetical
+learning.</p>
+<p><span class='pagenum'><a name="Page_75" id="Page_75">[Pg 75]</a></span></p>
+
+<h4>DESIRABLE BONDS NOW OFTEN NEGLECTED</h4>
+
+<p>As hitherto, I shall not try to list completely the elementary
+bonds that the course of study in arithmetic should
+provide for. The best means of preparing the student of
+this topic for sound criticism and helpful invention is to let
+him examine representative cases of bonds now often neglected
+which should be formed and representative cases of
+useless, or even harmful, bonds now often formed at considerable
+waste of time and effort.</p>
+
+<p>(1) <i>Numbers as measures of continuous quantities.</i>&mdash;The
+numbers one, two, three, 1, 2, 3, etc., should be connected
+soon after the beginning of arithmetic each with the appropriate
+amount of some continuous quantity like length or
+volume or weight, as well as with the appropriate sized
+collection of apples, counters, blocks, and the like. Lines
+should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two
+inches, three inches, etc.; weights should be lifted and called
+one pound, two pounds, etc.; things should be measured in
+glassfuls, handfuls, pints, and quarts. Otherwise the pupil
+is likely to limit the meaning of, say, <i>four</i> to four sensibly
+discrete things and to have difficulty in multiplication and
+division. Measuring, or counting by insensibly marked off
+repetitions of a unit, binds each number name to its meaning
+as &mdash;&mdash; <i>times whatever 1 is</i>, more surely than mere counting
+of the units in a collection can, and should re&euml;nforce the
+latter.</p>
+
+<p>(2) <i>Additions in the higher decades.</i>&mdash;In the case of all
+save the very gifted children, the additions with higher
+decades&mdash;that is, the bonds, 16 &#43; 7 = 23, 26 &#43; 7 = 33,
+36 &#43; 7 = 43, 14 &#43; 8 = 22, 24 &#43; 8 = 32, and the like&mdash;need to
+be specifically practiced until the tendency becomes generalized.
+'Counting' by 2s beginning with 1, and with 2,<span class='pagenum'><a name="Page_76" id="Page_76">[Pg 76]</a></span>
+counting by 3s beginning with 1, with 2, and with 3, counting
+by 4s beginning with 1, with 2, with 3, and with 4, and so on,
+make easy beginnings in the formation of the decade connections.
+Practice with isolated bonds should soon be added to
+get freer use of the bonds. The work of column addition
+should be checked for accuracy so that a pupil will continually
+get beneficial practice rather than 'practice in
+error.'</p>
+
+<p>(3) <i>The uneven divisions.</i>&mdash;The quotients with remainders
+for the divisions of every number to 19 by 2, every number
+to 29 by 3, every number to 39 by 4, and so on should be
+taught as well as the even divisions. A table like the
+following will be found a convenient means of making these
+connections:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>
+10 = .... 2s<br />
+10 = .... 3s and .... rem.<br />
+10 = .... 4s and .... rem.<br />
+10 = .... 5s<br />
+11 = .... 2s and .... rem.<br />
+11 = .... 3s and .... rem.<br />
+&nbsp;.<br />
+&nbsp;.<br />
+&nbsp;.<br />
+89 = .... 9s and .... rem.
+</td></tr>
+</table></div>
+
+<p>These bonds must be formed before short division can be
+efficient, are useful as a partial help toward selection of the
+proper quotient figures in long division, and are the chief
+instruments for one of the important problem series in
+applied arithmetic,&mdash;"How many <i>x</i>s can I buy for <i>y</i> cents
+at <i>z</i> cents per <i>x</i> and how much will I have left?" That
+these bonds are at present sadly neglected is shown by Kirby
+['13], who found that pupils in the last half of grade 3 and the<span class='pagenum'><a name="Page_77" id="Page_77">[Pg 77]</a></span>
+first half of grade 4 could do only about four such examples
+per minute (in a ten-minute test), and even at that rate
+made far from perfect records, though they had been taught
+the regular division tables. Sixty minutes of practice
+resulted in a gain of nearly 75 percent in number done per
+minute, with an increase in accuracy as well.</p>
+
+<p>(4) <i>The equation form</i>.&mdash;The equation form with an
+unknown quantity to be determined, or a missing number
+to be found, should be connected with its meaning and with
+the problem attitude long before a pupil begins algebra,
+and in the minds of pupils who never will study algebra.</p>
+
+<p>Children who have just barely learned to add and subtract
+learn easily to do such work as the following:&mdash;</p>
+
+<p>Write the missing numbers:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='left'>
+4 &#43; 8 = ....<br />
+5 &#43; .... = 14<br />
+.... &#43; 3 = 11<br />
+.... = 5 &#43; 2<br />
+16 = 7 &#43; ....<br />
+12 = .... &#43; 5
+</td></tr>
+</table></div>
+
+
+<p>The equation form is the simplest uniform way yet devised
+to state a quantitative issue. It is capable of indefinite
+extension if certain easily understood conventions about
+parentheses and fraction signs are learned. It should be
+employed widely in accounting and the treatment of commercial
+problems, and would be except for outworn conventions.
+It is a leading contribution of algebra to business and
+industrial life. Arithmetic can make it nearly as well.
+It saves more time in the case of drills on reducing fractions
+to higher and lower terms alone than is required to learn
+its meaning and use. To rewrite a quantitative problem<span class='pagenum'><a name="Page_78" id="Page_78">[Pg 78]</a></span>
+as an equation and then make the easy selection of the
+necessary technique to solve the equation is one of the most
+universally useful intellectual devices known to man. The
+words 'equals,' 'equal,' 'is,' 'are,' 'makes,' 'make,' 'gives,'
+'give,' and their rarer equivalents should therefore early
+give way on many occasions to the '=' which so far surpasses
+them in ultimate convenience and simplicity.</p>
+
+<p>(5) <i>Addition and subtraction facts in the case of fractions.</i>&mdash;In
+the case of adding and subtracting fractions, certain
+specific bonds&mdash;between the situation of halves and thirds
+to be added and the responses of thinking of the numbers
+as equal to so many sixths, between the situation thirds and
+fourths to be added and thinking of them as so many twelfths,
+between fourths and eighths to be added and thinking of
+them as eighths, and the like&mdash;should be formed separately.
+The general rule of thinking of fractions as their equivalents
+with some convenient denominator should come as an
+organization and extension of such special habits, not as
+an edict from the textbook or teacher.</p>
+
+<p>(6) <i>Fractional equivalents.</i>&mdash;Efficiency requires that in
+the end the much used reductions should be firmly connected
+with the situations where they are needed. They may as
+well, therefore, be so connected from the beginning, with
+the gain of making the general process far easier for the dull
+pupils to master. We shall see later that, for all save the
+very gifted pupils, the economical way to get an understanding
+of arithmetical principles is not, usually, to learn a rule
+and then apply it, but to perform instructive operations and,
+in the course of performing them, to get insight into the
+principles.</p>
+
+<p>(7) <i>Protective habits in multiplying and dividing with fractions.</i>&mdash;In
+multiplying and dividing with fractions special
+bonds should be formed to counteract the now harmful<span class='pagenum'><a name="Page_79" id="Page_79">[Pg 79]</a></span>
+influence of the 'multiply = get a larger number' and 'divide = get
+a smaller number' bonds which all work with integers
+has been re&euml;nforcing.</p>
+
+<p>For example, at the beginning of the systematic work
+with multiplication by a fraction, let the following be printed
+clearly at the top of every relevant page of the textbook and
+displayed on the blackboard:&mdash;</p>
+
+<p><i>When you multiply a number by anything more than 1 the
+result is larger than the number.</i></p>
+
+<p><i>When you multiply a number by 1 the result is the same as the
+number.</i></p>
+
+<p><i>When you multiply a number by anything less than 1 the
+result is smaller than the number.</i></p>
+
+<p>Let the pupils establish the new habit by many such
+exercises as:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="1" summary="" width="60%">
+<tr>
+<td align='left'>
+18 &times; 4 = ....<br />
+4 &times; 4 = ....<br />
+2 &times; 4 = ....<br />
+1 &times; 4 = ....<br />
+<sup>1</sup>&frasl;<sub>2</sub> &times; 4 = ....<br />
+<sup>1</sup>&frasl;<sub>4</sub> &times; 4 = ....<br />
+<sup>1</sup>&frasl;<sub>8</sub> &times; 4 = ....
+</td>
+<td align='left'>
+9 &times; 2 = ....<br />
+6 &times; 2 = ....<br />
+3 &times; 2 = ....<br />
+1 &times; 2 = ....<br />
+<sup>1</sup>&frasl;<sub>3</sub> &times; 2 = ....<br />
+<sup>1</sup>&frasl;<sub>6</sub> &times; 2 = ....<br />
+<sup>1</sup>&frasl;<sub>9</sub> &times; 2 = ....
+</td>
+</tr>
+</table></div>
+
+<p>In the case of division by a fraction the old harmful habit
+should be counteracted and refined by similar rules and
+exercises as follows:&mdash;</p>
+
+<p><i>When you divide a number by anything more than 1 the result
+is smaller than the number.</i></p>
+
+<p><i>When you divide a number by 1 the result is the same as the
+number.</i></p>
+
+<p><i>When you divide a number by anything less than 1 the result
+is larger than the number.</i><span class='pagenum'><a name="Page_80" id="Page_80">[Pg 80]</a></span></p>
+
+<p>State the missing numbers:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="1" summary="" width="60%">
+<tr><td align='left'>8 = .... 4s</td><td align='left'>12 = .... 6s</td><td align='left'>9 = .... 9s</td></tr>
+<tr><td align='left'>8 = .... 2s</td><td align='left'>12 = .... 4s</td><td align='left'>9 = .... 3s</td></tr>
+<tr><td align='left'>8 = .... 1s</td><td align='left'>12 = .... 3s</td><td align='left'>9 = .... 1s</td></tr>
+<tr><td align='left'>8 = .... <sup>1</sup>&frasl;<sub>2</sub>s</td><td align='left'>12 = .... 2s</td><td align='left'>9 = .... <sup>1</sup>&frasl;<sub>3</sub>s</td></tr>
+<tr><td align='left'>8 = .... <sup>1</sup>&frasl;<sub>4</sub>s</td><td align='left'>12 = .... 1s</td><td align='left'>9 = .... <sup>1</sup>&frasl;<sub>9</sub>s</td></tr>
+<tr><td align='left'>8 = .... <sup>1</sup>&frasl;<sub>8</sub>s</td><td align='left'>12 = .... <sup>1</sup>&frasl;<sub>2</sub>s</td></tr>
+<tr><td align='left'></td><td align='left'>12 = .... <sup>1</sup>&frasl;<sub>3</sub>s</td></tr>
+<tr><td align='left'></td><td align='left'>12 = .... <sup>1</sup>&frasl;<sub>4</sub>s</td></tr>
+</table></div>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="1" summary="" width="60%">
+<tr><td align='left'>16 &divide; 16 =</td><td align='left'>9 &divide; 9 =</td><td align='left'>10 &divide; 10 =</td><td align='left'>12 &divide; 6 =</td></tr>
+<tr><td align='left'>16 &divide; 8 =</td><td align='left'>9 &divide; 3 =</td><td align='left'>10 &divide; 5 =</td><td align='left'>12 &divide; 4 =</td></tr>
+<tr><td align='left'>16 &divide; 4 =</td><td align='left'>9 &divide; 1 =</td><td align='left'>10 &divide; 1 =</td><td align='left'>12 &divide; 3 =</td></tr>
+<tr><td align='left'>16 &divide; 2 =</td><td align='left'>9 &divide; <sup>1</sup>&frasl;<sub>3</sub> =</td><td align='left'>10 &divide; <sup>1</sup>&frasl;<sub>5</sub> =</td><td align='left'>12 &divide; 2 =</td></tr>
+<tr><td align='left'>16 &divide; 1 =</td><td align='left'>9 &divide; <sup>1</sup>&frasl;<sub>9</sub> =</td><td align='left'>10 &divide; <sup>1</sup>&frasl;<sub>10</sub> =</td><td align='left'>12 &divide; 1 =</td></tr>
+<tr><td align='left'>16 &divide; <sup>1</sup>&frasl;<sub>2</sub> =</td><td align='left'></td><td align='left'></td><td align='left'>12 &divide; <sup>1</sup>&frasl;<sub>2</sub> =</td></tr>
+<tr><td align='left'>16 &divide; <sup>1</sup>&frasl;<sub>4</sub> =</td><td align='left'></td><td align='left'></td><td align='left'>12 &divide; <sup>1</sup>&frasl;<sub>3</sub> =</td></tr>
+<tr><td align='left'>16 &divide; <sup>1</sup>&frasl;<sub>8</sub> =</td><td align='left'></td><td align='left'></td><td align='left'>12 &divide; <sup>1</sup>&frasl;<sub>4</sub> =</td></tr>
+<tr><td align='left'></td><td align='left'></td><td align='left'></td><td align='left'>12 &divide; <sup>1</sup>&frasl;<sub>6</sub> =</td></tr>
+</table></div>
+
+<p>(8) <i>'% of' means 'hundredths times.'</i>&mdash;In the case of percentage
+a series of bonds like the following should be
+formed:&mdash;</p>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="1" summary="">
+<tr><td align='left'></td><td align='left'>&nbsp;&nbsp;5</td><td align='left'>percent</td><td align='left'>of&nbsp;</td><td align='left'>= .05 times</td></tr>
+<tr><td align='left'></td><td align='left'>20</td><td align='left'>&nbsp;&nbsp; " &nbsp;&nbsp; "</td><td align='left'>"</td><td align='left'>= .20 &nbsp;&nbsp; "</td></tr>
+<tr><td align='left'></td><td align='left'>&nbsp;&nbsp;6</td><td align='left'>&nbsp;&nbsp; " &nbsp;&nbsp; "</td><td align='left'>"</td><td align='left'>= .06 &nbsp;&nbsp; "</td></tr>
+<tr><td align='left'></td><td align='left'>25</td><td align='left'>%</td><td align='left'>"</td><td align='left'>= .25 &times;</td></tr>
+<tr><td align='left'></td><td align='left'>12</td><td align='left'>%</td><td align='left'>"</td><td align='left'>= .12 &times;</td></tr>
+<tr><td align='left'></td><td align='left'>&nbsp;&nbsp;3</td><td align='left'>%</td><td align='left'>"</td><td align='left'>= .03 &times;</td></tr>
+</table></div>
+
+<p>Four five-minute drills on such connections between
+'<i>x</i> percent of' and 'its decimal equivalent times' are worth
+an hour's study of verbal definitions of the meaning of
+percent as per hundred or the like. The only use of the<span class='pagenum'><a name="Page_81" id="Page_81">[Pg 81]</a></span>
+study of such definitions is to facilitate the later formation
+of the bonds, and, with all save the brighter pupils, the
+bonds are more needed for an understanding of the definitions
+than the definitions are needed for the formation of the
+bonds.</p>
+
+<p>(9) <i>Habits of verifying results.</i>&mdash;Bonds should early be
+formed between certain manipulations of numbers and
+certain means of checking, or verifying the correctness of,
+the manipulation in question. The additions to 9 &#43; 9 and
+the subtractions to 18 &minus; 9 should be verified by objective
+addition and subtraction and counting until the pupil has
+sure command; the multiplications to 9 &times; 9 should be
+verified by objective multiplication and counting of the
+result (in piles of tens and a pile of ones) eight or ten times,<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a>
+and by addition eight or ten times;<a href="#Footnote_4_4" class="fnanchor">[4]</a> the divisions to 81 &divide; 9
+should be verified by multiplication and occasionally
+objectively until the pupil has sure command; column
+addition should be checked by adding the columns separately
+and adding the sums so obtained, and by making two
+shorter tasks of the given task and adding the two sums;
+'short' multiplication should be verified eight or ten times by
+addition; 'long' multiplication should be checked by
+reversing multiplier and multiplicand and in other ways;
+'short' and 'long' division should be verified by multiplication.</p>
+
+<p>These habits of testing an obtained result are of threefold
+value. They enable the pupil to find his own errors,
+and to maintain a standard of accuracy by himself. They
+give him a sense of the relations of the processes and the
+reasons why the right ways of adding, subtracting, multiplying,
+and dividing are right, such as only the very bright
+<span class='pagenum'><a name="Page_82" id="Page_82">[Pg 82]</a></span>
+pupils can get from verbal explanations. They put his
+acquisition of a certain power, say multiplication, to a
+real and intelligible use, in checking the results of his practice
+of a new power, and so instill a respect for arithmetical
+power and skill in general. The time spent in such verification
+produces these results at little cost; for the practice in
+adding to verify multiplications, in multiplying to verify
+divisions, and the like is nearly as good for general drill
+and review of the addition and multiplication themselves
+as practice devised for that special purpose.</p>
+
+<p>Early work in adding, subtracting, and reducing fractions
+should be verified by objective aids in the shape of lines and
+areas divided in suitable fractional parts. Early work with
+decimal fractions should be verified by the use of the equivalent
+common fractions for .25, .75, .125, .375, and the like.
+Multiplication and division with fractions, both common and
+decimal, should in the early stages be verified by objective
+aids. The placing of the decimal point in multiplication
+and division with decimal fractions should be verified by
+such exercises as:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="1" summary="">
+<tr><td align='right'>
+20<br />
+1.23 )<span class="overline"> 24.60</span><br />
+<span class="u"> 246</span> &nbsp;&nbsp;</td>
+<td class='spacer'>It cannot be 200; for 200 &times; 1.23 is much more than 24.6.<br />
+It cannot be 2; for 2 &times; 1.23 is much less than 24.6.</td>
+</tr>
+</table></div>
+
+
+
+<p>The establishment of habits of verifying results and their
+use is very greatly needed. The percentage of wrong
+answers in arithmetical work in schools is now so high that
+the pupils are often being practiced in error. In many
+cases they can feel no genuine and effective confidence in the
+processes, since their own use of the processes brings wrong
+answers as often as right. In solving problems they often
+cannot decide whether they have done the right thing or the
+wrong, since even if they have done the right thing, they may<span class='pagenum'><a name="Page_83" id="Page_83">[Pg 83]</a></span>
+have done it inaccurately. A wrong answer to a problem
+is therefore too often ambiguous and uninstructive to them.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a></p>
+
+<p>These illustrations of the last few pages are samples of
+the procedures recommended by a consideration of all the
+bonds that one might form and of the contribution that
+each would make toward the abilities that the study of
+arithmetic should develop and improve. It is by doing more
+or less at haphazard what psychology teaches us to do deliberately
+and systematically in this respect that many of the
+past advances in the teaching of arithmetic have been made.</p>
+
+
+<h4>WASTEFUL AND HARMFUL BONDS</h4>
+
+<p>A scrutiny of the bonds now formed in the teaching of
+arithmetic with questions concerning the exact service of
+each, results in a list of bonds of small value or even no value,
+so far as a psychologist can determine. I present here
+samples of such psychologically unjustifiable bonds with
+some of the reasons for their deficiencies.</p>
+
+<p>(1) <i>Arbitrary units.</i>&mdash;In drills intended to improve the
+ability to see and use the meanings of numbers as names for
+ratios or relative magnitudes, it is unwise to employ entirely
+arbitrary units. The procedure in II (on page 84) is better
+than that in I. Inches, half-inches, feet, and centimeters are
+better as units of length than arbitrary As. Square inches,
+square centimeters, and square feet are better for areas.
+Ounces and pounds should be lifted rather than arbitrary
+weights. Pints, quarts, glassfuls, cupfuls, handfuls, and
+cubic inches are better for volume.</p>
+
+<p>All the real merit in the drills on relative magnitude
+advocated by Speer, McLellan and Dewey, and others can
+be secured without spending time in relating magnitudes
+<span class='pagenum'><a name="Page_84" id="Page_84">[Pg 84]</a></span>
+for the sake of relative magnitude alone. The use of units
+of measure in drills which will never be used in <i>bona fide</i>
+measuring is like the use of fractions like sevenths, elevenths,
+and thirteenths. A very little of it is perhaps desirable to
+test the appreciation of certain general principles, but for
+regular training it should give place to the use of units of
+practical significance.</p>
+
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig3.jpg" width="640" height="240" alt="Fig. 3." title="" />
+<span class="caption smcap">Fig. 3.</span>
+<p class="pblockquot">I. If <i>A</i> is 1 which line is 2? Which line is 4? Which line is 3?
+<i>A</i> and <i>C</i> together equal what line? <i>A</i> and <i>B</i> together equal what
+line? How much longer is <i>B</i> than <i>A</i>? How much longer is <i>B</i>
+than <i>C</i>? How much longer is <i>D</i> than <i>A</i>?</p>
+</div>
+
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig4.jpg" width="800" height="300" alt="Fig. 4." title="" />
+<span class="caption smcap">Fig. 4.</span>
+<p class="pblockquot">II. <i>A</i> is 1 inch long. Which line is 2 inches long? Which line
+is 4 inches long? Which line is 3 inches long? <i>A</i> and <i>C</i> together
+make ... inches? <i>A</i> and <i>B</i> together make ... inches? <i>B</i> is ...
+... longer than <i>A</i>? <i>B</i> is ... ... longer than <i>C</i>? <i>D</i> is ...
+... longer than <i>A</i>?</p>
+</div>
+
+
+<p><span class='pagenum'><a name="Page_85" id="Page_85">[Pg 85]</a></span></p>
+
+<p>(2) <i>Multiples of 11.</i>&mdash;The multiplications of 2 to 12 by
+11 and 12 as single connections should be left for the pupil
+to acquire by himself as he needs them. These connections
+interfere with the process of learning two-place multiplication.
+The manipulations of numbers there required can
+be learned much more easily if 11 and 12 are used as multipliers
+in just the same way that 78 or 96 would be. Later
+the 12 &times; 2, 12 &times; 3, etc., may be taught. There is less reason
+for knowing the multiples of 11 than for knowing the
+multiples of 15, 16, or 25.</p>
+
+<p>(3) <i>Abstract and concrete numbers.</i>&mdash;The elaborate emphasis
+of the supposed fact that we cannot multiply 726
+by 8 dollars and the still more elaborate explanations of
+why nevertheless we find the cost of 726 articles at $8 each
+by multiplying 726 by 8 and calling the answer dollars are
+wasteful. The same holds of the corresponding pedantry
+about division. These imaginary difficulties should not be
+raised at all. The pupil should not think of multiplying
+or dividing men or dollars, but simply of the necessary
+equation and of the sort of thing that the missing number
+represents. "8 &times; 726 = .... Answer is dollars," or "8,
+726, multiply. Answer is dollars," is all that he needs to
+think, and is in the best form for his thought. Concerning
+the distinction between abstract and concrete numbers, both
+logic and common sense as well as psychology support the
+contention of McDougle ['14, p. 206f.], who writes:&mdash;</p>
+
+<p>"The most elementary counting, even that stage when the
+counts were not carried in the mind, but merely in notches
+on a stick or by DeMorgan's stones in a pot, requires some
+thought; and the most advanced counting implies memory
+of things. The terms, therefore, abstract and concrete number,
+have long since ceased to be used by thinking people.</p>
+
+<p>"Recently the writer visited an arithmetic class in a<span class='pagenum'><a name="Page_86" id="Page_86">[Pg 86]</a></span>
+State Normal School and saw a group of practically adult
+students confused about this very question concerning abstract
+and concrete numbers, according to their previous
+training in the conventionalities of the textbook. Their
+teacher diverted the work of the hour and she and the class
+spent almost the whole period in re&euml;stablishing the requirements
+'that the product must always be the same kind of
+unit as the multiplicand,' and 'addends must all be alike
+to be added.' This is not an exceptional case. Throughout
+the whole range of teaching arithmetic in the public schools
+pupils are obfuscated by the philosophical encumbrances
+which have been imposed upon the simplest processes of numerical
+work. The time is surely ripe, now that we are readjusting
+our ideas of the subject of arithmetic, to revise some
+of these wasteful and disheartening practices. Algebra
+historically grew out of arithmetic, yet it has not been
+laden with this distinction. No pupil in algebra lets <i>x</i>
+equal the horses; he lets <i>x</i> equal the <i>number</i> of horses, and
+proceeds to drop the idea of horses out of his consideration.
+He multiplies, divides, and extracts the root of the <i>number</i>,
+sometimes handling fractions in the process, and finally
+interprets the result according to the conditions of his
+problem. Of course, in the early number work there have
+been the sense-objects from which number has been perceived,
+but the mind retreats naturally from objectivity to
+the pure conception of number, and then to the number
+symbol. The following is taken from the appendix to
+Horn's thesis, where a seventh grade girl gets the population
+of the United States in 1820:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="5" cellspacing="0" summary="">
+<tr><td align='right'>
+7,862,166<br />
+233,634<br />
+1,538,022<br />
+<span class="overline">9,633,822</span>
+</td>
+<td align='left' valign='top'>whites<br />
+free negroes<br />
+slaves</td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_87" id="Page_87">[Pg 87]</a></span></p>
+
+<p class="noidt">In this problem three different kinds of addends are combined,
+if we accept the usual distinction. Some may say
+that this is a mistake,&mdash;that the pupil transformed the
+'whites,' 'free negroes,' and 'slaves' into a common
+unit, such as 'people' of 'population' and then added
+these common units. But this 'explanation' is entirely
+gratuitous, as one will find if he questions the pupil about
+the process. It will be found that the child simply added
+the figures as numbers only and then interpreted the result,
+according to the statement of the problem, without so much
+mental gymnastics. The writer has questioned hundreds
+of students in Normal School work on this point, and he
+believes that the ordinary mind-movement is correctly set
+forth here, no matter how well one may maintain as an
+academic proposition that this is not logical. Many classes
+in the Eastern Kentucky State Normal have been given
+this problem to solve, and they invariably get the same
+result:&mdash;</p>
+
+<p>'In a garden on the Summit are as many cabbage-heads
+as the total number of ladies and gentlemen in this class.
+How many cabbage-heads in the garden?'</p>
+
+<p>And the blackboard solution looks like this each time:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="5" cellspacing="0" summary="">
+<tr><td align='right'>
+29<br />
+15<br />
+<span class="overline">44</span>
+</td>
+<td align='left'>ladies<br />
+gentlemen<br />
+cabbage-heads</td></tr>
+</table></div>
+
+<p class="noidt">So, also, one may say: I have 6 times as many sheep as you
+have cows. If you have 5 cows, how many sheep have I?
+Here we would multiply the number of cows, which is 5,
+by 6 and call the result 30, which must be linked with the
+idea of sheep because the conditions imposed by the problem
+demand it. The mind naturally in this work separates the
+pure number from its situation, as in algebra, handles it<span class='pagenum'><a name="Page_88" id="Page_88">[Pg 88]</a></span>
+according to the laws governing arithmetical combinations,
+and labels the result as the statement of the problem demands.
+This is expressed in the following, which is tacitly accepted
+in algebra, and should be accepted equally in arithmetic:</p>
+
+<p>'In all computations and operations in arithmetic, all
+numbers are essentially abstract and should be so treated.
+They are concrete only in the thought process that attends
+the operation and interprets the result.'"</p>
+
+<p>(4) <i>Least common multiple.</i>&mdash;The whole set of bonds involved
+in learning 'least common multiple' should be left
+out. In adding and subtracting fractions the pupil should
+<i>not</i> find the least common multiple of their denominators
+but should find any common multiple that he can find
+quickly and correctly. No intelligent person would ever
+waste time in searching for the least common multiple of
+sixths, thirds, and halves except for the unfortunate traditions
+of an oversystematized arithmetic, but would think
+of their equivalents in sixths or twelfths or twenty-fourths
+or <i>any other convenient common multiple</i>. The process of
+finding the least common multiple is of such exceedingly
+rare application in science or business or life generally that
+the textbooks have to resort to purely fantastic problems
+to give drill in its use.</p>
+
+<p>(5) <i>Greatest common divisor.</i>&mdash;The whole set of bonds
+involved in learning 'greatest common divisor' should
+also be left out. In reducing fractions to lowest terms the
+pupil should divide by anything that he sees that he can
+divide by, favoring large divisors, and continue doing so
+until he gets the fraction in terms suitable for the purpose
+in hand. The reader probably never has had occasion to
+compute a greatest common divisor since he left school.
+If he has computed any, the chances are that he would have
+saved time by solving the problem in some other way!<span class='pagenum'><a name="Page_89" id="Page_89">[Pg 89]</a></span></p>
+
+<p>The following problems are taken at random from those
+given by one of the best of the textbooks that make the
+attempt to apply the facts of Greatest Common Divisor
+and Least Common Multiple to problems.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> Most of these
+problems are fantastic. The others are trivial, or are better
+solved by trial and adaptation.</p>
+
+<div class="pblockquot"><p><b>1.</b> A certain school consists of 132 pupils in the high school,
+154 in the grammar, and 198 in the primary grades. If each group
+is divided into sections of the same number containing as many
+pupils as possible, how many pupils will there be in each section?</p>
+
+<p><b>2.</b> A farmer has 240 bu. of wheat and 920 bu. of oats, which
+he desires to put into the least number of boxes of the same capacity,
+without mixing the two kinds of grain. Find how many
+bushels each box must hold.</p>
+
+<p><b>3.</b> Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively,
+and begin to toll at the same instant. When will they
+next toll together?</p>
+
+<p><b>4.</b> A, B, C, and D start together, and travel the same way
+around an island which is 600 mi. in circuit. A goes 20 mi. per
+day, B 30, C 25, and D 40. How long must their journeying continue,
+in order that they may all come together again?</p>
+
+<p><b>5.</b> The periods of three planets which move uniformly in circular
+orbits round the sun, are respectively 200, 250, and 300 da.
+Supposing their positions relatively to each other and the sun to
+be given at any moment, determine how many da. must elapse
+before they again have exactly the same relative positions.</p></div>
+
+<p>(6) <i>Rare and unimportant words.</i>&mdash;The bonds between
+rare or unimportant words and their meanings should not
+be formed for the mere sake of verbal variety in the problems
+of the textbook. A pupil should not be expected to solve
+a problem that he cannot read. He should not be expected
+in grades 2 and 3, or even in grade 4, to read words that he
+has rarely or never seen before. He should not be given
+elaborate drill in reading during the time devoted to the
+treatment of quantitative facts and relations.</p>
+
+<p><span class='pagenum'><a name="Page_90" id="Page_90">[Pg 90]</a></span>All
+this is so obvious that it may seem needless to relate.
+It is not. With many textbooks it is now necessary to
+give definite drill in reading the words in the printed problems
+intended for grades 2, 3, and 4, or to replace them by
+oral statements, or to leave the pupils in confusion concerning
+what the problems are that they are to solve. Many
+good teachers make a regular reading-lesson out of every
+page of problems before having them solved. There should
+be no such necessity.</p>
+
+<p>To define <i>rare</i> and <i>unimportant</i> concretely, I will say that
+for pupils up to the middle of grade 3, such words as the
+following are rare and unimportant (though each of them
+occurs in the very first fifty pages of some well-known beginner's
+book in arithmetic).</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="20" summary="" width="80%">
+<tr>
+<td align='left'>absentees<br />
+account<br />
+Adele<br />
+admitted<br />
+Agnes<br />
+agreed<br />
+Albany<br />
+Allen<br />
+allowed<br />
+alternate<br />
+Andrew<br />
+Arkansas<br />
+arrived<br />
+assembly<br />
+automobile<br />
+baking powder<br />
+balance<br />
+barley<br />
+beggar<br />
+Bertie<br />
+Bessie<br />
+bin<br />
+Boston<br />
+bouquet<br />
+bronze<br />
+buckwheat<br />
+Byron<br />
+camphor<br />
+Carl<br />
+Carrie<br />
+Cecil<br />
+Charlotte<br />
+charity<br />
+Chicago<br />
+cinnamon<br />
+Clara<br />
+clothespins<br />
+collect<br />
+comma<br />
+committee<br />
+concert<br />
+confectioner<br />
+cranberries<br />
+crane<br />
+currants<br />
+dairyman<br />
+Daniel<br />
+David<br />
+dealer<br />
+debt<br />
+delivered<br />
+Denver<br />
+</td>
+<td align='left'>department<br />
+deposited<br />
+dictation<br />
+discharged<br />
+discover<br />
+discovery<br />
+dish-water<br />
+drug<br />
+due<br />
+Edgar<br />
+Eddie<br />
+Edwin<br />
+election<br />
+electric<br />
+Ella<br />
+Emily<br />
+enrolled<br />
+entertainment<br />
+envelope<br />
+Esther<br />
+Ethel<br />
+exceeds<br />
+explanation<br />
+expression<br />
+generally<br />
+gentlemen<br />
+Gilbert<br />
+Grace<br />
+grading<br />
+Graham<br />
+grammar<br />
+Harold<br />
+hatchet<br />
+Heralds<br />
+hesitation<br />
+Horace Mann<br />
+impossible<br />
+income<br />
+indicated<br />
+inmost<br />
+inserts<br />
+installments<br />
+instantly<br />
+insurance<br />
+Iowa<br />
+Jack<br />
+Jennie<br />
+Johnny<br />
+Joseph<br />
+journey<br />
+Julia<br />
+Katherine<br />
+</td>
+<td align='left'>lettuce-plant<br />
+library<br />
+Lottie<br />
+Lula<br />
+margin<br />
+Martha<br />
+Matthew<br />
+Maud<br />
+meadow<br />
+mentally<br />
+mercury<br />
+mineral<br />
+Missouri<br />
+molasses<br />
+Morton<br />
+movements<br />
+muslin<br />
+Nellie<br />
+nieces<br />
+Oakland<br />
+observing<br />
+obtained<br />
+offered<br />
+office<br />
+onions<br />
+opposite<br />
+original<br />
+package<br />
+packet<br />
+palm<br />
+Patrick<br />
+Paul<br />
+payments<br />
+peep<br />
+Peter<br />
+perch<br />
+phaeton<br />
+photograph<br />
+piano<br />
+pigeons<br />
+Pilgrims<br />
+preserving<br />
+proprietor<br />
+purchased<br />
+Rachel<br />
+Ralph<br />
+rapidity<br />
+rather<br />
+readily<br />
+receipts<br />
+register<br />
+remanded<br />
+</td>
+<td align='left'>respectively<br />
+Robert<br />
+Roger<br />
+Ruth<br />
+rye<br />
+Samuel<br />
+San Francisco<br />
+seldom<br />
+sheared<br />
+shingles<br />
+skyrockets<br />
+sloop<br />
+solve<br />
+speckled<br />
+sponges<br />
+sprout<br />
+stack<br />
+Stephen<br />
+strap<br />
+successfully<br />
+suggested<br />
+sunny<br />
+supply<br />
+Susan<br />
+Susie's<br />
+syllable<br />
+talcum<br />
+term<br />
+test<br />
+thermometer<br />
+Thomas<br />
+torpedoes<br />
+trader<br />
+transaction<br />
+treasury<br />
+tricycle<br />
+tube<br />
+two-seated<br />
+united<br />
+usually<br />
+vacant<br />
+various<br />
+vase<br />
+velocipede<br />
+votes<br />
+walnuts<br />
+Walter<br />
+Washington<br />
+watched<br />
+whistle<br />
+woodland<br />
+worsted<br />
+</td>
+</tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_91" id="Page_91">[Pg 91]</a></span></p>
+
+<p>(7) <i>Misleading facts and procedures.</i>&mdash;Bonds should not
+be formed between articles of commerce and grossly inaccurate
+prices therefor, between events and grossly improbable
+consequences, or causes or accompaniments thereof,
+nor between things, qualities, and events which have no
+important connections one with another in the real world.
+In general, things should not be put together in the pupil's
+mind that do not belong together.</p>
+
+<p>If the reader doubts the need of this warning let him examine
+problems 1 to 5, all from reputable books that are
+in common use, or have been within a few years, and con<span class='pagenum'><a name="Page_92" id="Page_92">[Pg 92]</a></span>sider
+how addition, subtraction, and the habits belonging
+with each are confused by exercise 6.</p>
+
+<div class="pblockquot">
+<p><b>1.</b> If a duck flying <sup>3</sup>&frasl;<sub>5</sub> as fast as a hawk flies 90 miles in an hour,
+how fast does the hawk fly?</p>
+
+<p><b>2.</b> At <sup>5</sup>&frasl;<sub>8</sub> of a cent apiece how many eggs can I buy for $60?</p>
+
+<p><b>3.</b> At $.68 a pair how many pairs of overshoes can you buy for
+$816?</p>
+
+<p><b>4.</b> At $.13 a dozen how many dozen bananas can you buy for
+$3.12?</p>
+
+<p><b>5.</b> How many pecks of beans can be put into a box that will
+hold just 21 bushels?</p>
+
+<p><b>6.</b> Write answers:</p>
+
+<div class='center'>
+<table border="0" cellpadding="20" cellspacing="0" summary="">
+<tr><td align='right'>
+537<br />
+365<br />
+?<br />
+36<br />
+<span class="overline">1000</span>
+</td>
+<td align='left'>Beginning at the bottom say 11, 18, and 2 (writing it in
+its place) are 20. 5, 11, 14, and 6 (writing it) are 20, 5, 10. The number, omitted, is 62.
+</td></tr>
+</table></div>
+
+<div class='center'>
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr>
+<td align='left' valign='top'><i>a.</i></td><td align='right'>581<br />97<br />364<br />?<br /><span class="overline">1758</span></td>
+<td align='left' valign='top'><i>b.</i></td><td align='right'>625<br />?<br />90<br />417<br /><span class="overline">2050</span></td>
+<td align='left' valign='top'><i>c.</i></td><td align='right'>752<br />414<br />130<br />?<br /><span class="overline">2460</span></td>
+<td align='left' valign='top'><i>d.</i></td><td align='right'>314<br />429<br />?<br />76<br /><span class="overline">1000</span></td>
+<td align='left' valign='top'><i>e.</i></td><td align='right'>?<br />845<br />223<br />95<br /><span class="overline">2367</span></td>
+</tr>
+</table></div>
+</div>
+
+<p>(8) <i>Trivialities and absurdities.</i>&mdash;Bonds should not be
+formed between insignificant or foolish questions and the
+labor of answering them, nor between the general arithmetical
+work of the school and such insignificant or foolish
+questions. The following are samples from recent textbooks
+of excellent standing:&mdash;</p>
+
+<div class="pblockquot"><p>On one side of George's slate there are 32 words, and on the
+other side 26 words. If he erases 6 words from one side, and 8
+from the other, how many words remain on his slate?</p>
+
+<p>A certain school has 14 rooms, and an average of 40 children
+in a room. If every one in the school should make 500 straight
+marks on each side of his slate, how many would be made in all?</p>
+<p><span class='pagenum'><a name="Page_93" id="Page_93">[Pg 93]</a></span></p>
+<p>8 times the number of stripes in our flag is the number of years
+from 1800 until Roosevelt was elected President. In what year
+was he elected President?</p>
+
+<p>From the Declaration of Independence to the World's Fair in
+Chicago was 9 times as many years as there are stripes in the flag.
+How many years was it?</p></div>
+
+<p>(9) <i>Useless methods.</i>&mdash;Bonds should not be formed between
+a described situation and a method of treating the
+situation which would not be a useful one to follow in the case
+of the real situation. For example, "If I set 96 trees in
+rows, sixteen trees in a row, how many rows will I have?"
+forms the habit of treating by division a problem that in
+reality would be solved by counting the rows. So also "I
+wish to give 25 cents to each of a group of boys and find
+that it will require $2.75. How many boys are in the
+group?" forms the habit of answering a question by division
+whose answer must already have been present to give the
+data of the problem.</p>
+
+<p>(10) <i>Problems whose answers would, in real life, be already
+known.</i>&mdash;The custom of giving problems in textbooks which
+could not occur in reality because the answer has to be known
+to frame the problem is a natural result of the lazy author's
+tendency to work out a problem to fit a certain process and
+a certain answer. Such bogus problems are very, very
+common. In a random sampling of a dozen pages of
+"General Review" problems in one of the most widely
+used of recent textbooks, I find that about 6 percent of the
+problems are of this sort. Among the problems extemporized
+by teachers these bogus problems are probably still
+more frequent. Such are:&mdash;</p>
+
+<div class="pblockquot"><p>A clerk in an office addressed letters according to a given
+list. After she had addressed 2500, <sup>4</sup>&frasl;<sub>9</sub> of the names on the list
+had not been used; how many names were in the entire list?</p>
+
+<p>The Canadian power canal at  Sault Ste.  Marie  furnished
+<span class='pagenum'><a name="Page_94" id="Page_94">[Pg 94]</a></span>20,000 horse power. The canal on the Michigan side furnished
+2&frac12; times as much. How many horse power does the latter furnish?</p></div>
+
+<p>It may be asserted that the ideal of giving as described
+problems only problems that might occur and demand the
+same sort of process for solution with a real situation, is too
+exacting. If a problem is comprehensible and serves to
+illustrate a principle or give useful drill, that is enough,
+teachers may say. For really scientific teaching it is not
+enough. Moreover, if problems are given merely as tests
+of knowledge of a principle or as means to make some fact
+or principle clear or emphatic, and are not expected to be of
+direct service in the quantitative work of life, it is better
+to let the fact be known. For example, "I am thinking
+of a number. Half of this number is twice six. What is
+the number?" is better than "A man left his wife a certain
+sum of money. Half of what he left her was twice as much
+as he left to his son, who receives $6000. How much did
+he leave his wife?" The former is better because it makes
+no false pretenses.</p>
+
+<p>(11) <i>Needless linguistic difficulties.</i>&mdash;It should be unnecessary
+to add that bonds should not be formed between
+the pupil's general attitude toward arithmetic and needless,
+useless difficulty in language or needless, useless, wrong
+reasoning. Our teaching is, however, still tainted by both
+of these unfortunate connections, which dispose the pupil
+to think of arithmetic as a mystery and folly.</p>
+
+<p>Consider, for example, the profitless linguistic difficulty
+of problems 1-6, whose quantitative difficulties are simply
+those of:&mdash;</p>
+
+<div class="pblockquot">
+<div class="blockquot"><p class="noidt">
+<b>1.</b>  5 &#43; 8 &#43; 3 &#43; 7<br />
+<b>2.</b>  64 &divide; 8, and knowledge that 1 peck = 8 quarts<br />
+<b>3.</b>  12 &divide; 4<br />
+<span class='pagenum'><a name="Page_95" id="Page_95">[Pg 95]</a></span><b>4.</b>  6 &divide; 2<br />
+<b>5.</b>  3 &times; 2<br />
+<b>6.</b>  4 &times; 4
+</p></div>
+
+<p><b>1.</b> What amount should you obtain by putting together 5
+cents, 8 cents, 3 cents, and 7 cents? Did you find this result by
+adding or multiplying?</p>
+
+<p><b>2.</b> How many times must you empty a peck measure to fill a
+basket holding 64 quarts of beans?</p>
+
+<p><b>3.</b> If a girl commits to memory 4 pages of history in one day,
+in how many days will she commit to memory 12 pages?</p>
+
+<p><b>4.</b> If Fred had 6 chickens how many times could he give away
+2 chickens to his companions?</p>
+
+<p><b>5.</b> If a croquet-player drove a ball through 2 arches at each
+stroke, through how many arches will he drive it by 3 strokes?</p>
+
+<p><b>6.</b> If mamma cut the pie into 4 pieces and gave each person a
+piece, how many persons did she have for dinner if she used 4
+whole pies for dessert?</p></div>
+
+<p>Arithmetically this work belongs in the first or second
+years of learning. But children of grades 2 and 3, save a
+few, would be utterly at a loss to understand the language.</p>
+
+<p>We are not yet free from the follies illustrated in the lessons
+of pages 96 to 99, which mystified our parents.<span class='pagenum'><a name="Page_96" id="Page_96">[Pg 96]</a></span></p>
+
+<p>&nbsp;</p>
+<div class="figcenter" style="width: 734px;">
+<img src="images/fig5.jpg" width="734" height="600" alt="Fig. 5." title="Fig. 5." />
+<span class="caption smcap">Fig. 5.</span>
+</div>
+
+<p class="tabcap">LESSON I</p>
+<div class="pblockquot">
+
+<p><b>1.</b> In this picture, how many girls are in the swing?</p>
+
+<p><b>2.</b> How many girls are pulling the swing?</p>
+
+<p><b>3.</b> If you count both girls together, how many are they?<br />
+&nbsp; &nbsp; &nbsp; &nbsp; <b><i>One</i></b> girl and <b><i>one</i></b> other girl are how many?</p>
+
+<p><b>4.</b> How many kittens do you see on the stump?</p>
+
+<p><b>5.</b> How many on the ground?</p>
+
+<p><b>6.</b> How many kittens are in the picture?
+One kitten and one other kitten are how many?</p>
+
+<p><b>7.</b> If you should ask me how many girls are in the swing, or
+how many kittens are on the stump, I could answer aloud, <i>One</i>;
+or I could write <i>One</i>; or thus, <b><i>1</i></b>.</p>
+
+<p><b>8.</b> If I write <i>One</i>, this is called the <i>word One</i>.</p>
+
+<p><b>9.</b> This, <b><i>1</i></b>, is named a <b><i>figure One</i></b>, because it means the same
+as the word <i>One</i>, and stands for <i>One</i>.<span class='pagenum'><a name="Page_97" id="Page_97">[Pg 97]</a></span></p>
+
+<p><b>10.</b> Write 1. What is this named? Why?</p>
+
+<p><b>11.</b> A figure 1 may stand for <i>one</i> girl, <i>one</i> kitten, or <i>one</i> anything.</p>
+
+<p><b>12.</b> When children first attend school, what do they begin to
+learn? <i>Ans.</i>  Letters and words.</p>
+
+<p><b>13.</b> Could you read or write before you had learned either
+letters or words?</p>
+
+<p><b>14.</b> If we have all the <i>letters</i> together, they are named the
+Alphabet.</p>
+
+<p><b>15.</b> If we write or speak <i>words</i>, they are named Language.</p>
+
+<p><b>16.</b> You are commencing to study Arithmetic; and you can
+read and write in Arithmetic only as you learn the Alphabet and
+Language of Arithmetic. But little time will be required for this
+purpose.</p>
+</div>
+
+<p>&nbsp;</p>
+<div class="figcenter" style="width: 727px;">
+<img src="images/fig6.jpg" width="727" height="600" alt="Fig. 6." title="Fig. 6." />
+<span class="caption smcap">Fig. 6.</span>
+</div>
+
+<p class="tabcap">LESSON II</p>
+<div class="pblockquot">
+
+<p><b>1.</b> If we speak or write words, what do we name them, when
+taken together?</p>
+
+<p><b>2.</b> What are you commencing to study? <i>Ans.</i> Arithmetic.</p>
+
+<p><b>3.</b> What Language must you now learn?</p>
+
+<p><b>4.</b> What do we name this, <b>1</b>? Why?</p>
+
+<p><b>5.</b>  This figure, <b>1</b>, is part of the Language of Arithmetic.</p>
+
+<p><b>6.</b> If I should write something to stand for <b><i>Two</i></b>&mdash;<i>two</i> girls,
+<i>two</i> kittens, or <i>two</i> things of any kind&mdash;what do you think we
+would name it?</p>
+
+<p><b>7.</b> A <b><i>figure Two</i></b> is written thus: <b><i>2.</i></b>   Make a <i>figure two</i>.</p>
+
+<p><b>8.</b> Why do we name this a <i>figure two</i>?</p>
+
+<p><b>9.</b> This figure two (<b>2</b>) is part of the Language of Arithmetic.<span class='pagenum'><a name="Page_98" id="Page_98">[Pg 98]</a></span></p>
+
+<p><b>10.</b> In this picture one boy is sitting, playing a flageolet.
+What is the other boy doing? If the boy standing
+should sit down by the other, how many boys would be
+sitting together? One boy and one other boy are how many
+boys?</p>
+
+<p><b>11.</b> You see a flageolet and a violin. They are musical
+instruments. One musical instrument and one other musical
+instrument are how many?</p>
+
+<p><b>12.</b> I will write thus: 1 1 2. We say that 1 boy and 1 other
+boy, counted together, are 2 boys; or are equal to 2 boys. We
+will now write something to show that the first 1 and the other 1
+are to be counted together.</p>
+
+<div class="figright" style="width: 336px;">
+<img src="images/image04.jpg" width="336" height="344" alt="Plus" title="" />
+</div>
+
+<p><b>13.</b> We name a line drawn thus,&mdash;, a <b><i>horizontal line</i></b>. Draw
+such a line. Name it.</p>
+
+<p><b>14.</b> A line drawn thus, | , we name a <b><i>vertical line</i></b>. Draw such
+a line. Name it.<span class='pagenum'><a name="Page_99" id="Page_99">[Pg 99]</a></span></p>
+
+<p><b>15.</b> Now I will put two such lines together;
+thus, &#43;. What kind of a line do we name the
+first (&mdash;)? And what do we name the last?
+(|)? Are these lines long or short? Where
+do they cross each other?</p>
+
+<p><b>16.</b> Each of you write thus: &mdash;, | , &#43;.</p>
+
+
+<p><b>17.</b> This, &#43;, is named <b><i>Plus</i></b>. <i>Plus</i> means
+<i>more</i>; and &#43; also means <i>more</i>.</p>
+
+<p><b>18.</b> I will write.</p>
+
+<p class="center">
+<b><i>One and One More Equal Two</i></b>.<br />
+</p>
+
+<p><b>19.</b> Now I will write part of this in the Language of Arithmetic.
+I write the first <i>One</i> thus, 1; then the other <i>One</i> thus,
+1. Afterward I write, for the word <i>More</i>, thus, &#43;, placing the &#43;
+between 1 and 1, so that the whole stands thus: 1 &#43; 1. As I
+write, I say, <i>One and One more</i>.</p>
+
+<p><b>20.</b> Each of you write 1 &#43; 1. Read what you have written.</p>
+
+<p><b>21.</b> This &#43;, when written between the 1s, shows that they are
+to be put together, or counted together, so as to make 2.</p>
+
+<p><b>22.</b> Because &#43; shows what is to be done, it is called a <i>Sign</i>.
+If we take its name, <i>Plus</i>, and the word <i>Sign</i>, and put both words
+together, we have <i>Sign Plus</i>, or <i>Plus Sign</i>. In speaking of this
+we may call it <i>Sign Plus</i>, or <i>Plus Sign</i>, or <i>Plus</i>.</p>
+
+<p><b>23.</b> 1, 2, &#43;, are part of the Language of Arithmetic.</p>
+
+<p class="center">
+<b><i>Write the following in the Language of Arithmetic</i></b>:<br />
+</p>
+
+<p><b>24.</b> One and one more.</p>
+
+<p><b>25.</b> One and two more.</p>
+
+<p><b>26.</b> Two and one more.</p>
+</div>
+
+<p><span class='pagenum'><a name="Page_100" id="Page_100">[Pg 100]</a></span></p>
+
+<p>(12) <i>Ambiguities and falsities.</i>&mdash;Consider the ambiguities
+and false reasoning of these problems.</p>
+
+<div class="pblockquot">
+<p><b>1.</b> If you can earn 4 cents a day, how much can you earn in
+6 weeks? (Are Sundays counted? Should a child who earns 4
+cents some day expect to repeat the feat daily?)</p>
+
+<p><b>2.</b> How many lines must you make to draw ten triangles and
+five squares? (I can do this with 8 lines, though the answer the
+book requires is 50.)</p>
+
+<p><b>3.</b> A runner ran twice around an <sup>1</sup>&frasl;<sub>8</sub> mile track in two
+minutes. What distance did he run in <sup>2</sup>&frasl;<sub>3</sub> of a minute? (I do not
+know, but I do know that, save by chance, he did not run exactly
+<sup>2</sup>&frasl;<sub>3</sub> of <sup>1</sup>&frasl;<sub>8</sub> mile.)</p>
+
+<p><b>4.</b> John earned $4.35 in a week, and Henry earned $1.93.
+They put their money together and bought a gun. What did it
+cost? (Maybe $5, maybe $10. Did they pay for the whole of it?
+Did they use all their earnings, or less, or more?)</p>
+
+<p><b>5.</b> Richard has 12 nickels in his purse. How much more than
+50 cents would you give him for them? (Would a wise child
+give 60 cents to a boy who wanted to swap 12 nickels therefor, or
+would he suspect a trick and hold on to his own coins?)</p>
+
+<p><b>6.</b> If a horse trots 10 miles in one hour how far will he travel
+in 9 hours?</p>
+
+<p><b>7.</b> If a girl can pick 3 quarts of berries in 1 hour how many
+quarts can she pick in 3 hours?</p>
+
+<div class="blockquot"><p>(These last two, with a teacher insisting on the 90 and 9,
+might well deprive a matter-of-fact boy of respect for
+arithmetic for weeks thereafter.)</p></div>
+
+<p>The economics and physics of the next four problems speak for
+themselves.</p>
+
+<p><b>8.</b> I lost $15 by selling a horse for $85. What was the value
+of the horse?</p>
+
+<p><b>9.</b> If floating ice has 7 times as much of it under the surface
+of the water as above it, what part is above water? If an iceberg
+is 50 ft. above water, what is the entire height of the iceberg?
+How high above water would an iceberg 300 ft. high have to be?</p>
+
+<p><b>10.</b> A man's salary is $1000 a year and his expenses $625.
+<span class='pagenum'><a name="Page_101" id="Page_101">[Pg 101]</a></span>
+How many years will elapse before he is worth $10,000 if he is
+worth $2500 at the present time?</p>
+
+<p><b>11.</b> Sound travels 1120 ft. a second. How long after a cannon
+is fired in New York will the report be heard in Philadelphia, a
+distance of 90 miles?</p>
+</div>
+
+
+<h4>GUIDING PRINCIPLES</h4>
+
+<p>The reader may be wearied of these special details concerning
+bonds now neglected that should be formed and
+useless or harmful bonds formed for no valid reason. Any
+one of them by itself is perhaps a minor matter, but when we
+have cured all our faults in this respect and found all the
+possibilities for wiser selection of bonds, we shall have
+enormously improved the teaching of arithmetic. The
+ideal is such choice of bonds (and, as will be shown later,
+such arrangement of them) as will most improve the functions
+in question at the least cost of time and effort. The
+guiding principles may be kept in mind in the form of seven
+simple but golden rules:&mdash;</p>
+
+<p>1. Consider the situation the pupil faces.</p>
+
+<p>2. Consider the response you wish to connect with it.</p>
+
+<p>3. Form the bond; do not expect it to come by a miracle.</p>
+
+<p>4. Other things being equal, form no bond that will have
+to be broken.</p>
+
+<p>5. Other things being equal, do not form two or three
+bonds when one will serve.</p>
+
+<p>6. Other things being equal, form bonds in the way that
+they are required later to act.</p>
+
+<p>7. Favor, therefore, the situations which life itself will
+offer, and the responses which life itself will demand.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_102" id="Page_102">[Pg 102]</a></span></p>
+<h2><a name="CHAPTER_V" id="CHAPTER_V"></a>CHAPTER V</h2>
+
+<h3>THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE
+STRENGTH OF BONDS</h3>
+
+
+<p>An inventory of the bonds to be formed in learning arithmetic
+should be accompanied by a statement of how strong
+each bond is to be made and kept year by year. Since,
+however, the inventory itself has been presented here only
+in samples, the detailed statement of desired strength for
+each bond cannot be made. Only certain general facts
+will be noted here.</p>
+
+
+<h4>THE NEED OF STRONGER ELEMENTARY BONDS</h4>
+
+<p>The constituent bonds involved in the fundamental operations
+with numbers need to be much stronger than they now
+are. Inaccuracy in these operations means weakness of the
+constituent bonds. Inaccuracy exists, and to a degree that
+deprives the subject of much of its possible disciplinary
+value, makes the pupil's achievements of slight value for
+use in business or industry, and prevents the pupil from
+verifying his work with new processes by some previously
+acquired process.</p>
+
+<p>The inaccuracy that exists may be seen in the measurements
+made by the many investigators who have used
+arithmetical tasks as tests of fatigue, practice, individual
+differences and the like, and in the special studies of arith<span class='pagenum'><a name="Page_103" id="Page_103">[Pg 103]</a></span>metical
+achievements for their own sake made by Courtis
+and others.</p>
+
+<p>Burgerstein ['91], using such examples as</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='right'>28704516938276546397<br />
+<span class="u">&#43; 35869427359163827263</span></td></tr>
+</table></div>
+
+<p class="noidt">and similar long numbers to be multiplied by 2 or by 3 or
+by 4 or by 5 or by 6, found 851 errors in 28,267 answer-figures,
+or 3 per hundred answer-figures, or <sup>3</sup>&frasl;<sub>5</sub> of an error
+per example. The children were 9&frac12; to 15 years old. Laser
+['94], using the same sort of addition and multiplication,
+found somewhat over 3 errors per hundred answer-figures
+in the case of boys and girls averaging 11&frac12; years, during
+the period of their most accurate work. Holmes ['95],
+using addition of the sort just described, found 346 errors
+in 23,713 answer-figures or about 1&frac12; per hundred. The
+children were from all grades from the third to the eighth.
+In Laser's work, 21, 19, 13, and 10 answer-figures were obtained
+per minute. Friedrich ['97] with similar examples,
+giving the very long time of 20 minutes for obtaining about
+200 answer-figures, found from 1 to 2 per hundred wrong.
+King ['07] had children in grade 5 do sums, each consisting
+of 5 two-place numbers. In the most accurate work-period,
+they made 1 error per 20 columns. In multiplying
+a four-place by a four-place number they had less than one
+total answer right out of three. In New York City Courtis
+found ['11-'12] with his Test 7 that in 12 minutes the average
+achievement of fourth-grade children is 8.8 units attempted
+with 4.2 right. In grade 5 the facts are 10.9 attempts with
+5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade
+7, 15 attempts with 8.5 right; in grade 8, 15.7 attempts
+with 10.1 right. These results are near enough to those
+obtained from the country at large to serve as a text here.<span class='pagenum'><a name="Page_104" id="Page_104">[Pg 104]</a></span></p>
+
+<p>The following were set as official standards, in an excellent
+school system, Courtis Series B being used:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="5" summary="">
+<tr><th></th><th  class="smcap"> &nbsp; &nbsp; Grade. &nbsp; &nbsp;</th><th class="smcap">Speed<br />  &nbsp; &nbsp; Attempts. &nbsp; &nbsp; </th><th class="smcap"> &nbsp; &nbsp; Percent of  &nbsp; &nbsp; <br />Correct Answers.</th></tr>
+<tr><td align='left'>Addition</td><td align='center'>8</td><td align='center'>12</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>7</td><td align='center'>11</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>6</td><td align='center'>10</td><td align='center'>70</td></tr>
+<tr><td align='center'></td><td align='center'>5</td><td align='center'>9</td><td align='center'>70</td></tr>
+<tr><td align='center'></td><td align='center'>4</td><td align='center'>8</td><td align='center'>70</td></tr>
+<tr><td colspan='3'>&nbsp;</td></tr>
+<tr><td align='left'>Subtraction</td><td align='center'>8</td><td align='center'>12</td><td align='center'>90</td></tr>
+<tr><td align='center'></td><td align='center'>7</td><td align='center'>11</td><td align='center'>90</td></tr>
+<tr><td align='center'></td><td align='center'>6</td><td align='center'>10</td><td align='center'>90</td></tr>
+<tr><td align='center'></td><td align='center'>5</td><td align='center'>9</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>4</td><td align='center'>7</td><td align='center'>80</td></tr>
+<tr><td colspan='3'>&nbsp;</td></tr>
+<tr><td align='left'>Multiplication</td><td align='center'>8</td><td align='center'>11</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>7</td><td align='center'>10</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>6</td><td align='center'>9</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>5</td><td align='center'>7</td><td align='center'>70</td></tr>
+<tr><td align='center'></td><td align='center'>4</td><td align='center'>6</td><td align='center'>60</td></tr>
+<tr><td colspan='3'>&nbsp;</td></tr>
+<tr><td align='left'>Division</td><td align='center'>8</td><td align='center'>11</td><td align='center'>90</td></tr>
+<tr><td align='center'></td><td align='center'>7</td><td align='center'>10</td><td align='center'>90</td></tr>
+<tr><td align='center'></td><td align='center'>6</td><td align='center'>8</td><td align='center'>80</td></tr>
+<tr><td align='center'></td><td align='center'>5</td><td align='center'>6</td><td align='center'>70</td></tr>
+<tr><td align='center'></td><td align='center'>4</td><td align='center'>4</td><td align='center'>60</td></tr>
+</table></div>
+
+<p>Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding
+columns like those printed below, children in grade 4 got
+on the average less than 80 percent of correct answers.
+Their average speed was about 2 columns per minute. In
+doing division of the sort printed below children of grades
+3 <i>B</i> and 4 <i>A</i> got less than 95 percent of correct answers, the
+average speed being 4 divisions per minute. In both cases
+the slower computers were no more accurate than the faster
+ones. Practice improved the speed very rapidly, but the
+accuracy remained substantially unchanged. Brown ['11
+and '12] found a similar low status of ability and notable
+improvement from a moderate amount of special practice.<span class='pagenum'><a name="Page_105" id="Page_105">[Pg 105]</a></span></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="80%">
+<tr><td align='center'> 3</td><td align='center'>5</td><td align='center'>6</td><td align='center'>2</td><td align='center'>3</td><td align='center'>8</td><td align='center'>9</td><td align='center'>7</td><td align='center'>4</td><td align='center'>9</td></tr>
+<tr><td align='center'> 7</td><td align='center'>9</td><td align='center'>6</td><td align='center'>5</td><td align='center'>5</td><td align='center'>6</td><td align='center'>4</td><td align='center'>5</td><td align='center'>8</td><td align='center'>2</td></tr>
+<tr><td align='center'> 3</td><td align='center'>4</td><td align='center'>7</td><td align='center'>8</td><td align='center'>7</td><td align='center'>3</td><td align='center'>7</td><td align='center'>9</td><td align='center'>3</td><td align='center'>7</td></tr>
+<tr><td align='center'> 8</td><td align='center'>8</td><td align='center'>4</td><td align='center'>8</td><td align='center'>2</td><td align='center'>6</td><td align='center'>8</td><td align='center'>2</td><td align='center'>9</td><td align='center'>8</td></tr>
+<tr><td align='center'> 2</td><td align='center'>2</td><td align='center'>4</td><td align='center'>7</td><td align='center'>6</td><td align='center'>9</td><td align='center'>8</td><td align='center'>5</td><td align='center'>6</td><td align='center'>2</td></tr>
+<tr><td align='center'> 6</td><td align='center'>9</td><td align='center'>5</td><td align='center'>7</td><td align='center'>8</td><td align='center'>5</td><td align='center'>2</td><td align='center'>3</td><td align='center'>2</td><td align='center'>4</td></tr>
+<tr><td align='center'> 9</td><td align='center'>6</td><td align='center'>4</td><td align='center'>2</td><td align='center'>7</td><td align='center'>2</td><td align='center'>9</td><td align='center'>4</td><td align='center'>4</td><td align='center'>5</td></tr>
+<tr><td align='center'> 3</td><td align='center'>3</td><td align='center'>7</td><td align='center'>9</td><td align='center'>9</td><td align='center'>9</td><td align='center'>2</td><td align='center'>8</td><td align='center'>9</td><td align='center'>7</td></tr>
+<tr><td align='center'> 6</td><td align='center'>8</td><td align='center'>9</td><td align='center'>6</td><td align='center'>4</td><td align='center'>7</td><td align='center'>7</td><td align='center'>9</td><td align='center'>2</td><td align='center'>4</td></tr>
+<tr><td align='center'> 8</td><td align='center'>4</td><td align='center'>6</td><td align='center'>9</td><td align='center'>9</td><td align='center'>2</td><td align='center'>6</td><td align='center'>9</td><td align='center'>8</td><td align='center'>9</td></tr>
+<tr><td align='center'> &mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td><td align='center'>&mdash;</td></tr>
+</table></div>
+
+<div class="blockquot"><p class="noidt">
+<span style="margin-left: 0.5em;">20 = .... 5s</span><br />
+<span style="margin-left: 0.5em;">56 = .... 9s and .... <i>r.</i></span><br />
+<span style="margin-left: 0.5em;">30 = .... 7s and .... <i>r.</i></span><br />
+<span style="margin-left: 0.5em;">89 = .... 9s and .... <i>r.</i></span><br />
+<span style="margin-left: 0.5em;">20 = .... 8s and .... <i>r.</i></span><br />
+<span style="margin-left: 0.5em;">56 = .... 6s and .... <i>r.</i></span><br />
+<span style="margin-left: 0.5em;">31 = .... 4s and .... <i>r.</i></span><br />
+<span style="margin-left: 0.5em;">86 = .... 9s and .... <i>r.</i></span><br />
+</p></div>
+
+<p>It is clear that numerical work as inaccurate as this has
+little or no commercial or industrial value. If clerks got
+only six answers out of ten right as in the Courtis tests, one
+would need to have at least four clerks make each computation
+and would even then have to check many of their discrepancies
+by the work of still other clerks, if he wanted his
+accounts to show less than one error per hundred accounting
+units of the Courtis size.</p>
+
+<p>It is also clear that the "habits of ... absolute accuracy,
+and satisfaction in truth as a result" which arithmetic
+is supposed to further must be largely mythical in
+pupils who get right answers only from three to nine times
+out of ten!</p>
+
+
+<h4>EARLY MASTERY</h4>
+
+<p>The bonds in question clearly must be made far stronger
+than they now are. They should in fact be strong enough
+to abolish errors in computation, except for those due to<span class='pagenum'><a name="Page_106" id="Page_106">[Pg 106]</a></span>
+temporary lapses. It is much better for a child to know half
+of the multiplication tables, and to know that he does not
+know the rest, than to half-know them all; and this holds
+good of all the elementary bonds required for computation.
+Any bond should be made to work perfectly, though slowly,
+very soon after its formation is begun. Speed can easily
+be added by proper practice.</p>
+
+<p>The chief reasons why this is not done now seem to be the
+following: (1) Certain important bonds (like the additions
+with higher decades) are not given enough attention when
+they are first used. (2) The special training necessary
+when a bond is used in a different connection (as when the
+multiplications to 9 &times; 9 are used in examples like</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='right'>729<br />
+<span class="u"> &nbsp; &nbsp; 8</span>
+</td></tr>
+</table></div>
+
+<p class="noidt">where the pupil has also to choose the right number to multiply,
+keep in mind what is carried, use it properly, and write the
+right figure in the right place, and carry a figure, or remember
+that he carries none) is neglected. (3) The pupil is not
+taught to check his work. (4) He is not made responsible
+for substantially accurate results. Furthermore, the requirement
+of (4) without the training of (1), (2), and (3)
+will involve either a fruitless failure on the part of many
+pupils, or an utterly unjust requirement of time. The
+common error of supposing that the task of computation
+with integers consists merely in learning the additions to
+9 &#43; 9, the subtractions to 18 &minus; 9, the multiplications to 8 &times; 9,
+and the divisions to 81 &divide; 9, and in applying this knowledge
+in connection with the principles of decimal notation, has
+had a large share in permitting the gross inaccuracy of
+arithmetical work. The bonds involved in 'knowing the
+tables' do not make up one fourth of the bonds involved
+in real adding, subtracting, multiplying, and dividing (with
+integers alone).<span class='pagenum'><a name="Page_107" id="Page_107">[Pg 107]</a></span></p>
+
+<p>It should be noted that if the training mentioned in (1)
+and (2) is well cared for, the checking of results as recommended
+in (3) becomes enormously more valuable than it
+is under present conditions, though even now it is one of
+our soundest practices. If a child knows the additions to
+higher decades so that he can add a seen one-place number
+to a thought-of two-place number in three seconds or less
+with a correct answer 199 times out of 200, there is only
+an infinitesimal chance that a ten-figure column twice added
+(once up, once down) a few minutes apart with identical
+answers will be wrong. Suppose that, in long multiplication,
+a pupil can multiply to 9 &times; 9 while keeping
+his place and keeping track of what he is 'carrying'
+and of where to write the figure he writes, and can add
+what he carries without losing track of what he is to add
+it to, where he is to write the unit figure, what he is to
+multiply next and by what, and what he will then have
+to carry, in each case to a surety of 99 percent of
+correct responses. Then two identical answers got by
+multiplying one three-place number by another a few
+minutes apart, and with reversal of the numbers, will not
+be wrong more than twice in his entire school career.
+Checks approach proofs when the constituent bonds are
+strong.</p>
+
+<p>If, on the contrary, the fundamental bonds are so weak
+that they do not work accurately, checking becomes much
+less trustworthy and also very much more laborious. In
+fact, it is possible to show that below a certain point of
+strength of the fundamental bonds, the time required for
+checking is so great that part of it might better be spent in
+improving the fundamental bonds.</p>
+
+<p>For example, suppose that a pupil has to find the sum of
+five numbers like $2.49, $5.25, $6.50, $7.89, and $3.75.<span class='pagenum'><a name="Page_108" id="Page_108">[Pg 108]</a></span>
+Counting each act of holding in mind the number to be
+carried and each writing of a column's result as equivalent
+in difficulty to one addition, such a sum equals nineteen
+single additions. On this basis and with certain additional
+estimates<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> we can compute the practical consequences for
+a pupil's use of addition in life according to the mastery of
+it that he has gained in school.</p>
+
+<p>I have so computed the amount of checking a pupil will
+have to do to reach two agreeing numbers (out of two, or
+three, or four, or five, or whatever the number before he
+gets two that are alike), according to his mastery of the elementary
+processes. The facts appear in Table 1.</p>
+
+<p>It is obvious that a pupil whose mastery of the elements is
+that denoted by getting them right 96 times out of 100
+will require so much time for checking that, even if he were
+never to use this ability for anything save a few thousand
+sums in addition, he would do well to improve this ability
+before he tried to do the sums. An ability of 199 out of
+200, or 995 out of 1000, seems likely to save much more
+time than would be taken to acquire it, and a reasonable
+defense could be made for requiring 996 or 997 out of
+1000.</p>
+
+<p>A precision of from 995 to 997 out of 1000 being required,
+and ordinary sagacity being used in the teaching, speed will
+substantially take care of itself. Counting on the fingers
+or in words will not give that precision. Slow recourse to
+memory of serial addition tables will not give that precision.
+Nothing save sure memory of the facts operating under the
+conditions of actual examples will give it. And such memories
+will operate with sufficient speed.</p>
+
+<p><span class='pagenum'><a name="Page_109" id="Page_109">[Pg 109]</a></span></p>
+
+<p class="tabcap">TABLE 1</p>
+<p class="center"><span class="smcap">The Effect of Mastery of the Elementary Facts of Addition upon
+the Labor Required to Secure Two Agreeing Answers When
+Adding Five Three-figure Numbers</span></p>
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary="">
+<colgroup><col width="20%" /><col width="20%" /><col width="20%" /><col width="20%" /><col width="20%" /></colgroup>
+<tr><th class="smcap bbox">Mastery of the Elementary Additions Times Right in 1000</th>
+<th class="smcap bbox">Approximate Number of Wrong Answers in Sums of 5 Three-place Numbers per 1000</th>
+<th class="smcap bbox">Approximate Number of Agreeing Answers, after One Checking, per 1000</th>
+<th class="smcap bbox">Approximate Number of Agreeing Answers, after a Checking of the First Discrepancies</th>
+<th class="smcap bbox">Approximate Number of Checkings Required (over and above the First General Checking of the 100 Sums) to Secure Two Agreeing Results</th></tr>
+<tr><td align='center'> 960</td><td align='center'> 700</td><td align='center'> 90</td><td align='center'> 216</td><td align='center'> 4500</td></tr>
+<tr><td align='center'> 980</td><td align='center'> 380</td><td align='center'> 384</td><td align='center'> 676</td><td align='center'> 1200</td></tr>
+<tr><td align='center'> 990</td><td align='center'> 190</td><td align='center'> 656</td><td align='center'> 906</td><td align='center'> 470</td></tr>
+<tr><td align='center'> 995</td><td align='center'> 95</td><td align='center'> 819</td><td align='center'> 975</td><td align='center'> 210</td></tr>
+<tr><td align='center'> 996</td><td align='center'> 76</td><td align='center'> 854</td><td align='center'> 984</td><td align='center'> 165</td></tr>
+<tr><td align='center'> 997</td><td align='center'> 54</td><td align='center'> 895</td><td align='center'> 992</td><td align='center'> 115</td></tr>
+<tr><td align='center'> 998</td><td align='center'> 38</td><td align='center'> 925</td><td align='center'> 996</td><td align='center'> 80</td></tr>
+<tr><td align='center'> 999</td><td align='center'> 19</td><td align='center'> 962</td><td align='center'> 999</td><td align='center'> 40</td></tr>
+</table></div>
+
+<p>There is one intelligent objection to the special practice
+necessary to establish arithmetical connections so fully as
+to give the accuracy which both utilitarian and disciplinary
+aims require. It may be said that the pupils in grades 3,
+4, and 5 cannot appreciate the need and that consequently
+the work will be dull, barren, and alien, without close personal
+appropriation by the pupil's nature. It is true that no
+vehement life-purpose is directly involved by the problem
+of perfecting one's power to add 7 to 28 in grade 2, or by
+the problem of multiplying 253 by 8 accurately in grade 3<span class='pagenum'><a name="Page_110" id="Page_110">[Pg 110]</a></span>
+or by precise subtraction in long division in grade 4. It is
+also true, however, that the most humanly interesting of
+problems&mdash;one that the pupil attacks most whole-heartedly&mdash;will
+not be solved correctly unless the pupil has the
+necessary associative mechanisms in order; and the surer
+he is of them, the freer he is to think out the problem as
+such. Further, computation is not dull if the pupil can
+compute. He does not himself object to its barrenness of
+vital meaning, so long as the barrenness of failure is prevented.
+We must not forget that pupils like to learn. In
+teaching excessively dull individuals, who has not often
+observed the great interest which they display in anything
+that they are enabled to master? There is pathos in their
+joy in learning to recognize parts of speech, perform algebraic
+simplifications, or translate Latin sentences, and in other
+accomplishments equally meaningless to all their interests
+save the universal human interest in success and recognition.
+Still further, it is not very hard to show to pupils the imperative
+need of accuracy in scoring games, in the shop,
+in the store, and in the office. Finally, the argument
+that accurate work of this sort is alien to the pupil in
+these grades is still stronger against <i>inaccurate</i> work of
+the same sort. If we are to teach computation with
+two- and three- and four-place numbers at all, it should
+be taught as a reliable instrument, not as a combination
+of vague memories and faith. The author is ready
+to cut computation with numbers above 10 out of the
+curriculum of grades 1-6 as soon as more valuable educational
+instruments are offered in its place, but he is convinced
+that nothing in child-nature makes a large variety of
+inaccurate computing more interesting or educative or germane
+to felt needs, than a smaller variety of accurate
+computing!</p>
+<p><span class='pagenum'><a name="Page_111" id="Page_111">[Pg 111]</a></span></p>
+
+<h4>THE STRENGTH OF BONDS FOR TEMPORARY SERVICE</h4>
+
+<p>The second general fact is that certain bonds are of service
+for only a limited time and so need to be formed only to a
+limited and slight degree of strength. The data of problems
+set to illustrate a principle or improve some habit of computation
+are, of course, the clearest cases. The pupil needs
+to remember that John bought 3 loaves of bread and that
+they were 5-cent loaves and that he gave 25 cents to the
+baker only long enough to use the data to decide what change
+John should receive. The connections between the total
+described situation and the answer obtained, supposing
+some considerable computation to intervene, is a bond that
+we let expire almost as soon as it is born.</p>
+
+<p>It is sometimes assumed that the bond between a certain
+group of features which make a problem a 'Buy <i>a</i> things at
+<i>b</i> per thing, find total cost' problem or a 'Buy <i>a</i> things at <i>b</i>
+per thing, what change from <i>c</i>' problem or a 'What gain
+on buying for <i>a</i> and selling for <i>b</i>' problem or a 'How many
+things at <i>a</i> each can I buy for <i>b</i> cents' problem&mdash;it is assumed
+that the bond between these essential defining features
+and the operation or operations required for solution
+is as temporary as the bonds with the name of the buyer
+or the price of the thing. It is assumed that all problems
+are and should be solved by some pure act of reasoning
+without help or hindrance from bonds with the particular
+verbal structure and vocabulary of the problems. Whether
+or not they <i>should</i> be, they <i>are not</i>. Every time that a pupil
+solves a 'bought-sold' problem by subtraction he strengthens
+the tendency to respond to any problem whatsoever that
+contains the words 'bought for' and 'sold for' by subtraction;
+and he will by no means surely stop and survey
+every such problem in all its elements to make sure that no<span class='pagenum'><a name="Page_112" id="Page_112">[Pg 112]</a></span>
+other feature makes inapplicable the tendency to subtract
+which the 'bought sold' evokes.</p>
+
+<p>To prevent pupils from responding to the form of statement
+rather than the essential facts, we should then not
+teach them to forget the form of statement, but rather
+give them all the common forms of statement to which
+the response in question is an appropriate response, and
+only such. If a certain form of statement does in life
+always signify a certain arithmetical procedure, the bond
+between it and that procedure may properly be made very
+strong.</p>
+
+<p>Another case of the formation of bonds to only a slight
+degree of strength concerns the use of so-called 'crutches'
+such as writing &#43;, &minus;, and &times; in copying problems like those
+below:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="60%">
+<tr><td align='center'>Add</td><td align='center'>Subtract</td><td align='center'>Multiply</td></tr>
+<tr><td align='center'>23<br />61<br />&mdash;</td><td align='center'>79<br />24<br />&mdash;</td><td align='center'>32<br />&nbsp;&nbsp;3<br />&mdash;</td></tr>
+</table></div>
+
+<p class="noidt">or altering the figures when 'borrowing' in subtraction,
+and the like. Since it is undesirable that the pupil should
+regard the 'crutch' response as essential to the total procedure,
+or become so used to having it that he will be disturbed
+by its absence later, it is supposed that the bond between
+the situation and the crutch should not be fully
+formed. There is a better way out of the difficulty, in case
+crutches are used at all. This is to associate the crutch
+with a special 'set,' and its non-use with the general set
+which is to be the permanent one. For example, children
+may be taught from the start never to write the crutch
+sign or crutch figure unless the work is accompanied by
+"Write ... to help you to...."<span class='pagenum'><a name="Page_113" id="Page_113">[Pg 113]</a></span></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='justify' rowspan='2'>Write - to help you to remember that<br />
+you must subtract in this row.</td>
+<td align='left' colspan='5'>Find the differences:&mdash;</td></tr>
+<tr><td align='left'>39<br /><span class="u">23</span></td><td align='left'>67<br /><span class="u">44</span></td><td align='left'>78<br /><span class="u">36</span></td><td align='left'>56<br /><span class="u">26</span></td><td align='left'>45<br /><span class="u">24</span></td></tr>
+<tr><td>&nbsp;</td></tr>
+<tr><td align='justify' rowspan='2'>Remember that you must subtract<br />
+in this row.</td>
+<td align='left' colspan='5'>Find the differences:&mdash;</td></tr>
+<tr><td align='left'>85<br /><span class="u">63</span></td><td align='left'>27<br /><span class="u">14</span></td><td align='left'>96<br /><span class="u">51</span></td><td align='left'>38<br /><span class="u">45</span></td><td align='left'>78<br /><span class="u">32</span></td></tr>
+</table></div>
+
+<p>The bond evoking the use of the crutch may then be
+formed thoroughly enough so that there is no hesitation,
+insecurity, or error, without interfering to any harmful extent
+with the more general bond from the situation to work
+without the crutch.</p>
+
+
+<h4>THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS</h4>
+
+<p>Another instructive case concerns the bonds between certain
+words and their meanings, and between certain situations
+of commerce, industry, or agriculture and useful facts
+about these situations. Illustrations of the former are the
+bonds between <i>cube root</i>, <i>hectare</i>, <i>brokerage</i>, <i>commission</i>,
+<i>indorsement</i>, <i>vertex</i>, <i>adjacent</i>, <i>nonagon</i>, <i>sector</i>, <i>draft</i>, <i>bill
+of exchange</i>, and their meanings. Illustrations of the latter
+are the bonds from "Money being lent 'with interest' at no
+specified rate, what rate is charged?" to "The legal rate
+of the state," from "$<i>X</i> per M as a rate for lumber" to
+"Means $<i>X</i> per thousand board feet, a board foot being
+1 ft. by 1 ft. by 1 in."</p>
+
+<p>It is argued by many that such bonds are valuable for a
+short time; namely, while arithmetical procedures in connection
+with which they serve are learned, but that their
+value is only to serve as a means for learning these procedures
+and that thereafter they may be forgotten. "They
+are formed only as accessory means to certain more purely
+arithmetical knowledge or discipline; after this is acquired<span class='pagenum'><a name="Page_114" id="Page_114">[Pg 114]</a></span>
+they may be forgotten. Everybody does in fact forget
+them, relearning them later if life requires." So runs the
+argument.</p>
+
+<p>In some cases learning such words and facts only to use
+them in solving a certain sort of problems and then forget
+them may be profitable. The practice is, however, exceedingly
+risky. It is true that everybody does in fact
+forget many such meanings and facts, but this commonly
+means either that they should not have been learned at all
+at the time that they were learned, or that they should have
+been learned more permanently, or that details should have
+been learned with the expectation that they themselves
+would be forgotten but that a general fact or attitude would
+remain. For example, duodecagon should not be learned
+at all in the elementary school; indorsement should either
+not be learned at all there, or be learned for permanence of
+a year or more; the details of the metric system should be
+so taught as to leave for several years at least knowledge
+of the facts that there is a system so named that is important,
+whose tables go by tens, hundreds, or thousands,
+and a tendency (not necessarily strong) to connect meter,
+kilogram, and liter with measurement by the metric
+system and with approximate estimates of their several
+magnitudes.</p>
+
+<p>If an arithmetical procedure seems to require accessory
+bonds which are to be forgotten, once the procedure
+is mastered, we should be suspicious of the value
+of the procedure itself. If pupils forget what compound
+interest is, we may be sure that they will usually
+also have forgotten how to compute it. Surely there
+is waste if they have learned what it is only to learn
+how to compute it only to forget how to compute
+it!</p>
+<p><span class='pagenum'><a name="Page_115" id="Page_115">[Pg 115]</a></span></p>
+
+<h4>THE STRENGTH OF BONDS CONCERNING THE REASONS FOR
+ARITHMETICAL PROCESSES</h4>
+
+<p>The next case of the formation of bonds to slight strength
+is the problematic one of forming the bonds involved in
+understanding the reasons for certain processes only to
+forget them after the process has become a habit. Should
+a pupil, that is, learn why he inverts and multiplies, only to
+forget it as soon as he can be trusted to divide by a fraction?
+Should he learn why he puts the units figure of each
+partial product in multiplication under the figure that he
+multiplies by, only to forget the reason as soon as he has
+command of the process? Should he learn why he gets
+the number of square inches in a rectangle by multiplying
+the length by the width, both being expressed in linear
+inches, and forget why as soon as he is competent to make
+computations of the areas of rectangles?</p>
+
+<p>On general psychological grounds we should be suspicious
+of forming bonds only to let them die of starvation later,
+and tend to expect that elaborate explanations learned only
+to be forgotten either should not be learned at all, or should
+be learned at such a time and in such a way that they would
+not be forgotten. Especially we should expect that the
+general principles of arithmetic, the whys and wherefores
+of its fundamental ways of manipulating numbers, ought
+to be the last bonds of all to be forgotten. Details of <i>how</i>
+you arranged numbers to multiply might vanish, but the
+general reasons for the placing would be expected to persist
+and enable one to invent the detailed manipulations
+that had been forgotten.</p>
+
+<p>This suspicion is, I think, justified by facts. The doctrine
+that the customary deductive explanations of why we
+invert and multiply, or place the partial products as we do<span class='pagenum'><a name="Page_116" id="Page_116">[Pg 116]</a></span>
+before adding, may be allowed to be forgotten once the
+actual habits are in working order, has a suspicious source.
+It arose to meet the criticism that so much time and effort
+were required to keep these deductive explanations in memory.
+The fact was that the pupil learned to compute correctly
+<i>irrespective of</i> the deductive explanations. They
+were only an added burden. His inductive learning that
+the procedure gave the right answer really taught him. So
+he wisely shuffled off the extra burden of facts about the
+consequences of the nature of a fraction or the place values
+of our decimal notation. The bonds weakened because
+they were not used. They were not used because they were
+not useful in the shape and at the time that they were formed,
+or because the pupil was unable to understand the explanations
+so as to form them at all.</p>
+
+<p>The criticism was valid and should have been met in part
+by replacing the deductive explanations by inductive verifications,
+and in part by using the deductive reasoning as a
+check after the process itself is mastered. The very same
+discussions of place-value which are futile as proof that you
+must do a certain thing before you have done it, often become
+instructive as an explanation of why the thing that
+you have learned to do and are familiar with and have
+verified by other tests works as well as it does. The general
+deductive theory of arithmetic should not be learned only
+to be forgotten. Much of it should, by most pupils, not be
+learned at all. What is learned should be learned much
+later than now, as a synthesis and rationale of habits,
+not as their creator. What is learned of such deductive
+theory should rank among the most rather than least permanent
+of a pupil's stock of arithmetical knowledge and
+power. There are bonds which are formed only to be lost,
+and bonds formed only to be lost <i>in their first form</i>, being<span class='pagenum'><a name="Page_117" id="Page_117">[Pg 117]</a></span>
+used in a new organization as material for bonds of a higher
+order; but the bonds involved in deductive explanations
+of why certain processes are right are not such: they are
+not to be formed just to be forgotten, nor as mere prop&aelig;deutics
+to routine manipulations.</p>
+
+
+<h4>PROP&AElig;DEUTIC BONDS</h4>
+
+<p>The formation of bonds to a limited strength because they
+are to be lost in their first form, being worked over in different
+ways in other bonds to which they are prop&aelig;deutic or contributing
+is the most important case of low strength, or
+rather low permanence, in bonds.</p>
+
+<p>The bond between four 5s in a column to be added and
+the response of thinking '10, 15, 20' is worth forming, but
+it is displaced later by the multiplication bond or direct
+connection of 'four 5s to be added' with '20.' Counting
+by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc., forms
+serial bonds which as series might well be left to disappear.
+Their separate steps are kept as permanent bonds for use
+in column addition, but their serial nature is changed from
+2 (and 2) 4, (and 2) 6, (and 2) 8, etc., to two 2s = 4, three
+2s = 6, four 2s = 8, etc.; after playing their part in producing
+the bonds whereby any multiple of 2 by 2 to 9, can be got,
+the original serial bonds are, as series, needed no longer. The
+verbal response of saying 'and' in adding, after helping to establish
+the bonds whereby the general set of the mind toward
+adding co&ouml;perates with the numbers seen or thought of to
+produce their sum, should disappear; or remain so slurred
+in inner speech as to offer no bar to speed.</p>
+
+<p>The rule for such bonds is, of course, to form them strongly
+enough so that they work quickly and accurately for the
+time being and facilitate the bonds that are to replace them,<span class='pagenum'><a name="Page_118" id="Page_118">[Pg 118]</a></span>
+but not to overlearn them. There is a difference between
+learning something to be held for a short time, and the same
+amount of energy spent in learning for long retention. The
+former sort of learning is, of course, appropriate with many
+of these prop&aelig;deutic bonds.</p>
+
+<p>The bonds mentioned as illustrations are not <i>purely</i>
+prop&aelig;deutic, nor formed <i>only</i> to be transmuted into something
+else. Even the saying of 'and' in addition has
+some genuine, intrinsic value in distinguishing the process
+of addition, and may perhaps be usefully reviewed for a
+brief space during the first steps in adding common fractions.
+Some such prop&aelig;deutic bonds may be worth while
+apart from their value in preparing for other bonds. Consider,
+for example, exercises like those shown below which
+are prop&aelig;deutic to long division, giving the pupil some
+basis in experience for his selection of the quotient figures.
+These multiplications are intrinsically worth doing, especially
+the 12s and 25s. Whatever the pupil remembers
+of them will be to his advantage.</p>
+
+<div class="pblockquot">
+<p><b>1.</b> Count by 11s to 132, beginning 11, 22, 33.</p>
+
+<p><b>2.</b> Count by 12s to 144, beginning 12, 24, 36.</p>
+
+<p><b>3.</b> Count by 25s to 300, beginning 25, 50, 75.</p>
+
+<p><b>4.</b> State the missing numbers:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="10" summary="">
+<tr><th>A.</th><th> &nbsp; B.</th><th>C.</th><th>D.</th></tr>
+<tr><td align='right'> 3 11s =</td><td align='right'>5 11s =</td><td align='right'>8 ft. = .... in.</td><td align='right'>2 dozen =</td></tr>
+<tr><td align='right'> 4 12s =</td><td align='right'>3 12s =</td><td align='right'>10 ft. = .... in.</td><td align='right'>4 dozen =</td></tr>
+<tr><td align='right'> 5 12s =</td><td align='right'>6 12s =</td><td align='right'>7 ft. = .... in.</td><td align='right'>10 dozen =</td></tr>
+<tr><td align='right'> 6 11s =</td><td align='right'>12 11s =</td><td align='right'>4 ft. = .... in.</td><td align='right'>5 dozen =</td></tr>
+<tr><td align='right'> 9 11s =</td><td align='right'>2 12s =</td><td align='right'>6 ft. = .... in.</td><td align='right'>7 dozen =</td></tr>
+<tr><td align='right'> 7 12s =</td><td align='right'>9 12s =</td><td align='right'>9 ft. = .... in.</td><td align='right'>12 dozen =</td></tr>
+<tr><td align='right'> 8 12s =</td><td align='right'>7 11s =</td><td align='right'>11 ft. = .... in.</td><td align='right'>9 dozen =</td></tr>
+<tr><td align='right'>11 11s =</td><td align='right'>&nbsp; &nbsp; 12 12s =</td><td align='right'>&nbsp; &nbsp; 5 ft. = .... in.</td><td align='right'>&nbsp; &nbsp; 6 dozen =</td></tr>
+</table></div>
+<p><span class='pagenum'><a name="Page_119" id="Page_119">[Pg 119]</a></span></p>
+
+<p><b>5.</b> Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents,
+one dollar," and so on.</p>
+
+<p><b>6.</b> Count by 15s to $1.50.</p>
+
+<p><b>7.</b> Find the products. Do not use pencil. Think what they are.</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="10" summary="">
+<tr><th>A.</th><th>B.</th><th>C.</th><th>D.</th><th>E.</th></tr>
+<tr><td align='right'> 2 &times; 25</td><td align='right'>3 &times; 15</td><td align='right'>2 &times; 12</td><td align='right'>4 &times; 11</td><td align='right'>6 &times; 25</td></tr>
+<tr><td align='right'> 3 &times; 25</td><td align='right'>&nbsp; 10 &times; 15</td><td align='right'>2 &times; 15</td><td align='right'>4 &times; 15</td><td align='right'>6 &times; 15</td></tr>
+<tr><td align='right'> 5 &times; 25</td><td align='right'>4 &times; 15</td><td align='right'>2 &times; 25</td><td align='right'>4 &times; 12</td><td align='right'>6 &times; 12</td></tr>
+<tr><td align='right'>10 &times; 25</td><td align='right'>2 &times; 15</td><td align='right'>2 &times; 11</td><td align='right'>4 &times; 25</td><td align='right'>6 &times; 11</td></tr>
+<tr><td align='right'> 4 &times; 25</td><td align='right'>7 &times; 15</td><td align='right'>3 &times; 25</td><td align='right'>5 &times; 11</td><td align='right'>7 &times; 12</td></tr>
+<tr><td align='right'> 6 &times; 25</td><td align='right'>9 &times; 15</td><td align='right'>3 &times; 15</td><td align='right'>5 &times; 12</td><td align='right'>7 &times; 15</td></tr>
+<tr><td align='right'> 8 &times; 25</td><td align='right'>5 &times; 15</td><td align='right'>3 &times; 11</td><td align='right'>5 &times; 15</td><td align='right'>7 &times; 25</td></tr>
+<tr><td align='right'> 7 &times; 25</td><td align='right'>8 &times; 15</td><td align='right'>&nbsp; 3 &times; 12</td><td align='right'>5 &times; 25</td><td align='right'>7 &times; 11</td></tr>
+<tr><td align='right'> 9 &times; 25</td><td align='right'>6 &times; 15</td><td align='right'>8 &times; 12</td><td align='right'>&nbsp; 9 &times; 12</td><td align='right'>&nbsp; 8 &times; 25</td></tr>
+</table></div>
+
+<p>State the missing numbers:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="10" summary="">
+<tr><td align='right'>A. &nbsp; 36 = .... 12s</td><td align='right'>&nbsp; &nbsp; B. &nbsp; 44 = .... 11s</td><td align='right'>&nbsp; &nbsp; C. &nbsp; 50 = .... 25s</td></tr>
+<tr><td align='right'> 60 = .... 12s</td><td align='right'>88 = .... 11s</td><td align='right'>125 = .... 25s</td></tr>
+<tr><td align='right'> 24 = .... 12s</td><td align='right'>77 = .... 11s</td><td align='right'>75 = .... 25s</td></tr>
+<tr><td align='right'> 48 = .... 12s</td><td align='right'>55 = .... 11s</td><td align='right'>200 = .... 25s</td></tr>
+<tr><td align='right'>144 = .... 12s</td><td align='right'>99 = .... 11s</td><td align='right'>250 = .... 25s</td></tr>
+<tr><td align='right'>108 = .... 12s</td><td align='right'>110 = .... 11s</td><td align='right'>175 = .... 25s</td></tr>
+<tr><td align='right'> 72 = .... 12s</td><td align='right'>33 = .... 11s</td><td align='right'>225 = .... 25s</td></tr>
+<tr><td align='right'> 96 = .... 12s</td><td align='right'>66 = .... 11s</td><td align='right'>150 = .... 25s</td></tr>
+<tr><td align='right'> 84 = .... 12s</td><td align='right'>22 = .... 11s</td><td align='right'>100 = .... 25s</td></tr>
+</table></div>
+
+<p>Find the quotients and remainders. If you need to use paper
+and pencil to find them, you may. But find as many as you can
+without pencil and paper. Do Row A first. Then do Row B.
+Then Row C, etc.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="6" cellspacing="20" summary="">
+<tr><td align='left'>Row A.</td><td align='left'>11|<span class="overline">45</span></td><td align='left'>12|<span class="overline">45</span></td><td align='left'>25|<span class="overline">45</span></td><td align='left'>15|<span class="overline">45</span></td><td align='left'>21|<span class="overline">45</span></td><td align='left'>22|<span class="overline">45</span></td></tr>
+<tr><td align='left'>Row B.</td><td align='left'>25|<span class="overline">55</span></td><td align='left'>11|<span class="overline">55</span></td><td align='left'>12|<span class="overline">55</span></td><td align='left'>15|<span class="overline">55</span></td><td align='left'>22|<span class="overline">55</span></td><td align='left'>30|<span class="overline">55</span></td></tr>
+<tr><td align='left'>Row C.</td><td align='left'>12|<span class="overline">60</span></td><td align='left'>25|<span class="overline">60</span></td><td align='left'>15|<span class="overline">60</span></td><td align='left'>11|<span class="overline">60</span></td><td align='left'>30|<span class="overline">60</span></td><td align='left'>21|<span class="overline">60</span></td></tr>
+<tr><td align='left'>Row D.</td><td align='left'>12|<span class="overline">75</span></td><td align='left'>11|<span class="overline">75</span></td><td align='left'>15|<span class="overline">75</span></td><td align='left'>25|<span class="overline">75</span></td><td align='left'>30|<span class="overline">75</span></td><td align='left'>35|<span class="overline">75</span></td></tr>
+<tr><td align='left'>Row E.</td><td align='left'>11|<span class="overline">100</span></td><td align='left'>12|<span class="overline">100</span></td><td align='left'>25|<span class="overline">100</span></td><td align='left'>15|<span class="overline">100</span></td><td align='left'>30|<span class="overline">100</span></td><td align='left'>22|<span class="overline">100</span></td></tr>
+<tr><td align='left'>Row F.</td><td align='left'>11|<span class="overline">96</span></td><td align='left'>12|<span class="overline">96</span></td><td align='left'>25|<span class="overline">96</span></td><td align='left'>15|<span class="overline">96</span></td><td align='left'>30|<span class="overline">96</span></td><td align='left'>22|<span class="overline">96</span></td></tr>
+<tr><td align='left'>Row G.</td><td align='left'>25|<span class="overline">105</span></td><td align='left'>11|<span class="overline">105</span></td><td align='left'>15|<span class="overline">105</span></td><td align='left'>12|<span class="overline">105</span></td><td align='left'>22|<span class="overline">105</span></td><td align='left'>35|<span class="overline">105</span></td></tr>
+<tr><td align='left'>Row H.</td><td align='left'>12|<span class="overline">64</span></td><td align='left'>15|<span class="overline">64</span></td><td align='left'>25|<span class="overline">64</span></td><td align='left'>11|<span class="overline">64</span></td><td align='left'>22|<span class="overline">64</span></td><td align='left'>21|<span class="overline">64</span></td></tr>
+<tr><td align='left'>Row I.</td><td align='left'>11|<span class="overline">80</span></td><td align='left'>12|<span class="overline">80</span></td><td align='left'>15|<span class="overline">80</span></td><td align='left'>25|<span class="overline">80</span></td><td align='left'>35|<span class="overline">80</span></td><td align='left'>21|<span class="overline">80</span></td></tr>
+<tr><td align='left'>Row J.</td><td align='left'>25|<span class="overline">200</span></td><td align='left'>30|<span class="overline">200</span></td><td align='left'>75|<span class="overline">200</span></td><td align='left'>63|<span class="overline">200</span></td><td align='left'>65|<span class="overline">200</span></td><td align='left'>66|<span class="overline">200</span></td></tr>
+</table></div>
+
+<p>Do this section again. Do all the first column first. Then do
+the second column, then the third, and so on.</p></div>
+
+<p>Consider, from the same point of view, exercises like
+(3 &times; 4) &#43; 2, (7 &times; 6) &#43; 5, (9 &times; 4) &#43; 6, given as a preparation
+for written multiplication. The work of</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr>
+<td align='right'>&nbsp; &nbsp; 48<br /><span class="u"> &nbsp; 3</span></td>
+<td align='right'>&nbsp; &nbsp; 68<br /><span class="u"> &nbsp; 7</span></td>
+<td align='right'>&nbsp; &nbsp; 47<br /><span class="u"> &nbsp; 9</span></td>
+</tr>
+</table></div>
+
+<p class="noidt">and the like is facilitated if the pupil has easy control
+of the process of getting a product, and keeping it in mind
+while he adds a one-place number to it. The practice with
+(3 &times; 4) &#43; 2 and the like is also good practice intrinsically. So
+some teachers provide systematic preparatory drills of this
+type just before or along with the beginning of short multiplication.</p>
+
+<p>In some cases the bonds are purely prop&aelig;deutic or are
+formed <i>only</i> for later reconstruction. They then differ
+little from 'crutches.' The typical crutch forms a habit
+which has actually to be broken, whereas the purely prop&aelig;deutic
+bond forms a habit which is left to rust out from
+disuse.</p>
+
+<p>For example, as an introduction to long division, a pupil
+may be given exercises using one-figure divisors in the long
+form, as:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='right'>773</td><td align='left'>&nbsp; and 5 remainder</td></tr>
+<tr><td align='right'>7 )<span class="overline"> 5416</span></td></tr>
+<tr><td align='right'><span class="u"> 49</span> &nbsp; &nbsp;</td></tr>
+<tr><td align='right'>51&nbsp;&nbsp;</td></tr>
+<tr><td align='right'><span class="u"> 49</span>&nbsp;&nbsp;</td></tr>
+<tr><td align='right'>26</td></tr>
+<tr><td align='right'><span class="u">21</span></td></tr>
+<tr><td align='right'>5</td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_121" id="Page_121">[Pg 121]</a></span></p>
+<p>The important recommendation concerning these purely
+prop&aelig;deutic bonds, and bonds formed only for later reconstruction,
+is to be very critical of them, and not indulge in
+them when, by the exercise of enough ingenuity, some bond
+worthy of a permanent place in the individual's equipment
+can be devised which will do the work as well. Arithmetical
+teaching has done very well in this respect, tending
+to err by leaving out really valuable preparatory drills
+rather than by inserting uneconomical ones. It is in the
+teaching of reading that we find the formation of prop&aelig;deutic
+bonds of dubious value (with letters, phonograms,
+diacritical marks, and the like) often carried to demonstrably
+wasteful extremes.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_122" id="Page_122">[Pg 122]</a></span></p>
+<h2><a name="CHAPTER_VI" id="CHAPTER_VI"></a>CHAPTER VI</h2>
+
+<h3>THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE
+AMOUNT OF PRACTICE AND THE ORGANIZATION OF
+ABILITIES</h3>
+
+<h4>THE AMOUNT OF PRACTICE</h4>
+
+
+<p>It will be instructive if the reader will perform the following
+experiment as an introduction to the discussion of this
+chapter, before reading any of the discussion.</p>
+
+<p>Suppose that a pupil does all the work, oral and written,
+computation and problem-solving, presented for grades
+1 to 6 inclusive (that is, in the first two books of a three-book
+series) in the average textbook now used in the elementary
+school. How many times will he have exercised
+each of the various bonds involved in the four operations
+with integers shown below? That is, how many times will
+he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.?
+Every case of the action of each bond is to be counted.</p>
+
+
+<p class="center"><b>THE  FUNDAMENTAL  BONDS</b></p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="" width="80%">
+<tr><td align='left'>1 &#43; 1</td><td align='left'>2 &minus; 1</td><td align='left'>1 &times; 1</td><td align='left'>2 &divide; 1</td></tr>
+<tr><td align='left'>1 &#43; 2</td><td align='left'>2 &minus; 2</td><td align='left'>2 &times; 1</td><td align='left'>2 &divide; 2</td></tr>
+<tr><td align='left'>1 &#43; 3</td><td align='left'></td><td align='left'>3 &times; 1</td></tr>
+<tr><td align='left'>1 &#43; 4</td><td align='left'></td><td align='left'>4 &times; 1</td></tr>
+<tr><td align='left'>1 &#43; 5</td><td align='left'>3 &minus; 1</td><td align='left'>5 &times; 1</td><td align='left'>3 &divide; 1</td></tr>
+<tr><td align='left'>1 &#43; 6</td><td align='left'>3 &minus; 2</td><td align='left'>6 &times; 1</td><td align='left'>3 &divide; 2</td></tr>
+<tr><td align='left'>1 &#43; 7</td><td align='left'>3 &minus; 3</td><td align='left'>7 &times; 1</td><td align='left'>3 &divide; 3</td></tr>
+<tr><td align='left'>1 &#43; 8</td><td align='left'></td><td align='left'>8 &times; 1</td></tr>
+<tr><td align='left'>1 &#43; 9</td><td align='left'></td><td align='left'>9 &times; 1</td></tr>
+<tr><td align='left'><span class='pagenum'><a name="Page_123" id="Page_123">[Pg 123]</a></span></td><td align='left'>4 &minus; 1</td><td align='left'></td><td align='left'>4 &divide; 1</td></tr>
+<tr><td align='left'></td><td align='left'>4 &minus; 2</td><td align='left'></td><td align='left'>4 &divide; 2</td></tr>
+<tr><td align='left'>11 (or 21 or 31, etc.) &#43; 1</td><td align='left'>4 &minus; 3</td><td align='left'>1 &times; 2</td><td align='left'>4 &divide; 3</td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 2</td><td align='left'>4 &minus; 4</td><td align='left'>2 &times; 2</td><td align='left'>4 &divide; 4</td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 3</td><td align='left'></td><td align='left'>3 &times; 2</td><td align='left'></td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 4</td><td align='left'></td><td align='left'>4 &times; 2</td><td align='left'></td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 5</td><td align='left'>5 &minus; 1</td><td align='left'>5 &times; 2</td><td align='left'>5 &divide; 1</td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 6</td><td align='left'>5 &minus; 2</td><td align='left'>6 &times; 2</td><td align='left'>5 &divide; 2</td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 7</td><td align='left'>5 &minus; 3</td><td align='left'>7 &times; 2</td><td align='left'>5 &divide; 3</td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 8</td><td align='left'>5 &minus; 4</td><td align='left'>8 &times; 2</td><td align='left'>5 &divide; 4</td></tr>
+<tr><td align='left'>11 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 9</td><td align='left'>5 &minus; 5</td><td align='left'>9 &times; 2</td><td align='left'>5 &divide; 5</td></tr>
+<tr><td>&nbsp;</td></tr>
+<tr><td align='left'></td><td align='left'>6 &minus; 1</td><td align='left'>1 &times; 3</td><td align='left'>6 &divide; 1</td></tr>
+<tr><td align='left'>2 &#43; 1</td><td align='left'>6 &minus; 2</td><td align='left'>2 &times; 3</td><td align='left'>6 &divide; 2</td></tr>
+<tr><td align='left'>2 &#43; 2</td><td align='left'>6 &minus; 3</td><td align='left'>3 &times; 3</td><td align='left'>6 &divide; 3</td></tr>
+<tr><td align='left'>2 &#43; 3</td><td align='left'>6 &minus; 4</td><td align='left'>4 &times; 3</td><td align='left'>6 &divide; 4</td></tr>
+<tr><td align='left'>2 &#43; 4</td><td align='left'>6 &minus; 5</td><td align='left'>5 &times; 3</td><td align='left'>6 &divide; 5</td></tr>
+<tr><td align='left'>2 &#43; 5</td><td align='left'>6 &minus; 6</td><td align='left'>6 &times; 3</td><td align='left'>6 &divide; 6</td></tr>
+<tr><td align='left'>2 &#43; 6</td><td align='left'></td><td align='left'>7 &times; 3</td><td align='left'></td></tr>
+<tr><td align='left'>2 &#43; 7</td><td align='left'></td><td align='left'>8 &times; 3</td></tr>
+<tr><td align='left'>2 &#43; 8</td><td align='left'>7 &minus; 1</td><td align='left'>9 &times; 3</td><td align='left'>7 &divide; 1</td></tr>
+<tr><td align='left'>2 &#43; 9</td><td align='left'>7 &minus; 2</td><td align='left'></td><td align='left'>7 &divide; 2</td></tr>
+<tr><td align='left'></td><td align='left'>7 &minus; 3</td><td align='left'></td><td align='left'>7 &divide; 3</td></tr>
+<tr><td align='left'></td><td align='left'>7 &minus; 4</td><td align='left'>1 &times; 4</td><td align='left'>7 &divide; 4</td></tr>
+<tr><td align='left'>12 (or 22 or 32, etc.) &#43; 1</td><td align='left'>7 &minus; 5</td><td align='left'>2 &times; 4</td><td align='left'>7 &divide; 5</td></tr>
+<tr><td align='left'>12 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  " &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  &#43; 2</td><td align='left'>7 &minus; 6</td><td align='left'>and so on</td><td align='left'>7 &divide; 6</td></tr>
+<tr><td align='left'></td><td align='left'>7 &minus; 7</td><td align='left'>to 9 &times; 9</td><td align='left'>7 &divide; 7</td></tr>
+<tr><td align='left'>and so on to</td><td align='left'>and so on</td><td align='left'></td><td align='left'>and so on to</td></tr>
+<tr><td align='left'>9 &#43; 9</td><td align='left'>to 18 &minus; 9</td><td align='left'></td><td align='left'>82 &divide; 9 &nbsp; &nbsp; 83 &divide; 9, etc.</td></tr>
+<tr><td align='left'>19 (or 29 or 39, etc.) &#43; 9</td></tr>
+</table></div>
+
+<p>If estimating for the entire series is too long a task, it will
+be sufficient to use eight or ten from each, say:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="" width="80%">
+<colgroup><col width="25%" /><col width="25%" /><col width="25%" /><col width="25%" /></colgroup>
+<tr><td align='right'>3 &#43; 2</td><td align='right'>13, 23, etc. &#43; 2</td><td align='right'>7 &#43; 2</td><td align='right'>17, 27, etc. &#43; 2</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 3</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 3</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 3</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 3</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 4</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 4</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 4</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 4</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 5</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 5</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 5</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 5</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 6</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 6</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 6</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 6</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 7</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 7</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 7</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 7</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 8</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 8</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 8</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 8</td></tr>
+<tr><td align='right'>&nbsp; &nbsp;"&nbsp; 9</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 9</td>
+<td align='right'>&nbsp; &nbsp;"&nbsp; 9</td><td align='right'>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "&nbsp; &nbsp;&nbsp; &nbsp; &nbsp; 9</td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_124" id="Page_124">[Pg 124]</a></span></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="" width="80%">
+<colgroup><col width="25%" /><col width="25%" /><col width="25%" /><col width="25%" /></colgroup>
+<tr><td align='right'>3 &minus; 3</td><td align='right'>7 &minus; 7</td><td align='right'>9 &times; 7</td><td align='right'>63 &divide; 9</td></tr>
+<tr><td align='right'>4 &nbsp; " &nbsp; </td><td align='right'>8 &nbsp; " &nbsp; </td><td align='right'>7 &times; 9</td><td align='right'>64 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>5 &nbsp; " &nbsp; </td><td align='right'>9 &nbsp; " &nbsp; </td><td align='right'>8 &times; 6</td><td align='right'>65 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>6 &nbsp; " &nbsp; </td><td align='right'>10 &nbsp; " &nbsp; </td><td align='right'>6 &times; 8</td><td align='right'>66 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>7 &nbsp; " &nbsp; </td><td align='right'>11 &nbsp; " &nbsp; </td><td align='right'></td><td align='right'>67 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>8 &nbsp; " &nbsp; </td><td align='right'>12 &nbsp; " &nbsp; </td><td align='right'></td><td align='right'>68 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>9 &nbsp; " &nbsp; </td><td align='right'>13 &nbsp; " &nbsp; </td><td align='right'></td><td align='right'>69 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>10 &nbsp; " &nbsp; </td><td align='right'>14 &nbsp; " &nbsp; </td><td align='right'></td><td align='right'>70 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>11 &nbsp; " &nbsp; </td><td align='right'>15 &nbsp; " &nbsp; </td><td align='right'></td><td align='right'>71 &nbsp; " &nbsp; </td></tr>
+<tr><td align='right'>12 &nbsp; " &nbsp; </td><td align='right'>16 &nbsp; " &nbsp; </td><td align='right'></td></tr>
+</table></div>
+
+
+<p class="tabcap">TABLE 2</p>
+
+<p class="center"><span class="smcap">Estimates of the Amount of Practice Provided in Books I and II of
+the Average Three-Book Text in Arithmetic; by 50 Experienced
+Teachers</span></p>
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="1" cellspacing="1" summary="">
+<colgroup><col width="30%" /><col width="15%" /><col width="15%" /><col width="15%" /><col width="25%" /></colgroup>
+<tr><th class="smcap bbt">Arithmetical<br />Fact</th><th class="smcap bbt">Lowest<br />Estimate</th>
+<th class="smcap bbt">Median<br />Estimate</th><th class="smcap bbt">Highest<br />Estimate</th>
+<th class="smcap bbt">Range Required to<br />Include Half of<br />the Estimates</th></tr>
+<tr><td align='right'>3 or 13 or 23, etc. &#43; 2</td><td align='right'> 25</td><td align='right'> 1500</td><td align='right'>1,000,000</td><td align='right'> 800-5000</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 3</td><td align='right'> 24</td><td align='right'> 1450</td><td align='right'> 80,000</td><td align='right'> 475-5000</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 4</td><td align='right'> 23</td><td align='right'> 1150</td><td align='right'> 50,000</td><td align='right'> 750-5000</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 5</td><td align='right'> 22</td><td align='right'> 1400</td><td align='right'> 44,000</td><td align='right'> 700-5000</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 6</td><td align='right'> 21</td><td align='right'> 1350</td><td align='right'> 41,000</td><td align='right'> 700-4500</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 7</td><td align='right'> 21</td><td align='right'> 1500</td><td align='right'> 37,000</td><td align='right'> 600-4000</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 8</td><td align='right'> 20</td><td align='right'> 1400</td><td align='right'> 33,000</td><td align='right'> 550-4100</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 9</td><td align='right'> 20</td><td align='right'> 1150</td><td align='right'> 28,000</td><td align='right'> 650-4500</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='right'>7 or 17 or 27, etc. &#43; 2</td><td align='right'> 20</td><td align='right'> 1250</td><td align='right'>2,000,000</td><td align='right'> 600-5000</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 3</td><td align='right'> 19</td><td align='right'> 1100</td><td align='right'>1,000,000</td><td align='right'> 650-4900</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 4</td><td align='right'> 18</td><td align='right'> 1000</td><td align='right'> 80,000</td><td align='right'> 650-4900</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 5</td><td align='right'> 17</td><td align='right'> 1300</td><td align='right'> 80,000</td><td align='right'> 650-4400</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 6</td><td align='right'> 16</td><td align='right'> 1100</td><td align='right'> 29,000</td><td align='right'> 650-4500</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 7</td><td align='right'> 15</td><td align='right'> 1100</td><td align='right'> 25,000</td><td align='right'> 500-4500</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 8</td><td align='right'> 13</td><td align='right'> 1100</td><td align='right'> 21,000</td><td align='right'> 650-3800</td></tr>
+<tr><td align='right'>"  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  "  &nbsp; &nbsp; &nbsp; 9</td><td align='right'> 10</td><td align='right'> 1275</td><td align='right'> 17,000</td><td align='right'> 500-4000</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='right'> 3 &minus; 3</td><td align='right'> 25</td><td align='right'> 1000</td><td align='right'> 100,000</td><td align='right'> 500-4000</td></tr>
+<tr><td align='right'> 4 &minus; 3</td><td align='right'> 20</td><td align='right'> 1050</td><td align='right'> 500,000</td><td align='right'> 525-3000</td></tr>
+<tr><td align='right'> 5 &minus; 3</td><td align='right'> 20</td><td align='right'> 1100</td><td align='right'>2,500,000</td><td align='right'> 650-4200</td></tr>
+<tr><td align='right'> 6 &minus; 3</td><td align='right'> 10</td><td align='right'> 1050</td><td align='right'> 21,000</td><td align='right'> 650-3250</td></tr>
+<tr><td align='right'> 7 &minus; 3</td><td align='right'> 22</td><td align='right'> 1100</td><td align='right'> 15,000</td><td align='right'> 550-3050</td></tr>
+<tr><td align='right'> 8 &minus; 3</td><td align='right'> 21</td><td align='right'> 1075</td><td align='right'> 15,000</td><td align='right'> 650-3000</td></tr>
+<tr><td align='right'> 9 &minus; 3</td><td align='right'> 21</td><td align='right'> 1000</td><td align='right'> 15,000</td><td align='right'> 700-2600</td></tr>
+<tr><td align='right'> 10 &minus; 3</td><td align='right'> 20</td><td align='right'> 1000</td><td align='right'> 20,000</td><td align='right'> 600-2500</td></tr>
+<tr><td align='right'> 11 &minus; 3</td><td align='right'> 20</td><td align='right'> 1000</td><td align='right'> 15,000</td><td align='right'> 465-2550</td></tr>
+<tr><td align='right'> 12 &minus; 3</td><td align='right'> 18</td><td align='right'> 1000</td><td align='right'> 15,000</td><td align='right'> 650-2100</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='right'> 7 &minus; 7</td><td align='right'> 10</td><td align='right'> 1000</td><td align='right'> 18,000</td><td align='right'> 425-3000</td></tr>
+<tr><td align='right'> 8 &minus; 7</td><td align='right'> 15</td><td align='right'> 1000</td><td align='right'> 18,000</td><td align='right'> 413-3100</td></tr>
+<tr><td align='right'> 9 &minus; 7</td><td align='right'> 15</td><td align='right'> 950</td><td align='right'> 18,000</td><td align='right'> 550-3000</td></tr>
+<tr><td align='right'> 10 &minus; 7</td><td align='right'> 15</td><td align='right'> 950</td><td align='right'> 18,000</td><td align='right'> 600-3950</td></tr>
+<tr><td align='right'> 11 &minus; 7</td><td align='right'> 10</td><td align='right'> 900</td><td align='right'> 18,000</td><td align='right'> 550-3000</td></tr>
+<tr><td align='right'> 12 &minus; 7</td><td align='right'> 10</td><td align='right'> 925</td><td align='right'> 18,000</td><td align='right'> 525-3100</td></tr>
+<tr><td align='right'> 13 &minus; 7</td><td align='right'> 10</td><td align='right'> 900</td><td align='right'> 18,000</td><td align='right'> 500-2600</td></tr>
+<tr><td align='right'> 14 &minus; 7</td><td align='right'> 10</td><td align='right'> 900</td><td align='right'> 18,000</td><td align='right'> 500-3100</td></tr>
+<tr><td align='right'> 15 &minus; 7</td><td align='right'> 10</td><td align='right'> 925</td><td align='right'> 18,000</td><td align='right'> 500-3000</td></tr>
+<tr><td align='right'> 16 &minus; 7</td><td align='right'> 10</td><td align='right'> 875</td><td align='right'> 18,000</td><td align='right'> 500-2500</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='right'> 9 &times; 7</td><td align='right'> 10</td><td align='right'> 700</td><td align='right'> 20,000</td><td align='right'> 500-2000</td></tr>
+<tr><td align='right'> 7 &times; 9</td><td align='right'> 10</td><td align='right'> 700</td><td align='right'> 20,000</td><td align='right'> 500-1750</td></tr>
+<tr><td align='right'> 8 &times; 6</td><td align='right'> 10</td><td align='right'> 750</td><td align='right'> 20,000</td><td align='right'> 500-2500</td></tr>
+<tr><td align='right'> 6 &times; 8</td><td align='right'> 9</td><td align='right'> 700</td><td align='right'> 20,000</td><td align='right'> 500-2500</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='right'> 63 &divide; 9</td><td align='right'> 9</td><td align='right'> 500</td><td align='right'> 4,500</td><td align='right'> 300-2500</td></tr>
+<tr><td align='right'> 64 &divide; 9</td><td align='right'> 9</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 100- 700</td></tr>
+<tr><td align='right'> 65 &divide; 9</td><td align='right'> 8</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 100- 600</td></tr>
+<tr><td align='right'> 66 &divide; 9</td><td align='right'> 7</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 100- 550</td></tr>
+<tr><td align='right'> 67 &divide; 9</td><td align='right'> 7</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 75- 450</td></tr>
+<tr><td align='right'> 68 &divide; 9</td><td align='right'> 6</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 87- 575</td></tr>
+<tr><td align='right'> 69 &divide; 9</td><td align='right'> 6</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 87- 450</td></tr>
+<tr><td align='right'> 70 &divide; 9</td><td align='right'> 5</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 75- 575</td></tr>
+<tr><td align='right'> 71 &divide; 9</td><td align='right'> 5</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 75- 700</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='right'> <i>XX</i></td><td align='right'> 40</td><td align='right'> 550</td><td align='right'>1,000,000</td><td align='right'> 300-2000</td></tr>
+<tr><td align='right'> <i>XO</i></td><td align='right'> 20</td><td align='right'> 500</td><td align='right'> 11,500</td><td align='right'> 150-2000</td></tr>
+<tr><td align='right'> <i>XXX</i></td><td align='right'> 15</td><td align='right'> 450</td><td align='right'> 12,000</td><td align='right'> 100-1000</td></tr>
+<tr><td align='right'> <i>XXO</i></td><td align='right'> 25</td><td align='right'> 400</td><td align='right'> 15,000</td><td align='right'> 150-1000</td></tr>
+<tr><td align='right'> <i>XOO</i></td><td align='right'> 15</td><td align='right'> 400</td><td align='right'> 5,000</td><td align='right'> 100-1000</td></tr>
+<tr><td align='right'> <i>XOX</i></td><td align='right'> 10</td><td align='right'> 400</td><td align='right'> 10,000</td><td align='right'> 100- 975</td></tr>
+</table></div>
+
+<p>Having made his estimates the reader should compare
+them first with similar estimates made by experienced
+teachers (shown on page 124 f.), and then with the results of
+actual counts for representative textbooks in arithmetic
+(shown on pages 126 to 132).</p>
+
+<p>It will be observed in Table 2 that even experienced
+teachers vary enormously in their estimates of the amount
+<span class='pagenum'><a name="Page_126" id="Page_126">[Pg 126]</a></span>
+of practice given by an average textbook in arithmetic,
+and that most of them are in serious error by overestimating
+the amount of practice. In general it is the fact that
+we use textbooks in arithmetic with very vague and erroneous
+ideas of what is in them, and think they give much more
+practice than they do.</p>
+
+<p>The authors of the textbooks as a rule also probably had
+only very vague and erroneous ideas of what was in them.
+If they had known, they would almost certainly have revised
+their books. Surely no author would intentionally
+provide nearly four times as much practice on 2 &#43; 2 as on
+8 &#43; 8, or eight times as much practice on 2 &times; 2 as on 9 &times; 8,
+or eleven times as much practice on 2 &minus; 2 as on 17 &minus; 8, or
+over forty times as much practice on 2 &divide; 2 as on 75 &divide; 8 and
+75 &divide; 9, both together. Surely no author would have provided
+intentionally only twenty to thirty occurrences each
+of 16 &minus; 7, 16 &minus; 8, 16 &minus; 9, 17 &minus; 8, 17 &minus; 9, and 18 &minus; 9 for the
+entire course through grade 6; or have left the practice
+on 60 &divide; 7, 60 &divide; 8, 60 &divide; 9, 61 &divide; 7, 61 &divide; 8, 61 &divide; 9, and the like
+to occur only about once a year!</p>
+
+
+<p class="tabcap">TABLE 3</p>
+
+<p class="nblockquot"><span class="smcap">Amount of Practice: Addition Bonds in a Recent Textbook (A) of
+Excellent Repute. Books I and II, All Save Four Sections of
+Supplementary Material, to be Used at the Teacher's Discretion</span></p>
+
+<p><small>The Table reads: 2 &#43; 2 was used 226 times, 12 &#43; 2 was used 74 times,
+22 &#43; 2, 32 &#43; 2, 42 &#43; 2, and so on were used 50 times.</small></p>
+
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary="">
+<tr><th class="bbt"></th><th class="bbt">&nbsp; <b>2</b>&nbsp;</th><th class="bbt">&nbsp; <b>3</b>&nbsp;</th><th class="bbt">&nbsp; <b>4</b>&nbsp;</th><th class="bbt">&nbsp; <b>5</b>&nbsp;</th><th class="bbt">&nbsp; <b>6</b>&nbsp;</th><th class="bbt">&nbsp; <b>7</b>&nbsp;</th><th class="bbt">&nbsp; <b>8</b>&nbsp;</th><th class="bbt">&nbsp; <b>9</b></th><th class="smcap bbt"><b>Total</b></th></tr>
+<tr><td align='left'>&nbsp;&nbsp;<b>2</b></td><td align='right'> 226</td><td align='right'> 154</td><td align='right'> 162</td><td align='right'> 150</td><td align='right'> 97</td><td align='right'> 87</td><td align='right'> 66</td><td align='right'> 45</td><td></td></tr>
+<tr><td align='left'><b>12</b></td><td align='right'> 74</td><td align='right'> 53</td><td align='right'> 76</td><td align='right'> 46</td><td align='right'> 51</td><td align='right'> 37</td><td align='right'> 36</td><td align='right'> 33</td><td></td></tr>
+<tr><td align='left'><b>22</b>, etc.</td><td align='right'> 50</td><td align='right'> 60</td><td align='right'> 68</td><td align='right'> 63</td><td align='right'> 42</td><td align='right'> 50</td><td align='right'> 38</td><td align='right'> 26</td><td></td></tr>
+<tr><td align='left'><span class='pagenum'><a name="Page_127" id="Page_127">[Pg 127]</a></span>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'>&nbsp;&nbsp;<b>3</b></td><td align='right'> 216</td><td align='right'> 141</td><td align='right'> 127</td><td align='right'> 89</td><td align='right'> 82</td><td align='right'> 54</td><td align='right'> 58</td><td align='right'> 40</td><td></td></tr>
+<tr><td align='left'><b>13</b></td><td align='right'> 43</td><td align='right'> 43</td><td align='right'> 60</td><td align='right'> 70</td><td align='right'> 52</td><td align='right'> 30</td><td align='right'> 22</td><td align='right'> 18</td><td></td></tr>
+<tr><td align='left'><b>23</b>, etc.</td><td align='right'> 15</td><td align='right'> 30</td><td align='right'> 51</td><td align='right'> 50</td><td align='right'> 42</td><td align='right'> 32</td><td align='right'> 29</td><td align='right'> 30</td><td></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'>&nbsp;&nbsp;<b>7</b></td><td align='right'> 85</td><td align='right'> 90</td><td align='right'> 103</td><td align='right'> 103</td><td align='right'> 84</td><td align='right'> 81</td><td align='right'> 61</td><td align='right'> 47</td><td></td></tr>
+<tr><td align='left'><b>17</b></td><td align='right'> 35</td><td align='right'> 25</td><td align='right'> 42</td><td align='right'> 32</td><td align='right'> 35</td><td align='right'> 21</td><td align='right'> 29</td><td align='right'> 16</td><td></td></tr>
+<tr><td align='left'><b>27</b>, etc.</td><td align='right'> 30</td><td align='right'> 23</td><td align='right'> 32</td><td align='right'> 29</td><td align='right'> 24</td><td align='right'> 23</td><td align='right'> 25</td><td align='right'> 28</td><td></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'>&nbsp;&nbsp;<b>8</b></td><td align='right'> 185</td><td align='right'> 112</td><td align='right'> 146</td><td align='right'> 99</td><td align='right'> 75</td><td align='right'> 71</td><td align='right'> 73</td><td align='right'> 61</td><td></td></tr>
+<tr><td align='left'><b>18</b></td><td align='right'> 28</td><td align='right'> 35</td><td align='right'> 52</td><td align='right'> 46</td><td align='right'> 28</td><td align='right'> 29</td><td align='right'> 24</td><td align='right'> 14</td><td></td></tr>
+<tr><td align='left'><b>28</b>, etc.</td><td align='right'> 53</td><td align='right'> 36</td><td align='right'> 34</td><td align='right'> 38</td><td align='right'> 23</td><td align='right'> 36</td><td align='right'> 27</td><td align='right'> 27</td><td></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'>&nbsp;&nbsp;<b>9</b></td><td align='right'> 104</td><td align='right'> 81</td><td align='right'> 112</td><td align='right'> 96</td><td align='right'> 63</td><td align='right'> 74</td><td align='right'> 58</td><td align='right'> 57</td><td></td></tr>
+<tr><td align='left'><b>19</b></td><td align='right'> 13</td><td align='right'> 11</td><td align='right'> 31</td><td align='right'> 38</td><td align='right'> 25</td><td align='right'> 14</td><td align='right'> 22</td><td align='right'> 11</td><td></td></tr>
+<tr><td align='left'><b>29</b>, etc.</td><td align='right'> 19</td><td align='right'> 17</td><td align='right'> 27</td><td align='right'> 20</td><td align='right'> 32</td><td align='right'> 32</td><td align='right'> 19</td><td align='right'> 18</td><td></td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>2, 12, 22</b>, etc.</td><td align='right'> 350</td><td align='right'> 277</td><td align='right'> 306</td><td align='right'> 260</td><td align='right'> 190</td><td align='right'> 174</td><td align='right'> 140</td><td align='right'> 104</td><td align='right'>1801</td></tr>
+<tr><td align='left'><b>3, 13, 23</b>, etc.</td><td align='right'> 274</td><td align='right'> 214</td><td align='right'> 230</td><td align='right'> 209</td><td align='right'> 176</td><td align='right'> 116</td><td align='right'> 109</td><td align='right'> 88</td><td align='right'>1406</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'><b>7, 17, 27</b>, etc.</td><td align='right'> 148</td><td align='right'> 138</td><td align='right'> 187</td><td align='right'> 164</td><td align='right'> 141</td><td align='right'> 125</td><td align='right'> 115</td><td align='right'> 91</td><td align='right'>1109</td></tr>
+<tr><td align='left'><b>8, 18, 28</b>, etc.</td><td align='right'> 266</td><td align='right'> 183</td><td align='right'> 232</td><td align='right'> 185</td><td align='right'> 126</td><td align='right'> 136</td><td align='right'> 124</td><td align='right'> 102</td><td align='right'>1354</td></tr>
+<tr><td align='left'><b>9, 19, 29</b>, etc.</td><td align='right'> 136</td><td align='right'> 109</td><td align='right'> 170</td><td align='right'> 154</td><td align='right'> 120</td><td align='right'> 120</td><td align='right'> 99</td><td align='right'> 86</td><td align='right'> 994</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='center'>Totals</td><td align='right'>1164</td><td align='right'> 921</td><td align='right'>1125</td><td align='right'> 972</td><td align='right'> 753</td><td align='right'> 671</td><td align='right'> 687</td><td align='right'> 471</td><td></td></tr>
+</table></div>
+
+<p>&nbsp;</p>
+<p class="tabcap">TABLE 4</p>
+
+<p class="nblockquot"><span class="smcap">Amount of Practice: Subtraction Bonds in a Recent Textbook (A)
+of Excellent Repute. Books I and II, All Save Four Sections of
+Supplementary Material, to be Used at the Teacher's Discretion</span></p>
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary="">
+<tr><th class="bbt" rowspan='2'><span class="smcap">Minuends</span></th><th class="bbt" colspan='9'><span class="smcap">Subtrahends</span></th></tr>
+<tr><th class="bbt">&nbsp; <b>1</b>&nbsp;</th><th class="bbt">&nbsp; <b>2</b>&nbsp;</th><th class="bbt">&nbsp; <b>3</b>&nbsp;</th><th class="bbt">&nbsp; <b>4</b>&nbsp;</th><th class="bbt">&nbsp; <b>5</b>&nbsp;</th><th class="bbt">&nbsp; <b>6</b>&nbsp;</th><th class="bbt">&nbsp; <b>7</b>&nbsp;</th><th class="bbt">&nbsp; <b>8</b>&nbsp;</th><th class="bbt">&nbsp; <b>9</b></th></tr>
+<tr><td align='right'><b>1</b></td><td align='right'> 372</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><b>2</b></td><td align='right'> 214</td><td align='right'> 311</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><b>3</b></td><td align='right'> 136</td><td align='right'> 149</td><td align='right'> 189</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><b>4</b></td><td align='right'> 146</td><td align='right'> 142</td><td align='right'> 103</td><td align='right'> 205</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><b>5</b></td><td align='right'> 171</td><td align='right'> 91</td><td align='right'> 92</td><td align='right'> 164</td><td align='right'> 136</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><span class='pagenum'><a name="Page_128" id="Page_128">[Pg 128]</a></span><b>6</b></td><td align='right'> 80</td><td align='right'> 59</td><td align='right'> 69</td><td align='right'> 71</td><td align='right'> 81</td><td align='right'> 192</td><td align='right'></td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><b>7</b></td><td align='right'> 106</td><td align='right'> 57</td><td align='right'> 55</td><td align='right'> 67</td><td align='right'> 59</td><td align='right'> 156</td><td align='right'> 80</td><td align='right'></td><td align='right'></td></tr>
+<tr><td align='right'><b>8</b></td><td align='right'> 73</td><td align='right'> 50</td><td align='right'> 50</td><td align='right'> 75</td><td align='right'> 50</td><td align='right'> 62</td><td align='right'> 48</td><td align='right'> 152</td><td align='right'></td></tr>
+<tr><td align='right'><b>9</b></td><td align='right'> 71</td><td align='right'> 75</td><td align='right'> 54</td><td align='right'> 74</td><td align='right'> 48</td><td align='right'> 55</td><td align='right'> 55</td><td align='right'> 124</td><td align='right'> 133</td></tr>
+<tr><td align='right'><b>10</b></td><td align='right'> 261</td><td align='right'> 84</td><td align='right'> 63</td><td align='right'> 100</td><td align='right'> 193</td><td align='right'> 83</td><td align='right'> 57</td><td align='right'> 124</td><td align='right'> 91</td></tr>
+<tr><td align='right'></td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'></td></tr>
+<tr><td align='right'><b>11</b></td><td align='right'></td><td align='right'> 48</td><td align='right'> 31</td><td align='right'> 50</td><td align='right'> 36</td><td align='right'> 41</td><td align='right'> 32</td><td align='right'> 46</td><td align='right'> 35</td></tr>
+<tr><td align='right'><b>12</b></td><td align='right'></td><td align='right'></td><td align='right'> 48</td><td align='right'> 77</td><td align='right'> 57</td><td align='right'> 51</td><td align='right'> 35</td><td align='right'> 80</td><td align='right'> 30</td></tr>
+<tr><td align='right'><b>13</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 35</td><td align='right'> 22</td><td align='right'> 40</td><td align='right'> 29</td><td align='right'> 35</td><td align='right'> 28</td></tr>
+<tr><td align='right'><b>14</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 25</td><td align='right'> 37</td><td align='right'> 36</td><td align='right'> 49</td><td align='right'> 32</td></tr>
+<tr><td align='right'><b>15</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 33</td><td align='right'> 19</td><td align='right'> 48</td><td align='right'> 20</td></tr>
+<tr><td align='right'></td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'></td></tr>
+<tr><td align='right'><b>16</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 16</td><td align='right'> 36</td><td align='right'> 26</td></tr>
+<tr><td align='right'><b>17</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 27</td><td align='right'> 20</td></tr>
+<tr><td align='right'><b>18</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 19</td></tr>
+<tr><td align='right'></td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'>&nbsp; &nbsp;</td><td align='right'></td></tr>
+<tr><td align='right'>Total excluding<br /><b>1 &minus; 1</b>, <b>2 &minus; 2,</b> etc.</td><td align='right'>1258</td><td align='right'> 755</td><td align='right'> 565</td><td align='right'> 713</td><td align='right'> 571</td><td align='right'> 558</td><td align='right'> 327</td><td align='right'> 569</td><td align='right'> 301</td></tr>
+</table></div>
+
+<p>&nbsp;</p>
+<p class="tabcap">TABLE 5</p>
+
+<p class="center"><span class="smcap">Frequencies of Subtractions not Included in Table 4</span></p>
+
+<p><small>These are cases where the pupil would, by reason of his stage of advancement,
+probably operate 35 &minus; 30, 46 &minus; 46, etc., as one bond.</small></p>
+
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary="">
+<tr><th class="bbt" rowspan='2'><span class="smcap">Minuends</span></th><th class="bbt" colspan='10'><span class="smcap">Subtrahends</span></th></tr>
+<tr><td class="bbt">&nbsp;&nbsp;<b>1</b><br /><b>11</b><br /><b>21</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>2</b><br /><b>12</b><br /><b>22</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>3</b><br /><b>13</b><br /><b>23</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>4</b><br /><b>14</b><br /><b>24</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>5</b><br /><b>15</b><br /><b>25</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>6</b><br /><b>16</b><br /><b>26</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>7</b><br /><b>17</b><br /><b>27</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>8</b><br /><b>18</b><br /><b>28</b><br />etc.</td>
+<td class="bbt">&nbsp;&nbsp;<b>9</b><br /><b>19</b><br /><b>29</b><br />etc.</td>
+<td class="bbt">&nbsp;<br /><b>10</b><br /><b>20</b><br />etc.</td></tr>
+<tr><td align='right'><b>10</b>, <b>20</b>, <b>30</b>, <b>40</b>, etc.</td><td align='right'> 11</td><td align='right'> 29</td><td align='right'> 16</td><td align='right'> 52</td><td align='right'> 32</td><td align='right'> 51</td><td align='right'> 7</td><td align='right'> 30</td><td align='right'> 22</td><td align='right'> 60</td></tr>
+<tr><td align='right'><b>11</b>, <b>21</b>, <b>31</b>, <b>41</b>, etc.</td><td align='right'> 42</td><td align='right'> 14</td><td align='right'> 22</td><td align='right'> 32</td><td align='right'> 12</td><td align='right'> 26</td><td align='right'> 19</td><td align='right'> 52</td><td align='right'> 17</td><td align='right'> 10</td></tr>
+<tr><td align='right'><b>12</b>, <b>22</b>, <b>32</b>, <b>42</b>, etc.</td><td align='right'> 47</td><td align='right'> 97</td><td align='right'> 5</td><td align='right'> 13</td><td align='right'> 9</td><td align='right'> 21</td><td align='right'> 11</td><td align='right'> 24</td><td align='right'> 19</td><td align='right'> 17</td></tr>
+<tr><td align='right'><b>13</b>, <b>23</b>, <b>33</b>, <b>43</b>, etc.</td><td align='right'> 7</td><td align='right'> 40</td><td align='right'> 7</td><td align='right'> 14</td><td align='right'> 15</td><td align='right'> 13</td><td align='right'> 19</td><td align='right'> 19</td><td align='right'> 22</td><td align='right'> 3</td></tr>
+<tr><td align='right'><b>14</b>, <b>24</b>, <b>34</b>, <b>44</b>, etc.</td><td align='right'> 8</td><td align='right'> 28</td><td align='right'> 14</td><td align='right'> 58</td><td align='right'> 13</td><td align='right'> 16</td><td align='right'> 14</td><td align='right'> 26</td><td align='right'> 19</td><td align='right'> 7<span class='pagenum'><a name="Page_129" id="Page_129">[Pg 129]</a></span></td></tr>
+<tr><td align='right'><b>15, 25, 35, 45,</b> etc.</td><td align='right'> 21</td><td align='right'> 28</td><td align='right'> 29</td><td align='right'> 54</td><td align='right'> 51</td><td align='right'> 15</td><td align='right'> 21</td><td align='right'> 12</td><td align='right'> 24</td><td align='right'> 8</td></tr>
+<tr><td align='right'><b>16, 26, 36, 46,</b> etc.</td><td align='right'> 5</td><td align='right'> 18</td><td align='right'> 12</td><td align='right'> 27</td><td align='right'> 35</td><td align='right'> 69</td><td align='right'> 13</td><td align='right'> 17</td><td align='right'> 19</td><td align='right'> 2</td></tr>
+<tr><td align='right'><b>17, 27, 37, 47,</b> etc.</td><td align='right'> 5</td><td align='right'> 9</td><td align='right'> 12</td><td align='right'> 40</td><td align='right'> 32</td><td align='right'> 54</td><td align='right'> 24</td><td align='right'> 12</td><td align='right'> 12</td><td align='right'> 1</td></tr>
+<tr><td align='right'><b>18, 28, 38, 48,</b> etc.</td><td align='right'> 2</td><td align='right'> 16</td><td align='right'> 10</td><td align='right'> 23</td><td align='right'> 22</td><td align='right'> 36</td><td align='right'> 18</td><td align='right'> 47</td><td align='right'> 16</td><td align='right'> 0</td></tr>
+<tr><td align='left'><b>19, 29, 39,</b> etc.</td><td align='right'> 5</td><td align='right'> 7</td><td align='right'> 7</td><td align='right'> 10</td><td align='right'> 13</td><td align='right'> 28</td><td align='right'> 14</td><td align='right'> 23</td><td align='right'> 16</td><td align='right'> 0</td></tr>
+<tr><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='left'>Totals</td><td align='right'>153</td><td align='right'>286</td><td align='right'>134</td><td align='right'>323</td><td align='right'>234</td><td align='right'>329</td><td align='right'>160</td><td align='right'>262</td><td align='right'>186</td><td align='right'>108</td></tr>
+</table></div>
+
+<p>&nbsp;</p>
+<p class="tabcap">TABLE 6</p>
+
+<p class="center"><span class="smcap">Amount of Practice: Multiplication Bonds in Another Recent
+Textbook (B) of Excellent Repute. Books I and II</span></p>
+
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary="">
+<tr><th class="bbt" rowspan='2'><span class="smcap">Multipliers</span></th><th class="bbt" colspan='11'><span class="smcap">Multiplicands</span></th></tr>
+<tr><th class="bbt">&nbsp; <b>0</b>&nbsp;</th><th class="bbt">&nbsp; <b>1</b>&nbsp;</th><th class="bbt">&nbsp; <b>2</b>&nbsp;</th><th class="bbt">&nbsp; <b>3</b>&nbsp;</th><th class="bbt">&nbsp; <b>4</b>&nbsp;</th><th class="bbt">&nbsp; <b>5</b>&nbsp;</th><th class="bbt">&nbsp; <b>6</b>&nbsp;</th><th class="bbt">&nbsp; <b>7</b>&nbsp;</th><th class="bbt">&nbsp; <b>8</b>&nbsp;</th><th class="bbt">&nbsp; <b>9</b></th><th class="bbt">Totals</th></tr>
+<tr><td align='center'><b>1</b></td><td align='right'> 299</td><td align='right'> 534</td><td align='right'> 472</td><td align='right'> 271</td><td align='right'> 310</td><td align='right'> 293</td><td align='right'> 261</td><td align='right'> 178</td><td align='right'> 195</td><td align='right'> 99</td><td align='right'> 2912</td></tr>
+<tr><td align='center'><b>2</b></td><td align='right'> 350</td><td align='right'> 644</td><td align='right'> 668</td><td align='right'> 480</td><td align='right'> 458</td><td align='right'> 377</td><td align='right'> 332</td><td align='right'> 238</td><td align='right'> 239</td><td align='right'> 155</td><td align='right'> 3941</td></tr>
+<tr><td align='center'><b>3</b></td><td align='right'> 280</td><td align='right'> 487</td><td align='right'> 509</td><td align='right'> 388</td><td align='right'> 318</td><td align='right'> 302</td><td align='right'> 247</td><td align='right'> 199</td><td align='right'> 227</td><td align='right'> 152</td><td align='right'> 3109</td></tr>
+<tr><td align='center'><b>4</b></td><td align='right'> 186</td><td align='right'> 375</td><td align='right'> 398</td><td align='right'> 242</td><td align='right'> 203</td><td align='right'> 265</td><td align='right'> 197</td><td align='right'> 163</td><td align='right'> 159</td><td align='right'> 93</td><td align='right'> 2281</td></tr>
+<tr><td align='center'><b>5</b></td><td align='right'> 268</td><td align='right'> 359</td><td align='right'> 393</td><td align='right'> 234</td><td align='right'> 263</td><td align='right'> 243</td><td align='right'> 217</td><td align='right'> 192</td><td align='right'> 197</td><td align='right'> 114</td><td align='right'> 2480</td></tr>
+<tr><td align='center'><b>6</b></td><td align='right'> 180</td><td align='right'> 284</td><td align='right'> 265</td><td align='right'> 199</td><td align='right'> 196</td><td align='right'> 191</td><td align='right'> 168</td><td align='right'> 169</td><td align='right'> 165</td><td align='right'> 106</td><td align='right'> 1923</td></tr>
+<tr><td align='center'><b>7</b></td><td align='right'> 135</td><td align='right'> 283</td><td align='right'> 277</td><td align='right'> 176</td><td align='right'> 187</td><td align='right'> 158</td><td align='right'> 155</td><td align='right'> 121</td><td align='right'> 145</td><td align='right'> 118</td><td align='right'> 1755</td></tr>
+<tr><td align='center'><b>8</b></td><td align='right'> 137</td><td align='right'> 272</td><td align='right'> 292</td><td align='right'> 175</td><td align='right'> 192</td><td align='right'> 164</td><td align='right'> 158</td><td align='right'> 157</td><td align='right'> 126</td><td align='right'> 126</td><td align='right'> 1799</td></tr>
+<tr><td align='center'><b>9</b></td><td align='right'> 71</td><td align='right'> 173</td><td align='right'> 140</td><td align='right'> 122</td><td align='right'> 97</td><td align='right'> 102</td><td align='right'> 101</td><td align='right'> 100</td><td align='right'> 82</td><td align='right'> 110</td><td align='right'> 1098</td></tr>
+<tr><td align='center'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td><td align='left'>&nbsp;</td></tr>
+<tr><td align='center'>Totals</td><td align='right'>1906</td><td align='right'>3411</td><td align='right'>3414</td><td align='right'>2287</td><td align='right'>2224</td><td align='right'>2095</td><td align='right'>1836</td><td align='right'>1517</td><td align='right'>1535</td><td align='right'>1073</td><td align='left'>&nbsp;</td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_130" id="Page_130">[Pg 130]</a></span></p>
+
+<p>&nbsp;</p>
+<p class="tabcap">TABLE 7</p>
+<p class="center"><span class="smcap">Amount Of Practice: Divisions Without Remainder In Textbook B,
+Parts I And II</span></p>
+
+<div class='center'>
+<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary="">
+<tr><th class="bbt" rowspan='2'><span class="smcap">Dividends</span></th><th class="bbt" colspan='11'><span class="smcap">Divisors</span></th></tr>
+<tr><th class="bbt">&nbsp; <b>2</b>&nbsp;</th><th class="bbt">&nbsp; <b>3</b>&nbsp;</th><th class="bbt">&nbsp; <b>4</b>&nbsp;</th><th class="bbt">&nbsp; <b>5</b>&nbsp;</th><th class="bbt">&nbsp; <b>6</b>&nbsp;</th><th class="bbt">&nbsp; <b>7</b>&nbsp;</th><th class="bbt">&nbsp; <b>8</b>&nbsp;</th><th class="bbt">&nbsp; <b>9</b></th><th class="bbt">Totals</th></tr>
+<tr><td align='left' rowspan='6'>Integral multiples of <b>2</b> to <b>9</b><br />
+in sequence; <i>i.e.</i>,<br />
+<b>4 &divide; 2</b> occurred <b>397</b> times,<br />
+<b>6 &divide; 2</b> occurred <b>256</b> times,<br />
+<b>6 &divide; 3, 224</b> times,<br />
+<b>9 &divide; 3, 124</b> times.</td>
+<td align='right'> 397</td><td align='right'> 224</td><td align='right'> 250</td><td align='right'> 130</td><td align='right'> 93</td><td align='right'> 44</td><td align='right'> 98</td><td align='right'> 23</td><td align='right'> 1259</td></tr>
+<tr><td align='right'> 256</td><td align='right'> 124</td><td align='right'> 152</td><td align='right'> 79</td><td align='right'> 28</td><td align='right'> 43</td><td align='right'> 61</td><td align='right'> 25</td><td align='right'> 768</td></tr>
+<tr><td align='right'> 318</td><td align='right'> 123</td><td align='right'> 130</td><td align='right'> 65</td><td align='right'> 50</td><td align='right'> 19</td><td align='right'> 39</td><td align='right'> 19</td><td align='right'> 763</td></tr>
+<tr><td align='right'> 258</td><td align='right'> 98</td><td align='right'> 86</td><td align='right'> 105</td><td align='right'> 25</td><td align='right'> 24</td><td align='right'> 34</td><td align='right'> 20</td><td align='right'> 650</td></tr>
+<tr><td align='right'> 198</td><td align='right'> 49</td><td align='right'> 76</td><td align='right'> 27</td><td align='right'> 22</td><td align='right'> 30</td><td align='right'> 33</td><td align='right'> 16</td><td align='right'> 451</td></tr>
+<tr><td align='right'>77<br />180<br />69</td><td align='right'> 54<br />91<br />46</td><td align='right'> 36<br />50<br />37</td><td align='right'> 31<br />38<br />24</td><td align='right'> 28<br />17<br />12</td><td align='right'> 27<br />13<br />17</td><td align='right'> 16<br />22<br />16</td><td align='right'> 9<br />16<br />15</td><td align='right'> 278<br />427<br />236</td></tr>
+<tr><td align='left'>Totals</td><td align='right'>1753</td><td align='right'> 809</td><td align='right'> 817</td><td align='right'> 499</td><td align='right'> 275</td><td align='right'> 217</td><td align='right'> 319</td><td align='right'> 142</td><td align='right'> </td></tr>
+</table></div>
+
+<p>&nbsp;</p>
+<p class="tabcap">TABLE 8</p>
+<p class="center"><span class="smcap">Division Bonds, With And Without Remainders.&nbsp; Book B</span></p>
+
+
+
+<div class='pblockquot'>
+<p>All work through grade 6, except estimates of quotient figures in long division.</p>
+
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'>Dividend</td><td align='right'>2</td><td align='right' colspan='2'>3</td><td align='right' colspan='3'>4</td><td align='right' colspan='4'>5</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>1</td><td align='right'>2</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>41</td><td align='right'>386</td><td align='right'>27</td><td align='right'>189</td><td align='right'>240</td><td align='right'>26</td><td align='right'>397</td><td align='right'>66</td><td align='right'>185</td><td align='right'>23</td><td align='right'>136</td><td align='right'>43</td><td align='right'>53</td><td align='right'>135</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>6</td><td align='right' colspan='6'>7</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>21</td><td align='right'>256</td><td align='right'>224</td><td align='right'>68</td><td align='right'>43</td><td align='right'>83</td><td align='right'>23</td><td align='right'>72</td><td align='right'>55</td><td align='right'>38</td><td align='right'>46</td><td align='right'>32</td><td align='right'>54</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>8</td><td align='right' colspan='8'>9</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>17</td><td align='right'>318</td><td align='right'>30</td><td align='right'>250</td><td align='right'>22</td><td align='right'>28</td><td align='right'>39</td><td align='right'>91</td><td align='right'>19</td><td align='right'>50</td><td align='right'>124</td><td align='right'>49</td><td align='right'>25</td><td align='right'>15</td><td align='right'>18</td><td align='right'>30</td><td align='right'>38</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>10</td><td align='right' colspan='8'>11</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>258</td><td align='right'>38</td><td align='right'>46</td><td align='right'>120</td><td align='right'>19</td><td align='right'>9</td><td align='right'>24</td><td align='right'>24</td><td align='right'>32</td><td align='right'>21</td><td align='right'>16</td><td align='right'>3</td><td align='right'>7</td><td align='right'>11</td><td align='right'>14</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>12</td><td align='right' colspan='8'>13</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>198</td><td align='right'>123</td><td align='right'>152</td><td align='right'>29</td><td align='right'>93</td><td align='right'>9</td><td align='right'>16</td><td align='right'>7</td><td align='right'>45</td><td align='right'>16</td><td align='right'>15</td><td align='right'>11</td><td align='right'>7</td><td align='right'>4</td><td align='right'>5</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>14</td><td align='right' colspan='8'>15</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>77</td><td align='right'>20</td><td align='right'>13</td><td align='right'>5</td><td align='right'>8</td><td align='right'>44</td><td align='right'>8</td><td align='right'>6</td><td align='right'>69</td><td align='right'>98</td><td align='right'>16</td><td align='right'>79</td><td align='right'>8</td><td align='right'>8</td><td align='right'>4</td><td align='right'>6</td></tr>
+<tr><td align='right'><span class='pagenum'><a name="Page_131" id="Page_131">[Pg 131]</a></span>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>16</td><td align='right' colspan='8'>17</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>180</td><td align='right'>19</td><td align='right'>130</td><td align='right'>14</td><td align='right'>6</td><td align='right'>9</td><td align='right'>98</td><td align='right'>3</td><td align='right'>61</td><td align='right'>9</td><td align='right'>15</td><td align='right'>14</td><td align='right'>6</td><td align='right'>6</td><td align='right'>12</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>18</td><td align='right' colspan='8'>19</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>69</td><td align='right'>49</td><td align='right'>13</td><td align='right'>6</td><td align='right'>28</td><td align='right'>7</td><td align='right'>7</td><td align='right'>23</td><td align='right'>21</td><td align='right'>6</td><td align='right'>10</td><td align='right'>5</td><td align='right'>3</td><td align='right'>4</td><td align='right'>10</td><td align='right'>4</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>20</td><td align='right' colspan='7'>21</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>24</td><td align='right'>86</td><td align='right'>65</td><td align='right'>11</td><td align='right'>3</td><td align='right'>23</td><td align='right'>5</td><td align='right'>54</td><td align='right'>12</td><td align='right'>8</td><td align='right'>5</td><td align='right'>43</td><td align='right'>10</td><td align='right'>5</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>22</td><td align='right' colspan='7'>23</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>17</td><td align='right'>16</td><td align='right'>15</td><td align='right'>8</td><td align='right'>13</td><td align='right'>6</td><td align='right'>15</td><td align='right'>7</td><td align='right'>8</td><td align='right'>11</td><td align='right'>8</td><td align='right'>6</td><td align='right'>3</td><td align='right'>2</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>24</td><td align='right' colspan='7'>25</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>91</td><td align='right'>76</td><td align='right'>18</td><td align='right'>50</td><td align='right'>5</td><td align='right'>61</td><td align='right'>1</td><td align='right'>11</td><td align='right'>13</td><td align='right'>105</td><td align='right'>5</td><td align='right'>6</td><td align='right'>5</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>26</td><td align='right' colspan='7'>27</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>5</td><td align='right'>6</td><td align='right'>3</td><td align='right'>3</td><td align='right'>4</td><td align='right'>6</td><td align='right'>3</td><td align='right'>46</td><td align='right'>8</td><td align='right'>10</td><td align='right'>4</td><td align='right'>2</td><td align='right'>6</td><td align='right'>25</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>28</td><td align='right' colspan='7'>29</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>36</td><td align='right'>8</td><td align='right'>3</td><td align='right'>19</td><td align='right'>3</td><td align='right'>7</td><td align='right'>6</td><td align='right'>8</td><td align='right'>0</td><td align='right'>5</td><td align='right'>11</td><td align='right'>2</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>30</td><td align='right' colspan='6'>31</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>21</td><td align='right'>27</td><td align='right'>25</td><td align='right'>6</td><td align='right'>7</td><td align='right'>13</td><td align='right'>4</td><td align='right'>3</td><td align='right'>1</td><td align='right'>1</td><td align='right'>4</td><td align='right'>2</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>32</td><td align='right' colspan='6'>33</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>50</td><td align='right'>11</td><td align='right'>3</td><td align='right'>6</td><td align='right'>39</td><td align='right'>5</td><td align='right'>8</td><td align='right'>7</td><td align='right'>7</td><td align='right'>2</td><td align='right'>6</td><td align='right'>1</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>34</td><td align='right' colspan='6'>35</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>8</td><td align='right'>3</td><td align='right'>5</td><td align='right'>2</td><td align='right'>1</td><td align='right'>1</td><td align='right'>10</td><td align='right'>31</td><td align='right'>5</td><td align='right'>24</td><td align='right'>5</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>36</td><td align='right' colspan='6'>37</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>37</td><td align='right'>16</td><td align='right'>22</td><td align='right'>2</td><td align='right'>6</td><td align='right'>19</td><td align='right'>12</td><td align='right'>8</td><td align='right'>7</td><td align='right'>5</td><td align='right'>3</td><td align='right'>9</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>38</td><td align='right' colspan='6'>39</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>7</td><td align='right'>8</td><td align='right'>7</td><td align='right'>1</td><td align='right'>1</td><td align='right'>5</td><td align='right'>4</td><td align='right'>3</td><td align='right'>7</td><td align='right'>4</td><td align='right'>3</td><td align='right'>1</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>40</td><td align='right' colspan='5'>41</td><td align='right' colspan='5'>42</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>38</td><td align='right'>9</td><td align='right'>2</td><td align='right'>34</td><td align='right'>2</td><td align='right'>6</td><td align='right'>6</td><td align='right'>3</td><td align='right'>7</td><td align='right'>5</td><td align='right'>7</td><td align='right'>28</td><td align='right'>30</td><td align='right'>10</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>43</td><td align='right' colspan='5'>44</td><td align='right' colspan='5'>45</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>7</td><td align='right'>5</td><td align='right'>10</td><td align='right'>3</td><td align='right'>2</td><td align='right'>7</td><td align='right'>6</td><td align='right'>4</td><td align='right'>5</td><td align='right'>0</td><td align='right'>24</td><td align='right'>6</td><td align='right'>7</td><td align='right'>10</td><td align='right'>20</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>46</td><td align='right' colspan='5'>47</td><td align='right' colspan='5'>48</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>3</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>2</td><td align='right'>6</td><td align='right'>2</td><td align='right'>2</td><td align='right'>0</td><td align='right'>3</td><td align='right'>7</td><td align='right'>17</td><td align='right'>4</td><td align='right'>33</td><td align='right'>2</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>49</td><td align='right' colspan='5'>50</td><td align='right' colspan='4'>51</td><td align='right' colspan='4'>52</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>7</td><td align='right'>27</td><td align='right'>9</td><td align='right'>2</td><td align='right'>4</td><td align='right'>6</td><td align='right'>3</td><td align='right'>8</td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>5</td><td align='right'>5</td><td align='right'>5</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>53</td><td align='right' colspan='4'>54</td><td align='right' colspan='4'>55</td><td align='right' colspan='4'>56</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>12</td><td align='right'>5</td><td align='right'>1</td><td align='right'>16</td><td align='right'>5</td><td align='right'>3</td><td align='right'>4</td><td align='right'>2</td><td align='right'>0</td><td align='right'>13</td><td align='right'>16</td><td align='right'>8</td></tr>
+<tr><td align='right'>&nbsp;<span class='pagenum'><a name="Page_132" id="Page_132">[Pg 132]</a></span></td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>57</td><td align='right' colspan='4'>58</td><td align='right' colspan='4'>59</td><td align='right' colspan='4'>60</td><td align='right' colspan='3'>61</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>0</td><td align='right'>3</td><td align='right'>1</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>0</td><td align='right'>3</td><td align='right'>3</td><td align='right'>9</td><td align='right'>1</td><td align='right'>1</td><td align='right'>2</td><td align='right'>5</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>62</td><td align='right' colspan='3'>63</td><td align='right' colspan='3'>64</td><td align='right' colspan='3'>65</td><td align='right' colspan='3'>66</td><td align='right' colspan='3'>67</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>6</td><td align='right'>1</td><td align='right'>17</td><td align='right'>5</td><td align='right'>9</td><td align='right'>5</td><td align='right'>22</td><td align='right'>0</td><td align='right'>1</td><td align='right'>10</td><td align='right'>1</td><td align='right'>2</td><td align='right'>1</td><td align='right'>4</td><td align='right'>0</td><td align='right'>1</td><td align='right'>1</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>68</td><td align='right' colspan='3'>69</td><td align='right' colspan='3'>70</td><td align='right' colspan='2'>71</td><td align='right' colspan='2'>72</td><td align='right' colspan='2'>73</td><td align='right' colspan='2'>74</td><td align='right' colspan='2'>75</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>1</td><td align='right'>3</td><td align='right'>2</td><td align='right'>0</td><td align='right'>6</td><td align='right'>1</td><td align='right'>6</td><td align='right'>2</td><td align='right'>1</td><td align='right'>0</td><td align='right'>16</td><td align='right'>10</td><td align='right'>7</td><td align='right'>5</td><td align='right'>3</td><td align='right'>3</td><td align='right'>5</td><td align='right'>3</td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='left'>Dividend</td><td align='right'>76</td><td align='right' colspan='2'>77</td><td align='right' colspan='2'>78</td><td align='right' colspan='2'>79</td><td align='right' colspan='2'>80</td><td align='right'>81</td><td align='right'>82</td><td align='right'>83</td><td align='right'>84</td><td align='right'>85</td><td align='right'>86</td><td align='right'>87</td><td align='right'>88</td><td align='right'>89</td></tr>
+<tr><td align='left'>Divisor</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td></tr>
+<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>3</td><td align='right'>2</td><td align='right'>3</td><td align='right'>0</td><td align='right'>4</td><td align='right'>1</td><td align='right'>0</td><td align='right'>2</td><td align='right'>4</td><td align='right'>15</td><td align='right'>2</td><td align='right'>4</td><td align='right'>1</td><td align='right'>2</td><td align='right'>0</td><td align='right'>3</td><td align='right'>2</td><td align='right'>7</td></tr>
+</table></div>
+
+<p>Tables 3 to 8 show that even gifted authors make instruments
+for instruction in arithmetic which contain much less
+practice on certain elementary facts than teachers suppose;
+and which contain relatively much more practice on the
+more easily learned facts than on those which are harder
+to learn.</p>
+
+<p>How much practice should be given in arithmetic? How
+should it be divided among the different bonds to be
+formed? Below a certain amount there is waste because,
+as has been shown in Chapter VI, the pupil will need more
+time to detect and correct his errors than would have been
+required to give him mastery. Above a certain amount
+there is waste because of unproductive overlearning. If
+668 is just enough for 2 &times; 2, 82 is not enough for 9 &times; 8. If
+82 is just enough for 9 &times; 8, 668 is too much for 2 &times; 2.</p>
+
+<p>It is possible to find the answers to these questions for
+the pupil of median ability (or any stated ability) by suitable
+experiments. The amount of practice will, of course,
+<span class='pagenum'><a name="Page_133" id="Page_133">[Pg 133]</a></span>
+vary according to the ability of the pupil. It will also vary
+according to the interest aroused in him and the satisfaction
+he feels in progress and mastery. It will also vary according
+to the amount of practice of other related bonds; 7 &#43; 7 = 14 and 60 &divide; 7 = 8 and 4 remainder will help the formation
+of 7 &#43; 8 = 15 and 61 &divide; 7 = 8 and 5 remainder. It will also, of
+course, vary with the general difficulty of the bond, 17 &minus; 8 = 9
+being under ordinary conditions of teaching harder to form
+than 7 &minus; 2 = 5.</p>
+
+<p>Until suitable experiments are at hand we may estimate
+for the fundamental bonds as follows, assuming that by
+the end of grade 6 a strength of 199 correct out of 200 is
+to be had, and that the teaching is by an intelligent person
+working in accord with psychological principles as to both
+ability and interest.</p>
+
+<p>For one of the easier bonds, most facilitated by other
+bonds (such as 2 &times; 5 = 10, or 10 &minus; 2 = 8, or the double bond
+7 = two 3s and 1 remainder) in the case of the median or
+average pupil, twelve practices in the week of first learning,
+supported by twenty-five practices during the two months
+following, and maintained by thirty practices well spread
+over the later periods should be enough. For the more
+gifted pupils lesser amounts down to six, twelve, and fifteen
+may suffice. For the less gifted pupils more may be required
+up to thirty, fifty, and a hundred. It is to be doubted,
+however, whether pupils requiring nearly two hundred repetitions
+of each of these easy bonds should be taught arithmetic
+beyond a few matters of practical necessity.</p>
+
+<p>For bonds of ordinary difficulty, with average facilitation
+from other bonds (such as 11 &minus; 3, 4 &times; 7, or 48 &divide; 8 = 6) in the
+case of the median or average pupil, we may estimate twenty
+practices in the week of first learning, supported by thirty,
+and maintained by fifty practices well spread over the later<span class='pagenum'><a name="Page_134" id="Page_134">[Pg 134]</a></span>
+periods. Gifted pupils may gain and keep mastery with
+twelve, fifteen, and twenty practices respectively. Pupils dull
+at arithmetic may need up to twenty, sixty, and two hundred.
+Here, again, it is to be doubted whether a pupil for whom
+arithmetical facts, well taught and made interesting, are
+so hard to acquire as this, should learn many of them.</p>
+
+<p>For bonds of greater difficulty, less facilitated by other
+bonds (such as 17 &minus; 9, 8 &times; 7, or 12&frac12;% of = <sup>1</sup>&frasl;<sub>8</sub> of), the practice
+may be from ten to a hundred percent more than the above.</p>
+
+
+<h4>UNDERLEARNING AND OVERLEARNING</h4>
+
+<p>If we accept the above provisional estimates as reasonable,
+we may consider the harm done by giving less and by giving
+more than these reasonable amounts. Giving less is indefensible.
+The pupil's time is wasted in excessive checking
+to find his errors. He is in danger of being practiced in error.
+His attention is diverted from the learning of new facts
+and processes by the necessity of thinking out these supposedly
+mastered facts. All new bonds are harder to learn
+than they should be because the bonds which should facilitate
+them are not strong enough to do so. Giving more does
+harm to some extent by using up time that could be spent
+better for other purposes, and (though not necessarily) by
+detracting from the pupil's interest in arithmetic. In
+certain cases, however, such excess practice and overlearning
+are actually desirable. Three cases are of special importance.</p>
+
+<p>The first is the case of a bond operating under a changed
+mental set or adjustment. A pupil may know 7 &times; 8 adequately
+as a thing by itself, but need more practice to operate
+it in</p>
+
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr>
+<td align='right'>285<br />
+7<br />
+&mdash;&mdash;</td>
+</tr>
+</table>
+
+<p class="noidt">where he has to remember that 3 is to be added to the<span class='pagenum'><a name="Page_135" id="Page_135">[Pg 135]</a></span>
+56 when he obtains it, and that only the 9 is to be written
+down, the 5 to be held in mind for later use. The practice
+required to operate the bond efficiently in this new set is
+desirable, even though it is excess from a narrower point of
+view, and causes the straightforward 'seven eights are fifty-six'
+to be overlearned. So also a pupil's work with 24,
+34, 44, etc., &#43; 9 may react to give what would be excess
+practice from the point of view of 4 &#43; 9 alone; his work in
+estimating approximate quotient figures in long division
+may give excess practice on the division tables. There are
+many such cases. Even adding the 5 and 7 in <sup>5</sup>&frasl;<sub>12</sub> &#43; <sup>7</sup>&frasl;<sub>12</sub> is
+not quite the same task as adding 5 and 7 undisturbed by
+the fact that they are twelfths. We know far too little
+about the amount of practice needed to adapt arithmetical
+bonds to efficient operation in these more complicated conditions
+to estimate even approximately the allowances to
+be made. But some allowance, and often a rather large
+allowance, must be made.</p>
+
+<p>The second is the case where the computation in general
+should be made very easy and sure for the pupil except for
+some one new element that is being learned. For example,
+in teaching the meaning and uses of 'Averages' and of
+uneven division, we may deliberately use 2, 3, and 4 as divisors
+rather than 7 and 9, so as to let all the pupil's energy be
+spent in learning the new facts, and so that the fraction in
+the quotient may be something easily understood, real, and
+significant. In teaching the addition of mixed numbers,
+we may use, in the early steps,</p>
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr>
+<td align='left'>11&frac12;<br />
+13&frac12;<br />
+24<br />
+&mdash;&mdash;</td>
+<td align='center'>rather than</td>
+<td align='left'>79&frac12;<br />
+98&frac12;<br />
+67<br />
+&mdash;&mdash;</td>
+</tr>
+</table></div>
+
+<p class="noidt">so as to save attention for the new process itself. In cancellation,
+we may give excess practice to divisions by 2, 3, 4,
+and 5 in order to make the transfer to the new habits of con<span class='pagenum'><a name="Page_136" id="Page_136">[Pg 136]</a></span>sidering
+two numbers together from the point of view of their
+divisibility by some number. In introducing trade discount,
+we may give excess practice on '5% of' and '10% of'
+deliberately, so that the meaning of discount may not be
+obscured by difficulties in the computation itself. Excess
+practice on, and overlearning of, certain bonds is thus very
+often justifiable.</p>
+
+<p>The third case concerns bonds whose importance for
+practical uses in life or as notable facilitators of other bonds
+is so great that they may profitably be brought to a greater
+strength than 199 correct out of 200 at a speed of 2 sec. or less,
+or be brought to that degree of strength very early. Examples
+of bonds of such special practical use are the subtractions
+from 10, &frac12; &#43; &frac12;, &frac12; &#43; &frac14;, &frac12; of 60, &frac14; of 60, and the fractional parts
+of 12 and of $1.00. Examples of notable facilitating bonds
+are ten 10s = 100, ten 100s = 1000, additions like 2 &#43; 2,
+3 &#43; 3, and 4 &#43; 4, and all the multiplication tables to 9 &times; 9.</p>
+
+<p>In consideration of these three modifying cases or principles,
+a volume could well be written concerning just how
+much practice to give to each bond, in each of the types of
+complex situations where it has to operate. There is evidently
+need for much experimentation to expose the facts,
+and for much sagacity and inventiveness in making sure of
+effective learning without wasteful overlearning.</p>
+
+<p>The facts of primary importance are:&mdash;</p>
+
+<p class="nnblockquot">(1) The textbook or other instrument of instruction which
+is a teacher's general guide may give far too little
+practice on certain bonds.</p>
+
+<p class="nnblockquot">(2) It may divide the practice given in ways that are
+apparently unjustifiable.</p>
+
+<p class="nnblockquot">(3) The teacher needs therefore to know how much practice
+it does give, where to supplement it, and what to
+omit.</p>
+
+<p><span class='pagenum'><a name="Page_137" id="Page_137">[Pg 137]</a></span></p>
+
+<p class="nnblockquot">(4) The omissions, on grounds of apparent excess practice,
+should be made only after careful consideration
+of the third principle described above.</p>
+
+<p class="nnblockquot">(5) The amount of practice should always be considered
+in the light of its interest and appeal to the pupil's
+tendency to work with full power and zeal. Mere
+repetition of bonds when the learner does not care
+whether he is improving is rarely justifiable on any
+grounds.</p>
+
+<p class="nnblockquot">(6) Practice that is actually in excess is not a very grave
+defect if it is enjoyed and improves the pupil's attitude
+toward arithmetic. Not much time is lost; a
+hundred practices for each of a thousand bonds after
+mastery to 199 in 200 at 2 seconds will use up less than
+60 hours, or 15 hours per year in grades 3 to 6.</p>
+
+<p class="nnblockquot">(7) By the proper division of practice among bonds,
+the arrangement of learning so that each bond helps
+the others, the adroit shifting of practice of a bond
+to each new type of situation requiring it to operate
+under changed conditions, and the elimination of
+excess practice where nothing substantial is gained,
+notable improvements over the past hit-and-miss
+customs may be expected.</p>
+
+<p class="nnblockquot">(8) Unless the material for practice is adequate, well
+balanced and sufficiently motivated, the teacher must
+keep close account of the learning of pupils. Otherwise
+disastrous underlearning of many bonds is
+almost sure to occur and retard the pupil's development.</p>
+
+
+<h4>THE ORGANIZATION OF ABILITIES</h4>
+
+<p>There is danger that the need of brevity and simplicity
+which has made us speak so often of a bond or an ability,<span class='pagenum'><a name="Page_138" id="Page_138">[Pg 138]</a></span>
+and of the amount of practice it requires, may mislead the
+reader into thinking that these bonds and abilities are to be
+formed each by itself alone and kept so. They should rarely
+be formed so and never kept so. This we have indicated from
+time to time by references to the importance of forming a
+bond in the way in which it is to be used, to the action of
+bonds in changed situations, to facilitation of one bond by
+others, to the co&ouml;peration of abilities, and to their integration
+into a total arithmetical ability.</p>
+
+<p>As a matter of fact, only a small part of drill work in
+arithmetic should be the formation of isolated bonds. Even
+the very young pupil learning 5 and 3 are 8 should learn it
+with '5 and 5 = 10,' '5 and 2 = 7,' at the back of his mind,
+so to speak. Even so early, 5 &#43; 3 = 8 should be part of an
+organized, co&ouml;perating system of bonds. Later 50 &#43; 30 = 80
+should become allied to it. Each bond should be considered,
+not simply as a separate tool to be put in a compartment
+until needed, but also as an improvement of one total tool
+or machine, arithmetical ability.</p>
+
+<p>There are differences of course. Knowledge of square
+root can be regarded somewhat as a separate tool to be
+sharpened, polished, and used by itself, whereas knowledge
+of the multiplication tables cannot. Yet even square root
+is probably best made more closely a part of the total ability,
+being taught as a special case of dividing where divisor is
+to be the same as quotient, the process being one of estimating
+and correcting.</p>
+
+<p>In general we do not wish the pupil to be a repository
+of separated abilities, each of which may operate only if you
+ask him the sort of questions which the teacher used to ask
+him, or otherwise indicate to him which particular arithmetical
+tool he is to use. Rather he is to be an effective
+organization of abilities, co&ouml;perating in useful ways to meet<span class='pagenum'><a name="Page_139" id="Page_139">[Pg 139]</a></span>
+the quantitative problems life offers. He should not as a
+rule have to think in such fashion as: "Is this interest or
+discount? Is it simple interest or compound interest?
+What did I do in compound interest? How do I multiply
+by 2 percent?" The situation that calls up interest should
+also call up the kind of interest that is appropriate, and the
+technique of operating with percents should be so welded
+together with interest in his mind that the right co&ouml;peration
+will occur almost without supervision by him.</p>
+
+<p>As each new ability is acquired, then, we seek to have it
+take its place as an improvement of a thinking being, as a
+co&ouml;perative member of a total organization, as a soldier
+fighting together with others, as an element in an educated
+personality. Such an organization of bonds will not form
+itself any more than any one bond will create itself. If the
+elements of arithmetical ability are to act together as a
+total organized unified force they must be made to act together
+in the course of learning. What we wish to have
+work together we must put together and give practice in
+teamwork.</p>
+
+<p>We can do much to secure such co&ouml;perative action when
+and where and as it is needed by a very simple expedient;
+namely, to give practice with computation and problems
+such as life provides, instead of making up drills and problems
+merely to apply each fact or principle by itself. Though a
+pupil has solved scores of problems reading, "A triangle
+has a base of <i>a</i> feet and an altitude of <i>b</i> feet, what is its
+area?" he may still be practically helpless in finding the
+area of a triangular plot of ground; still more helpless in
+using the formula for a triangle which is one of two into which
+a trapezoid is divided. Though a pupil has learned to solve
+problems in trade discount, simple interest, compound
+interest, and bank discount one at a time, stated in a few<span class='pagenum'><a name="Page_140" id="Page_140">[Pg 140]</a></span>
+set forms, he may be practically helpless before the actual
+series of problems confronting him in starting in business,
+and may take money out of the savings bank when he ought
+to borrow on a time loan, or delay payment on his bills when
+by paying cash he could save money as well as improve his
+standing with the wholesaler.</p>
+
+<p>Instead of making up problems to fit the abilities given
+by school instruction, we should preferably modify school
+instruction so that arithmetical abilities will be organized
+into an effective total ability to meet the problems that life
+will offer. Still more generally, <i>every bond formed should be
+formed with due consideration of every other bond that has been
+or will be formed; every ability should be practiced in the most
+effective possible relations with other abilities</i>.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_141" id="Page_141">[Pg 141]</a></span></p>
+<h2><a name="CHAPTER_VII" id="CHAPTER_VII"></a>CHAPTER VII</h2>
+
+<h3>THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION
+OF BONDS</h3>
+
+
+<p>The bonds to be formed having been chosen, the next
+step is to arrange for their most economical order of formation&mdash;to
+arrange to have each help the others as much as
+possible&mdash;to arrange for the maximum of facilitation and
+the minimum of inhibition.</p>
+
+<p>The principle is obvious enough and would probably be
+admitted in theory by any intelligent teacher, but in practice
+we are still wedded to conventional usages which arose
+long before the psychology of arithmetic was studied. For
+example, we inherit the convention of studying addition of
+integers thoroughly, and then subtraction, and then multiplication,
+and then division, and many of us follow it though
+nobody has ever given a proof that this is the best order
+for arithmetical learning. We inherit also the opposite
+convention of studying in a so-called "spiral" plan, a little
+addition, subtraction, multiplication, and division, and then
+some more of each, and then some more, and many of us
+follow this custom, with an unreasoned faith that changing
+about from one process to another is <i>per se</i> helpful.</p>
+
+<p>Such conventions are very strong, illustrating our common
+tendency to cherish most those customs which we cannot
+justify! The reductions of denominate numbers ascending
+and descending were, until recently, in most courses of study,<span class='pagenum'><a name="Page_142" id="Page_142">[Pg 142]</a></span>
+kept until grade 4 or grade 5 was reached, although this material
+is of far greater value for drills on the multiplication and
+division tables than the customary problems about apples,
+eggs, oranges, tablets, and penholders. By some historical
+accident or for good reasons the general treatment of denominate
+numbers was put late; by our na&iuml;ve notions of
+order and system we felt that any use of denominate numbers
+before this time was heretical; we thus became blind to the
+advantages of quarts and pints for the tables of 2s; yards
+and feet for the tables of 3s; gallons and quarts for the tables
+of 4s; nickels and cents for the 5s; weeks and days for the
+7s; pecks and quarts for the 8s; and square yards and
+square feet for the 9s. Problems like 5 yards = __ feet or
+15 feet = __ yards have not only the advantages of brevity,
+clearness, practical use, real reference, and ready variation,
+but also the very great advantage that part of the data have
+to be <i>thought of</i> in a useful way instead of <i>read off</i> from the
+page. In life, when a person has twenty cents with which
+to buy tablets of a certain sort, he <i>thinks of</i> the price
+in making his purchase, asking it of the clerk only in
+case he does not know it, and in planning his purchases
+beforehand he <i>thinks of</i> prices as a rule. In spite of these
+and other advantages, not one textbook in ten up to 1900
+made early use of these exercises with denominate numbers.
+So strong is mere use and wont.</p>
+
+<p>Besides these conventional customs, there has been, in
+those responsible for arithmetical instruction, an admiration
+for an arrangement of topics that is easy for a person, after
+he knows the subject, to use in thinking of its constituent
+parts and their relations. Such arrangements are often
+called 'logical' arrangements of subject matter, though
+they are often far from logical in any useful sense. Now
+the easiest order in which to think of a hierarchy of habits<span class='pagenum'><a name="Page_143" id="Page_143">[Pg 143]</a></span>
+after you have formed them all may be an extremely difficult
+order in which to form them. The criticism of other orders
+as 'scrappy,' or 'unsystematic,' valid enough if the course
+of study is thought of as an object of contemplation, may be
+foolish if the course of study is regarded as a working instrument
+for furthering arithmetical learning.</p>
+
+<p>We must remember that all our systematizing and labeling
+is largely without meaning to the pupils. They cannot at
+any point appreciate the system as a progression from that
+point toward this and that, since they have no knowledge
+of the 'this or that.' They do not as a rule think of their
+work in grade 4 as an outcome of their work in grade
+3 with extensions of <i>a</i> to <i>a</i><sub>1</sub>, and additions of <i>b</i><sub>2</sub> and <i>b</i><sub>3</sub> to
+<i>b</i> and <i>b</i><sub>1</sub>, and refinements of <i>c</i> and <i>d</i> by <i>c</i><sub>4</sub> and <i>d</i><sub>5</sub>. They
+could give only the vaguest account of what they did in
+grade 3, much less of why it should have been done then.
+They are not much disturbed by a lack of so-called 'system'
+and 'logical' progression for the same reason that they
+are not much helped by their presence. What they need
+and can use is a <i>dynamically</i> effective system or order, one
+that they can learn easily and retain long by, regardless of
+how it would look in a museum of arithmetical systems.
+Unless their actual arithmetical habits are usefully related
+it does no good to see the so-called logical relations; and
+if their habits are usefully related, it does not very much
+matter whether or not they do see these; finally, they can be
+brought to see them best by first acquiring the right habits
+in a dynamically effective order.</p>
+
+
+<h4>DECREASING INTERFERENCE AND INCREASING FACILITATION</h4>
+
+<p>Psychology offers no single, easy, royal road to discovering
+this dynamically best order. It can only survey the
+bonds, think what each demands as prerequisite and offers<span class='pagenum'><a name="Page_144" id="Page_144">[Pg 144]</a></span>
+as future help, recommend certain orders for trial, and
+measure the efficiency of each order as a means of attaining
+the ends desired. The ingenious thought and careful experimentation
+of many able workers will be required for
+many years to come.</p>
+
+<p>Psychology can, however, even now, give solid constructive
+help in many instances, either by recommending
+orders that seem almost certainly better than those in
+vogue, or by proposing orders for trial which can be justified
+or rejected by crucial tests.</p>
+
+<p>Consider, for example, the situation, 'a column of one-place
+numbers to be added, whose sum is over 9,' and the
+response 'writing down the sum.' This bond is commonly
+firmly fixed before addition with two-place numbers is
+undertaken. As a result the pupil has fixed a habit that
+he has to break when he learns two-place addition. If <i>oral</i>
+answers only are given with such single columns until
+two-place addition is well under way, the interference is
+avoided.</p>
+
+<p>In many courses of study the order of systematic formation
+of the multiplication table bonds is : 1 &times; 1, 2 &times; 1, etc., 1 &times; 2,
+2 &times; 2, etc., 1 &times; 3, 2 &times; 3, etc., 1 &times; 9, 2 &times; 9, etc. This is probably
+wrong in two respects. There is abundant reason to believe
+that the &times; 5s should be learned first, since they are easier to
+learn than the 1s or the 2s, and give the idea of multiplying
+more emphatically and clearly. There is also abundant
+reason to believe that the 1 &times; 5, 1 &times; 2, 1 &times; 3, etc., should
+be put very late&mdash;after at least three or four tables are
+learned, since the question "What is 1 times 2?" (or 3 or 5)
+is unnecessary until we come to multiplication of two- and
+three-place numbers, seems a foolish question until
+then, and obscures the notion of multiplication if put
+early. Also the facts are best learned once for all as the
+<span class='pagenum'><a name="Page_145" id="Page_145">[Pg 145]</a></span>
+habits "1 times <i>k</i> is the same as <i>k</i>," and "<i>k</i> times 1 is the
+same as <i>k</i>."<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a></p>
+
+<p>In another connection it was recommended that the
+divisions to 81 &divide; 9 be learned by selective thinking or reasoning
+from the multiplications. This determines the order
+of bonds so far as to place the formation of the division
+bonds soon after the learning of the multiplications. For
+other reasons it is well to make the proximity close.</p>
+
+<p>One of the arbitrary systematizations of the order of
+formation of bonds restricts operations at first to the numbers
+1 to 10, then to numbers under 100, then to numbers
+under 1000, then to numbers under 10,000. Apart from the
+avoidance of unreal and pedantic problems in applied
+arithmetic to which work with large numbers in low grades
+does somewhat predispose a teacher, there is little merit
+in this restriction of the order of formation of bonds. Its
+demerits are many. For example, when the pupil is learning
+to 'carry' in addition he can be given better practice
+by soon including tasks with sums above 100, and can get
+a valuable sense of the general use of the process by being
+given a few examples with three- and four-place numbers
+to be added. The same holds for subtraction. Indeed,
+there is something to be said in favor of using six- or seven-place
+numbers in subtraction, enforcing the 'borrowing'
+process by having it done again and again in the same example,
+and putting it under control by having the decision
+between 'borrowing' and 'not borrowing' made again
+and again in the same example. When the multiplication
+<span class='pagenum'><a name="Page_146" id="Page_146">[Pg 146]</a></span>
+tables are learned the most important use for them is not
+in tedious reviews or trivial problems with answers under
+100, but in regular 'short' multiplication of two- and three- and
+even four-place numbers. Just as the addition combinations
+function mainly in the higher-decade modifications
+of them, so the multiplication combinations function chiefly
+in the cases where the bond has to operate while the added
+tasks of keeping one's place, adding what has been carried,
+writing down the right figure in the right place, and holding
+the right number for later addition, are also taken care of.
+It seems best to introduce such short multiplication as soon
+as the &times; 5s, &times; 2s, &times; 3s, and &times; 4s are learned and to put the
+&times; 6s, &times; 7s, and the rest to work in such short multiplication
+as soon as each is learned.</p>
+
+<p>Still surer is the need for four-, five-, and six-place numbers
+when two-place numbers are used in multiplying. When
+the process with a two-place multiplier is learned, multiplications
+by three-place numbers should soon follow. They
+are not more difficult then than later. On the contrary, if
+the pupil gets used to multiplying only as one does with
+two-place multipliers, he will suffer more by the resulting
+interference than he does from getting six- or seven-place
+answers whose meaning he cannot exactly realize. They
+teach the rationale and the manipulations of long multiplication
+with especial economy because the principles and the
+procedures are used two or three times over and the contrasts
+between the values which the partial products have
+in adding become three instead of one.</p>
+
+<p>The entire matter of long multiplication with integers
+and United States money should be treated as a teaching
+unit and the bonds formed in close organization, even though
+numbers as large as 900,000 are occasionally involved. The
+reason is not that it is more logical, or less scrappy, but
+<span class='pagenum'><a name="Page_147" id="Page_147">[Pg 147]</a></span>
+that each of the bonds in question thus gets much help from,
+and gives much help to, the others.</p>
+
+<p>In sharp contrast to a topic like 'long multiplication'
+stands a topic like denominate numbers. It most certainly
+should not be treated as a large teaching unit, and all the
+bonds involved in adding, subtracting, multiplying, and
+dividing with all the ordinary sorts of measures should certainly
+not be formed in close sequence. The reductions
+ascending and descending for many of the measures should
+be taught as drills on the appropriate multiplication and
+division tables. The reduction of feet and inches to inches,
+yards and feet to yards, gallons and quarts to quarts, and
+the like are admirable exercises in connection with the
+(<i>a</i> &times; <i>b</i>) &#43; <i>c</i> = .... problems,&mdash;the 'Bought 3 lbs. of sugar
+at 7 cents and 5 cents worth of matches' problems. The
+reductions of inches to feet and inches and the like are admirable
+exercises in the <i>d</i> = (.... &times; <i>b</i>) &#43; <i>c</i> or 'making change'
+problem, which in its small-number forms is an excellent
+preparatory step for short division. They are also of great
+service in early work with fractions. The feet-mile, square-foot-square-inch,
+and other simple relations give a genuine
+and intelligible demand for multiplication with large numbers.</p>
+
+<p>Knowledge of the metric system for linear and square
+measure would perhaps, as an introduction to decimal fractions,
+more than save the time spent to learn it. It would
+even perhaps be worth while to invent a measure (call it
+the <i>twoqua</i>) midway between the quart and gallon and
+teach carrying in addition and borrowing in subtraction
+by teaching first the addition and subtraction of 'gallon,
+twoqua, quart, and pint' series! Many of the bonds which
+a system-made tradition huddled together uselessly in a
+chapter on denominate numbers should thus be formed<span class='pagenum'><a name="Page_148" id="Page_148">[Pg 148]</a></span>
+as helpful preparations for and applications of other bonds
+all the way from the first to the eighth half-year of instruction
+in arithmetic.</p>
+
+<p>The bonds involved in the ability to respond correctly to
+the series:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='right'> 5 = .... 2s and .... remainder</td></tr>
+<tr><td align='right'> 5 = .... 3s and .... remainder</td></tr>
+<tr><td align='right'>88 = .... 9s and .... remainder</td></tr>
+</table></div>
+
+<p class="noidt">should be formed before, not during, the training in short
+division. They are admirable at that point as practice on
+the division tables; are of practical service in the making-change
+problems of the small purchase and the like; and
+simplify the otherwise intricate task of keeping one's place,
+choosing the quotient figure, multiplying by it, subtracting
+and holding in mind the new number to be divided, which is
+composed half of the remainder and half of a figure in the
+written dividend. This change of order is a good illustration
+of the nearly general rule that "<i>When the practice or review
+required to perfect or hold certain bonds can, by an inexpensive
+modification, be turned into a useful preparation for
+new bonds, that modification should be made.</i>"</p>
+
+<p>The bonds involved in the four operations with United
+States money should be formed in grades 3 and 4
+along with or very soon after the corresponding bonds with
+three-place and four-place integers. This statement would
+have seemed preposterous to the pedagogues of fifty years
+ago. "United States money," they would have said, "is
+an application of decimals. How can it be learned until the
+essentials of decimal fractions are known? How will the
+child understand when multiplying $.75 by 3 that 3 times
+5 cents is 1 dime and 5 cents, or that 3 times 70 cents is
+2 dollars and 1 dime? Why perplex the young pupils with
+the difficulties of placing the decimal point? Why disturb
+<span class='pagenum'><a name="Page_149" id="Page_149">[Pg 149]</a></span>
+the learning of the four operations with integers by adding
+at each step a second 'procedure with United States
+money'?"</p>
+
+<p>The case illustrates very well the error of the older oversystematic
+treatment of the order of topics and the still
+more important error of confusing the logic of proof with
+the psychology of learning. To prove that 3 &times; $.75 = $2.25
+to the satisfaction of certain arithmeticians, you may need
+to know the theory of decimal fractions; but to do such
+multiplication all a child needs is to do just what he has
+been doing with integers and then "Put a $ before the answer
+to show that it means dollars and cents, and put a decimal
+point in the answer to show which figures mean dollars and
+which figures mean cents." And this is general. The
+ability to operate with integers plus the two habits of prefixing
+$ and separating dollars from cents in the result will
+enable him to operate with United States money.</p>
+
+<p>Consequently good practice came to use United States
+money not as a consequence of decimal fractions, learned
+by their aid, but as an introduction to decimal fractions
+which aids the pupil to learn them. So it has gradually
+pushed work with United States money further and further
+back, though somewhat timidly.</p>
+
+<p>We need not be timid. The pupil will have no difficulty
+in adding, subtracting, multiplying, and dividing with
+United States money&mdash;unless we create it by our explanations!
+If we simply form the two bonds described above
+and show by proper verification that the procedure always
+gives the right answer, the early teaching of the four operations
+with United States money will in fact actually show a
+learning profit! It will save more time in the work with
+integers than was spent in teaching it! For, in the first
+place, it will help to make work with four-place and five<span class='pagenum'><a name="Page_150" id="Page_150">[Pg 150]</a></span>-place
+numbers more intelligible and vital. A pupil can
+understand $16.75 or $28.79 more easily than 1675 or 2879.
+The former may be the prices of a suit or sewing machine or
+bicycle. In the second place, it permits the use of a large
+stock of genuine problems about spending, saving, sharing,
+and the like with advertisements and catalogues and school
+enterprises. In the third place, it permits the use of common-sense
+checks. A boy may find one fourth of 3000 as
+7050 or 75 and not be disturbed, but he will much more
+easily realize that one fourth of $30.00 is not over $70 or
+less than $1. Even the decimal point of which we used to
+be so afraid may actually help the eye to keep its place in
+adding.</p>
+
+
+<h4>INTEREST</h4>
+
+<p>So far, the illustrations of improvements in the order of
+bonds so as to get less interference and more facilitation
+than the customary orders secure have sought chiefly to
+improve the mechanical organization of the bonds. Any
+gain in interest which the changes described effected would
+be largely due to the greater achievement itself. Dewey
+and others have emphasized a very different principle of
+improving the order of formation of bonds&mdash;the principle
+of determination of the bonds to be formed by some vital,
+engaging problem which arouses interest enough to lighten
+the labor and which goes beyond or even against cut-and-dried
+plans for sequences in order to get effective problems.
+For example, the work of the first month in grade 2B might
+sacrifice facilitations of the mechanical sort in order to put
+arithmetic to use in deciding what dimensions a rabbit's
+cage should have to give him 12 square feet of floor space,
+how much bread he should have per meal to get 6 ounces
+a day, how long a ten-cent loaf would last, how many loaves<span class='pagenum'><a name="Page_151" id="Page_151">[Pg 151]</a></span>
+should be bought per week, how much it costs to feed the
+rabbit, how much he has gained in weight since he was
+brought to the school, and so on.</p>
+
+<p>Such sacrifices of the optimal order if interest were equal,
+in order to get greater interest or a healthier interest, are
+justifiable. Vital problems as nuclei around which to organize
+arithmetical learning are of prime importance. It
+is even safe probably to insist that some genuine problem-situation
+requiring a new process, such as addition with
+carrying, multiplication by two-place numbers, or division
+with decimals, be provided in every case as a part of the
+introduction to that process. The sacrifice should not
+be too great, however; the search for vital problems that
+fit an economical order of subject matter is as much needed
+as the amendment of that order to fit known interests;
+and the assurance that a problem helps the pupil to learn
+arithmetic is as important as the assurance that arithmetic
+is used to help the pupil solve his personal problems.</p>
+
+<p>Much ingenuity and experimentation will be required
+to find the order that is satisfactory in both quality and
+quantity of interest or motive and helpfulness of the bonds
+one to another. The difficulty of organizing arithmetic
+around attractive problems is much increased by the fact
+of class instruction. For any one pupil vital, personal
+problems or projects could be found to provide for many
+arithmetical abilities; and any necessary knowledge and
+technique which these projects did not develop could be
+somehow fitted in along with them. But thirty children,
+half boys and half girls, varying by five years in age, coming
+from different homes, with different native capacities, will
+not, in September, 1920, unanimously feel a vital need to
+solve any one problem, and then conveniently feel another
+on, say, October 15! In the mechanical laws of learning<span class='pagenum'><a name="Page_152" id="Page_152">[Pg 152]</a></span>
+children are much alike, and the gain we may hope to make
+from reducing inhibitions and increasing facilitations is,
+for ordinary class-teaching, probably greater than that
+to be made from the discovery of attractive central problems.
+We should, however, get as much as possible of both.</p>
+
+
+<h4>GENERAL PRINCIPLES</h4>
+
+<p>The reader may by now feel rather helpless before the
+problem of the arrangement of arithmetical subject matter.
+"Sometimes you complete a topic, sometimes you take it
+piecemeal months or years apart, often you make queer
+twists and shifts to get a strategic advantage over the
+enemy," he may think, "but are there no guiding principles,
+no general rules?" There is only one that is absolutely
+general, to <i>take the order that works best for arithmetical learning</i>.
+There are particular rules, but there are so many
+and they are so limited by an 'other things being equal'
+clause, that probably a general eagerness to think out the
+<i>pros</i> and <i>cons</i> for any given proposal is better than a stiff
+attempt to adhere to these rules. I will state and illustrate
+some of them, and let the reader judge.</p>
+
+<p><i>Other things being equal, one new sort of bonds should not
+be started until the previous set is fairly established, and two
+different sets should not be started at once.</i> Thus, multiplication
+of two- and three-place numbers by 2, 3, 4, and 5 will
+first use numbers such that no carrying is required, and no
+zero difficulties are encountered, then introduce carrying,
+then introduce multiplicands like 206 and 320. If other
+things were equal, the carrying would be split into two steps&mdash;first
+drills with (4 &times; 6) &#43; 2, (3 &times; 7) &#43; 3, (5 &times; 4) &#43; 1, and
+the like, and second the actual use of these habits in the
+multiplication. The objection to this separation of the
+double habit is that the first part of it in isolation is too
+<span class='pagenum'><a name="Page_153" id="Page_153">[Pg 153]</a></span>
+artificial&mdash;that it may be better to suffer the extra difficulty
+of forming the two together than to teach so rarely
+used habits as the (<i>a</i> &times; <i>b</i>) &#43; <i>c</i> series. Experimental tests are
+needed to decide this point.</p>
+
+<p><i>Other things being equal, bonds should be formed in such
+order that none will have to be broken later.</i> For example,
+there is a strong argument for teaching long division first,
+or very early, with remainders, letting the case of zero remainder
+come in as one of many. If the pupils have been
+familiarized with the remainder notion by the drills recommended
+as preparation for short division,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> the use of remainders
+in long division will offer little difficulty. The
+exclusive use of examples without remainders may form the
+habit of not being exact in computation, of trusting to
+'coming out even' as a sole check, and even of writing down
+a number to fit the final number to be divided instead of
+obtaining it by honest multiplication.</p>
+
+<p>For similar reasons additions with 2 and 3 as well as 1
+to be 'carried' have much to recommend them in the very
+first stages of column addition with carrying. There is
+here the added advantage that a pupil will be more likely to
+remember to carry if he has to think <i>what</i> to carry. The
+present common practice of using small numbers for ease
+in the addition itself teaches many children to think of
+carrying as adding one.</p>
+
+<p><i>Other things being equal, arrange to have variety.</i> Thus
+it is probably, though not surely, wise to interrupt the
+monotony of learning the multiplication and division tables,
+by teaching the fundamentals of 'short' multiplication
+and perhaps of division after the 5s, 2s, 3s, and 4s are learned.
+This makes a break of several weeks. The facts for the 6s,
+7s, 8s, and 9s can then be put to varied use as fast as learned.
+<span class='pagenum'><a name="Page_154" id="Page_154">[Pg 154]</a></span>
+It is almost certainly wise to interrupt the first half-year's
+work with addition and subtraction, by teaching 2 &times; 2, 2 &times; 3,
+3 &times; 2, 2 &times; 4, 4 &times; 2, 2 &times; 5, later by 2 &times; 10, 3 &times; 10,
+4 &times; 10, 5 &times; 10,
+later by &frac12; &#43; &frac12;, 1&frac12; &#43; &frac12;, &frac12; of 2, &frac12; of 4, &frac12; of 6, and at some time
+by certain profitable exercises wherein a pupil tells all he
+knows about certain numbers which may be made nuclei of
+important facts (say, 5, 8, 10, 12, 15, and 20).</p>
+
+<p><i>Other things being equal, use objective aids to verify an
+arithmetical process or inference after it is made, as well as to
+provoke it.</i> It is well at times to let pupils do everything
+that they can with relations abstractly conceived, testing
+their results by objective counting, measuring, adding, and
+the like. For example, an early step in adding should be
+to show three things, put them under a book, show two
+more, put these under the book, and then ask how many
+there are under the book, letting the objective counting
+come later as the test of the correctness of the addition.</p>
+
+<p><i>Other things being equal, reserve all explanations of why a
+process must be right until the pupils can use the process accurately,
+and have verified the fact that it is right.</i> Except for
+the very gifted pupils, the ordinary preliminary deductive
+explanations of what must be done are probably useless as
+means of teaching the pupils what to do. They use up
+much time and are of so little permanent effect that, as we
+have seen, the very arithmeticians who advocate making
+them, admit that after a pupil has mastered the process he
+may be allowed to forget the reasons for it. I am not sure
+that the deductive proofs of why we place the decimal point
+as we do in division by a decimal, or invert and multiply
+in dividing by a fraction, and the like, are worth teaching at
+all. If they are to be taught at all, the time to teach them
+is (except for the very gifted) after the pupil has mastered
+the process and has confidence in it. He then at least<span class='pagenum'><a name="Page_155" id="Page_155">[Pg 155]</a></span>
+knows what process he is to prove is right, and that it is
+right, and has had some chance of seeing <i>why</i> it is right from
+his experience with it.</p>
+
+<p>One more principle may be mentioned without illustration.
+<i>Arrange the order of bonds with due regard for the aims
+of the other studies of the curriculum and the practical needs of
+the pupil outside of school.</i> Arithmetic is not a book or a
+closed system of exercises. It is the quantitative work of
+the pupils in the elementary school. No narrower view of
+it is adequate.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_156" id="Page_156">[Pg 156]</a></span></p>
+<h2><a name="CHAPTER_VIII" id="CHAPTER_VIII"></a>CHAPTER VIII</h2>
+
+<h2>THE DISTRIBUTION OF PRACTICE</h2>
+
+
+<h3>THE PROBLEM</h3>
+
+<p>The same amount of practice may be distributed in various
+ways. Figures 7 to 10, for example, show 200 practices with
+division by a fraction distributed over three and a half years
+of 10 months in four different ways. In Fig. 7, practice is
+somewhat equally distributed over the whole period. In
+Fig. 8 the practice is distributed at haphazard. In Fig. 9
+there is a first main learning period, a review after about
+ten weeks, a review at the beginning of the seventh grade,
+another review at the beginning of the eighth grade,
+and some casual practice rather at random. In Fig. 10
+there is a main learning period, with reviews diminishing
+in length and separated by wider and wider intervals, with
+occasional practice thereafter to keep the ability alive
+and healthy.</p>
+
+<p>Plans I and II are obviously inferior to Plans III and IV;
+and Plan IV gives promise of being more effective than
+Plan III, since there seems danger that the pupil working
+by Plan III might in the ten weeks lose too much of what
+he had gained in the initial practice, and so again in the next
+ten weeks.</p>
+
+<p><span class='pagenum'><a name="Page_157" id="Page_157">[Pg 157]</a></span></p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig07.jpg" width="800" height="109" alt="Fig. 7." title="" />
+<p class="nblockquot"><b><span class="smcap">Fig. 7.</span>&mdash;Plan I. 200 practices distributed somewhat evenly over 3&frac12; years of 10
+months. In Figs. 7, 8, 9, and 10, each tenth of an inch along the base line
+represents one month. Each hundredth of a square inch represents four practices,
+a little square <sup>1</sup>&frasl;<sub>20</sub> of an inch wide and <sup>1</sup>&frasl;<sub>20</sub> inch high representing one practice.</b></p>
+</div>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig08.jpg" width="800" height="134" alt="Fig. 8." title="" />
+<span class="caption"><span class="smcap">Fig. 8.</span>&mdash;Plan II. 200 practices distributed haphazard over 3&frac12; years of 10 months.</span>
+</div>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig09.jpg" width="800" height="489" alt="Fig. 9." title="" />
+<span class="caption"><span class="smcap">Fig. 9.</span>&mdash;Plan III. A learning period, three reviews, and incidental practice.</span>
+</div>
+
+
+<p>It is not wise, however, to try now to make close decisions
+in the case of practice with division by a fraction; or to
+determine what the best distribution of practice is for that
+or any other ability to be improved. The facts of psychology
+are as yet not adequate for very close decisions, nor are the<span class='pagenum'><a name="Page_158" id="Page_158">[Pg 158]</a></span>
+types of distribution of practice that are best adapted to
+different abilities even approximately worked out.</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig10.jpg" width="800" height="466" alt="Fig. 10." title="" />
+<span class="caption"><span class="smcap">Fig. 10.</span>&mdash;Plan IV.  A learning period with reviews of decreasing length at increasing intervals.</span>
+</div>
+
+
+
+<h3>SAMPLE DISTRIBUTIONS</h3>
+
+<p>Let us rather examine some actual cases of distribution
+of practice found in school work and consider, not the
+attainment of the best possible distribution, but simply
+the avoidance of gross blunders and the attainment of
+reasonable, defensible procedures in this regard.</p>
+
+<p>Figures 11 to 18 show the distribution of examples in multiplication
+with multipliers of various sorts. <i>X</i> stands for any
+digit except zero. <i>O</i> stands for 0. <i>XXO</i> thus means a multiplier
+like 350 or 270 or 160; <i>XOX</i> means multipliers like 407,
+905, or 206; <i>XX</i> means multipliers like 25, 17, 38. Each of
+these diagrams covers approximately 3&frac12; years of school work,
+or from about the middle of grade 3 to the end of grade 6.
+They are made from counts of four textbooks (A, B, C, and
+D), the count being taken for each successive 8 pages.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a>
+<span class='pagenum'><a name="Page_159" id="Page_159">[Pg 159]</a></span>
+Each tenth of an inch along the base line equals 8 pages of the
+text in question.  Each .01 sq. in. equals one example. The
+books, it will be observed, differ in the amount of practice
+given, as well as in the way in which it is distributed.</p>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig11.jpg" width="800" height="380" alt="Fig. 11." title="" />
+<span class="caption"><span class="smcap">Fig. 11.</span>&mdash;Distribution of practise with multipliers of the <i>XX</i> type in the first two books of the three-book text A.</span>
+</div>
+
+
+<p><span class='pagenum'><a name="Page_160" id="Page_160">[Pg 160]</a></span></p>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig12.jpg" width="800" height="427" alt="Fig. 12." title="" />
+<p class="nblockquot"><b><span class="smcap">Fig. 12.</span>&mdash;Same as Fig. 11, but for text B. Following this period come certain pages
+of computation to be used by the teacher at her discretion, containing 24 <i>XX</i>
+multiplications.</b></p>
+</div>
+
+
+<p>These distributions are worthy of careful study; we shall
+note only a few salient facts about them here. Of the distributions
+of multiplications with multipliers of the <i>XX</i>
+type, that of book D (Fig. 14) is perhaps the best. A
+(Fig. 11) has too much of the practice too late; B (Fig. 12)
+gives too little practice in the first learning; C (Fig. 13)
+gives too much in the first learning and in grade 6. Among
+the distributions of multiplication with multipliers of the
+<i>XOX</i> type, that of book D (Fig. 18) is again probably the
+best. A, B, and C (Figs. 15, 16, and 17) have too much
+practice early and too long intervals between reviews.
+Book C (Fig. 17) by a careless oversight has one case of<span class='pagenum'><a name="Page_161" id="Page_161">[Pg 161]</a></span>
+this very difficult process, without any explanation, weeks
+before the process is taught!</p>
+
+<div class="figcenter" style="width: 768px;">
+<img src="images/fig13.jpg" width="768" height="795" alt="Fig. 13." title="" />
+<span class="caption"><span class="smcap">Fig. 13.</span>&mdash;Same as Fig. 11, but for text C.</span>
+</div><p><span class='pagenum'><a name="Page_162" id="Page_162">[Pg 162]</a></span></p>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig14.jpg" width="800" height="333" alt="Fig. 14." title="" />
+<span class="caption"><span class="smcap">Fig. 14.</span>&mdash;Same as Fig. 11, but for text D.</span>
+</div>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig15.jpg" width="800" height="412" alt="Fig. 15." title="" />
+<span class="caption"><span class="smcap">Fig. 15.</span>&mdash;Distribution of practice with multipliers of the <i>XOX</i> type in the first
+two books of the three-book text A.</span>
+</div><p><span class='pagenum'><a name="Page_163" id="Page_163">[Pg 163]</a></span></p>
+
+<div class="figcenter" style="width: 768px;">
+<img src="images/fig16.jpg" width="768" height="804" alt="Fig. 16." title="" />
+<p class="nblockquot"><b><span class="smcap">Fig. 16.</span>&mdash;Same as Fig. 15, but for text B. Following this period come certain
+pages of computation to be used by the teacher at her discretion, containing
+17 <i>XOX</i> multiplications.</b></p>
+</div><p><span class='pagenum'><a name="Page_164" id="Page_164">[Pg 164]</a></span></p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig17.jpg" width="800" height="584" alt="Fig. 17." title="" />
+<span class="caption"><span class="smcap">Fig. 17.</span>&mdash;Same as Fig. 16, but for text C.</span>
+</div>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig18.jpg" width="800" height="220" alt="Fig. 18." title="" />
+<span class="caption"><span class="smcap">Fig. 18.</span>&mdash;Same as Fig. 16, but for text D.</span>
+</div>
+
+
+
+<p>Figures 19, 20, 21, 22, and 23 all concern the first two books
+of the three-book text E.</p>
+
+<p>Figure 19 shows the distribution of practice on 5 &times; 5 in
+the first two books of text E. The plan is the same as in
+Figs. 11 to 18, except that each tenth of an inch along the
+base line represents ten pages. Figure 20 shows the dis<span class='pagenum'><a name="Page_165" id="Page_165">[Pg 165]</a></span>tribution
+of practice on 7 &times; 7; Fig. 21 shows it for 6 &times; 7 and
+7 &times; 6 together. In Figs. 20 and 21 also, 0.1 inch along the
+base line equals ten pages.</p>
+
+<p>Figures 22 and 23 show the distribution of practice on the
+divisions of 72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or
+9, and on the divisions of 81, 82 ... 89 by 9. Each tenth
+of an inch along the base line represents ten pages here
+also.</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig19.jpg" width="800" height="559" alt="Fig. 19." title="" />
+<span class="caption"><span class="smcap">Fig. 19.</span>&mdash;Distribution of practice with 5 &times; 5 in the first two books of the three-book
+text E.</span>
+</div>
+
+<p>Figures 19 to 23 show no consistent plan for distributing
+practice. With 5 &times; 5 (Fig. 19) the amount of practice increases
+from the first treatment in grade 3 to the end of
+grade 6, so that the distribution would be better if the pupil
+began at the end and went backward! With 7 &times; 7 (Fig. 20)<span class='pagenum'><a name="Page_166" id="Page_166">[Pg 166]</a></span>
+the practice is distributed rather evenly and in small doses.
+With 6 &times; 7 and 7 &times; 6 (Fig. 21) much of it is in very large
+doses. With the divisions (Figs. 22 and 23) the practice is
+distributed more suitably, though in Fig. 23 there is too
+much of it given at one time in the middle of the period.</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig20.jpg" width="800" height="239" alt="Fig. 20." title="" />
+<span class="caption"><span class="smcap">Fig. 20.</span>&mdash;Distribution of practice with 7 &times; 7 in the first two books of text E.</span>
+</div>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig21.jpg" width="800" height="527" alt="Fig. 21." title="" />
+<span class="caption"><span class="smcap">Fig. 21.</span>&mdash;Distribution of practice with 6 &times; 7 or 7 &times; 6 in the first two books of text E.</span>
+</div><p><span class='pagenum'><a name="Page_167" id="Page_167">[Pg 167]</a></span></p>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig22.jpg" width="800" height="374" alt="Fig. 22." title="" />
+<span class="caption"><span class="smcap">Fig. 22.</span>&mdash;Distribution of practice with 72, 73 ... 79 &divide; 8 or 9 in the first two
+books of text E.</span>
+</div>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig23.jpg" width="800" height="233" alt="Fig. 23." title="" />
+<span class="caption"><span class="smcap">Fig. 23.</span>&mdash;Distribution of practice with 81, 82 ... 89 &divide; 9 in the first two books
+of text E.</span>
+</div>
+
+
+<h4>POSSIBLE IMPROVEMENTS</h4>
+
+<p>Even if we knew what the best distribution of practice
+was for each ability of the many to be inculcated by arithmetical
+instruction, we could perhaps not provide it for all
+of them. For, in the first place, the allotments for some of<span class='pagenum'><a name="Page_168" id="Page_168">[Pg 168]</a></span>
+them might interfere with those for others. In the second
+place, there are many other considerations of importance
+in the ordering of topics besides giving the optimal distribution
+of practice to each ability. Such are considerations of
+interest, of welding separate abilities into an integrated
+total ability, and of the limitations due to the school schedule
+with its Saturdays, Sundays, holidays, and vacations.</p>
+
+<p>Improvement can, however, be made over present practice
+in many respects. A scientific examination of the teaching
+of almost any class for a year, or of many of our standard
+instruments of instruction, will reveal opportunities for
+improving the distribution of practice with no sacrifice of
+interest, and with an actual gain in integrated functioning
+arithmetical power. In particular it will reveal cases where
+an ability is given practice and then, never being used again,
+left to die of inactivity. It will reveal cases where an ability
+is given practice and then left so long without practice that
+the first effect is nearly lost. There will be cases where
+practice is given and reviews are given, but all in such isolation
+from everything else in arithmetic that the ability, though
+existent, does not become a part of the pupil's general working
+equipment. There will be cases where more practice
+is given in the late than the earlier periods for no apparent
+extrinsic advantage; and cases where the practice is put
+where it is for no reason that is observable save that the
+teacher or author in question has decided to have some drill
+work at that time!</p>
+
+<p>Each ability has its peculiar needs in this matter, and no
+set rules are at present of much value. It will be enough
+for the present if we are aroused to the problem of distribution,
+avoid obvious follies like those just noted, and exercise
+what ingenuity we have.</p>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_169" id="Page_169">[Pg 169]</a></span></p>
+<h2><a name="CHAPTER_IX" id="CHAPTER_IX"></a>CHAPTER IX</h2>
+
+<h3>THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND
+GENERAL NOTIONS IN ARITHMETIC<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a></h3>
+
+
+<h4>RESPONSES TO ELEMENTS AND CLASSES</h4>
+
+<p>The plate which you see, the egg before you at the breakfast
+table, and this page are concrete things, but whiteness, whether
+of plate, egg, or paper, is, we say, an abstract quality. To be able
+to think of whiteness irrespective of any concrete white object is
+to be able to have an abstract idea or notion of white; to be able
+to respond to whiteness, irrespective of whether it is a part of
+china, eggshell, paper or whatever object, is to be able to respond
+to the abstract element of whiteness.</p>
+
+<p>Learning arithmetic involves the formation of very many such
+ideas, the acquisition of very many such powers of response to
+elements regardless of the gross total situations in which they
+appear. To appreciate the fiveness of five boys, five pencils, five
+inches, five rings of a bell; to understand the division into eight
+equal parts of 40 cents, 32 feet, 64 minutes, or 16 ones; to respond
+correctly to the fraction relation in <sup>2</sup>&frasl;<sub>3</sub>, <sup>5</sup>&frasl;<sub>6</sub>,
+<sup>3</sup>&frasl;<sub>4</sub>, <sup>7</sup>&frasl;<sub>12</sub>, <sup>1</sup>&frasl;<sub>8</sub>, or any other; to
+be sensitive to the common element of 9 = 3 &times; 3, 16 = 4 &times; 4,
+625 = 25 &times; 25, .04 = .2 &times; .2,  &frac14; = &frac12; &times; &frac12;,&mdash;these are obvious illustrations.
+All the numbers which the pupil learns to understand and manipulate
+are in fact abstractions; all the operations are abstractions;
+percent, discount, interest, height, length, area, volume, are abstractions;
+sum, difference, product, quotient, remainder, average, are
+facts that concern elements or aspects which may appear with
+countless different concrete surroundings or concomitants.</p>
+
+<p>Towser is a particular dog; your house lot on Elm Street is a
+particular rectangle; Mr. and Mrs. I.S. Peterson and their
+daughter Louise are a particular family of three. In contrast to
+<span class='pagenum'><a name="Page_170" id="Page_170">[Pg 170]</a></span>
+these particulars, we mean by a dog, a rectangle, and a family of
+three, <i>any</i> specimens of these classes of facts. The idea of a dog,
+of rectangles in general, of any family of three is a general notion,
+a concept or idea of a class or species. The ability to respond to
+any dog, or rectangle, or family of three, regardless of which
+particular one it may be, is the general notion in action.</p>
+
+<p>Learning arithmetic involves the formation of very many such
+general notions, such powers of response to any member of a
+certain class. Thus a hundred different sized lots may all be
+responded to as rectangles; <sup>9</sup>&frasl;<sub>18</sub>, <sup>12</sup>&frasl;<sub>27</sub>,
+<sup>15</sup>&frasl;<sub>24</sub>, and <sup>27</sup>&frasl;<sub>36</sub> may all be responded
+to as members of the class, 'both members divisible by 3.' The
+same fact may be responded to in different ways according to the
+class to which it is assigned. Thus 4 in <sup>3</sup>&frasl;<sub>4</sub>, <sup>4</sup>&frasl;<sub>5</sub>, 45, 54, and 405 is
+classed respectively as 'a certain sized part of unity,' 'a certain
+number of parts of the size shown by the 5,' 'a certain number
+of tens,' 'a certain number of ones,' and 'a certain number of
+hundreds.' Each abstract quality may become the basis of a
+class of facts. So fourness as a quality corresponds to the class
+'things four in number or size'; the fractional quality or relation
+corresponds to the class 'fractions.' The bonds formed with
+classes of facts and with elements or features by which one whole
+class of facts is distinguished from another, are in fact, a chief concern
+of arithmetical learning.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a></p>
+
+
+<h4>FACILITATING THE ANALYSIS OF ELEMENTS</h4>
+
+<p>Abstractions and generalizations then depend upon analysis
+and upon bonds formed with more or less subtle elements rather
+than with gross total concrete situations. The process involved is
+most easily understood by considering the means employed to
+facilitate it.</p>
+
+<p>The first of these is having the learner respond to the total
+situations containing the element in question with the attitude
+of piecemeal examination, and with attentiveness to one element
+after another, especially to so near an approximation to the
+element in question as he can already select for attentive examination.
+This attentiveness to one element after another serves to
+<span class='pagenum'><a name="Page_171" id="Page_171">[Pg 171]</a></span>
+emphasize whatever appropriate minor bonds from the element
+in question the learner already possesses. Thus, in teaching
+children to respond to the 'fiveness' of various collections, we
+show five boys or five girls or five pencils, and say, "See how
+many boys are standing up. Is Jack the only boy that is standing
+here? Are there more than two boys standing? Name the boys
+while I point at them and count them. (Jack) is one, and (Fred)
+is one more, and (Henry) is one more. Jack and Fred make (two)
+boys. Jack and Fred and Henry make (three) boys." (And so on
+with the attentive counting.) The mental set or attitude is
+directed toward favoring the partial and predominant activity
+of 'how-many-ness' as far as may be; and the useful bonds that
+the 'fiveness,' the 'one and one and one and one and one-ness,'
+already have, are emphasized as far as may be.</p>
+
+<p>The second of the means used to facilitate analysis is having
+the learner respond to many situations each containing the element
+in question (call it A), but with varying concomitants (call these
+V. C.) his response being so directed as, so far as may be, to separate
+each total response into an element bound to the A and an
+element bound to the V. C.</p>
+
+<p>Thus the child is led to associate the responses&mdash;'Five boys,'
+'Five girls,' 'Five pencils,' 'Five inches,' 'Five feet,' 'Five books,'
+'He walked five steps,' 'I hit my desk five times,' and the like&mdash;each
+with its appropriate situation. The 'Five' element of the
+response is thus bound over and over again to the 'fiveness'
+element of the situation, the mental set being 'How many?,' but
+is bound only once to any one of the concomitants. These concomitants
+are also such as have preferred minor bonds of their
+own (the sight of a row of boys <i>per se</i> tends strongly to call up the
+'Boys' element of the response). The other elements of the
+responses (boys, girls, pencils, etc.) have each only a slight connection
+with the 'fiveness' element of the situations. These slight
+connections also in large part<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a> counteract each other, leaving the
+field clear for whatever uninhibited bond the 'fiveness' has.</p>
+
+<p>The third means used to facilitate analysis is having the learner
+respond to situations which, pair by pair, present the element
+in a certain context and present that same context with <i>the opposite
+of the element in question</i>, or with something at least very unlike the
+element. Thus, a child who is being taught to respond to 'one
+fifth' is not only led to respond to 'one fifth of a cake,' 'one
+<span class='pagenum'><a name="Page_172" id="Page_172">[Pg 172]</a></span>
+fifth of a pie,' 'one fifth of an apple,' 'one fifth of ten inches,'
+'one fifth of an army of twenty soldiers,' and the like; he is also
+led to respond to each of these <i>in contrast with</i> 'five cakes,' 'five
+pies,' 'five apples,' 'five times ten inches,' 'five armies of
+twenty soldiers.' Similarly the 'place values' of tenths,
+hundredths, and the rest are taught by contrast with the tens,
+hundreds, and thousands.</p>
+
+<p>These means utilize the laws of connection-forming to disengage
+a response element from gross total responses and attach
+it to some situation element. The forces of use, disuse, satisfaction,
+and discomfort are so maneuvered that an element which
+never exists by itself in nature can influence man almost as if it
+did so exist, bonds being formed with it that act almost or quite
+irrespective of the gross total situation in which it inheres. What
+happens can be most conveniently put in a general statement by
+using symbols.</p>
+
+<p>Denote by <i>a &#43; b</i>, <i>a &#43; g</i>, <i>a &#43; l</i>, <i>a &#43; q</i>, <i>a &#43; v</i>, and <i>a &#43; B</i> certain
+situations alike in the element <i>a</i> and different in all else. Suppose
+that, by original nature or training, a child responds to these
+situations respectively by <i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>7</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>12</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>17</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>22</sub>,
+<i>r</i><sub>1</sub> &#43; <i>r</i><sub>27</sub>.  Suppose that man's neurones are capable of such action
+that <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>7</sub>, <i>r</i><sub>12</sub>, <i>r</i><sub>22</sub>, and <i>r</i><sub>27</sub>, can each be made singly.</p>
+
+
+<p class="center">Case I. Varying Concomitants</p>
+
+<p>Suppose that <i>a</i> &#43; <i>b</i>, <i>a</i> &#43; <i>g</i>, <i>a</i> &#43; <i>l</i>, etc., occur once each.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'>We have</td><td align='left'><i>a</i> &#43; <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>g</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>7</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>l</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>12</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>q</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>17</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>v</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>22</sub>, and</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>B</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>27</sub>, as shown in Scheme I.</td></tr>
+</table></div>
+
+<p class="center">Scheme I</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='center'></td><td align='center'><i>a</i></td><td align='center'><i>b</i></td><td align='center'><i>g</i></td><td align='center'><i>l</i></td><td align='center'><i>q</i></td><td align='center'><i>v</i></td><td align='center'><i>B</i></td></tr>
+<tr><td align='center'><i>r</i><sub>1</sub></td><td align='center'>6</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr>
+<tr><td align='center'><i>r</i><sub>2</sub></td><td align='center'>1</td><td align='center'>1</td></tr>
+<tr><td align='center'><i>r</i><sub>7</sub></td><td align='center'>1</td><td></td><td align='center'>1</td></tr>
+<tr><td align='center'><i>r</i><sub>12</sub></td><td align='center'>1</td><td></td><td></td><td align='center'>1</td></tr>
+<tr><td align='center'><i>r</i><sub>17</sub></td><td align='center'>1</td><td></td><td></td><td></td><td align='center'>1</td></tr>
+<tr><td align='center'><i>r</i><sub>22</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr>
+<tr><td align='center'><i>r</i><sub>27</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr>
+</table></div>
+<p><span class='pagenum'><a name="Page_173" id="Page_173">[Pg 173]</a></span></p>
+
+<p class="noidt"><i>a</i> is thus responded to by <i>r</i><sub>1</sub> (that is, connected with <i>r</i><sub>1</sub>) each time,
+or six in all, but only once each with <i>b</i>, <i>g</i>, <i>l</i>, <i>q</i>, <i>v</i>, and <i>B</i>. <i>b</i>, <i>g</i>, <i>l</i>, <i>q</i>, <i>v</i>,
+and <i>B</i> are connected once each with <i>r</i><sub>1</sub> and once respectively with
+<i>r</i><sub>2</sub>, <i>r</i><sub>7</sub>, <i>r</i><sub>12</sub>, etc. The bond from <i>a</i> to <i>r</i><sub>1</sub>, has had six times as much
+exercise as the bond from <i>a</i> to <i>r</i><sub>2</sub>, or from <i>a</i> to <i>r</i><sub>7</sub>, etc. In any new
+gross situation, <i>a</i> 0, <i>a</i> will be more predominant in determining
+response than it would otherwise have been; and <i>r</i><sub>1</sub> will be more
+likely to be made than <i>r</i><sub>2</sub>, <i>r</i><sub>7</sub>, <i>r</i><sub>12</sub>, etc., the other previous associates
+in the response to a situation containing <i>a</i>. That is, the bond
+from the element <i>a</i> to the response <i>r</i><sub>1</sub> has been notably strengthened.</p>
+
+
+<p class="center">Case II. Contrasting Concomitants</p>
+
+<p>Now suppose that <i>b</i> and <i>g</i> are very dissimilar elements (<i>e.g.</i>,
+white and black), that <i>l</i> and <i>q</i> are very dissimilar (<i>e.g.</i>, long and
+short), and that <i>v</i> and <i>B</i> are also very dissimilar. To be very
+dissimilar means to be responded to very differently, so that <i>r</i><sub>7</sub>,
+the response to <i>g</i>, will be very unlike <i>r</i><sub>2</sub>, the response to <i>b</i>. So <i>r</i><sub>7</sub>
+may be thought of as <i>r</i><sub>not 2</sub> or <i>r</i><sub>-2</sub>. In the same way <i>r</i><sub>12</sub> may be
+thought of as <i>r</i><sub>not 12</sub> or <i>r</i><sub>-12</sub>, and <i>r</i><sub>27</sub> may be called <i>r</i><sub>not 22</sub> or <i>r</i><sub>-22</sub>.</p>
+
+<p>Then, if the situations <i>a b</i>, <i>a g</i>, <i>a l</i>, <i>a q</i>, <i>a v</i>, and <i>a B</i> are responded
+to, each once, we have:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>g</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>not 2</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>l</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>12</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>q</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>not 12</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>v</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>22</sub>, and</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>B</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>not 22</sub>, as shown in Scheme II.</td></tr>
+</table></div>
+
+<p class="center">Scheme II</p>
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='center'></td><td align='center'><i>a</i></td><td align='center'><i>b</i></td><td align='center'><i>g</i></td><td align='center'><i>l</i></td><td align='center'><i>q</i></td><td align='center'><i>v</i></td><td align='center'><i>B</i></td></tr>
+<tr><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'>(opp. of <i>b</i>)</td><td align='center'></td><td align='center'>(opp. of <i>l</i>)</td><td align='center'></td><td align='center'>(opp. of <i>v</i>)</td></tr>
+<tr><td align='left'><i>r</i><sub>1</sub></td><td align='center'>6</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>not 1</sub></td></tr>
+<tr><td align='left'><i>r</i><sub>2</sub></td><td align='center'>1</td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>not 2</sub></td><td align='center'>1</td><td></td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>12</sub></td><td align='center'>1</td><td></td><td></td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>not 12</sub></td><td align='center'>1</td><td></td><td></td><td></td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>22</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>not 22</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr>
+</table></div>
+
+
+<p class="noidt"><i>r</i><sub>1</sub> is connected to <i>a</i> by 6 repetitions. <i>r</i><sub>2</sub> and <i>r</i><sub>not 2</sub> are each connected
+to <i>a</i> by 1 repetition, but since they interfere, canceling each<span class='pagenum'><a name="Page_174" id="Page_174">[Pg 174]</a></span>
+other so to speak, the net result is for <i>a</i> to have zero tendency to
+call up <i>r</i><sub>2</sub> or <i>r</i><sub>not 2</sub>. <i>r</i><sub>12</sub> and <i>r</i><sub>not 12</sub> are each connected to <i>a</i> by 1
+repetition, but they interfere with or cancel each other with the
+net result that <i>a</i> has zero tendency to call up <i>r</i><sub>12</sub> or <i>r</i><sub>not 12</sub>. So
+with <i>r</i><sub>22</sub> and <i>r</i><sub>not 22</sub>. Here then the net result of the six connections
+of <i>a b</i>, <i>a g</i>, <i>a l</i>, <i>a q</i>, <i>a v</i>, and <i>a B</i> is to connect <i>a</i> with <i>r</i>, and with
+nothing else.</p>
+
+
+<p class="center">Case III. Contrasting Concomitants and Contrasting Element</p>
+
+<p>Suppose now that the facts are as in Case II, but with the
+addition of six experiences where a certain element which is the
+opposite of, or very dissimilar to, <i>a</i> is connected with the response
+<i>r</i><sub>not 1</sub>, or <i>r</i><sub>-1</sub> which is opposite to, or very dissimilar to <i>r</i><sub>1</sub>. Call
+this opposite of <i>a</i>, &minus; <i>a</i>.</p>
+
+<p>That is, we have not only</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>g</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>not 2</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>l</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>12</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>q</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>not 12</sub>,</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>v</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>22</sub>, and</td></tr>
+<tr><td align='left'></td><td align='left'><i>a</i> &#43; <i>B</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>1</sub> &#43; <i>r</i><sub>not 22</sub>,</td></tr>
+</table></div>
+
+<p class="noidt">but also</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'>&minus; <i>a</i> &#43; <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>not 1</sub> &#43; <i>r</i><sub>2</sub>,</td></tr>
+<tr><td align='left'>&minus; <i>a</i> &#43; <i>g</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>not 1</sub> &#43; <i>r</i><sub>not 2</sub>,</td></tr>
+<tr><td align='left'>&minus; <i>a</i> &#43; <i>l</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>not 1</sub> &#43; <i>r</i><sub>12</sub>,</td></tr>
+<tr><td align='left'>&minus; <i>a</i> &#43; <i>q</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>not 1</sub> &#43; <i>r</i><sub>not 12</sub>,</td></tr>
+<tr><td align='left'>&minus; <i>a</i> &#43; <i>v</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>not 1</sub> &#43; <i>r</i><sub>22</sub>, and</td></tr>
+<tr><td align='left'>&minus; <i>a</i> &#43; <i>B</i></td><td align='center'>" &nbsp; &nbsp; &nbsp; &nbsp; "</td><td align='left'><i>r</i><sub>not 1</sub> &#43; <i>r</i><sub>not 22</sub>, as shown in Scheme III.</td></tr>
+</table></div>
+
+
+<p class="center">Scheme III</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='center'></td><td align='center'><i>a</i></td><td align='center'><i>opp.</i></td><td align='center'><i>b</i></td><td align='center'><i>g</i></td><td align='center'><i>l</i></td><td align='center'><i>q</i></td><td align='center'><i>v</i></td><td align='center'><i>B</i></td></tr>
+<tr><td align='center'></td><td align='center'></td><td align='center'><i>of a</i></td><td align='center' colspan='2'>(opp. of <i>b</i>)</td><td align='center' colspan='2'>(opp. of <i>l</i>)</td><td align='center' colspan='2'>(opp. of <i>v</i>)</td></tr>
+<tr><td align='left'><i>r</i><sub>1</sub></td><td align='center'>6</td><td></td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>not 1</sub></td><td></td><td align='center'>6</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr>
+<tr><td align='left'><i>r</i><sub>2</sub></td><td align='center'>1</td><td align='center'>1</td><td align='center'>2</td></tr>
+<tr><td align='left'><i>r</i><sub>not 2</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td align='center'>2</td></tr>
+<tr><td align='left'><i>r</i><sub>12</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td align='center'>2<span class='pagenum'><a name="Page_175" id="Page_175">[Pg 175]</a></span></td></tr>
+<tr><td align='left'><i>r</i><sub>not 12</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td></td><td align='center'>2</td></tr>
+<tr><td align='left'><i>r</i><sub>22</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td></td><td></td><td align='center'>2</td></tr>
+<tr><td align='left'><i>r</i><sub>not 22</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td></td><td></td><td></td><td align='center'>2</td></tr>
+</table></div>
+
+
+
+<p>In this series of twelve experiences <i>a</i> connects with <i>r</i><sub>1</sub> six times
+and the opposite of <i>a</i> connects with <i>r</i><sub>not 1</sub> six times. <i>a</i> connects
+equally often with three pairs of mutual destructives <i>r</i><sub>2</sub> and <i>r</i><sub>not 2</sub>,
+<i>r</i><sub>12</sub> and <i>r</i><sub>not 12</sub>, <i>r</i><sub>22</sub> and <i>r</i><sub>not 22</sub>, and so has zero tendency to call
+them up. &minus; <i>a</i> has also zero tendency to call up any of these
+responses except its opposite, <i>r</i><sub>not 1</sub>. <i>b</i>, <i>g</i>, <i>l</i>, <i>q</i>, <i>v</i>, and <i>B</i> are made
+to connect equally often with <i>r</i><sub>1</sub> and <i>r</i><sub>not 1</sub>. So, of these elements,
+<i>a</i> is the only one left with a tendency to call up <i>r</i><sub>1</sub>.</p>
+
+<p>Thus, by the mere action of frequency of connection, <i>r</i><sub>1</sub> is connected
+with <i>a</i>; the bonds from <i>a</i> to anything except <i>r</i><sub>1</sub> are being
+counteracted, and the slight bonds from anything except <i>a</i> to <i>r</i><sub>1</sub>
+are being counteracted. The element <i>a</i> becomes predominant
+in situations containing it; and its bond toward <i>r</i><sub>1</sub> becomes
+relatively enormously strengthened and freed from competition.</p>
+
+<p>These three processes occur in a similar, but more complicated,
+form if the situations <i>a</i> &#43; <i>b</i>, <i>a</i> &#43; <i>g</i>, etc., are replaced by <i>a</i> &#43; <i>b</i> &#43; <i>c</i> &#43; <i>d</i> &#43; <i>e</i> &#43; <i>f</i>,
+<i>a</i> &#43; <i>g</i> &#43; <i>h</i> &#43; <i>i</i> &#43; <i>j</i> &#43; <i>k</i>, etc., and the responses <i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>7</sub>,
+<i>r</i><sub>1</sub> &#43; <i>r</i><sub>12</sub>, etc., are replaced by <i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub> &#43; <i>r</i><sub>3</sub> &#43; <i>r</i><sub>4</sub> &#43; <i>r</i><sub>5</sub> &#43; <i>r</i><sub>6</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>7</sub> &#43; <i>r</i><sub>8</sub> &#43; <i>r</i><sub>9</sub> &#43; <i>r</i><sub>10</sub> &#43; <i>r</i><sub>11</sub>,
+etc.&mdash;<i>provided the r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub>, etc., <i>can be made singly</i>.
+In so far as any one of the responses is necessarily co-active with
+any one of the others (so that, for example, <i>r</i><sub>13</sub> always brings <i>r</i><sub>26</sub>
+with it and <i>vice versa</i>), the exact relations of the numbers recorded
+in schemes like schemes I, II, and III on pages 172 to 174 will
+change; but, unless <i>r</i><sub>1</sub> has such an inevitable co-actor, the general
+results of schemes I, II, and III will hold good. If <i>r</i><sub>1</sub> does have
+such an inseparable co-actor, say <i>r</i><sub>2</sub>, then, of course, <i>a</i> can never
+acquire bonds with <i>r</i><sub>1</sub> alone, but everywhere that <i>r</i><sub>1</sub> or <i>r</i><sub>2</sub> appears
+in the preceding schemes the other element must appear also.
+<i>r</i><sub>1</sub> <i>r</i><sub>2</sub> would then have to be used as a unit in analysis.</p>
+
+<p>The '<i>a</i> &#43; <i>b</i>,' '<i>a</i> &#43; <i>g</i>,' '<i>a</i> &#43; <i>l</i>,' ... '<i>a</i> &#43; <i>B</i>' situations may occur
+unequal numbers of times, altering the exact numerical relations
+of the connections formed and presented in schemes I, II, and III;
+but the process in general remains the same.</p>
+
+<p>So much for the effect of use and disuse in attaching appropriate
+response elements to certain subtle elements of situations. There
+are three main series of effects of satisfaction and discomfort.<span class='pagenum'><a name="Page_176" id="Page_176">[Pg 176]</a></span>
+They serve, first, to emphasize, from the start, the desired bonds
+leading to the responses <i>r</i><sub>1</sub> &#43; <i>r</i><sub>2</sub>, <i>r</i><sub>1</sub> &#43; <i>r</i><sub>7</sub>, etc., to the total situations,
+and to weed out the undesirable ones. They also act to emphasize,
+in such comparisons and contrasts as have been described, every
+action of the bond from <i>a</i> to <i>r</i><sub>1</sub>; and to eliminate every tendency
+of <i>a</i> to connect with aught save <i>r</i><sub>1</sub>, and of aught save <i>a</i> to
+connect with <i>r</i><sub>1</sub>. Their third service is to strengthen the bonds
+produced of appropriate responses to <i>a</i> wherever it occurs,
+whether or not any formal comparisons and contrasts take place.</p>
+
+<p>The process of learning to respond to the difference of pitch
+in tones from whatever instrument, to the 'square-root-ness' of
+whatever number, to triangularity in whatever size or combination
+of lines, to equality of whatever pairs, or to honesty in whatever
+person or instance, is thus a consequence of associative learning,
+requiring no other forces than those of use, disuse, satisfaction,
+and discomfort. "What happens in such cases is that the response,
+by being connected with many situations alike in the
+presence of the element in question and different in other respects,
+is bound firmly to that element and loosely to each of its concomitants.
+Conversely any element is bound firmly to any one
+response that is made to all situations containing it and very, very
+loosely to each of those responses that are made to only a few
+of the situations containing it. The element of triangularity, for
+example, is bound firmly to the response of saying or thinking
+'triangle' but only very loosely to the response of saying or
+thinking white, red, blue, large, small, iron, steel, wood, paper,
+and the like. A situation thus acquires bonds not only with some
+response to it as a gross total, but also with responses to any of
+its elements that have appeared in any other gross totals. Appropriate
+response to an element regardless of its concomitants is a
+necessary consequence of the laws of exercise and effect if an animal
+learns to make that response to the gross total situations that
+contain the element and not to make it to those that do not.
+Such prepotent determination of the response by one or another
+element of the situation is no transcendental mystery, but, given
+the circumstances, a general rule of all learning." Such are at
+bottom only extreme cases of the same learning as a cat exhibits
+that depresses a platform in a certain box whether it faces north
+or south, whether the temperature is 50 or 80 degrees, whether one
+or two persons are in sight, whether she is exceedingly or moderately
+hungry, whether fish or milk is outside the box. All learning is
+analytic, representing the activity of elements within a total
+situation. In man, by virtue of certain instincts and the course<span class='pagenum'><a name="Page_177" id="Page_177">[Pg 177]</a></span>
+of his training, very subtle elements of situations can so operate.</p>
+
+<hr style='width: 45%;' />
+
+<p>Learning by analysis does not often proceed in the carefully
+organized way represented by the most ingenious
+marshaling of comparing and contrasting activities. The
+associations with gross totals, whereby in the end an element
+is elevated to independent power to determine response,
+may come in a haphazard order over a long interval
+of time. Thus a gifted three-year-old boy will have the
+response element of 'saying or thinking <i>two</i>,' bound to the
+'two-ness' element of very many situations in connection
+with the 'how-many' mental set; and he will have made
+this analysis without any formal, systematic training. An
+imperfect and inadequate analysis already made is indeed
+usually the starting point for whatever systematic abstraction
+the schools direct. Thus the kindergarten exercises
+in analyzing out number, color, size, and shape commonly
+assume that 'one-ness' <i>versus</i> 'more-than-one-ness,' black
+and white, big and little, round and not round are, at least
+vaguely, active as elements responded to in some independence
+of their contexts. Moreover, the tests of actual
+trial and success in further undirected exercises usually
+co&ouml;perate to confirm and extend and refine what the systematic
+drills have given. Thus the ordinary child in school
+is left, by the drills on decimal notation, with only imperfect
+power of response to the 'place-values.' He continues to
+learn to respond properly to them by finding that 4 &times; 40
+= 160, 4 &times; 400 = 1600, 800 &minus; 80 = 720, 800 &minus; 8 = 792, 800 &minus; 800
+= 0, 42 &times; 48 = 2016, 24 &times; 48 = 1152, and the like, are
+satisfying; while 4 &times; 40 = 16, 23 &times; 48 = 832, 800 &minus; 8 = 0,
+and the like, are not. The process of analysis is
+the same in such casual, unsystematized formation of
+connections with elements as in the deliberately man<span class='pagenum'><a name="Page_178" id="Page_178">[Pg 178]</a></span>aged,
+piecemeal inspection, comparison, and contrast described
+above.</p>
+
+
+<h4>SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS</h4>
+
+<p>The arrangement of a pupil's experiences so as to direct
+his attention to an element, vary its concomitants instructively,
+stimulate comparison, and throw the element into relief
+by contrast may be by fixed, formal, systematic exercises.
+Or it may be by much less formal exercises, spread
+over a longer time, and done more or less incidentally in
+other connections. We may call these two extremes the
+'systematic' and 'opportunistic,' since the chief feature of
+the former is that it systematically provides experiences designed
+to build up the power of correct response to the element,
+whereas the chief feature of the latter is that it uses
+especially such opportunities as occur by reason of the
+pupil's activities and interests.</p>
+
+<p>Each method has its advantages and disadvantages.
+The systematic method chooses experiences that are specially
+designed to stimulate the analysis; it provides these at a
+certain fixed time so that they may work together; it can
+then and there test the pupils to ascertain whether they
+really have the power to respond to the element or aspect
+or feature in question. Its disadvantages are, first, that
+many of the pupils will feel no need for and attach no interest
+or motive to these formal exercises; second, that some of the
+pupils may memorize the answers as a verbal task instead
+of acquiring insight into the facts; third, that the ability
+to respond to the element may remain restricted to the
+special cases devised for the systematic training, and not be
+available for the genuine uses of arithmetic.</p>
+
+<p>The opportunistic method is strong just where the systematic
+is weak. Since it seizes upon opportunities created<span class='pagenum'><a name="Page_179" id="Page_179">[Pg 179]</a></span>
+by the pupil's abilities and interests, it has the attitude of
+interest more often. Since it builds up the experiences
+less formally and over a wider space of time, the pupils are
+less likely to learn verbal answers. Since its material comes
+more from the genuine uses of life, the power acquired is
+more likely to be applicable to life.</p>
+
+<p>Its disadvantage is that it is harder to manage. More
+thought and experimentation are required to find the best
+experiences; greater care is required to keep track of the
+development of an abstraction which is taught not in two
+days, but over two months; and one may forget to test
+the pupils at the end. In so far as the textbook and teacher
+are able to overcome these disadvantages by ingenuity and
+care, the opportunistic method is better.</p>
+
+
+<h4>ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS</h4>
+
+<p>We may expect much improvement in the formation of
+abstract and general ideas in arithmetic from the application
+of three principles in addition to those already described.
+They are: (1) Provide enough actual experiences before
+asking the pupil to understand and use an abstract or general
+idea. (2) Develop such ideas gradually, not attempting to
+give complete and perfect ideas all at once. (3) Develop
+such ideas so far as possible from experiences which will be
+valuable to the pupil in and of themselves, quite apart from
+their merit as aids in developing the abstraction or general
+notion. Consider these three principles in order.</p>
+
+<p>Children, especially the less gifted intellectually, need
+more experiences as a basis for and as applications of an
+arithmetical abstraction or concept than are usually given
+them. For example, in paving the way for the principle,
+"Any number times 0 equals 0," it is not safe to say, "John
+worked 8 days for 0 minutes per day. How many minutes<span class='pagenum'><a name="Page_180" id="Page_180">[Pg 180]</a></span>
+did he work?" and "How much is 0 times 4 cents?" It
+will be much better to spend ten or fifteen minutes as follows:<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a>
+"What does zero mean? (Not any. No.) How
+many feet are there in eight yards? In 5 yards? In 3
+yards? In 2 yards? In 1 yard? In 0 yard? How many
+inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = ....
+qt. 5 pk. = .... qt. 0 pk. = .... qt. A boy receives
+60 cents an hour when he works. How much does he receive
+when he works 3 hr.? 8 hr.? 6 hr.? 0 hr.? A
+boy received 60 cents a day for 0 days. How much did he
+receive? How much is 0 times $600? How much is 0
+times $5000? How much is 0 times a million dollars? 0
+times any number equals....</p>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'>&nbsp; 232</td><td align='left'>(At the blackboard.) 0 time 232 equals what?</td></tr>
+<tr><td align='left'> &nbsp; &nbsp; 30</td><td align='left'>I write 0 under the 0.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> 3 times 232 equals what?</td></tr>
+<tr><td align='left'>&mdash;&mdash;</td></tr>
+<tr><td align='left'>6960</td><td align='left'>Continue at the blackboard with</td></tr>
+</table></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='right'></td><td align='right'>734</td><td align='right'>321</td><td align='right'>312</td><td align='right'>41</td></tr>
+<tr><td align='right'></td><td align='right'>20</td><td align='right'>40</td><td align='right'>30</td><td align='right'>60</td><td align='right'>etc."</td></tr>
+<tr><td align='right'></td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td><td align='right'>&mdash;&mdash;</td></tr>
+</table></div>
+
+<p>Pupils in the elementary school, except the most gifted,
+should not be expected to gain mastery over such concepts
+as <i>common fraction</i>, <i>decimal fraction</i>, <i>factor</i>, and <i>root</i> quickly.
+They can learn a definition quickly and learn to use it in
+very easy cases, where even a vague and imperfect understanding
+of it will guide response correctly. But complete
+<span class='pagenum'><a name="Page_181" id="Page_181">[Pg 181]</a></span>
+and exact understanding commonly requires them to take,
+not one intellectual step, but many; and mastery in use
+commonly comes only as a slow growth. For example,
+suppose that pupils are taught that .1, .2, .3, etc., mean <sup>1</sup>&frasl;<sub>10</sub>,
+<sup>2</sup>&frasl;<sub>10</sub>, <sup>3</sup>&frasl;<sub>10</sub>, etc., that .01, .02, .03, etc., mean <sup>1</sup>&frasl;<sub>100</sub>,
+<sup>2</sup>&frasl;<sub>100</sub>, <sup>3</sup>&frasl;<sub>100</sub>,
+etc., that .001, .002, .003, etc., mean <sup>1</sup>&frasl;<sub>1000</sub>,
+<sup>2</sup>&frasl;<sub>1000</sub>, <sup>3</sup>&frasl;<sub>1000</sub>, etc.,
+and that .1, .02, .001, etc., are decimal fractions. They
+may then respond correctly when asked to write a decimal
+fraction, or to state which of these,&mdash;<sup>1</sup>&frasl;<sub>4</sub>, .4, <sup>3</sup>&frasl;<sub>8</sub>, .07, .002,
+<sup>5</sup>&frasl;<sub>6</sub>,&mdash;are
+common fractions and which are decimal fractions.
+They may be able, though by no means all of them will be,
+to write decimal fractions which equal <sup>1</sup>&frasl;<sub>2</sub> and <sup>1</sup>&frasl;<sub>5</sub>, and the
+common fractions which equal .1 and .09. Most of them
+will not, however, be able to respond correctly to "Write
+a decimal mixed number"; or to state which of these,&mdash;<sup>1</sup>&frasl;<sub>100</sub>
+.4&frac12;, <sup>.007</sup>&frasl;<sub>350</sub>, $.25,&mdash;are common fractions, and which are
+decimals; or to write the decimal fractions which equal <sup>3</sup>&frasl;<sub>4</sub>
+and <sup>1</sup>&frasl;<sub>3</sub>.</p>
+
+<p>If now the teacher had given all at once the additional
+experiences needed to provide the ability to handle these
+more intricate and subtle features of decimal-fraction-ness,
+the result would have been confusion for most pupils. The
+general meaning of .32, .14, .99, and the like requires some
+understanding of .30, .10, .90, and .02, .04, .08; but it is
+not desirable to disturb the child with .30 while he is trying
+to master 2.3, 4.3, 6.3, and the like. Decimals in general
+require connection with place value and the contrasts of
+.41 with 41, 410, 4.1, and the like, but if the relation to place
+values in general is taught in the same lesson with the relation
+to &frasl;<sub>10</sub>s, &frasl;<sub>100</sub>s, &frasl;<sub>1000</sub>s, the mind will suffer from
+violent indigestion.</p>
+
+<p>A wise pedagogy in fact will break up the process of learn<span class='pagenum'><a name="Page_182" id="Page_182">[Pg 182]</a></span>ing
+the meaning and use of decimal fractions into many
+teaching units, for example, as follows:&mdash;</p>
+
+<p>(1) Such familiarity with fractions with large denominators
+as is desirable for pupils to have, as by an exercise
+in reducing to lowest terms,
+<sup>8</sup>&frasl;<sub>10</sub>, <sup>36</sup>&frasl;<sub>64</sub>,
+<sup>20</sup>&frasl;<sub>25</sub>, <sup>18</sup>&frasl;<sub>24</sub>,
+<sup>24</sup>&frasl;<sub>32</sub>, <sup>21</sup>&frasl;<sub>30</sub>,
+<sup>25</sup>&frasl;<sub>100</sub>, <sup>40</sup>&frasl;<sub>100</sub>,
+and the like. This is good as a review of cancellation, and
+as an extension of the idea of a fraction.</p>
+
+<p>(2) Objective work, showing <sup>1</sup>&frasl;<sub>10</sub> sq. ft., <sup>1</sup>&frasl;<sub>50</sub> sq. ft., <sup>1</sup>&frasl;<sub>100</sub>
+sq. ft., and <sup>1</sup>&frasl;<sub>1000</sub> sq. ft., and having these identified and the
+forms <sup>1</sup>&frasl;<sub>10</sub> sq. ft., <sup>1</sup>&frasl;<sub>100</sub> sq. ft., and <sup>1</sup>&frasl;<sub>1000</sub> sq. ft. learned. Finding
+how many feet = <sup>1</sup>&frasl;<sub>10</sub> mile and <sup>1</sup>&frasl;<sub>100</sub> mile.</p>
+
+<p>(3) Familiarity with &nbsp; &frasl;<sub>100</sub>s and &nbsp; &frasl;<sub>1000</sub>s by reductions of
+<sup>750</sup>&frasl;<sub>1000</sub>, <sup>50</sup>&frasl;<sub>100</sub>, etc., to lowest terms and by writing the missing
+numerators in <sup>500</sup>&frasl;<sub>1000</sub> = &nbsp;&frasl;<sub>100</sub> = &nbsp;&frasl;<sub>10</sub> and the like, and by finding
+<sup>1</sup>&frasl;<sub>10</sub>, <sup>1</sup>&frasl;<sub>100</sub>, and <sup>1</sup>&frasl;<sub>1000</sub> of 3000, 6000, 9000, etc.</p>
+
+<p>(4) Writing <sup>1</sup>&frasl;<sub>10</sub> as .1 and <sup>1</sup>&frasl;<sub>100</sub> as .01,
+<sup>11</sup>&frasl;<sub>100</sub>, <sup>12</sup>&frasl;<sub>100</sub>, <sup>13</sup>&frasl;<sub>100</sub>, etc., as
+.11, .12, .13. United States money is used as the introduction.
+Application is made to miles.</p>
+
+<p>(5) Mixed numbers with a first decimal place. The
+cyclometer or speedometer. Adding numbers like 9.1,
+14.7, 11.4, etc.</p>
+
+<p>(6) Place value in general from thousands to hundredths.</p>
+
+<p>(7) Review of (1) to (6).</p>
+
+<p>(8) Tenths and hundredths of a mile, subtraction when
+both numbers extend to hundredths, using a railroad table
+of distances.</p>
+
+<p>(9) Thousandths. The names 'decimal fractions or
+decimals,' and 'decimal mixed numbers or decimals.'
+Drill in reading any number to thousandths. The work
+will continue with gradual extension and refinement of the
+understanding of decimals by learning how to operate with
+them in various ways.</p>
+
+<p>Such may seem a slow progress, but in fact it is not, and<span class='pagenum'><a name="Page_183" id="Page_183">[Pg 183]</a></span>
+many of these exercises whereby the pupil acquires his
+mastery of decimals are useful as organizations and applications
+of other arithmetical facts.</p>
+
+<p>That, it will be remembered, was the third principle:&mdash;"Develop
+abstract and general ideas by experiences which
+will be intrinsically valuable." The reason is that, even
+with the best of teaching, some pupils will not, within any
+reasonable limits of time expended, acquire ideas that are
+fully complete, rigorous when they should be, flexible when
+they should be, and absolutely exact. Many children (and
+adults, for that matter) could not within any reasonable
+limits of time be so taught the nature of a fraction that
+they could decide unerringly in original exercises like:&mdash;</p>
+
+<p>Is <sup>2.75</sup>&frasl;<sub>25</sub> a common fraction?</p>
+
+<p>Is $.25 a decimal fraction?</p>
+
+<p>Is one <i>x</i>th of <i>y</i> a fraction?</p>
+
+<p>Can the same words mean both a common fraction and a
+decimal fraction?</p>
+
+<p>Express 1 as a common fraction.</p>
+
+<p>Express 1 as a decimal fraction.</p>
+
+<p>These same children can, however, be taught to operate
+correctly with fractions in the ordinary uses thereof. And
+that is the chief value of arithmetic to them. They should
+not be deprived of it because they cannot master its subtler
+principles. So we seek to provide experiences that will
+teach all pupils something of value, while stimulating in
+those who have the ability the growth of abstract ideas and
+general principles.</p>
+
+<p>Finally, we should bear in mind that working with qualities
+and relations that are only partly understood or even
+misunderstood does under certain conditions give control
+over them. The general process of analytic learning in<span class='pagenum'><a name="Page_184" id="Page_184">[Pg 184]</a></span>
+life is to respond as well as one can; to get a clearer idea
+thereby; to respond better the next time; and so on. For
+instance, one gets some sort of notion of what <sup>1</sup>&frasl;<sub>5</sub> means;
+he then answers such questions as <sup>1</sup>&frasl;<sub>5</sub> of 10 = ? <sup>1</sup>&frasl;<sub>5</sub> of 5 = ? <sup>1</sup>&frasl;<sub>5</sub>
+of 20 = ?; by being told when he is right and when he is
+wrong, he gets from these experiences a better idea of <sup>1</sup>&frasl;<sub>5</sub>;
+again he does his best with <sup>1</sup>&frasl;<sub>5</sub> =  &nbsp;&frasl;<sub>10</sub>, <sup>1</sup>&frasl;<sub>5</sub> = &nbsp;&frasl;<sub>15</sub>, etc., and as before
+refines and enlarges his concept of <sup>1</sup>&frasl;<sub>5</sub>. He adds <sup>1</sup>&frasl;<sub>5</sub> to <sup>2</sup>&frasl;<sub>5</sub>, etc.,
+<sup>1</sup>&frasl;<sub>5</sub> to <sup>3</sup>&frasl;<sub>10</sub>, etc., <sup>1</sup>&frasl;<sub>5</sub> to <sup>1</sup>&frasl;<sub>2</sub>, etc., and thereby gains still further,
+and so on.</p>
+
+<p>What begins as a blind habit of manipulation started by
+imitation may thus grow into the power of correct response
+to the essential element. The pupil who has at the start
+no notion at all of 'multiplying' may learn what multiplying
+is by his experience that '4 6 multiplying gives 24';
+'3 9 multiplying gives 27,' etc. If the pupil keeps on
+doing something with numbers and differentiates right
+results, he will often reach in the end the abstractions which
+he is supposed to need in the beginning. It may even be
+the case with some of the abstractions required in arithmetic
+that elaborate provision for comprehension beforehand is
+not so efficient as the same amount of energy devoted partly
+to provision for analysis itself beforehand and partly to
+practice in response to the element in question without full
+comprehension.</p>
+
+<p>It certainly is not the best psychology and not the best
+educational theory to think that the pupil first masters a
+principle and then merely applies it&mdash;first does some thinking
+and then computes by mere routine. On the contrary,
+the applications should help to establish, extend, and refine
+the principle&mdash;the work a pupil does with numbers should
+be a main means of increasing his understanding of the
+principles of arithmetic as a science.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_185" id="Page_185">[Pg 185]</a></span></p>
+<h2><a name="CHAPTER_X" id="CHAPTER_X"></a>CHAPTER X</h2>
+
+<h3>THE PSYCHOLOGY OF THINKING: REASONING IN
+ARITHMETIC</h3>
+
+
+<h4>THE ESSENTIALS OF ARITHMETICAL REASONING</h4>
+
+<p>We distinguish aimless reverie, as when a child dreams
+of a vacation trip, from purposive thinking, as when he tries
+to work out the answer to "How many weeks of vacation
+can a family have for $120 if the cost is $22 a week for board,
+$2.25 a week for laundry, and $1.75 a week for incidental
+expenses, and if the railroad fares for the round trip are
+$12?" We distinguish the process of response to familiar
+situations, such as five integral numbers to be added, from
+the process of response to novel situations, such as (for a
+child who has not been trained with similar problems):&mdash;"A
+man has four pieces of wire. The lengths are 120 yd.,
+132 meters, 160 feet, and <sup>1</sup>&frasl;<sub>8</sub> mile. How much more does
+he need to have 1000 yd. in all?" We distinguish 'thinking
+things together,' as when a diagram or problem or proof
+is understood, from thinking of one thing after another as
+when a number of words are spelled or a poem in an unknown
+tongue is learned. In proportion as thinking is purposive,
+with selection from the ideas that come up, and in
+proportion as it deals with novel problems for which no
+ready-made habitual response is available, and in proportion
+as many bonds act together in an organized way to
+produce response, we call it reasoning.<span class='pagenum'><a name="Page_186" id="Page_186">[Pg 186]</a></span></p>
+
+<p>When the conclusion is reached as the effect of many
+particular experiences, the reasoning is called inductive.
+When some principle already established leads to another
+principle or to a conclusion about some particular fact, the
+reasoning is called deductive. In both cases the process
+involves the analysis of facts into their elements, the selection
+of the elements that are deemed significant for the
+question at hand, the attachment of a certain amount of
+importance or weight to each of them, and their use in the
+right relations. Thought may fail because it has not suitable
+facts, or does not select from them the right ones, or
+does not attach the right amount of weight to each, or does
+not put them together properly.</p>
+
+<p>In the world at large, many of our failures in thinking
+are due to not having suitable facts. Some of my readers,
+for example, cannot solve the problem&mdash;"What are the
+chances that in drawing a card from an ordinary pack of
+playing-cards four times in succession, the same card will be
+drawn each time?" And it will be probably because they do
+not know certain facts about the theory of probabilities.
+The good thinkers among such would look the matter up
+in a suitable book. Similarly, if a person did not happen
+to know that there were fifty-two cards in all and that no
+two were alike, he could not reason out the answer, no matter
+what his mastery of the theory of probabilities. If a competent
+thinker, he would first ask about the size and nature of
+the pack. In the actual practice of reasoning, that is, we have
+to survey our facts to see if we lack any that are necessary.
+If we do, the first task of reasoning is to acquire those
+facts.</p>
+
+<p>This is specially true of the reasoning about arithmetical
+facts in life. "Will 3&frac12; yards of this be enough for a dress?"
+Reason directs you to learn how wide it is, what style of<span class='pagenum'><a name="Page_187" id="Page_187">[Pg 187]</a></span>
+dress you intend to make of it, how much material that
+style normally calls for, whether you are a careful or a
+wasteful cutter, and how big the person is for whom the
+dress is to be made. "How much cheaper as a diet is bread
+alone, than bread with butter added to the extent of 10% of
+the weight of the bread?" Reason directs you to learn the
+cost of bread, the cost of butter, the nutritive value of bread,
+and the nutritive value of butter.</p>
+
+<p>In the arithmetic of the school this feature of reasoning
+appears in cases where some fact about common measures
+must be brought to bear, or some table of prices or discounts
+must be consulted, or some business custom must be remembered
+or looked up.</p>
+
+<p>Thus "How many badges, each 9 inches long, can be
+made from 2&frac12; yd. ribbon?" cannot be solved without
+getting into mind 1 yd. = 36 inches. "At Jones' prices,
+which costs more, 3&frac34; lb. butter or 6&frac12; lb. lard? How much
+more?" is a problem which directs the thinker to ascertain
+Jones' prices.</p>
+
+<p>It may be noted that such problems are, other things
+being equal, somewhat better training in thinking than
+problems where all the data are given in the problem itself
+(<i>e.g.</i>, "Which costs more, 3&frac34; lb. butter at 48&cent; per lb. or
+6&frac12; lb. lard at 27&cent; per lb.? How much more?"). At least
+it is unwise to have so many problems of the latter sort that
+the pupil may come to think of a problem in applied arithmetic
+as a problem where everything is given and he has
+only to manipulate the data. Life does not present its
+problems so.</p>
+
+<p>The process of selecting the right elements and attaching
+proper weight to them may be illustrated by the following
+problem:&mdash;"Which of these offers would you take, supposing
+that you wish a D.C.K. upright piano, have $50 saved,<span class='pagenum'><a name="Page_188" id="Page_188">[Pg 188]</a></span>
+can save a little over $20 per month, and can borrow from
+your father at 6% interest?"</p>
+
+<div class="pblockquot">
+<p class="center">A</p>
+
+<p>A Reliable Piano. The Famous D.C.K. Upright. You pay
+$50 cash down and $21 a month for only a year and a half. <i>No
+interest</i> to pay. We ask you to pay only for the piano and allow
+you plenty of time.</p>
+
+<p class="center">B</p>
+
+<p>We offer the well-known D.C.K. Piano for $390. $50 cash
+and $20 a month thereafter. Regular interest at 6%. The
+interest soon is reduced to less than $1 a month.</p>
+
+<p class="center">C</p>
+
+<p>The D.C.K. Piano. Special Offer, $375, cash. Compare our
+prices with those of any reliable firm.</p></div>
+
+<p>If you consider chiefly the "only," "No interest to pay,"
+"only," and "plenty of time" in offer A, attaching much
+weight to them and little to the thought, "How much will
+$50 plus <span class="inline_eqn">(18 &times; $21)</span> be?", you will probably decide wrongly.</p>
+
+<p>The situations of life are often complicated by many elements
+of little or even of no relevance to the correct solution.
+The offerer of A may belong to your church; your dearest
+friend may urge you to accept offer B; you may dislike
+to talk with the dealer who makes offer C; you may have
+a prejudice against owing money to a relative; that prejudice
+may be wise or foolish; you may have a suspicion that the
+B piano is shopworn; that suspicion may be well-founded
+or groundless; the salesman for C says, "You don't want
+your friends to say that you bought on the installment plan.
+Only low-class persons do that," etc. The statement of
+arithmetical problems in school usually assists the pupil to the
+extent of ruling out all save definitely quantitative elements,<span class='pagenum'><a name="Page_189" id="Page_189">[Pg 189]</a></span>
+and of ruling out all quantitative elements except those
+which should be considered. The first of the two simplifications
+is very beneficial, on the whole, since otherwise there
+might be different correct solutions to a problem according
+to the nature and circumstances of the persons involved.
+The second simplification is often desirable, since it will often
+produce greater improvement in the pupils, per hour of
+time spent, than would be produced by the problems requiring
+more selection. It should not, however, be a universal
+custom; for in that case the pupils are tempted to
+think that in every problem they must use all the quantities
+given, as one must use all the pieces in a puzzle picture.</p>
+
+<p>It is obvious that the elements selected must not only be
+right but also be in the right relations to one another. For
+example, in the problems below, the 6 must be thought of
+in relation to a dozen and as being half of a dozen, and also
+as being 6 times 1. 1 must be mentally tied to "each."
+The 6 as half of a dozen must be related to the $1.00, $1.60,
+etc. The 6 as 6 times 1 must be related to the $.09, $.14,
+etc.</p>
+
+<div class="pblockquot">
+<p class="center"><b>Buying in Quantity</b></p>
+
+<p>These are a grocer's prices for
+certain things by the dozen
+and for a single one. He sells
+a half dozen at half the price
+of a dozen. Find out how
+much you save by buying 6
+all at one time instead of buying
+them one at a time.</p></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><th>&nbsp;</th><th>Doz.</th><th>Each</th></tr>
+<tr><td align='left'><b>1.</b> Evaporated Milk</td><td align='right'>$1.00</td><td align='right'>$.09</td></tr>
+<tr><td align='left'><b>2.</b> Puffed Rice</td><td align='right'>1.60</td><td align='right'>.14</td></tr>
+<tr><td align='left'><b>3.</b> Puffed Wheat</td><td align='right'>1.10</td><td align='right'>.10</td></tr>
+<tr><td align='left'><b>4.</b> Canned Soup</td><td align='right'>1.90</td><td align='right'>.17</td></tr>
+<tr><td align='left'><b>5.</b> Sardines</td><td align='right'>1.80</td><td align='right'>.16</td></tr>
+<tr><td align='left'><b>6.</b> Beans (No. 2 cans)</td><td align='right'>1.50</td><td align='right'>.13</td></tr>
+<tr><td align='left'><b>7.</b> Pork and Beans</td><td align='right'>1.70</td><td align='right'>.15</td></tr>
+<tr><td align='left'><b>8.</b> Peas (No. 2 cans)</td><td align='right'>1.40</td><td align='right'>.12</td></tr>
+<tr><td align='left'><b>9.</b> Tomatoes (extra cans)</td><td align='right'>3.20</td><td align='right'>.28</td></tr>
+<tr><td align='left'><b>10.</b> Ripe olives (qt. cans)</td><td align='right'>7.20</td><td align='right'>.65</td></tr>
+</table></div>
+
+<p>It is obvious also that in such arithmetical work as we<span class='pagenum'><a name="Page_190" id="Page_190">[Pg 190]</a></span>
+have been describing, the pupil, to be successful, must
+'think things together.' Many bonds must co&ouml;perate to
+determine his final response.</p>
+
+<p>As a preface to reasoning about a problem we often have
+the discovery of the problem and the classification of just
+what it is, and as a postscript we have the critical inspection
+of the answer obtained to make sure that it is verified by
+experiment or is consistent with known facts. During the
+process of searching for, selecting, and weighting facts, there
+may be similar inspection and validation, item by item.</p>
+
+
+<h4>REASONING AS THE CO&Ouml;PERATION OF ORGANIZED HABITS</h4>
+
+<p>The pedagogy of the past made two notable errors in
+practice based on two errors about the psychology of reasoning.
+It considered reasoning as a somewhat magical power
+or essence which acted to counteract and overrule the ordinary
+laws of habit in man; and it separated too sharply
+the 'understanding of principles' by reasoning from the
+'mechanical' work of computation, reading problems, remembering
+facts and the like, done by 'mere' habit and
+memory.</p>
+
+<p>Reasoning or selective, inferential thinking is not at all
+opposed to, or independent of, the laws of habit, but really
+is their necessary result under the conditions imposed by
+man's nature and training. A closer examination of selective
+thinking will show that no principles beyond the
+laws of readiness, exercise, and effect are needed to explain
+it; that it is only an extreme case of what goes on in associative
+learning as described under the 'piecemeal'
+activity of situations; and that attributing certain features
+of learning to mysterious faculties of abstraction or reasoning
+gives no real help toward understanding or controlling
+them.<span class='pagenum'><a name="Page_191" id="Page_191">[Pg 191]</a></span></p>
+
+<p>It is true that man's behavior in meeting novel problems
+goes beyond, or even against, the habits represented by
+bonds leading from gross total situations and customarily
+abstracted elements thereof. One of the two reasons therefor,
+however, is simply that the finer, subtle, preferential
+bonds with subtler and less often abstracted elements go
+beyond, and at times against, the grosser and more usual
+bonds. One set is as much due to exercise and effect as the
+other. The other reason is that in meeting novel problems
+the mental set or attitude is likely to be one which rejects
+one after another response as their unfitness to satisfy a
+certain desideratum appears. What remains as the apparent
+course of thought includes only a few of the many
+bonds which did operate, but which, for the most part, were
+unsatisfying to the ruling attitude or adjustment.</p>
+
+<p>Successful responses to novel data, associations by similarity
+and purposive behavior are in only apparent opposition
+to the fundamental laws of associative learning. Really
+they are beautiful examples of it. Man's successful responses
+to novel data&mdash;as when he argues that the diagonal
+on a right triangle of 796.278 mm. base and 137.294 mm.
+altitude will be 808.022 mm., or that Mary Jones, born this
+morning, will sometime die&mdash;are due to habits, notably
+the habits of response to certain elements or features, under
+the laws of piecemeal activity and assimilation.</p>
+
+<p>Nothing is less like the mysterious operations of a faculty
+of reasoning transcending the laws of connection-forming,
+than the behavior of men in response to novel situations.
+Let children who have hitherto confronted only such arithmetical
+tasks, in addition and subtraction with one- and
+two-place numbers and multiplication with one-place numbers,
+as those exemplified in the first line below, be told to
+do the examples shown in the second line.<span class='pagenum'><a name="Page_192" id="Page_192">[Pg 192]</a></span></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" width="90%">
+<tr><td align='center'><span class="smcap">Add</span></td><td align='center'><span class="smcap">Add</span></td><td align='center'><span class="smcap">Add</span></td><td align='center'><span class="smcap">Subt.</span></td><td align='center'><span class="smcap">Subt.</span></td><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td></tr>
+<tr>
+  <td align='center'>8<br />5<br />&mdash;</td>
+  <td align='center'>37<br />24<br />&mdash;</td>
+  <td align='center'>35<br />68<br />23<br />19<br />&mdash;</td>
+  <td align='center'>8<br />5<br />&mdash;</td>
+  <td align='center'>37<br />24<br />&mdash;</td>
+  <td align='center'>8<br />5<br />&mdash;</td>
+  <td align='center'>9<br />7<br />&mdash;</td>
+  <td align='center'>6<br />3<br />&mdash;</td>
+</tr>
+<tr><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td></tr>
+<tr>
+  <td align='center'>32<br />23<br />&mdash;</td>
+  <td align='center'>43<br />22<br />&mdash;</td>
+  <td align='center'>34<br />26<br />&mdash;</td>
+</tr>
+</table></div>
+
+<p>They will add the numbers, or subtract the lower from the
+upper number, or multiply <span class="inline_eqn">3 &times; 2</span> and <span class="inline_eqn">2 &times; 3,</span> etc., getting 66,
+86, and 624, or respond to the element of 'Multiply' attached
+to the two-place numbers by "I can't" or "I don't
+know what to do," or the like; or, if one is a child of great
+ability, he may consider the 'Multiply' element and the
+bigness of the numbers, be reminded by these two aspects
+of the situation of the fact that</p>
+
+<p class="noidt">
+'9<br />
+&nbsp;9 multiply'<br />
+&mdash;
+</p>
+
+<p class="noidt">gave only 81, and that</p>
+
+<p class="noidt">
+'10<br />
+&nbsp;10 multiply'<br />
+&mdash;&mdash;
+</p>
+
+<p class="noidt">gave only 100, or the like; and
+so may report an intelligent and justified "I can't," or reject
+the plan of <span class="inline_eqn">3 &times; 2</span> and <span class="inline_eqn">2 &times; 3,</span>
+with 66, 86, and 624 for answers,
+as unsatisfactory. What the children will do will, in every
+case, be a product of the elements in the situation that are
+potent with them, the responses which these evoke, and the
+further associates which these responses in turn evoke. If
+the child were one of sufficient genius, he might infer the
+procedure to be followed as a result of his knowledge of the
+principles of decimal notation and the meaning of 'Multiply,'
+responding correctly to the 'place-value' element
+of each digit and adding his 6 tens and 9 tens, 20 twos and
+3 thirties; but if he did thus invent the shorthand addition
+of a collection of twenty-three collections, each of 32 units,
+he would still do it by the operation of bonds, subtle but real.<span class='pagenum'><a name="Page_193" id="Page_193">[Pg 193]</a></span></p>
+
+<p>Association by similarity is, as James showed long ago,
+simply the tendency of an element to provoke the responses
+which have been bound to it. <i>abcde</i> leads to <i>vwxyz</i> because
+<i>a</i> has been bound to <i>vwxyz</i> by original nature, exercise, or
+effect.</p>
+
+<p>Purposive behavior is the most important case of the
+influence of the attitude or set or adjustment of an organism
+in determining (1) which bonds shall act, and (2) which
+results shall satisfy. James early described the former fact,
+showing that the mechanism of habit can give the directedness
+or purposefulness in thought's products, provided that
+mechanism includes something paralleling the problem, the
+aim, or need, in question.</p>
+
+<p>The second fact, that the set or attitude of the man helps
+to determine which bonds shall satisfy, and which shall
+annoy, has commonly been somewhat obscured by vague
+assertions that the selection and retention is of what is
+"in point," or is "the right one," or is "appropriate," or the
+like. It is thus asserted, or at least hinted, that "the will,"
+"the voluntary attention," "the consciousness of the
+problem," and other such entities are endowed with magic
+power to decide what is the "right" or "useful" bond and
+to kill off the others. The facts are that in purposive thinking
+and action, as everywhere else, bonds are selected and
+retained by the satisfyingness, and are killed off by the discomfort,
+which they produce; and that the potency of the
+man's set or attitude to make this satisfy and that annoy&mdash;to
+put certain conduction-units in readiness to act and others
+in unreadiness&mdash;is in every way as important as its potency
+to set certain conduction-units in actual operation.</p>
+
+<p>Reasoning is not a radically different sort of force operating
+against habit but the organization and co&ouml;peration of
+many habits, thinking facts together. Reasoning is not<span class='pagenum'><a name="Page_194" id="Page_194">[Pg 194]</a></span>
+the negation of ordinary bonds, but the action of many of
+them, especially of bonds with subtle elements of the situation.
+Some outside power does not enter to select and
+criticize; the pupil's own total repertory of bonds relevant
+to the problem is what selects and rejects. An unsuitable
+idea is not killed off by some <i>actus purus</i> of intellect, but by
+the ideas which it itself calls up, in connection with the total
+set of mind of the pupil, and which show it to be inadequate.</p>
+
+<p>Almost nothing in arithmetic need be taught as a matter
+of mere unreasoning habit or memory, nor need anything,
+first taught as a principle, ever become a matter of mere
+habit or memory. <span class="inline_eqn">5 &times; 4 = 20</span> should not be learned as an
+isolated fact, nor remembered as we remember that Jones'
+telephone number is 648 J 2. Almost everything in arithmetic
+should be taught as a habit that has connections with
+habits already acquired and will work in an organization
+with other habits to come. The use of this organized
+hierarchy of habits to solve novel problems is reasoning.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_195" id="Page_195">[Pg 195]</a></span></p>
+<h2><a name="CHAPTER_XI" id="CHAPTER_XI"></a>CHAPTER XI</h2>
+
+<h3>ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE
+SCHOOL</h3>
+
+
+<h4>THE UTILIZATION OF INSTINCTIVE INTERESTS</h4>
+
+<p>The activities essential to acquiring ability in arithmetic
+can rely on little in man's instinctive equipment beyond the
+purely intellectual tendencies of curiosity and the satisfyingness
+of thought for thought's sake, and the general enjoyment
+of success rather than failure in an enterprise to
+which one sets oneself. It is only by a certain amount
+of artifice that we can enlist other vehement inborn interests
+of childhood in the service of arithmetical knowledge
+and skill. When this can be done at no cost the
+gain is great. For example, marching in files of two, in files
+of three, in files of four, etc., raising the arms once, two
+times, three times, showing a foot, a yard, an inch with
+the hands, and the like are admirable because learning
+the meanings of numbers thus acquires some of the zest
+of the passion for physical action. Even in late grades
+chances to make pictures showing the relations of fractional
+parts, to cut strips, to fold paper, and the like will
+be useful.</p>
+
+<p>Various social instincts can be utilized in matches after
+the pattern of the spelling match, contests between rows,
+certain number games, and the like. The scoring of both<span class='pagenum'><a name="Page_196" id="Page_196">[Pg 196]</a></span>
+the play and the work of the classroom is a useful field for
+control by the teacher of arithmetic.</p>
+
+<p>Hunt ['12] has noted the more important games which
+have some considerable amount of arithmetical training as
+a by-product and which are more or less suitable for class
+use. Flynn ['12] has described games, most of them for home
+use, which give very definite arithmetical drill, though in
+many cases the drills are rather behind the needs of children
+old enough to understand and like the game itself.</p>
+
+<p>It is possible to utilize the interests in mystery, tricks,
+and puzzles so as to arouse a certain form of respect for
+arithmetic and also to get computational work done. I
+quote one simple case from Miss Selkin's admirable collection
+['12, p. 69 f.]:&mdash;</p>
+
+
+<p class="tabcap">I. <span class="smcap">ADDITION</span></p>
+<div class="pblockquot">
+
+<p>"We must admit that there is nothing particularly interesting
+in a long column of numbers to be added. Let the teacher, however,
+suggest that he can write the answer at sight, and the task
+will assume a totally different aspect.</p>
+
+<p>"A very simple number trick of this kind can be performed by
+making use of the principle of complementary addition. The
+arithmetical complement of a number with respect to a larger
+number is the difference between these two numbers. Most
+interesting results can be obtained by using complements with
+respect to 9.</p>
+
+<p>"The children may be called upon to suggest several numbers of
+two, three, or more digits. Below these write an equal number of
+addends and immediately announce the answer. The children,
+impressed by this apparently rapid addition, will set to work
+enthusiastically to test the results of this lightning calculation.</p>
+
+<p>"Example:&mdash;</p></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='right'>357 </td><td align='left' rowspan='3' valign='middle'><span class="sz30">}</span></td><td></td><td align='right'>999</td></tr>
+<tr><td align='right'>682 </td><td align='left'>A &nbsp; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp; &nbsp;</td><td align='right'>&times; 3</td></tr>
+<tr><td align='right'>793 </td><td align='left'></td><td align='right'><span class="overline">2997</span></td></tr>
+<tr><td align='right'>&nbsp;</td></tr>
+<tr><td align='right'>642 </td><td align='left' rowspan='3' valign='middle'><span class="sz30">}</span></td></tr>
+<tr><td align='right'>317 </td><td align='left'>B</td></tr>
+<tr><td align='right'>206 </td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_197" id="Page_197">[Pg 197]</a></span></p>
+<div class="pblockquot">
+<p>"Explanation:&mdash;The addends in group A are written down at
+random or suggested by the class. Those in group B are their
+complements. To write the first number in group B we look at
+the first number in group A and, starting at the left write 6, the
+complement of 3 with respect to 9; 4, the complement of 5; 2,
+the complement of 7. The second and third addends in group B
+are derived in the same way. Since we have three addends in
+each group, the problem reduces itself to multiplying 999 by 3,
+or to taking 3000 &minus; 3. Any number of addends may be used and
+each addend may consist of any number of digits."</p></div>
+
+<p>Respect for arithmetic as a source of tricks and magic is
+very much less important than respect for its everyday
+services; and computation to test such tricks is likely to be
+undertaken zealously only by the abler pupils. Consequently
+this source of interest should probably be used only
+sparingly, and perhaps the teacher should give such exhibitions
+only as a reward for efficiency in the regular work.
+For example, if the work for a week is well done in four
+days the fifth day might be given up to some semi-arithmetical
+entertainment, such as the demonstration of an
+adding machine, the story of primitive methods of counting,
+team races in computation, an exhibition of lightning
+calculation and intellectual sleight-of-hand by the teacher,
+or the voluntary study of arithmetical puzzles.</p>
+
+<p>The interest in achievement, in success, mentioned above
+is stronger in children than is often realized and makes advisable
+the systematic use of the practice experiment as a
+method of teaching much of arithmetic. Children who thus
+compete with their own past records, keeping an exact score
+from week to week, make notable progress and enjoy hard
+work in making it.</p>
+
+
+<h4>THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES</h4>
+
+<p>Negatively the difficulty of the work that pupils should
+be expected to do is conditioned by the gradual maturing<span class='pagenum'><a name="Page_198" id="Page_198">[Pg 198]</a></span>
+of their capacities. Other things being equal, the common
+custom of reserving hard things for late in the elementary
+school course is, of course, sound. It seems probable that
+little is gained by using any of the child's time for arithmetic
+before grade 2, though there are many arithmetical facts
+that he can learn in grade 1. Postponement of systematic
+work in arithmetic to grade 3 or even grade 4 is allowable
+if better things are offered. With proper textbooks and
+oral and written exercises, however, a child in grades 2 and
+3 can spend time profitably on arithmetical work. When
+all children can be held in school through the eighth grade
+it does not much matter whether arithmetic is begun early
+or late. If, however, many children are to leave in grades 5
+and 6 as now, we may think it wise to provide somehow that
+certain minima of arithmetical ability be given them.</p>
+
+<p>There are, so far as is known, no special times and seasons
+at which the human animal by inner growth is specially ripe
+for one or another section or aspect of arithmetic, except
+in so far as the general inner growth of intellectual powers
+makes the more abstruse and complex tasks suitable to
+later and later years.</p>
+
+<p>Indeed, very few of even the most enthusiastic devotees
+of the recapitulation theory or culture-epoch theory have
+attempted to apply either to the learning of arithmetic, and
+Branford is the only mathematician, so far as I know, who
+has advocated such application, even tempered by elaborate
+shiftings and reversals of the racial order. He says:&mdash;</p>
+
+<div class="pblockquot"><p>"Thus, for each age of the individual life&mdash;infancy, childhood,
+school, college&mdash;may be selected from the racial history
+the most appropriate form in which mathematical experience can
+be assimilated. Thus the capacity of the infant and early childhood
+is comparable with the capacity of animal consciousness
+and primitive man. The mathematics suitable to later childhood
+and boyhood (and, of course, girlhood) is comparable with Ar<span class='pagenum'><a name="Page_199" id="Page_199">[Pg 199]</a></span>ch&aelig;an
+mathematics passing on through Greek and Hindu to medi&aelig;val
+European mathematics; while the student is become sufficiently
+mature to begin the assimilation of modern and highly
+abstract European thought. The filling in of details must necessarily
+be left to the individual teacher, and also, within some such
+broadly marked limits, the precise order of the marshalling of the
+material for each age. For, though, on the whole, mathematical
+development has gone forward, yet there have been lapses from
+advances already made. Witness the practical world-loss of much
+valuable Hindu thought, and, for long centuries, the neglect of
+Greek thought: witness the world-loss of the invention by the
+Babylonians of the Zero, until re-invented by the Hindus, passed
+on by them to the Arabs, and by these to Europe.</p>
+
+<p>"Moreover, many blunders and false starts and false principles
+have marked the whole course of development. In a phrase,
+rivers have their backwaters. But it is precisely the teacher's
+function to avoid such racial mistakes, to take short cuts ultimately
+discovered, and to guide the young along the road ultimately
+found most accessible with such halts and retracings&mdash;returns
+up side-cuts&mdash;as the mental peculiarities of the pupils
+demand.</p>
+
+<p>"All this, the practical realization of the spirit of the principle,
+is to be wisely left to the mathematical teacher, familiar with the
+history of mathematical science and with the particular limitations
+of his pupils and himself." ['08, p. 245.]</p></div>
+
+<p>The latitude of modification suggested by Branford reduces
+the guidance to be derived from racial history to
+almost <i>nil</i>. Also it is apparent that the racial history in
+the case of arithmetical achievement is entirely a matter of
+acquisition and social transmission. Man's original nature
+is destitute of all arithmetical ideas. The human germs
+do not know even that one and one make two!</p>
+
+
+<h4>INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL</h4>
+
+<p>A scientific plan for teaching arithmetic would begin with
+an exact inventory of the knowledge and skill which the
+pupils already possessed. Our ordinary notions of what a
+child knows at entrance to grade 1, or grade 2, or grade 3,<span class='pagenum'><a name="Page_200" id="Page_200">[Pg 200]</a></span>
+and of what a first-grade child or second-grade child can do,
+are not adequate. If they were, we should not find reputable
+textbooks arranging to teach elaborately facts already
+sufficiently well known to over three quarters of the pupils
+when they enter school. Nor should we find other textbooks
+presupposing in their first fifty pages a knowledge
+of words which not half of the children can read even
+at the end of the 2 B grade.</p>
+
+<p>We do find just such evidence that ordinary ideas about
+the abilities of children at the beginning of systematic school
+training in arithmetic may be in gross error. For example, a
+reputable and in many ways admirable recent book has
+fourteen pages of exercises to teach the meaning of two and
+the fact that one and one make two! As an example of the
+reverse error, consider putting all these words in the first
+twenty-five pages of a beginner's book:&mdash;<i>absentees, attendance,
+blanks, continue, copy, during, examples, grouped, memorize,
+perfect, similar, splints, therefore, total</i>!</p>
+
+<p>Little, almost nothing, has been done toward providing
+an exact inventory compared with what needs to be done.
+We may note here (1) the facts relevant to arithmetic found
+by Stanley Hall, Hartmann, and others in their general investigations
+of the knowledge possessed by children at entrance
+to school, (2) the facts concerning the power of children
+to perceive differences in length, area, size of collection,
+and organization within a collection such as is shown in
+Fig. 24, and certain facts and theories about early awareness
+of number.</p>
+
+<p>In the Berlin inquiry of 1869, knowledge of the meaning
+of two, three, and four appeared in 74, 74, and 73 percent of
+the children upon entrance to school. Some of those recorded
+as ignorant probably really knew, but failed to understand
+that they were expected to reply or were shy. Only<span class='pagenum'><a name="Page_201" id="Page_201">[Pg 201]</a></span>
+85 percent were recorded as knowing their fathers' names.
+Seven eighths as many children knew the meanings of two,
+three, and four as knew their fathers' names. In a similar
+but more careful experiment with Boston children in September,
+1880, Stanley Hall found that 92 percent knew three,
+83 percent knew four, and 71&frac12; percent knew five. Three
+was known about as well as the color red; four was known
+about as well as the color blue or yellow or green. Hartmann
+['90] found that two thirds of the children entering school
+in Annaberg could count from one to ten. This is about as
+many as knew money, or the familiar objects of the town,
+or could repeat words spoken to them.</p>
+
+<div class="figcenter" style="width: 661px;">
+<img src="images/fig24.jpg" width="661" height="600" alt="Fig. 24." title="Fig. 24." />
+<span class="caption"><span class="smcap">Fig. 24.</span>&mdash;Objective presentation.</span>
+</div>
+
+<p>In the Stanford form of the Binet tests counting four
+pennies is given as an ability of the typical four-year-old.
+Counting 13 pennies correctly in at least one out of two<span class='pagenum'><a name="Page_202" id="Page_202">[Pg 202]</a></span>
+trials, and knowing three of the four coins,&mdash;penny, nickel,
+dime, and quarter,&mdash;are given as abilities of the typical
+six-year-old.</p>
+
+
+<h4>THE PERCEPTION OF NUMBER AND QUANTITY</h4>
+
+<p>We know that educated adults can tell how many lines or
+dots, etc., they see in a single glance (with an exposure too
+short for the eye to move) up to four or more, according to
+the clearness of the objects and their grouping. For example,
+Nanu ['04] reports that when a number of bright circles on
+a dark background are shown to educated adults for only
+.033 second, ten can be counted when arranged to form a
+parallelogram, but only five when arranged in a row. With
+certain groupings, of course, their 'perception' involves
+much inference, even conscious addition and multiplication.
+Similarly they can tell, up to twenty and beyond, the
+number of taps, notes, or other sounds in a series too rapid
+for single counting if the sounds are grouped in a convenient
+rhythm.</p>
+
+<p>These abilities are, however, the product of a long and
+elaborate learning, including the learning of arithmetic
+itself. Elementary psychology and common experience
+teach us that the mere observation of groups or quantities,
+no matter how clear their number quality appears to the
+person who already knows the meanings of numbers, does
+not of itself create the knowledge of the meanings of numbers
+in one who does not. The experiments of Messenger ['03]
+and Burnett ['06] showed that there is no direct intuitive
+apprehension even of two as distinct from one. We have to
+<i>learn</i> to feel the two touches or see the two dots or lines as two.</p>
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/img220.jpg" width="640" height="445" alt="" title="" />
+</div>
+
+<p>We do not know by exact measurements the growth in
+children of this ability to count or infer the number of elements
+in a collection seen or series heard. Still less do we<span class='pagenum'><a name="Page_203" id="Page_203">[Pg 203]</a></span>
+know what the growth would be without the influence of
+school training in counting, grouping, adding, and multiplying.
+Many textbooks and teachers seem to overestimate
+it greatly. Not all educated adults can, apart from measurement,
+decide with surety which of these lines is the longer,
+or which of these areas is the larger, or whether this is a
+ninth or a tenth or an eleventh of a circle.</p>
+
+
+<div class="figleft" style="width: 480px;">
+<img src="images/img221.jpg" width="480" height="594" alt="" title="" />
+</div>
+
+<p>Children upon entering school have not been tested carefully
+in respect to judgments of length and area, but we
+know from such studies as Gilbert's ['94] that the difference
+required in their case is probably over twice that required
+for children of 13 or 14. In judging weights, for example,
+a difference of 6 is perceived as easily by children 13 to
+15 years of age as a difference of 15 by six-year-olds.</p>
+
+<p>A teacher who has adult powers of estimating length or
+area or weight and who also knows already which of the two<span class='pagenum'><a name="Page_204" id="Page_204">[Pg 204]</a></span>
+is longer or larger or heavier, may use two lines to illustrate
+a difference which they really hide from the child.
+It is unlikely, for example, that the first of these lines
+______________    ________________  would be recognized as
+shorter than the second by every child in a fourth-grade class,
+and it is extremely unlikely that it would be recognized as
+being <sup>7</sup>&frasl;<sub>8</sub> of the length of the latter, rather than <sup>3</sup>&frasl;<sub>4</sub> of it or <sup>5</sup>&frasl;<sub>6</sub> of
+it or <sup>9</sup>&frasl;<sub>10</sub> of it or <sup>11</sup>&frasl;<sub>12</sub> of it. If the two were shown to a second
+grade, with the question, "The first line is 7. How long
+is the other line?" there would be very many answers
+of 7 or 9; and these might be entirely correct arithmetically,
+the pupils' errors being all due to their inability
+to compare the lengths accurately.</p>
+
+<p>The quantities used
+should be such that their
+mere discrimination offers
+no difficulty even
+to a child of blunted
+sense powers. If <sup>7</sup>&frasl;<sub>8</sub> and
+1 are to be compared,
+<i>A</i> and <i>B</i> are not allowable.
+<i>C</i>, <i>D</i>, and <i>E</i> are
+much better.</p>
+
+
+<p>Teachers probably often
+underestimate or
+neglect the sensory
+difficulties of the tasks
+they assign and of the
+material they use to
+illustrate absolute
+and relative magnitudes.
+The result may be more pernicious when the pupils
+answer correctly than when they fail. For their correct
+<span class='pagenum'><a name="Page_205" id="Page_205">[Pg 205]</a></span>
+answering may be due to their divination of what the
+teacher wants; and they may call a thing an inch larger
+to suit her which does not really seem larger to them
+at all. This, of course, is utterly destructive of their respect
+for arithmetic as an exact and matter-of-fact instrument.
+For example, if a teacher drew a series of lines 20, 21, 22, 23,
+24, and 25 inches long on the blackboard in this form&mdash;
+_____ ________ and asked, "This is 20 inches long, how long
+is this?" she might, after some errors and correction thereof,
+finally secure successful response to all the lines by all the
+children. But their appreciation of the numbers 20, 21, 22,
+23, 24, and 25 would be actually damaged by the exercise.</p>
+
+
+<h4>THE EARLY AWARENESS OF NUMBER</h4>
+
+<p>There has been some disagreement concerning the origin
+of awareness of number in the individual, in particular concerning
+the relative importance of the perception of how-many-ness
+and that of how-much-ness, of the perception
+of a defined aggregate and the perception of a defined ratio.
+(See McLellan and Dewey ['95], Phillips ['97 and '98], and
+Decroly and Degand ['12].)</p>
+
+<p>The chief facts of significance for practice seem to be these:
+(1) Children with rare exceptions hear the names <i>one</i>, <i>two</i>,
+<i>three</i>, <i>four</i>, <i>half</i>, <i>twice</i>, <i>two times</i>, <i>more</i>, <i>less</i>, <i>as many as</i>, <i>again</i>,
+<i>first</i>, <i>second</i>, and <i>third</i>, long before they have analyzed out
+the qualities and relations to which these words refer so as
+to feel them at all clearly. (2) Their knowledge of the qualities
+and relations is developed in the main in close association
+with the use of these words to the child and by the child.
+(3) The ordinary experiences of the first five years so develop
+in the child awareness of the 'how many somethings' in
+various groups, of the relative magnitudes of two groups or
+quantities of any sort, and of groups and magnitudes as<span class='pagenum'><a name="Page_206" id="Page_206">[Pg 206]</a></span>
+related to others in a series. For instance, if fairly gifted,
+a child comes, by the age of five, to see that a row of four
+cakes is an aggregate of four, seeing each cake as a part of
+the four and the four as the sum of its parts, to know that
+two of them are as many as the other two, that half of them
+would be two, and to think, when it is useful for him to do
+so, of four as a step beyond three on the way to five, or to
+think of hot as a step from warm on the way to very hot.
+The degree of development of these abilities depends upon
+the activity of the law of analysis in the individual and the
+character of his experiences.</p>
+
+<p>(4) He gets certain bad habits of response from the
+ambiguity of common usage of 2, 3, 4, etc., for second, third,
+fourth. Thus he sees or hears his parents or older children
+or others count pennies or rolls or eggs by saying one, two,
+three, four, and so on. He himself is perhaps misled into so
+counting. Thus the names properly belonging to a series
+of aggregations varying in amount come to be to him the
+names of the positions of the parts in a counted whole.
+This happens especially with numbers above 3 or 4, where the
+correct experience of the number as a name for the group
+has rarely been present. This attaching to the cardinal
+numbers above three or four the meanings of the ordinal
+numbers seems to affect many children on entrance to
+school. The numbering of pages in books, houses, streets,
+etc., and bad teaching of counting often prolong this error.</p>
+
+<p>(5) He also gets the habit, not necessarily bad, but often
+indirectly so, of using many names such as eight, nine, ten,
+eleven, fifteen, a hundred, a million, without any meaning.</p>
+
+<p>(6) The experiences of half, twice, three times as many,
+three times as long, etc., are rarer; even if they were not,
+they would still be less easily productive of the analysis of
+the proper abstract element than are the experiences of<span class='pagenum'><a name="Page_207" id="Page_207">[Pg 207]</a></span>
+two, three, four, etc., in connection with aggregates of things
+each of which is usually called one, such as boys, girls, balls,
+apples. Experiences of the names, two, three, and four, in
+connection with two twos, two threes, two fours, are very
+rare.</p>
+
+<p>Hence, the names, two, three, etc., mean to these children
+in the main, "one something and one something," "one
+something usually called one, and one something usually
+called one, and another something usually called one," and
+more rarely and imperfectly "two times anything," "three
+times anything," etc.</p>
+
+<p>With respect to Mr. Phillips' emphasis of the importance
+of the series-idea in children's minds, the matters of importance
+are: first, that the knowledge of a series of number
+names in order is of very little consequence to the teaching
+of arithmetic and of still less to the origin of awareness of
+number. Second, the habit of applying this series of words
+in counting in such a way that 8 is associated with the eighth
+thing, 9 with the ninth thing, etc., is of consequence because
+it does so much mischief. Third, the really valuable idea of
+the number series, the idea of a series of groups or of magnitudes
+varying by steps, is acquired later, as a result, not a
+cause, of awareness of numbers.</p>
+
+<p>With respect to the McLellan-Dewey doctrine, the ratio
+aspect of numbers should be emphasized in schools, not
+because it is the main origin of the child's awareness of
+number, but because it is <i>not</i>, and because the ordinary practical
+issues of child life do <i>not</i> adequately stimulate its action.
+It also seems both more economical and more scientific to
+introduce it through multiplication, division, and fractions
+rather than to insist that 4 and 5 shall from the start mean
+4 or 5 times anything that is called 1, for instance, that 8
+inches shall be called 4 two-inches, or 10 cents, 5 two-cents.
+<span class='pagenum'><a name="Page_208" id="Page_208">[Pg 208]</a></span>
+If I interpret Professor Dewey's writings correctly, he would
+agree that the use of inch, foot, yard, pint, quart, ounce,
+pound, glassful, cupful, handful, spoonful, cent, nickel,
+dime, and dollar gives a sufficient range of units for the
+first two school years. Teaching the meanings of &frac12; of 4,
+&frac12; of 6, &frac12; of 8, &frac12; of 10, &frac12; of 20,
+<sup>1</sup>&frasl;<sub>3</sub> of 6, <sup>1</sup>&frasl;<sub>3</sub> of 9, <sup>1</sup>&frasl;<sub>3</sub> of 30, &frac14; of 8,
+two 2s, five 2s, and the like, in early grades, each in connection
+with many different units of measure, provides a
+sufficient assurance that numbers will connect with relationships
+as well as with collections.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_209" id="Page_209">[Pg 209]</a></span></p>
+<h2><a name="CHAPTER_XII" id="CHAPTER_XII"></a>CHAPTER XII</h2>
+
+<h3>INTEREST IN ARITHMETIC</h3>
+
+
+<h4>CENSUSES OF PUPILS' INTERESTS</h4>
+
+<p>Arithmetic, although it makes little or no appeal to
+collecting, muscular manipulation, sensory curiosity, or
+the potent original interests in things and their mechanisms
+and people and their passions, is fairly well liked by children.
+The censuses of pupils' likes and dislikes that have been
+made are not models of scientific investigation, and the
+resulting percentages should not be used uncritically.
+They are, however, probably not on the average over-favorable
+to arithmetic in any unfair way. Some of their
+results are summarized below. In general they show arithmetic
+to be surpassed in interest clearly by only the manual
+arts (shopwork and manual training for boys, cooking and
+sewing for girls), drawing, certain forms of gymnastics, and
+history. It is about on a level with reading and science.
+It clearly surpasses grammar, language, spelling, geography,
+and religion.</p>
+
+<p>Lobsien ['03], who asked one hundred children in
+each of the first five grades (<i>Stufen</i>) of the elementary
+schools of Kiel, "Which part of the school work (literally,
+'which instruction period') do you like best?" found
+arithmetic led only by drawing and gymnastics in the
+case of the boys, and only by handwork in the case of the
+girls.<span class='pagenum'><a name="Page_210" id="Page_210">[Pg 210]</a></span></p>
+
+<p>This is an exaggerated picture of the facts, since no count
+is made of those who especially dislike arithmetic. Arithmetic
+is as unpopular with some as it is popular with others.
+When full allowance is made for this, arithmetic still has
+popularity above the average. Stern ['05] asked, "Which
+subject do you like most?" and "Which subject do you
+like least?" The balance was greatly in favor of gymnastics
+for boys (28-1), handwork for girls (32-1&frac12;), and drawing
+for both (16&frac12;-6). Writing (6&frac12;-4), arithmetic (14&frac12;-13),
+history (9-6&frac12;), reading (8&frac12;-8), and singing (6-7&frac12;) come
+next. Religion, nature study, physiology, geography, geometry,
+chemistry, language, and grammar are low.</p>
+
+<p>McKnight ['07] found with boys and girls in grades 7 and
+8 of certain American cities that arithmetic was liked better
+than any of the school subjects except gymnastics and
+manual training. The vote as compared with history
+was:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="4" summary="">
+<tr><td align='left'>Arithmetic</td><td align='left'>327 liked greatly,</td><td align='left'>&nbsp;96 disliked greatly.</td></tr>
+<tr><td align='left'>History</td><td align='left'>164 liked greatly,</td><td align='left'>113 disliked greatly.</td></tr>
+</table></div>
+
+<p>In a later study Lobsien ['09] had 6248 pupils from 9 to
+15 years old representing all grades of the elementary school
+report, so far as they could, the subject most disliked, the
+subject most liked, the subject next most liked, and the
+subject next in order. No child was forced to report all of
+these four judgments, or even any of them. Lobsien counts
+the likes and the dislikes for each subject. Gymnastics,
+handwork, and cooking are by far the most popular. History
+and drawing are next, followed by arithmetic and reading.
+Below these are geography, writing, singing, nature study,
+biblical history, catechism, and three minor subjects.</p>
+
+<p>Lewis ['13] secured records from English children in elementary
+schools of the order of preference of all the studies<span class='pagenum'><a name="Page_211" id="Page_211">[Pg 211]</a></span>
+listed below. He reports the results in the following
+table of percents:</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="5" summary="" rules="cols">
+<tr><th class="bbt">&nbsp;</th>
+<th class="bbt"><span class="smcap">Top Third<br />of Studies for Interest</span></th>
+<th class="bbt"><span class="smcap">Middle Third<br />of Studies for Interest</span></th>
+<th class="bbt"><span class="smcap">Lowest Third<br />of Studies for Interest</span></th>
+</tr>
+<tr><td align='left'>Drawing</td><td align='center'>78</td><td align='center'>20</td><td align='center'>&nbsp; 2</td></tr>
+<tr><td align='left'>Manual Subjects</td><td align='center'>66</td><td align='center'>26</td><td align='center'>&nbsp; 8</td></tr>
+<tr><td align='left'>History</td><td align='center'>64</td><td align='center'>24</td><td align='center'>12</td></tr>
+<tr><td align='left'>Reading</td><td align='center'>53</td><td align='center'>38</td><td align='center'>&nbsp; 9</td></tr>
+<tr><td align='left'>Singing</td><td align='center'>32</td><td align='center'>48</td><td align='center'>20</td></tr>
+<tr><td></td></tr>
+<tr><td align='left'>Drill</td><td align='center'>20</td><td align='center'>55</td><td align='center'>25</td></tr>
+<tr><td align='left'>Arithmetic</td><td align='center'>16</td><td align='center'>53</td><td align='center'>31</td></tr>
+<tr><td align='left'>Science</td><td align='center'>23</td><td align='center'>37</td><td align='center'>40</td></tr>
+<tr><td align='left'>Nature Study</td><td align='center'>16</td><td align='center'>36</td><td align='center'>48</td></tr>
+<tr><td align='left'>Dictation</td><td align='center'>&nbsp; 4</td><td align='center'>57</td><td align='center'>39</td></tr>
+<tr><td></td></tr>
+<tr><td align='left'>Composition</td><td align='center'>18</td><td align='center'>28</td><td align='center'>54</td></tr>
+<tr><td align='left'>Scripture</td><td align='center'>&nbsp; 4</td><td align='center'>38</td><td align='center'>58</td></tr>
+<tr><td align='left'>Recitation</td><td align='center'>&nbsp; 9</td><td align='center'>23</td><td align='center'>68</td></tr>
+<tr><td align='left'>Geography</td><td align='center'>&nbsp; 4</td><td align='center'>24</td><td align='center'>72</td></tr>
+<tr><td align='left'>Grammar</td><td align='center'>&mdash;</td><td align='center'>&nbsp; 6</td><td align='center'>94</td></tr>
+<tr><td class="bb"></td><td class="bb"></td><td class="bb"></td><td class="bb"></td></tr>
+</table></div>
+
+<p>Brandell ['13] obtained data from 2137 Swedish children
+in Stockholm (327), Norrk&ouml;ping (870), and Gothenburg
+(940).</p>
+
+<p>In general he found, as others have, that handwork, shopwork
+for boys and household work for girls, and drawing
+were reported as much better liked than arithmetic. So also
+was history, and (in this he differs from most students of
+this matter) so were reading and nature study. Gymnastics
+he finds less liked than arithmetic. Religion, geography,
+language, spelling, and writing are, as in other studies, much
+less popular than arithmetic.<span class='pagenum'><a name="Page_212" id="Page_212">[Pg 212]</a></span></p>
+
+<p>Other studies are by Lilius ['11] in Finland, Walsemann
+['07], Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern
+['13 and '14], in Germany. They confirm the general results
+stated.</p>
+
+<p>The reasons for the good showing that arithmetic makes
+are probably the strength of its appeal to the interest in
+definite achievement, success, doing what one attempts to
+do; and of its appeal, in grades 5 to 8, to the practical
+interest of getting on in the world, acquiring abilities that
+the world pays for. Of these, the former is in my opinion
+much the more potent interest. Arithmetic satisfies it especially
+well, because, more than any other of the 'intellectual'
+studies of the elementary school, it permits the
+pupil to see his own progress and determine his own success
+or failure.</p>
+
+<p>The most important applications of the psychology of
+satisfiers and annoyers to arithmetic will therefore be in
+the direction of utilizing still more effectively this interest
+in achievement. Next in importance come the plans to
+attach to arithmetical learning the satisfyingness of bodily
+action, play, sociability, cheerfulness, and the like, and of
+significance as a means of securing other desired ends than
+arithmetical abilities themselves. Next come plans to relieve
+arithmetical learning from certain discomforts such as
+the eyestrain of some computations and excessive copying
+of figures. These will be discussed here in the inverse
+order.</p>
+
+
+<h4>RELIEVING EYESTRAIN</h4>
+
+<p>At present arithmetical work is, hour for hour, probably
+more of a tax upon the eyes than reading. The task of
+copying numbers from a book to a sheet of paper is one of
+the very hardest tasks that the eyes of a pupil in the ele<span class='pagenum'><a name="Page_213" id="Page_213">[Pg 213]</a></span>mentary
+schools have to perform. A certain amount of
+such work is desirable to teach a child to write numbers, to
+copy exactly, and to organize material in shape for computation.
+But beyond that, there is no more reason for a pupil
+to copy every number with which he is to compute than for
+him to copy every word he is to read. The meaningless
+drudgery of copying figures should be mitigated by arranging
+much work in the form of exercises like those shown on pages
+216, 217, and 218, and by having many of the textbook
+examples in addition, subtraction, and multiplication done
+with a slip of paper laid below the numbers, the answers
+being written on it. There is not only a resulting gain in
+interest, but also a very great saving of time for the pupil
+(very often copying an example more than quadruples the
+time required to get its answer), and a much greater efficiency
+in supervision. Arithmetical errors are not confused with
+errors of copying,<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a> and the teacher's task of following a
+pupil's work on the page is reduced to a minimum, each
+pupil having put the same part of the day's work in just the
+same place. The use of well-printed and well-spaced pages
+of exercises relieves the eyestrain of working with badly
+made gray figures, unevenly and too closely or too widely
+spaced. I reproduce in Fig. 25 specimens taken at random
+from one hundred random samples of arithmetical work by
+pupils in grade 8. Contrast the task of the eyes in working
+with these and their task in working with pages 216 to 218.
+The customary method of always copying the numbers
+to be used in computation from blackboard or book to
+a sheet of paper is an utterly unjustifiable cruelty and
+waste.<span class='pagenum'><a name="Page_214" id="Page_214">[Pg 214]</a></span></p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/fig25a.jpg" width="600" height="794" alt="Fig. 25a." title="Fig. 25a." />
+<p class="nblockquot"><b><span class="smcap">Fig.</span> 25<i>a</i>.&mdash;Specimens taken at random from the computation work of eighth-grade
+pupils. This computation occurred in a genuine test. In the original gray of
+the pencil marks the work is still harder to make out.</b></p>
+</div>
+
+
+<p>&nbsp;<span class='pagenum'><a name="Page_215" id="Page_215">[Pg 215]</a></span></p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/fig25b.jpg" width="577" height="800" alt="Fig. 25b." title="Fig. 25b." /><br />
+<p class="nblockquot"><b><span class="smcap">Fig.</span> 25<i>b</i>.&mdash;Specimens taken at random from the computation work of eighth-grade
+pupils. This computation occurred in a genuine test. In the original gray of
+the pencil marks the work is still harder to make out.</b></p>
+</div>
+
+<p><span class='pagenum'><a name="Page_216" id="Page_216">[Pg 216]</a></span></p>
+
+<p>Write the products:&mdash;</p>
+
+<div class="dblockquot">
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='right'><small>A.</small> 3 4s =</td><td align='right'><small>B.</small> 5 7s =</td><td align='right'><small>C.</small> 9 2s =</td></tr>
+<tr><td align='right'>5 2s =</td><td align='right'>8 3s =</td><td align='right'>4 4s =</td></tr>
+<tr><td align='right'>7 2s =</td><td align='right'>4 2s =</td><td align='right'>2 7s =</td></tr>
+<tr><td align='right'>1 6 &nbsp;=</td><td align='right'>4 5s =</td><td align='right'>6 4s =</td></tr>
+<tr><td align='right'>1 3 &nbsp;=</td><td align='right'>4 7s =</td><td align='right'>5 5s =</td></tr>
+<tr><td align='right'>3 7s =</td><td align='right'>5 9s =</td><td align='right'>3 6s =</td></tr>
+<tr><td align='right'>4 1s =</td><td align='right'>7 5s =</td><td align='right'>3 2s =</td></tr>
+<tr><td align='right'>6 8s =</td><td align='right'>7 1s =</td><td align='right'>3 9s =</td></tr>
+<tr><td align='right'>9 8s =</td><td align='right'>6 3s =</td><td align='right'>5 1s =</td></tr>
+<tr><td align='right'>4 3s =</td><td align='right'>4 9s =</td><td align='right'>8 6s =</td></tr>
+<tr><td align='right'>2 4s =</td><td align='right'>3 5s =</td><td align='right'>8 4s =</td></tr>
+<tr><td align='right'>2 2s =</td><td align='right'>9 6s =</td><td align='right'>8 5s =</td></tr>
+<tr><td align='right'>8 7s =</td><td align='right'>2 5s =</td><td align='right'>7 9s =</td></tr>
+<tr><td align='right'>5 8s =</td><td align='right'>5 4s =</td><td align='right'>6 2s =</td></tr>
+<tr><td align='right'>7 6s =</td><td align='right'>8 2s =</td><td align='right'>7 4s =</td></tr>
+<tr><td align='right'>7 3s =</td><td align='right'>8 9s =</td><td align='right'>9 3s =</td></tr>
+</table></div>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr><td align='right'><small>D.</small> 4 &nbsp; &nbsp; 20s =</td><td align='right'><small>E.</small> 9 &nbsp; &nbsp; 60s =</td><td align='left'><small>F.</small> 40 &times; 2 = 80</td></tr>
+<tr><td align='right'> 4 &nbsp; 200s =</td><td align='right'>9 &nbsp; 600s =</td><td align='left'> &nbsp; &nbsp; 20 &times; 2 =</td></tr>
+<tr><td align='right'> 6 &nbsp; &nbsp; 30s =</td><td align='right'>5 &nbsp; &nbsp; 30s =</td><td align='left'> &nbsp; &nbsp; 30 &times; 2 =</td></tr>
+<tr><td align='right'> 6 &nbsp; 300s =</td><td align='right'>5 &nbsp; 300s =</td><td align='left'> &nbsp; &nbsp; 40 &times; 2 =</td></tr>
+<tr><td align='right'> 7 &times; &nbsp; 50 =</td><td align='right'>8 &times; &nbsp; 20 =</td><td align='left'> &nbsp; &nbsp; 20 &times; 3 =</td></tr>
+<tr><td align='right'> 7 &times; 500 =</td><td align='right'>8 &times; 200 =</td><td align='left'> &nbsp; &nbsp; 30 &times; 3 =</td></tr>
+<tr><td align='right'> 3 &times; &nbsp; 40 =</td><td align='right'>2 &times; &nbsp; 70 =</td><td align='left'> &nbsp; 300 &times; 3 = 900</td></tr>
+<tr><td align='right'> 3 &times; 400 =</td><td align='right'>2 &times; 700 =</td><td align='left'> &nbsp;  300 &times; 2 =</td></tr>
+</table></div>
+</div>
+
+<p><span class='pagenum'><a name="Page_217" id="Page_217">[Pg 217]</a></span></p>
+
+<p>Write the missing numbers: (<i>r</i> stands for remainder.)</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='left'>25 = .... 3s and .... <i>r</i>.</td><td align='left'>30 = .... 4s and .... <i>r</i>.</td></tr>
+<tr><td align='left'>25 = .... 4s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>30 = .... 5s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>25 = .... 5s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>30 = .... 6s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>25 = .... 6s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>30 = .... 7s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>25 = .... 7s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>30 = .... 8s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>25 = .... 8s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>30 = .... 9s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>25 = .... 9s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td>&nbsp;</td></tr>
+<tr><td align='left'>26 = .... 3s and .... <i>r</i>.</td><td align='left'>31 = .... 4s and .... <i>r</i>.</td></tr>
+<tr><td align='left'>26 = .... 4s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>31 = .... 5s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>26 = .... 5s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>31 = .... 6s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>26 = .... 6s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>31 = .... 7s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>26 = .... 7s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>31 = .... 8s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>26 = .... 8s &nbsp; " &nbsp; .... <i>r</i>.</td><td align='left'>31 = .... 9s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+<tr><td align='left'>26 = .... 9s &nbsp; " &nbsp; .... <i>r</i>.</td></tr>
+</table></div>
+
+<p>Write the whole numbers or mixed numbers which these
+fractions equal:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr>
+  <td align='center'>5<br /><span class="overline">4</span></td>
+  <td align='center'>4<br /><span class="overline">3</span></td>
+  <td align='center'>9<br /><span class="overline">5</span></td>
+  <td align='center'>4<br /><span class="overline">2</span></td>
+  <td align='center'>7<br /><span class="overline">3</span></td>
+</tr>
+<tr>
+  <td align='center'>7<br /><span class="overline">4</span></td>
+  <td align='center'>5<br /><span class="overline">3</span></td>
+  <td align='center'>11<br /><span class="overline">&nbsp;8&nbsp;</span></td>
+  <td align='center'>3<br /><span class="overline">2</span></td>
+  <td align='center'>8<br /><span class="overline">8</span></td>
+</tr>
+<tr>
+  <td align='center'>8<br /><span class="overline">4</span></td>
+  <td align='center'>6<br /><span class="overline">3</span></td>
+  <td align='center'>9<br /><span class="overline">8</span></td>
+  <td align='center'>9<br /><span class="overline">4</span></td>
+  <td align='center'>16<br /><span class="overline">&nbsp;8&nbsp;</span></td>
+</tr>
+<tr>
+  <td align='center'>11<br /><span class="overline">&nbsp;4&nbsp;</span></td>
+  <td align='center'>7<br /><span class="overline">5</span></td>
+  <td align='center'>13<br /><span class="overline">&nbsp;8&nbsp;</span></td>
+  <td align='center'>8<br /><span class="overline">5</span></td>
+  <td align='center'>6<br /><span class="overline">6</span></td>
+</tr>
+</table></div>
+
+<p>Write the missing figures:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr>
+  <td align='right'>6<br /><span class="overline">8</span></td><td>=</td><td align='left'>&nbsp;<br /><span class="overline">4</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='right'>2<br /><span class="overline">4</span></td><td>=</td><td align='left'>&nbsp;<br /><span class="overline">2</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='right'>8<br /><span class="overline">10</span></td><td>=</td><td align='left'>&nbsp;<br /><span class="overline">5</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='right'>1<br /><span class="overline">5</span></td><td>=</td><td align='left'>&nbsp;<br /><span class="overline">10</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='right'>2<br /><span class="overline">3</span></td><td>=</td><td align='left'>&nbsp;<br /><span class="overline">6</span></td>
+</tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_218" id="Page_218">[Pg 218]</a></span></p>
+
+<p>Write the missing numerators:&mdash;</p>
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr>
+  <td align='right'>1<br /><span class="overline">2</span></td><td>=</td>
+  <td align='left'>&nbsp;<br /><span class="overline">12</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">8</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">10</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">4</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">16</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">6</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">14</span></td>
+</tr>
+<tr>
+  <td align='right'>1<br /><span class="overline">3</span></td><td>=</td>
+  <td align='left'>&nbsp;<br /><span class="overline">12</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">9</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">18</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">6</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">15</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">24</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">21</span></td>
+</tr>
+<tr>
+  <td align='right'>1<br /><span class="overline">4</span></td><td>=</td>
+  <td align='left'>&nbsp;<br /><span class="overline">12</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">16</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">8</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">24</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">20</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">28</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">32</span></td>
+</tr>
+<tr>
+  <td align='right'>1<br /><span class="overline">5</span></td><td>=</td>
+  <td align='left'>&nbsp;<br /><span class="overline">10</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">20</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">15</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">25</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">40</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">35</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">30</span></td>
+</tr>
+<tr>
+  <td align='right'>2<br /><span class="overline">3</span></td><td>=</td>
+  <td align='left'>&nbsp;<br /><span class="overline">12</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">18</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">21</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">6</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">15</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">24</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">9</span></td>
+</tr>
+<tr>
+  <td align='right'>3<br /><span class="overline">4</span></td><td>=</td>
+  <td align='left'>&nbsp;<br /><span class="overline">8</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">16</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">12</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">20</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">24</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">32</span></td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='left'>&nbsp;<br /><span class="overline">28</span></td>
+</tr>
+</table></div>
+
+<p>Find the products.  Cancel when you can:&mdash;</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr>
+  <td align='center'>&nbsp;5&nbsp;<br /><span class="overline">16</span></td><td>&times; 4 =</td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='center'>11<br /><span class="overline">12</span></td><td>&times; 3 =</td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='center'>2<br /><span class="overline">3</span></td><td>&times; 5 =</td>
+</tr>
+<tr>
+  <td align='center'>&nbsp;7&nbsp;<br /><span class="overline">12</span></td><td>&times; 8 =</td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='center'>8<br /><span class="overline">5</span></td><td>&times; 15 =</td>
+  <td>&nbsp; &nbsp; &nbsp; &nbsp; </td>
+  <td align='center'>1<br /><span class="overline">6</span></td><td>&times; 8 =</td>
+</tr>
+</table></div>
+
+
+<p><span class='pagenum'><a name="Page_219" id="Page_219">[Pg 219]</a></span></p>
+
+<h4>SIGNIFICANCE FOR RELATED ACTIVITIES</h4>
+
+<p>The use of bodily action, social games, and the like was discussed
+in the section on original tendencies. "Significance as a
+means of securing other desired ends than arithmetical learning
+itself" is therefore our next topic. Such significance can
+be given to arithmetical work by using that work as a means
+to present and future success in problems of sports, housekeeping,
+shopwork, dressmaking, self-management, other
+school studies than arithmetic, and general school life and
+affairs. Significance as a means to future ends alone can also
+be more clearly and extensively attached to it than it now is.</p>
+
+<p>Whatever is done to supply greater strength of motive
+in studying arithmetic must be carefully devised so as not
+to get a strong but wrong motive, so as not to get abundant
+interest but in something other than arithmetic, and so as
+not to kill the goose that after all lays the golden eggs&mdash;the
+interest in intellectual activity and achievement itself.
+It is easy to secure an interest in laying out a baseball
+diamond, measuring ingredients for a cake, making a balloon
+of a certain capacity, or deciding the added cost of an extra
+trimming of ribbon for one's dress. The problem is to
+<i>attach</i> that interest to arithmetical learning. Nor should a
+teacher be satisfied with attaching the interest as a mere
+tail that steers the kite, so long as it stays on, or as a sugar-coating
+that deceives the pupil into swallowing the pill, or
+as an anodyne whose dose must be increased and increased
+if it is to retain its power. Until the interest permeates the
+arithmetical activity itself our task is only partly done, and
+perhaps is made harder for the next time.</p>
+
+<p>One important means of really interfusing the arithmetical
+learning itself with these derived interests is to lead
+the pupil to seek the help of arithmetic himself&mdash;to lead
+him, in Dewey's phrase, to 'feel the need'&mdash;to take the<span class='pagenum'><a name="Page_220" id="Page_220">[Pg 220]</a></span>
+'problem' attitude&mdash;and thus appreciate the technique
+which he actively hunts for to satisfy the need. In so far
+as arithmetical learning is organized to satisfy the practical
+demands of the pupil's life at the time, he should, so to speak,
+come part way to get its help.</p>
+
+<p>Even if we do not make the most skillful use possible of
+these interests derived from the quantitative problems of
+sports, housekeeping, shopwork, dressmaking, self-management,
+other school studies, and school life and affairs, the
+gain will still be considerable. To have them in mind will
+certainly preserve us from giving to children of grades 3 and
+4 problems so devoid of relation to their interests as those
+shown below, all found (in 1910) in thirty successive pages
+of a book of excellent repute:&mdash;</p>
+
+<div class="pblockquot">
+<p>A chair has 4 legs. How many legs have 8 chairs? 5 chairs?</p>
+
+<p>A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?</p>
+
+<p class="center">(Eight more of the same sort.)</p>
+
+<p>In 1890 New York had 1,513,501 inhabitants, Milwaukee had
+206,308, Boston had 447,720, San Francisco 297,990. How many
+had these cities together?</p>
+
+<p class="center">(Five more of the same sort.)</p>
+
+<p>Milton was born in 1608 and died in 1674. How many years
+did he live?</p>
+
+<p class="center">(Several others of the same sort.)</p>
+
+<p>The population of a certain city was 35,629 in 1880 and 106,670
+in 1890. Find the increase.</p>
+
+<p class="center">(Several others of this sort.)</p>
+
+<p>A number of others about the words in various inaugural addresses
+and the Psalms in the Bible.</p></div>
+
+<p>It also seems probable that with enough care other systematic
+plans of textbooks can be much improved in this respect.
+From every point of view, for example, the early work in arithmetic
+should be adapted to some extent to the healthy childish
+interests in home affairs, the behavior of other children,
+and the activities of material things, animals, and plants.<span class='pagenum'><a name="Page_221" id="Page_221">[Pg 221]</a></span></p>
+
+<p class="tabcap">TABLE 9</p>
+
+<p class="center"><span class="smcap">Frequency of Appearance of Certain Words about Family Life,
+Play, and Action in Eight Elementary Textbooks in Arithmetic</span>,
+pp. 1-50.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="80%" rules="cols">
+<tr><th class='bbt'></th><th class='bbt'> A</th><th class='bbt'> B</th><th class='bbt'> C</th><th class='bbt'> D</th><th class='bbt'> E</th><th class='bbt'> F</th><th class='bbt'> G</th><th class='bbt'> H</th></tr>
+<tr><td align='left'>baby</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 4</td><td align='center'></td><td></td></tr>
+<tr><td align='left'>brother</td><td align='center'> 2</td><td align='center'></td><td align='center'> 6</td><td align='center'> 1</td><td align='center'> 1</td><td align='center'></td><td align='center'> 1</td><td></td></tr>
+<tr><td align='left'>family</td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 4</td><td></td></tr>
+<tr><td align='left'>father</td><td align='center'> 1</td><td align='center'></td><td align='center'> 3</td><td align='center'> 5</td><td align='center'></td><td align='center'> 2</td><td align='center'> 1</td><td></td></tr>
+<tr><td align='left'>help</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>home</td><td align='center'> 2</td><td align='center'></td><td align='center'> 4</td><td align='center'> 4</td><td align='center'> 2</td><td align='center'> 2</td><td align='center'> 7</td><td align='center'> 1</td></tr>
+<tr><td align='left'>mother</td><td align='center'> 4</td><td align='center'> 2</td><td align='center'> 9</td><td align='center'> 5</td><td align='center'></td><td align='center'> 5</td><td align='center'> 1</td><td align='center'> 7</td></tr>
+<tr><td align='left'>sister</td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'> 2</td><td align='center'> 2</td><td align='center'> 9</td><td align='center'> 1</td><td align='center'> 1</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='left'>fork</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>knife</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>plate</td><td align='center'> 4</td><td align='center'> 2</td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td></td></tr>
+<tr><td align='left'>spoon</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='left'>doll</td><td align='center'> 10</td><td align='center'> 1</td><td align='center'> 10</td><td align='center'> 6</td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td align='center'> 9</td></tr>
+<tr><td align='left'>game</td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'> 3</td><td align='center'></td><td align='center'></td><td align='center'> 5</td><td align='center'> 5</td></tr>
+<tr><td align='left'>jump</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 4</td></tr>
+<tr><td align='left'>marbles</td><td align='center'> 10</td><td align='center'> 4</td><td align='center'> 10</td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td align='center'> 1</td><td></td></tr>
+<tr><td align='left'>play</td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'> 3</td><td align='center'></td><td></td></tr>
+<tr><td align='left'>run</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td align='center'> 3</td></tr>
+<tr><td align='left'>sing</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>tag</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>toy</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td align='left'>car</td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'> 4</td><td align='center'></td><td align='center'> 2</td><td align='center'> 3</td><td align='center'> 1</td></tr>
+<tr><td align='left'>cut</td><td align='center'></td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td align='center'> 6</td><td align='center'> 2</td><td align='center'></td><td align='center'> 8</td></tr>
+<tr><td align='left'>dig</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td></td></tr>
+<tr><td align='left'>flower</td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'> 4</td><td align='center'> 1</td><td align='center'> 1</td><td align='center'> 2</td><td></td></tr>
+<tr><td align='left'>grow</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>plant</td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr>
+<tr><td align='left'>seed</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 3</td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td></td></tr>
+<tr><td align='left'>string</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'> 10</td><td align='center'> 1</td><td align='center'> 1</td></tr>
+<tr><td align='left'>wheel</td><td align='center'> 5</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td></td></tr>
+<tr><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td></tr>
+</table></div>
+
+<p><span class='pagenum'><a name="Page_222" id="Page_222">[Pg 222]</a></span></p>
+
+<p>The words used by textbooks give some indication of how
+far this aim is being realized, or rather of how far short we
+are of realizing it. Consider, for example, the words home,
+mother, father, brother, sister, help, plate, knife, fork,
+spoon, play, game, toy, tag, marbles, doll, run, jump, sing,
+plant, seed, grow, flower, car, wheel, string, cut, dig. The
+frequency of appearance in the first fifty pages of eight beginners'
+arithmetics was as shown in Table 9. The eight
+columns refer to the eight books (the first fifty pages of
+each). The numbers refer to the number of times the word
+in question appeared, the number 10 meaning 10 <i>or more</i>
+times in the fifty pages. Plurals, past tenses, and the like
+were counted. <i>Help</i>, <i>fork</i>, <i>knife</i>, <i>spoon</i>, <i>jump</i>, <i>sing</i>, and <i>tag</i>
+did not appear at all! <i>Toy</i> and <i>grow</i> appeared each once
+in the 400 pages! <i>Play</i>, <i>run</i>, <i>dig</i>, <i>plant</i>, and <i>seed</i> appeared
+once in a hundred or more pages. <i>Baby</i> did not appear as
+often as <i>buggy</i>. <i>Family</i> appeared no oftener than <i>fence</i> or
+<i>Friday</i>. <i>Father</i> appears about a third as often as <i>farmer</i>.</p>
+
+<p>Book A shows only 10 of these thirty words in the fifty
+pages; book B only 4; book C only 12; and books D, E, F,
+G, and H only 13, 8, 14, 13, 10, respectively. The total number
+of appearances (counting the 10s as only 10 in each case)
+is 40 for A, 9 for B, 60 for C, 42 for D, 25 for E, 62 for F, 30
+for G, and 37 for H. The five words&mdash;apple, egg, Mary,
+milk, and orange&mdash;are used oftener than all these thirty
+together.</p>
+
+<p>If it appeared that this apparent neglect of childish affairs
+and interests was deliberate to provide for a more systematic
+treatment of pure arithmetic, a better gradation of problems,
+and a better preparation for later genuine use than could be
+attained if the author of the textbook were tied to the child's
+apron strings, the neglect could be defended. It is not at all
+certain that children in grade 2 get much more enjoyment<span class='pagenum'><a name="Page_223" id="Page_223">[Pg 223]</a></span>
+or ability from adding the costs of purchases for Christmas
+or Fourth of July, or multiplying the number of cakes each
+child is to have at a party by the number of children who
+are to be there, than from adding gravestones or multiplying
+the number of hairs of bald-headed men. When, however,
+there is nothing gained by substituting remote facts for
+those of familiar concern to children, the safe policy is surely
+to favor the latter. In general, the neglect of childish
+data does not seem to be due to provision for some other
+end, but to the same inertia of tradition which has carried
+over the problems of laying walls and digging wells into city
+schools whose children never saw a stone wall or dug well.</p>
+
+<hr style='width: 45%;' />
+
+<p>I shall not go into details concerning the arrangement of
+courses of study, textbooks, and lesson-plans to make desirable
+connections between arithmetical learning and sports,
+housework, shopwork, and the rest. It may be worth while,
+however, to explain the term <i>self-management</i>, since this
+source of genuine problems of real concern to the pupils
+has been overlooked by most writers.</p>
+
+<p>By self-management is meant the pupil's use of his time,
+his abilities, his knowledge, and the like. By the time he
+reaches grade 5, and to some extent before then, a boy
+should keep some account of himself, of how long it takes
+him to do specified tasks, of how much he gets done in a
+specified time at a certain sort of work and with how many
+errors, of how much improvement he makes month by
+month, of which things he can do best, and the like. Such
+objective, matter-of-fact, quantitative study of one's behavior
+is not a stimulus to morbid introspection or egotism;
+it is one of the best preventives of these. To treat oneself
+impersonally is one of the essential elements of mental
+balance and health. It need not, and should not, encourage<span class='pagenum'><a name="Page_224" id="Page_224">[Pg 224]</a></span>
+priggishness. On the contrary, this matter-of-fact study of
+what one is and does may well replace a certain amount of
+the exhortations and admonitions concerning what one ought
+to do and be. All this is still truer for a girl.</p>
+
+<p>The demands which such an accounting of one's own
+activities make of arithmetic have the special value of connecting
+directly with the advanced work in computation.
+They involve the use of large numbers, decimals, averaging,
+percentages, approximations, and other facts and processes
+which the pupil has to learn for later life, but to which his
+childish activities as wage-earner, buyer and seller, or shopworker
+from 10 to 14 do not lead. Children have little
+money, but they have time in thousands of units! They do
+not get discounts or bonuses from commercial houses, but
+they can discount their quantity of examples done for the
+errors made, and credit themselves with bonuses of all sorts
+for extra achievements.</p>
+
+
+<h4>INTRINSIC INTEREST IN ARITHMETICAL LEARNING</h4>
+
+<p>There remains the most important increase of interest in
+arithmetical learning&mdash;an increase in the interest directly
+bound to achievement and success in arithmetic itself.
+"Arithmetic," says David Eugene Smith, "is a game and
+all boys and girls are players." It should not be a <i>mere</i>
+game for them and they should not <i>merely</i> play, but their
+unpractical interest in doing it because they can do it and can
+see how well they do do it is one of the school's most precious
+assets. Any healthy means to give this interest more and
+better stimulus should therefore be eagerly sought and
+cherished.</p>
+
+<p>Two such means have been suggested in other connections.
+The first is the extension of training in checking and verifying
+work so that the pupil may work to a standard of ap<span class='pagenum'><a name="Page_225" id="Page_225">[Pg 225]</a></span>proximately
+100% success, and may know how nearly he is
+attaining it. The second is the use of standardized practice
+material and tests, whereby the pupil may measure himself
+against his own past, and have a clear, vivid, and trustworthy
+idea of just how much better or faster he can do the
+same tasks than he could do a month or a year ago, and of
+just how much harder things he can do now than then.</p>
+
+<p>Another means of stimulating the essential interest in
+quantitative thinking itself is the arrangement of the work
+so that real arithmetical thinking is encouraged more than
+mere imitation and assiduity. This means the avoidance
+of long series of applied problems all of one type to be
+solved in the same way, the avoidance of miscellaneous series
+and review series which are almost verbatim repetitions of
+past problems, and in general the avoidance of excessive
+repetition of any one problem-situation. Stimulation to
+real arithmetical thinking is weak when a whole day's
+problem work requires no choice of methods, or when a
+review simply repeats without any step of organization or
+progress, or when a pupil meets a situation (say the 'buy <i>x</i>
+things at <i>y</i> per thing, how much pay' situation) for the five-hundredth
+time.</p>
+
+<p>Another matter worthy of attention in this connection is
+the unwise tendency to omit or present in diluted form some
+of the topics that appeal most to real intellectual interests,
+just because they are hard. The best illustration, perhaps,
+is the problem of ratio or "How many times as large (long,
+heavy, expensive, etc.) as <i>x</i> is <i>y</i>?" Mastery of the 'times
+as' relation is hard to acquire, but it is well worth acquiring,
+not only because of its strong intellectual appeal, but also
+because of its prime importance in the applications of
+arithmetic to science. In the older arithmetics it was confused
+by pedantries and verbal difficulties and penalized<span class='pagenum'><a name="Page_226" id="Page_226">[Pg 226]</a></span>
+by unreal problems about fractions of men doing parts of a
+job in strange and devious times. Freed from these, it
+should be reinstated, beginning as early as grade 5 with such
+simple exercises as those shown below and progressing to the
+problems of food values, nutritive ratios, gears, speeds, and
+the like in grade 8.</p>
+
+
+<div class="blockquot"><p><br />
+John is 4 years old.<br />
+Fred is 6 years old.<br />
+Mary is 8 years old.<br />
+Nell is 10 years old.<br />
+Alice is 12 years old.<br />
+Bert is 15 years old.<br />
+<br />
+Who is twice as old as John?<br />
+Who is half as old as Alice?<br />
+Who is three times as old as John?<br />
+Who is one and one half times as old as Nell?<br />
+Who is two thirds as old as Fred?<br />
+<span style="margin-left: 3em;">etc., etc., etc.</span><br />
+<br />
+Alice is .... times as old as John.<br />
+John is .... as old as Mary.<br />
+Fred is .... times as old as John.<br />
+Alice is .... times as old as Fred.<br />
+Fred is .... as old as Mary.<br />
+<span style="margin-left: 3em;">etc., etc., etc.</span><br />
+</p></div>
+
+<p>Finally it should be remembered that all improvements
+in making arithmetic worth learning and helping the pupil
+to learn it will in the long run add to its interest. Pupils
+like to learn, to achieve, to gain mastery. Success is interesting.
+If the measures recommended in the previous chapters
+are carried out, there will be little need to entice pupils to
+take arithmetic or to sugar-coat it with illegitimate attractions.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_227" id="Page_227">[Pg 227]</a></span></p>
+<h2><a name="CHAPTER_XIII" id="CHAPTER_XIII"></a>CHAPTER XIII</h2>
+
+<h3>THE CONDITIONS OF LEARNING</h3>
+
+
+<p>We shall consider in this chapter the influence of time of
+day, size of class, and amount of time devoted to arithmetic
+in the school program, the hygiene of the eyes in arithmetical
+work, the use of concrete objects, and the use of
+sounds, sights, and thoughts as situations and of speech and
+writing and thought as responses.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a></p>
+
+
+<h4>EXTERNAL CONDITIONS</h4>
+
+<p>Computation of one or another sort has been used by
+several investigators as a test of efficiency at different times
+in the day. When freed from the effects of practice on the
+one hand and lack of interest due to repetition on the other,
+the results uniformly show an increase in speed late in the
+school session with a falling off in accuracy that about
+balances it.<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a> There is no wisdom in putting arithmetic
+early in the session because of its <i>difficulty</i>. Lively and
+sociable exercises in mental arithmetic with oral answers in
+fact seem to be admirably fitted for use late in the session.
+Except for the general principles (1) of starting the day with
+work that will set a good standard of cheerful, efficient
+<span class='pagenum'><a name="Page_228" id="Page_228">[Pg 228]</a></span>
+production and (2) of getting the least interesting features of
+the day's work done fairly early in the day, psychology
+permits practical exigencies to rule the program, so far as
+present knowledge extends. Adequate measurements of the
+effect of time of day on <i>improvement</i> have not been made,
+but there is no reason to believe that any one time between
+9 <span class="smcap">a.m.</span> and 4 <span class="smcap">p.m.</span> is appreciably more favorable to arithmetical
+learning than to learning geography, history,
+spelling, and the like.</p>
+
+<p>The influence of size of class upon progress in school
+studies is very difficult to measure because (1) within the
+same city system the average of the six (or more) sizes of
+class that a pupil has experienced will tend to approximate
+closely to the corresponding average for any other child;
+because further (2) there may be a tendency of supervisory
+officers to assign more pupils to the better teachers; and
+because (3) separate systems which differ in respect to size
+of class probably differ in other respects also so that their
+differences in achievement may be referable to totally
+different differences.</p>
+
+<p>Elliott ['14] has made a beginning by noting size of class
+during the year of test in connection with his own measures
+of the achievements of seventeen hundred pupils, supplemented
+by records from over four hundred other classes.
+As might be expected from the facts just stated, he finds no
+appreciable difference between classes of different sizes
+within the same school system, the effect of the few months
+in a small class being swamped by the antecedents or concomitants
+thereof.</p>
+
+<p>The effect of the amount of time devoted to arithmetic
+in the school program has been studied extensively by Rice
+['02 and '03] and Stone ['08].</p>
+
+<p>Dr. Rice ['02] measured the arithmetical ability of some<span class='pagenum'><a name="Page_229" id="Page_229">[Pg 229]</a></span>
+6000 children in 18 different schools in 7 different cities.
+The results of these measurements are summarized in Table
+10. This table "gives two averages for each grade as well
+as for each school as a whole. Thus, the school at the top
+shows averages of 80.0 and 83.1, and the one at the bottom,
+25.3 and 31.5. The first represents the percentage of
+answers which were absolutely correct; the second shows
+what per cent of the problems were correct in principle, <i>i.e.</i>
+the average that would have been received if no mechanical
+errors had been made."</p>
+
+<p>The facts of Dr. Rice's table show that there is a positive
+relation between the general standing of a school system
+in the tests and the amount of time devoted to arithmetic
+by its program. The relation is not close, however, being
+that expressed by a correlation coefficient of .36&frac12;. Within
+any one school system there is no relation between the
+standing of a particular school and the amount of time devoted
+to arithmetic in that school's program. It must be
+kept in mind that the amount of time given in the school
+program may be counterbalanced by emphasizing work at
+home and during study periods, or, on the other hand,
+may be a symptom of correspondingly small or great emphasis
+on arithmetic in work set for the study periods at
+home.</p>
+
+<p>A still more elaborate investigation of this same topic was
+made by Stone ['08]. I quote somewhat fully from it, since
+it is an instructive sample of the sort of studies that will
+doubtless soon be made in the case of every elementary
+school subject. He found that school systems differed notably
+in the achievements made by their sixth-grade pupils
+in his tests of computation (the so-called 'fundamentals')
+and of the solution of verbally described problems (the
+so-called 'reasoning'). The facts were as shown in Table 11.<span class='pagenum'><a name="Page_230" id="Page_230">[Pg 230]</a></span></p>
+
+<p class="tabcap">TABLE  10</p>
+
+<p class="tabcap"><span class="smcap">Averages for Individual Schools in Arithmetic</span></p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="" rules="cols">
+<tr><th class="bbox1" rowspan='2'><span class="smcap">City</span></th><th  class="bbox1" rowspan='2'><span class="smcap">School</span></th><th colspan='2' class='bbox1'><span class="smcap">6th Year</span></th><th colspan='2' class='bbox1'><span class="smcap">7th Year</span></th><th colspan='2' class='bbox1'><span class="smcap">8th Year</span></th><th colspan='3' class='bbox1'><span class="smcap">School Average</span></th><th class='bbox1'></th></tr>
+<tr><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class='bbox1'>Percent of<br />Mechanical Errors</th><th class='bbox1'>Minutes<br />Daily</th></tr>
+<tr><td align='right'>III</td><td align='center'> 1</td><td align='center'>79.3</td><td align='center'>80.3</td><td align='center'>81.1</td><td align='center'>82.3</td><td align='center'>91.7</td><td align='center'>93.9</td><td align='center'>80.0</td><td align='center'>83.1</td><td align='center'> 3.7</td><td align='center'> 53</td></tr>
+<tr><td align='right'> I</td><td align='center'> 1</td><td align='center'>80.4</td><td align='center'>81.5</td><td align='center'>64.2</td><td align='center'>67.2</td><td align='center'>80.9</td><td align='center'>82.8</td><td align='center'>76.6</td><td align='center'>80.3</td><td align='center'> 4.6</td><td align='center'> 60</td></tr>
+<tr><td align='right'> I</td><td align='center'> 2</td><td align='center'>80.9</td><td align='center'>83.4</td><td align='center'>43.5</td><td align='center'>50.9</td><td align='center'>72.7</td><td align='center'>79.1</td><td align='center'>69.3</td><td align='center'>75.1</td><td align='center'> 7.7</td><td align='center'> 25</td></tr>
+<tr><td align='right'> I</td><td align='center'> 3</td><td align='center'>72.2</td><td align='center'>74.0</td><td align='center'>63.5</td><td align='center'>66.2</td><td align='center'>74.5</td><td align='center'>76.6</td><td align='center'>67.8</td><td align='center'>72.2</td><td align='center'> 6.1</td><td align='center'> 45</td></tr>
+<tr><td align='right'> I</td><td align='center'> 4</td><td align='center'>69.9</td><td align='center'>72.2</td><td align='center'>54.6</td><td align='center'>57.8</td><td align='center'>66.5</td><td align='center'>69.1</td><td align='center'>64.3</td><td align='center'>70.3</td><td align='center'> 8.5</td><td align='center'> 45</td></tr>
+<tr><td align='right'> II</td><td align='center'> 1</td><td align='center'>71.2</td><td align='center'>75.3</td><td align='center'>33.6</td><td align='center'>35.7</td><td align='center'>36.8</td><td align='center'>40.0</td><td align='center'>60.2</td><td align='center'>64.8</td><td align='center'> 7.1</td><td align='center'> 60</td></tr>
+<tr><td align='right'>III</td><td align='center'> 2</td><td align='center'>43.7</td><td align='center'>45.0</td><td align='center'>53.9</td><td align='center'>56.7</td><td align='center'>51.1</td><td align='center'>53.1</td><td align='center'>54.5</td><td align='center'>58.9</td><td align='center'> 7.4</td><td align='center'> 60</td></tr>
+<tr><td align='right'> IV</td><td align='center'> 1</td><td align='center'>58.9</td><td align='center'>60.4</td><td align='center'>31.2</td><td align='center'>34.1</td><td align='center'>41.6</td><td align='center'>43.5</td><td align='center'>55.1</td><td align='center'>58.4</td><td align='center'> 5.6</td><td align='center'> 60</td></tr>
+<tr><td align='right'> IV</td><td align='center'> 2</td><td align='center'>59.8</td><td align='center'>63.1</td><td align='center'> &mdash;</td><td align='center'> &mdash;</td><td align='center'>22.5</td><td align='center'>22.5</td><td align='center'>53.9</td><td align='center'>58.8</td><td align='center'> 8.3</td><td align='center'> &mdash;</td></tr>
+<tr><td align='right'> IV</td><td align='center'> 3</td><td align='center'>54.9</td><td align='center'>58.1</td><td align='center'>35.2</td><td align='center'>38.6</td><td align='center'>43.5</td><td align='center'>45.0</td><td align='center'>51.5</td><td align='center'>57.6</td><td align='center'>10.5</td><td align='center'> 60</td></tr>
+<tr><td align='right'> IV</td><td align='center'> 4</td><td align='center'>42.3</td><td align='center'>45.1</td><td align='center'>16.1</td><td align='center'>19.2</td><td align='center'>48.7</td><td align='center'>48.7</td><td align='center'>42.8</td><td align='center'>48.2</td><td align='center'>11.2</td><td align='center'> &mdash;</td></tr>
+<tr><td align='right'> V</td><td align='center'> 1</td><td align='center'>44.1</td><td align='center'>48.7</td><td align='center'>29.2</td><td align='center'>32.5</td><td align='center'>51.1</td><td align='center'>58.3</td><td align='center'>45.9</td><td align='center'>51.3</td><td align='center'>10.5</td><td align='center'> 40</td></tr>
+<tr><td align='right'> VI</td><td align='center'> 1</td><td align='center'>68.3</td><td align='center'>71.3</td><td align='center'>33.5</td><td align='center'>36.6</td><td align='center'>26.9</td><td align='center'>30.7</td><td align='center'>39.0</td><td align='center'>42.9</td><td align='center'> 9.0</td><td align='center'> 33</td></tr>
+<tr><td align='right'> VI</td><td align='center'> 2</td><td align='center'>46.1</td><td align='center'>49.5</td><td align='center'>19.5</td><td align='center'>24.2</td><td align='center'>30.2</td><td align='center'>40.6</td><td align='center'>36.5</td><td align='center'>43.6</td><td align='center'>16.2</td><td align='center'> 30</td></tr>
+<tr><td align='right'> VI</td><td align='center'> 3</td><td align='center'>34.5</td><td align='center'>36.4</td><td align='center'>30.5</td><td align='center'>35.1</td><td align='center'>23.3</td><td align='center'>24.1</td><td align='center'>36.0</td><td align='center'>42.5</td><td align='center'>15.2</td><td align='center'> 48</td></tr>
+<tr><td align='right'>VII</td><td align='center'> 1</td><td align='center'>35.2</td><td align='center'>37.7</td><td align='center'>29.1</td><td align='center'>32.5</td><td align='center'>25.1</td><td align='center'>27.2</td><td align='center'>40.5</td><td align='center'>45.9</td><td align='center'>11.7</td><td align='center'> 42</td></tr>
+<tr><td align='right'>VII</td><td align='center'> 2</td><td align='center'>35.2</td><td align='center'>38.7</td><td align='center'>15.0</td><td align='center'>16.4</td><td align='center'>19.6</td><td align='center'>21.2</td><td align='center'>36.5</td><td align='center'>40.6</td><td align='center'>10.1</td><td align='center'> 75</td></tr>
+<tr><td align='right'>VII</td><td align='center'> 3</td><td align='center'>27.6</td><td align='center'>33.7</td><td align='center'> 8.9</td><td align='center'>10.1</td><td align='center'>11.3</td><td align='center'>11.3</td><td align='center'>25.3</td><td align='center'>31.5</td><td align='center'>19.6</td><td align='center'> 45</td></tr>
+<tr><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td></tr>
+</table></div>
+
+<p>High achievement by a system in computation went with
+high achievement in solving the problems, the correlation
+being about .50; and the system that scored high in addition
+or subtraction or multiplication or division usually showed
+closely similar excellence in the other three, the correlations
+being about .90.</p>
+
+<p class="tabcap">TABLE 11</p>
+
+<p class="center"><span class="smcap">Scores Made by the Sixth-Grade Pupils of Each of Twenty-Six School
+Systems</span></p>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="1" cellspacing="0" summary="" rules="cols">
+<tr><th class="bbt smcap">System</th><th class="bbt smcap">Score in Tests<br />with Problems</th><th class="bbt smcap">Score in Tests<br />in Computing</th></tr>
+<tr><td align='center'>23</td><td align='center'> 356</td><td align='center'> 1841</td></tr>
+<tr><td align='center'>24</td><td align='center'> 429</td><td align='center'> 3513</td></tr>
+<tr><td align='center'>17</td><td align='center'> 444</td><td align='center'> 3042</td></tr>
+<tr><td align='center'>&nbsp; 4</td><td align='center'> 464</td><td align='center'> 3563</td></tr>
+<tr><td align='center'>25</td><td align='center'> 464</td><td align='center'> 2167</td></tr>
+<tr><td align='center'>22</td><td align='center'> 468</td><td align='center'> 2311</td></tr>
+<tr><td align='center'>16</td><td align='center'> 469</td><td align='center'> 3707</td></tr>
+<tr><td align='center'>20</td><td align='center'> 491</td><td align='center'> 2168</td></tr>
+<tr><td align='center'>18</td><td align='center'> 509</td><td align='center'> 3758</td></tr>
+<tr><td align='center'>15</td><td align='center'> 532</td><td align='center'> 2779</td></tr>
+<tr><td align='center'>&nbsp; 3</td><td align='center'> 533</td><td align='center'> 2845</td></tr>
+<tr><td align='center'>&nbsp; 8</td><td align='center'> 538</td><td align='center'> 2747</td></tr>
+<tr><td align='center'>&nbsp; 6</td><td align='center'> 550</td><td align='center'> 3173</td></tr>
+<tr><td align='center'>&nbsp; 1</td><td align='center'> 552</td><td align='center'> 2935</td></tr>
+<tr><td align='center'>10</td><td align='center'> 601</td><td align='center'> 2749</td></tr>
+<tr><td align='center'>&nbsp; 2</td><td align='center'> 615</td><td align='center'> 2958</td></tr>
+<tr><td align='center'>21</td><td align='center'> 627</td><td align='center'> 2951</td></tr>
+<tr><td align='center'>13</td><td align='center'> 636</td><td align='center'> 3049</td></tr>
+<tr><td align='center'>14</td><td align='center'> 661</td><td align='center'> 3561</td></tr>
+<tr><td align='center'>&nbsp; 9</td><td align='center'> 691</td><td align='center'> 3404</td></tr>
+<tr><td align='center'>&nbsp; 7</td><td align='center'> 734</td><td align='center'> 3782</td></tr>
+<tr><td align='center'>12</td><td align='center'> 736</td><td align='center'> 3410</td></tr>
+<tr><td align='center'>11</td><td align='center'> 759</td><td align='center'> 3261</td></tr>
+<tr><td align='center'>26</td><td align='center'> 791</td><td align='center'> 3682</td></tr>
+<tr><td align='center'>19</td><td align='center'> 848</td><td align='center'> 4099</td></tr>
+<tr><td align='center'>&nbsp; 5</td><td align='center'> 914</td><td align='center'> 3569</td></tr>
+<tr><td class="bb"></td><td class="bb"></td><td class="bb"></td></tr>
+</table></div>
+
+<p>Of the conditions under which arithmetical learning took
+place, the one most elaborately studied was the amount of
+time devoted to arithmetic. On the basis of replies by
+principals of schools to certain questions, he gave each of<span class='pagenum'><a name="Page_231" id="Page_231">[Pg 231]</a></span>
+the twenty-six school systems a measure for the probable
+time spent on arithmetic up through grade 6. Leaving
+home study out of account, there seems to be little or no
+correlation between the amount of time a system devotes
+to arithmetic and its score in problem-solving, and not
+much more between time expenditure and score in computation.
+With home study included there is little relation to<span class='pagenum'><a name="Page_232" id="Page_232">[Pg 232]</a></span>
+the achievement of the system in solving problems, but
+there is a clear effect on achievement in computation.
+The facts as given by Stone are:&mdash;</p>
+
+<p class="tabcap">TABLE 12</p>
+
+<p class="center"><span class="smcap">Correlation of Time Expenditures with Abilities</span></p>
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><td rowspan='2'>Without Home Study <span class="sz30">{</span></td><td align='left'> Reasoning and Time Expenditure</td><td align='right'>&minus;.01</td></tr>
+<tr><td align='left'> Fundamentals and Time Expenditure</td><td align='right'>.09</td></tr>
+<tr><td colspan='3'>&nbsp;</td></tr>
+<tr><td rowspan='2'>Including Home Study <span class="sz30">{</span></td><td align='left'> Reasoning and Time Expenditure</td><td align='right'>.13</td></tr>
+<tr><td align='left'> Fundamentals and Time Expenditure</td><td align='right'>.49</td></tr>
+</table></div>
+
+
+<p>These correlations, it should be borne in mind, are for
+school systems, not for individual pupils. It might be that,
+though the system which devoted the most time to arithmetic
+did not show corresponding superiority in the product over
+the system devoting only half as much time, the pupils
+within the system did achieve in exact proportion to the
+time they gave to study. Neither correlation would permit
+inference concerning the effect of different amounts of time
+spent by the same pupil.</p>
+
+<p>Stone considered also the printed announcements of the
+courses of study in arithmetic in these twenty-six systems.
+Nineteen judges rated these announced courses of study for
+excellence according to the instructions quoted below:&mdash;</p>
+
+
+<h4>CONCERNING THE RATING OF COURSES OF STUDY</h4>
+
+<p class="center">Judges please read before scoring</p>
+
+<p class="noidt">I. Some Factors Determining Relative Excellence.</p>
+
+<p>(N. B. The following enumeration is meant to be suggestive
+rather than complete or exclusive. And each scorer is urged to
+rely primarily on his own judgment.)</p>
+
+<div class="sblockquot">
+
+<p class="nblockquot">1. Helpfulness to the teacher in teaching the subject matter
+outlined.</p>
+<p><span class='pagenum'><a name="Page_233" id="Page_233">[Pg 233]</a></span></p>
+<p class="nblockquot">2. Social value or concreteness of sources of problems.</p>
+
+<p class="nblockquot">3. The arrangement of subject matter.</p>
+
+<p class="nblockquot">4. The provision made for adequate drill.</p>
+
+<p class="nblockquot">5. A reasonable minimum requirement with suggestions for
+valuable additional work.</p>
+
+<p class="nblockquot">6. The relative values of any predominating so-called methods&mdash;such
+as Speer, Grube, etc.</p>
+
+<p class="nblockquot">7. The place of oral or so-called mental arithmetic.</p>
+
+<p class="nblockquot">8. The merit of textbook references.</p>
+</div>
+
+<p class="noidt">II. Cautions and Directions.</p>
+
+<p>(Judges please follow as implicitly as possible.)</p>
+
+<div class="sblockquot">
+<p class="nblockquot">1. Include references to textbooks as parts of the Course of Study.</p>
+
+<p>This necessitates judging the parts of the texts referred to.</p>
+
+<p class="nblockquot">2. As far as possible become equally familiar with all courses
+before scoring any.</p>
+
+<p class="nblockquot">3. When you are ready to begin to score, (1) arrange in serial
+order according to excellence, (2) starting with the
+middle one score it 50, then score above and below 50
+according as courses are better or poorer, indicating relative
+differences in excellence by relative differences in
+scores, <i>i.e.</i> in so far as you find that the courses differ
+by about equal steps, score those better than the middle
+one 51, 52, etc., and those poorer 49, 48, etc., but if you
+find that the courses differ by unequal steps show these
+inequalities by omitting numbers.</p>
+
+<p class="nblockquot">4. Write ratings on the slip of paper attached to each course.</p>
+</div>
+
+<p>The systems whose courses of study were thus rated highest
+did not manifest any greater achievement in Stone's tests
+than the rest. The thirteen with the most approved announcements
+of courses of study were in fact a little inferior
+in achievement to the other thirteen, and the correlation
+coefficients were slightly negative.</p>
+
+<p>Stone also compared eighteen systems where there was
+supervision of the work by superintendents or supervisors
+as well as by principals with four systems where the principals
+and teachers had no such help. The scores in his tests
+were very much lower in the four latter cities.</p>
+<p><span class='pagenum'><a name="Page_234" id="Page_234">[Pg 234]</a></span></p>
+
+<h4>THE HYGIENE OF THE EYES IN ARITHMETIC</h4>
+
+
+<div class="figcenter" style="width: 336px;">
+<img src="images/fig26.jpg" width="336" height="388" alt="Fig. 26." title="Fig. 26." />
+<span class="caption"><span class="smcap">Fig. 26.</span>&mdash;Type too large.</span>
+</div>
+
+<p>We have already noted that the task of reading and copying
+numbers is one of the hardest that the eyes have to perform
+in the elementary school, and that it should be alleviated
+by arranging much of the work so that only answers need
+be written by the pupil. The figures to be read and copied<span class='pagenum'><a name="Page_235" id="Page_235">[Pg 235]</a></span>
+should obviously be in type of suitable size and style, so
+arranged and spaced on the page or blackboard as to cause
+a minimum of effort and strain.</p>
+
+<div class="figcenter" style="width: 448px;">
+<img src="images/fig27.jpg" width="448" height="160" alt="Fig. 27." title="Fig. 27." />
+<span class="caption"><span class="smcap">Fig. 27.</span>&mdash;12-point, 11-point, and 10-point type.</span>
+</div>
+
+<p><i>Size.</i>&mdash;Type may be too large as well as too small, though
+the latter is the commoner error. If it is too large, as in
+Fig. 26, which is a duplicate of type actually used in a form
+of practice pad, the eye has to make too many fixations to
+take in a given content. All things considered, 12-point
+type in grades 3 and 4, 11-point in grades 5 and 6, and
+10-point in grades 7 and 8 seem the most desirable
+sizes. These are shown in Fig. 27. Too small type occurs
+oftenest in fractions and in the dimension-numbers or scale
+numbers of drawings. Figures 28, 29, and 30 are samples
+from actual school practice. Samples of the desirable size
+are shown in Figs. 31 and 32. The technique of modern
+typesetting makes it very difficult and expensive to make
+fractions of the horizontal type</p>
+
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="0" summary="">
+<tr>
+  <td>(</td>
+  <td class="ft05">1<br /><span class="overline">4</span></td>
+  <td>,</td>
+  <td class="ft05">3<br /><span class="overline">8</span></td>
+  <td>,</td>
+  <td class="ft05">5<br /><span class="overline">6</span></td>
+  <td>)</td>
+</tr>
+</table></div>
+
+<p class="noidt">large enough without
+making the whole-number figures with which they are
+mingled too large or giving an uncouth appearance to the
+total. Consequently fractions somewhat smaller than are
+desirable may have to be used occasionally in textbooks.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a>
+There is no valid excuse, however, for the excessively small
+fractions which often are made in blackboard work.</p>
+
+<p><span class='pagenum'><a name="Page_236" id="Page_236">[Pg 236]</a></span></p>
+<div class="figcenter" style="width: 600px;">
+<div class="figleft" style="width: 300px;">
+<img src="images/fig28.jpg" width="225" height="200" alt="Fig. 28." title="Fig. 28." /><br />
+<span class="caption"><span class="smcap">Fig. 28.</span>&mdash;Type of measurements too small.</span><br />
+<br />
+</div>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<p>This is a picture of Mary's garden.<br />
+How many feet is it around the garden?</p>
+</div>
+
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig29.png" width="640" height="296" alt="Fig. 29." title="Fig. 29." />
+<span class="caption"><span class="smcap">Fig. 29.</span>&mdash;Type too small.</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_237" id="Page_237">[Pg 237]</a></span></p>
+
+<div class="figcenter" style="width: 725px;">
+<img src="images/fig30.png" width="725" height="600" alt="Fig. 30." title="Fig. 30." />
+<span class="caption"><span class="smcap">Fig. 30.</span>&mdash;Numbers too small and badly designed.</span>
+</div>
+
+<div class="figcenter" style="width: 350px;">
+<img src="images/fig31.jpg" width="255" height="235" alt="Fig. 31." title="Fig. 31." /><br />
+<span class="caption"><span class="smcap">Fig. 31.</span>&mdash;Figure 28 with suitable numbers.</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_238" id="Page_238">[Pg 238]</a></span></p>
+
+<div class="figcenter" style="width: 751px;">
+<img src="images/fig32.png" width="751" height="600" alt="Fig. 32." title="Fig. 32." />
+<span class="caption"><span class="smcap">Fig. 32.</span>&mdash;Figure 30 with suitable numbers.</span>
+</div>
+
+<p><i>Style.</i>&mdash;The ordinary type forms often have 3 and 8 so
+made as to require strain to distinguish them. 5 is sometimes
+easily confused with 3 and even with 8. 1, 4, and 7
+may be less easily distinguishable than is desirable. Figure
+33 shows a specially good type in which each figure is represented
+by its essential<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a> features without any distracting
+shading or knobs or turns. Figure 34 shows some of the types
+in common use. There are no demonstrably great differences
+amongst these. In fractions there is a notable gain
+from using the slant form (<sup>2</sup>&frasl;<sub>3</sub>, <sup>3</sup>&frasl;<sub>4</sub>) for exercises in addition
+<span class='pagenum'><a name="Page_239" id="Page_239">[Pg 239]</a></span>
+and subtraction, and for almost all mixed numbers. This
+appears clearly to the eye in the comparison of Fig. 35 below,
+where the same fractions all in 10-point type are displayed
+in horizontal and in slant form. The figures in the
+slant form are in general larger and the space between them
+and the fraction-line is wider. Also the slant form makes
+it easier for the eye to examine the denominators to see
+whether reductions are necessary. Except for a few cases
+to show that the operations can be done just as truly with
+the horizontal forms, the book and the blackboard should
+display mixed numbers and fractions to be added or subtracted
+in the slant form. The slant line should be at an
+angle of approximately 45 degrees. Pupils should be taught
+to use this form in their own work of this sort.</p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/fig33.jpg" width="448" height="200" alt="Fig. 33." title="Fig. 33." /><br />
+<span class="caption"><span class="smcap">Fig. 33.</span>&mdash;Block type; a very desirable type except that it is somewhat too
+heavy.</span>
+</div>
+
+<p>&nbsp;</p>
+<div class="figcenter" style="width: 400px;">
+<img src="images/fig34.png" width="294" height="448" alt="Fig. 34." title="Fig. 34." /><br />
+<span class="caption"><span class="smcap">Fig. 34.</span>&mdash;Common styles of printed numbers.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_240" id="Page_240">[Pg 240]</a></span></p>
+
+
+<p>When script figures are presented they should be of simple
+design, showing clearly the essential features of the figure,
+the line being everywhere of equal or nearly equal width
+(that is, without shading, and without ornamentation or
+eccentricity of any sort). The opening of the 3 should be
+wide to prevent confusion with 8; the top of the 3 should be
+curved to aid its differentiation from 5; the down stroke
+of the 9 should be almost or quite straight; the 1, 4, 7, and 9
+should be clearly distinguishable. There are many ways of
+distinguishing them clearly, the best probably being to use
+the straight line for 1, the open 4 with clear angularity, a
+wide top to the 7, and a clearly closed curve for the top of the 9.</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig35.jpg" width="448" height="140" alt="Fig. 35." title="Fig. 35." /><br />
+<span class="caption"><span class="smcap">Fig. 35.</span>&mdash;Diagonal and horizontal fractions compared.</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_241" id="Page_241">[Pg 241]</a></span></p>
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/p241.png" width="480" height="598" alt="Figs. 36, 37." title="Fig. 36, 37." /><br />
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><td align='left'><b><span class="smcap">Fig. 36.</span>&mdash;Good vertical spacing.</b></td>
+<td align='right'><b><span class="smcap">Fig. 37.</span>&mdash;Bad vertical spacing.</b></td>
+</tr>
+</table>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_242" id="Page_242">[Pg 242]</a></span></p>
+
+<div class="figcenter" style="width: 480px;">
+<img src="images/p242.png" width="480" height="612" alt="Figs. 38, 39." title="Figs. 38, 39." />
+<span class="caption"><span class="smcap">Figs.</span> 38 (above) and 39 (below).&mdash;Good and bad left-right spacing.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_243" id="Page_243">[Pg 243]</a></span></p>
+
+<p>The pupil's writing of figures should be clear. He will
+thereby be saved eyestrain and errors in his school work as
+well as given a valuable ability for life. Handwriting of
+figures is used enormously in spite of the development of
+typewriters; illegible figures are commonly more harmful
+than illegible letters or words, since the context far less often
+tells what the figure is intended to be; the habit of making
+clear figures is not so hard to acquire, since they are written
+unjoined and require only the automatic action of ten minor
+acts of skill. The schools have missed a great opportunity in
+this respect. Whereas the hand writing of words is often better
+than it needs to be for life's purposes, the writing of figures is
+usually much worse. The figures presented in books on penmanship
+are also commonly bad, showing neglect or misunderstanding
+of the matter on the part of leaders in penmanship.</p>
+
+<p><i>Spacing.</i>&mdash;Spacing up and down the column is rarely too
+wide, but very often too narrow. The specimens shown in
+Figs. 36 and 37 show good practice contrasted with the
+common fault.</p>
+
+<p>Spacing from right to left is generally fairly satisfactory
+in books, though there is a bad tendency to adopt some one
+routine throughout and so to miss chances to use reductions
+and increases of spacing so as to help the eye and the mind
+in special cases. Specimens of good and bad spacing are
+shown in Figs. 38 and 39. In the work of the pupils, the
+spacing from right to left is often too narrow. This crowding
+of letters, together with unevenness of spacing, adds
+notably to the task of eye and mind.</p>
+
+<p><i>The composition or make-up of the page.</i>&mdash;Other things
+being equal, that arrangement of the page is best which
+helps a child most to keep his place on a page and to find
+it after having looked away to work on the paper on which
+he computes, or for other good reasons. A good page and
+a bad page in this respect are shown in Figs. 40 and 41.<span class='pagenum'><a name="Page_244" id="Page_244">[Pg 244]</a></span></p>
+
+
+<div class="figcenter" style="width: 480px;">
+<img src="images/fig40.png" width="480" height="635" alt="Fig. 40." title="Fig. 40." />
+<span class="caption"><span class="smcap">Fig. 40.</span>&mdash;A page well made up to suit the action of the eye.</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_245" id="Page_245">[Pg 245]</a></span></p>
+
+<div class="figcenter" style="width: 480px;">
+<img src="images/fig41.png" width="477" height="439" alt="Fig. 41." title="Fig. 41." /><br />
+<span class="caption"><span class="smcap">Fig. 41.</span>&mdash;The same matter as in Fig. 40, much
+less well made up.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_246" id="Page_246">[Pg 246]</a></span></p>
+
+<p><i>Objective presentations.</i>&mdash;Pictures, diagrams, maps, and
+other presentations should not tax the eye unduly,</p>
+
+<div class="blockquot">
+<p class="nblockquot">(<i>a</i>) by requiring too fine distinctions, or</p>
+
+<p class="nblockquot">(<i>b</i>) by inconvenient arrangement of the data, preventing
+easy counting, measuring, comparison, or whatever
+the task is, or</p>
+
+<p class="nblockquot">(<i>c</i>) by putting too many facts in one picture so that the
+eye and mind, when trying to make out any one, are
+confused by the others.</p></div>
+
+<p>Illustrations of bad practices in these respects are shown
+in Figs. 42 to 52. A few specimens of work well arranged for
+the eye are shown in Figs. 53 to 56.</p>
+
+<p>Good rules to remember are:&mdash;</p>
+
+<p>Other things being equal, make distinctions by the clearest
+method, fit material to the tendency of the eye to see an
+'eyeful' at a time (roughly 1&frac12; inch by &frac12; inch in a book;
+1&frac12; ft. by &frac12; ft. on the blackboard), and let one picture teach
+only one fact or relation, or such facts and relations as do
+not interfere in perception.</p>
+
+<p>The general conditions of seating, illumination, paper, and
+the like are even more important when the eyes are used
+with numbers than when they are used with words.</p>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig42.jpg" width="800" height="321" alt="Fig. 42." title="Fig. 42." />
+<span class="caption"><span class="smcap">Fig. 42.</span>&mdash;Try to count the rungs on the ladder, or the shocks in the wagon.</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_247" id="Page_247">[Pg 247]</a></span></p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig43.jpg" width="800" height="386" alt="Fig. 43." title="Fig. 43." />
+<span class="caption"><span class="smcap">Fig. 43.</span>&mdash;How many oars do you see? How many birds? How many fish?</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig44.jpg" width="800" height="271" alt="Fig. 44." title="Fig. 44." />
+<span class="caption"><span class="smcap">Fig. 44.</span>&mdash;Count the birds in each of the three flocks of birds.</span>
+</div>
+
+
+<p><span class='pagenum'><a name="Page_248" id="Page_248">[Pg 248]</a></span></p>
+
+<div class="figcenter" style="width: 720px;">
+<img src="images/fig45.png" width="720" height="660" alt="Fig. 45." title="Fig. 45." />
+<span class="caption"><span class="smcap">Fig. 45.</span>&mdash;Note the lack of clear division of the hundreds. Consider the difficulty
+of counting one of these columns of dots.</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_249" id="Page_249">[Pg 249]</a></span></p>
+
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig46.png" width="640" height="360" alt="Fig. 46." title="Fig. 46." />
+<span class="caption"><span class="smcap">Fig. 46.</span>&mdash;What do you suppose these pictures are intended to show?</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig47.png" width="800" height="500" alt="Fig. 47." title="Fig. 47." />
+<p class="nblockquot"><b><span class="smcap">Fig. 47.</span>&mdash;Would a beginner know that after THIRTEEN he was to switch around
+and begin at the other end? Could you read the SIX of TWENTY-SIX if you
+did not already know what it ought to be? What meaning would all the
+brackets have for a little child in grade 2? Does this picture illustrate or
+obfuscate?</b></p>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_250" id="Page_250">[Pg 250]</a></span></p>
+
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig48.jpg" width="800" height="700" alt="Fig. 48." title="Fig. 48." />
+<span class="caption"><span class="smcap">Fig. 48.</span>&mdash;How long did it take you to find out what these pictures mean?</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig49.png" width="640" height="250" alt="Fig. 49." title="Fig. 49." /><br />
+<p class="nblockquot"><b><span class="smcap">Fig. 49.</span>&mdash;Count the figures in the first row, using your eyes alone; have some one
+make lines of 10, 11, 12, 13, and more repetitions of this figure spaced closely as
+here. Count 20 or 30 such lines, using the eye unaided by fingers, pencil, etc.</b></p>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_251" id="Page_251">[Pg 251]</a></span></p>
+
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig50.png" width="640" height="147" alt="Fig. 50." title="Fig. 50." />
+<span class="caption"><span class="smcap">Fig. 50.</span>&mdash;Can you answer the question without measuring? Could a child of
+seven or eight?</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig51.png" width="640" height="320" alt="Fig. 51." title="Fig. 51." /><br />
+<span class="caption"><span class="smcap">Fig. 51.</span>&mdash;What are these drawings intended to show? Why do they show the
+facts only obscurely and dubiously?</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 579px;">
+<img src="images/fig52.png" width="579" height="480" alt="Fig. 52." title="Fig. 52." />
+<span class="caption"><span class="smcap">Fig. 52.</span>&mdash;What are these drawings intended to show? What simple change would
+make them show the facts much more clearly?</span>
+</div>
+
+<p>&nbsp;<span class='pagenum'><a name="Page_252" id="Page_252">[Pg 252]</a></span></p>
+
+<div class="figcenter" style="width: 683px;">
+<img src="images/fig53.png" width="683" height="600" alt="Fig. 53." title="Fig. 53." />
+<span class="caption"><span class="smcap">Fig. 53.</span>&mdash;Arranged in convenient "eye-fulls."</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 600px;">
+<div class="figleft" style="width: 336px;">
+<img src="images/fig54.png" width="235" height="235" alt="Fig. 54." title="Fig. 54." /><br />
+<span class="caption"><span class="smcap">Fig. 54.</span>&mdash;Clear, simple, and easy of comparison.</span><br />
+<br />
+</div>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<p class="nblockquot"> &nbsp; &nbsp; Tell which bar has&mdash;<br />
+<small>
+<b>1.</b> About &nbsp; 5 percent of its length black.<br />
+<b>2.</b> About 10 percent of its length black.<br />
+<b>3.</b> About 25 percent of its length black.<br />
+<b>4.</b> About 75 percent of its length black.<br />
+<b>5.</b> About 90 percent of its length black.<br />
+<b>6.</b> About 95 percent of its length black.<br />
+</small></p>
+</div>
+<p>&nbsp;<span class='pagenum'><a name="Page_253" id="Page_253">[Pg 253]</a></span></p>
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig55.png" width="640" height="155" alt="Fig. 55." title="Fig. 55." />
+<span class="caption"><span class="smcap">Fig. 55.</span>&mdash;Clear, simple, and well spaced.</span>
+</div>
+
+<p>&nbsp;</p>
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig56.png" width="640" height="206" alt="Fig. 56." title="Fig. 56." />
+<span class="caption"><span class="smcap">Fig. 56.</span>&mdash;Well arranged, though a little wider spacing between the squares would
+make it even better.</span>
+</div>
+
+
+<h4>THE USE OF CONCRETE OBJECTS IN ARITHMETIC</h4>
+
+<p>We mean by concrete objects actual things, events, and
+relations presented to sense, in contrast to words and numbers
+and symbols which mean or stand for these objects or
+for more abstract qualities and relations. Blocks, tooth-picks,
+coins, foot rules, squared paper, quart measures,
+bank books, and checks are such concrete things. A foot
+rule put successively along the three thirds of a yard rule,
+a bell rung five times, and a pound weight balancing sixteen
+ounce weights are such concrete events. A pint beside
+a quart, an inch beside a foot, an apple shown cut in halves
+display such concrete relations to a pupil who is attentive
+to the issue.</p>
+
+<p>Concrete presentations are obviously useful in arithmetic
+to teach meanings under the general law that a word or
+number or sign or symbol acquires meaning by being connected
+with actual things, events, qualities, and relations.<span class='pagenum'><a name="Page_254" id="Page_254">[Pg 254]</a></span>
+We have also noted their usefulness as means to verifying
+the results of thinking and computing, as when a pupil,
+having solved, "How many badges each 5 inches long can
+be made from 3<small><sup>1</sup>&frasl;<sub>3</sub></small> yd. of ribbon?" by using 10 &times; <sup>12</sup>&frasl;<sub>5</sub>, draws a
+line 3<small><sup>1</sup>&frasl;<sub>3</sub></small> yd. long and divides it into 5-inch lengths.</p>
+
+<p>Concrete experiences are useful whenever the meaning of a
+number, like 9 or <sup>7</sup>&frasl;<sub>8</sub> or .004, or of an operation, like multiplying
+or dividing or cubing, or of some term, like rectangle or
+hypothenuse or discount, or some procedure, like voting or
+insuring property against fire or borrowing money from a
+bank, is absent or incomplete or faulty. Concrete work thus
+is by no means confined to the primary grades but may be
+appropriate at all stages when new facts, relations, and procedures
+are to be taught.</p>
+
+<p>How much concrete material shall be presented will depend
+upon the fact or relation or procedure which is to be
+made intelligible, and the ability and knowledge of the
+pupil. Thus 'one half' will in general require less concrete
+illustration than 'five sixths'; and five sixths will require
+less in the case of a bright child who already knows
+<sup>2</sup>&frasl;<sub>3</sub>, <sup>3</sup>&frasl;<sub>4</sub>, <sup>3</sup>&frasl;<sub>8</sub>, <sup>5</sup>&frasl;<sub>8</sub>, <sup>7</sup>&frasl;<sub>8</sub>, <sup>2</sup>&frasl;<sub>5</sub>,
+<sup>3</sup>&frasl;<sub>5</sub>, and <sup>4</sup>&frasl;<sub>5</sub>
+than in the case of a dull child or one who
+only knows <sup>2</sup>&frasl;<sub>3</sub> and <sup>3</sup>&frasl;<sub>4</sub>. As a general rule the same topic will
+require less concrete material the later it appears in the school
+course. If the meanings of the numbers are taught in
+grade 2 instead of grade 1, there will be less need of blocks,
+counters, splints, beans, and the like. If 1&frac12; &#43; &frac12; = 2 is
+taught early in grade 3, there will be more gain from the
+use of 1&frac12; inches and &frac12; inch on the foot rule than if the same
+relations were taught in connection with the general addition
+of like fractions late in grade 4. Sometimes the understanding
+can be had either by connecting the idea with the
+reality directly, or by connecting the two indirectly <i>via</i>
+some other idea. The amount of concrete material to be<span class='pagenum'><a name="Page_255" id="Page_255">[Pg 255]</a></span>
+used will depend on its relative advantage per unit of time
+spent. Thus it might be more economical to connect <sup>5</sup>&frasl;<sub>12</sub>,
+<sup>7</sup>&frasl;<sub>12</sub>, and <sup>11</sup>&frasl;<sub>12</sub> with real meanings indirectly by calling up the
+resemblance to the <sup>2</sup>&frasl;<sub>3</sub>, <sup>3</sup>&frasl;<sub>4</sub>, <sup>3</sup>&frasl;<sub>8</sub>, <sup>5</sup>&frasl;<sub>8</sub>, <sup>7</sup>&frasl;<sub>8</sub>, <sup>2</sup>&frasl;<sub>5</sub>,
+<sup>3</sup>&frasl;<sub>5</sub>, <sup>4</sup>&frasl;<sub>5</sub>, and <sup>5</sup>&frasl;<sub>6</sub> already studied,
+than by showing <sup>5</sup>&frasl;<sub>12</sub> of an apple, <sup>7</sup>&frasl;<sub>12</sub> of a yard, <sup>11</sup>&frasl;<sub>12</sub> of a foot,
+and the like.</p>
+
+<p>In general the economical course is to test the understanding
+of the matter from time to time, using more concrete
+material if it is needed, but being careful to encourage
+pupils to proceed to the abstract ideas and general principles
+as fast as they can. It is wearisome and debauching to
+pupils' intellects for them to be put through elaborate concrete
+experiences to get a meaning which they could have
+got themselves by pure thought. We should also remember
+that the new idea, say of the meaning of decimal fractions,
+will be improved and clarified by using it (see page 183 f.), so
+that the attainment of a <i>perfect</i> conception of decimal fractions
+before doing anything with them is unnecessary and
+probably very wasteful.</p>
+
+<p>A few illustrations may make these principles more instructive.</p>
+
+<p>(<i>a</i>) Very large numbers, such as 1000, 10,000, 100,000, and
+1,000,000, need more concrete aids than are commonly given.
+Guessing contests about the value in dollars of the school
+building and other buildings, the area of the schoolroom
+floor and other surfaces in square inches, the number of
+minutes in a week, and year, and the like, together with
+proper computations and measurements, are very useful to
+re&euml;nforce the concrete presentations and supply genuine problems
+in multiplication and subtraction with large numbers.</p>
+
+<p>(<i>b</i>) Numbers very much smaller than one, such as <sup>1</sup>&frasl;<sub>32</sub>, <sup>1</sup>&frasl;<sub>64</sub>,
+.04, and .002, also need some concrete aids. A diagram like
+that of Fig. 57 is useful.<span class='pagenum'><a name="Page_256" id="Page_256">[Pg 256]</a></span></p>
+
+<p>(<i>c</i>) <i>Majority</i> and <i>plurality</i> should be understood by every
+citizen. They can be understood without concrete aid, but
+an actual vote is well worth while for the gain in vividness
+and surety.</p>
+
+<div class="figcenter" style="width: 640px;">
+<img src="images/fig57.png" width="480" height="480" alt="Fig. 57." title="Fig. 57." /><br />
+<span class="caption"><span class="smcap">Fig. 57.</span>&mdash;Concrete aid to understanding fractions with large denominators.<br />
+A = <sup>1</sup>&frasl;<sub>1000</sub> sq. ft.; B = <sup>1</sup>&frasl;<sub>100</sub> sq. ft.; C = <sup>1</sup>&frasl;<sub>50</sub> sq. ft.; D = <sup>1</sup>&frasl;<sub>10</sub> sq. ft.</span>
+</div>
+
+<p>(<i>d</i>) Insurance against loss by fire can be taught by explanation
+and analogy alone, but it will be economical to
+have some actual insuring and payment of premiums and
+a genuine loss which is reimbursed.<span class='pagenum'><a name="Page_257" id="Page_257">[Pg 257]</a></span></p>
+
+<p>(<i>e</i>) Four play banks in the corners of the room, receiving
+deposits, cashing checks, and later discounting notes will
+give good educational value for the time spent.</p>
+
+<p>(<i>f</i>) Trade discount, on the contrary, hardly requires more
+concrete illustration than is found in the very problems to
+which it is applied.</p>
+
+<p>(<i>g</i>) The process of finding the number of square units in a
+rectangle by multiplying with the appropriate numbers
+representing length and width is probably rather hindered
+than helped by the ordinary objective presentation as an
+introduction. The usual form of objective introduction is
+as follows:&mdash;</p>
+
+<div class="figcenter" style="width: 448px;">
+<img src="images/fig58.png" width="448" height="336" alt="Fig. 58." title="Fig. 58." /><br />
+<span class="caption"><span class="smcap">Fig. 58.</span></span>
+</div>
+
+<div class="pblockquot"><p>How long is this rectangle? How large is each square? How
+many square inches are there in the top row? How many rows are<span class='pagenum'><a name="Page_258" id="Page_258">[Pg 258]</a></span>
+there? How many square inches are there in the whole rectangle?
+Since there are three rows each containing 4 square inches,
+we have 3 &times; 4 square inches = 12 square inches.</p>
+
+<p>Draw a rectangle 7 inches long and 2 inches wide. If you divide
+it into inch squares how many rows will there be? How many
+inch squares will there be in each row? How many square inches
+are there in the rectangle?</p></div>
+
+<div class="figcenter" style="width: 525px;">
+<img src="images/fig59.png" width="525" height="403" alt="Fig. 59." title="Fig. 59." /><br />
+<span class="caption"><span class="smcap">Fig. 59.</span></span>
+</div>
+
+<p>It is better actually to hide the individual square units as
+in Fig. 59. There are four reasons: (1) The concrete rows
+and columns rather distract attention from the essential
+thing to be learned. This is not that "<i>x</i> rows one square
+wide, <i>y</i> squares in a row will make <i>xy</i> squares in all," but
+that "by using proper units and the proper operation the
+area of any rectangle can be found from its length and
+width." (2) Children have little difficulty in learning to<span class='pagenum'><a name="Page_259" id="Page_259">[Pg 259]</a></span>
+multiply rather than add, subtract, or divide when computing
+area. (3) The habit so formed holds good for areas like
+1<small><sup>2</sup>&frasl;<sub>3</sub></small> by 4&frac12;, with fractional dimensions, in which any effort
+to count up the areas of rows is very troublesome and confusing.
+(4) The notion that a square inch is an area 1' by 1'
+rather than &frac12;' by 2' or <sup>1</sup>&frasl;<sub>3</sub> in. by 3 in. or 1&frac12; in. by <sup>2</sup>&frasl;<sub>3</sub> in. is
+likely to be formed too emphatically if much time is spent
+upon the sort of concrete presentation shown above. It is
+then better to use concrete counting of rows of small areas
+as a means of <i>verification after</i> the procedure is learned, than
+as a means of deriving it.</p>
+
+<p>There has been, especially in Germany, much argument
+concerning what sort of number-pictures (that is, arrangement
+of dots, lines, or the like, as shown in Fig. 60) is best
+for use in connection with the number names in the early
+years of the teaching of arithmetic.</p>
+
+<p>Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell
+['14], and others have measured the accuracy of children in
+estimating the number of dots in arrangements of one or
+more of these different types.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a> Many writers interpret a
+difference in favor of estimating, say, the square arrangements
+of Born or Lay as meaning that such is the best
+arrangement to use in teaching. The inference is, however,
+unjustified. That certain number-pictures are easier to
+estimate numerically does not necessarily mean that they
+are more instructive in learning. One set may be easier
+to estimate just because they are more familiar, having
+been oftener experienced. Even if the favored set was so
+after equal experience with all sets, accuracy of estimation
+would be a sign of superiority for use in instruction only
+if all other things were equal (or in favor of the arrangement
+<span class='pagenum'><a name="Page_260" id="Page_260">[Pg 260]</a></span>
+in question). Obviously the way to decide which of these
+is best to use in teaching is by using them in teaching and
+measuring all relevant results, not by merely recording which
+of them are most accurately estimated in certain time exposures.</p>
+
+<div class="figcenter" style="width: 1024px;">
+<img src="images/fig60.png" width="1024" height="565" alt="Fig. 60." title="Fig. 60." />
+<span class="caption"><span class="smcap">Fig. 60.</span>&mdash;Various proposed arrangements of dots for use in teaching the meanings of the numbers 1 to 10.</span>
+</div>
+
+<p><span class='pagenum'><a name="Page_261" id="Page_261">[Pg 261]</a></span></p>
+
+
+<p>It may be noted that the Born, Lay, and Freeman pictures
+have claims for special consideration on grounds of probable
+instructiveness. Since they are also superior in the tests in
+respect to accuracy of estimate, choice should probably be
+made from these three by any teacher who wishes to connect
+one set of number-pictures systematically with the number
+names, as by drills with the blackboard or with cards.</p>
+
+<p>Such drills are probably useful if undertaken with zeal,
+and if kept as supplementary to more realistic objective
+work with play money, children marching, material to be
+distributed, garden-plot lengths to be measured, and the
+like, and if so administered that the pupils soon get the
+generalized abstract meaning of the numbers freed from
+dependence on an inner picture of any sort. This freedom
+is so important that it may make the use of many types of
+number-pictures advisable rather than the use of the one
+which in and of itself is best.</p>
+
+<p>As Meumann says: "Perceptual reckoning can be overdone.
+It had its chief significance for the surety and clearness
+of the first foundation of arithmetical instruction. If,
+however, it is continued after the first operations become
+familiar to the child, and extended to operations which develop
+from these elementary ones, it necessarily works as a
+retarding force and holds back the natural development of
+arithmetic. This moves on to work with abstract number
+and with mechanical association and reproduction." ['07,
+Vol. 2, p. 357.]</p>
+
+<p>Such drills are commonly overdone by those who make<span class='pagenum'><a name="Page_262" id="Page_262">[Pg 262]</a></span>
+use of them, being given too often, and continued after their
+instructiveness has waned, and used instead of more significant,
+interesting, and varied work in counting and estimating
+and measuring real things. Consequently, there is now
+rather a prejudice against them in our better schools. They
+should probably be reinstated but to a moderate and judicious
+use.</p>
+
+
+<h4>ORAL, MENTAL, AND WRITTEN ARITHMETIC</h4>
+
+<p>There has been much dispute over the relative merits of
+oral and written work in arithmetic&mdash;a question which is
+much confused by the different meanings of 'oral' and
+'written.' <i>Oral</i> has meant (1) work where the situations
+are presented orally and the pupil's final responses are given
+orally, or (2) work where the situations are presented orally
+and the pupils' final responses are written or partly written
+and partly oral, or (3) work where the situations are presented
+in writing or print and the final responses are oral. <i>Written</i>
+has meant (1) work where the situations are presented in
+writing or print and the final responses are made in writing,
+or (2) work where also many of the intermediate responses
+are written, or (3) work where the situations are presented
+orally but the final responses and a large percentage of the
+intermediate computational responses are written. There
+are other meanings than these.</p>
+
+<p>It is better to drop these very ambiguous terms and ask
+clearly what are the merits and demerits, in the case of any
+specified arithmetical work, of auditory and of visual presentation
+of the situations, and of saying and of writing each
+specified step in the response.</p>
+
+<p>The disputes over mental <i>versus</i> written arithmetic are
+also confused by ambiguities in the use of 'mental.' Mental
+has been used to mean "done without pencil and paper"<span class='pagenum'><a name="Page_263" id="Page_263">[Pg 263]</a></span>
+and also "done with few overt responses, either written or
+spoken, between the setting of the task and the announcement
+of the answer." In neither case is the word <i>mental</i>
+specially appropriate as a description of the total fact. As
+before, we should ask clearly, "What are the merits and
+demerits of making certain specified intermediate responses in
+inner speech or imaged sounds or visual images or imageless
+thought&mdash;that is, <i>without</i> actual writing or overt speech?"</p>
+
+<p>It may be said at the outset that oral, written, and inner
+presentations of initial situations, oral, written, and inner
+announcements of final responses, and oral, written, and
+inner management of intermediate processes have varying
+degrees of merit according to the particular arithmetical
+exercise, pupil, and context. Devotion to oralness or mentalness
+as such is simply fanatical. Various combinations, such
+as the written presentation of the situation with inner
+management of the intermediate responses and oral announcement
+of the final response have their special merits
+for particular cases.</p>
+
+<p>These merits the reader can evaluate for himself for any
+given sort of work for a given class by considering: (1) The
+amount of practice received by the class per hour spent;
+(2) the ease of correction of the work; (3) the ease of understanding
+the tasks; (4) the prevention of cheating; (5) the
+cheerfulness and sociability of the work; (6) the freedom
+from eyestrain, and other less important desiderata.</p>
+
+<p>It should be noted that the stock schemes A, B, C, and D
+below are only a few of the many that are possible and that
+schemes E, F, G, and H have special merits.</p>
+
+<p>The common practice of either having no use made of
+pencil and paper or having all computations and even much
+verbal analysis written out elaborately for examination is
+unfavorable for learning. The demands which life itself<span class='pagenum'><a name="Page_264" id="Page_264">[Pg 264]</a></span>
+will make of arithmetical knowledge and skill will range from
+tasks done with every percentage of written work from zero
+up to the case where every main result obtained by thought
+is recorded for later use by further thought. In school the
+best way is that which, for the pupils in question, has the best
+total effect upon quality of product, speed, and ease of production,
+re&euml;nforcement of training already given, and preparation
+for training to be given. There is nothing intellectually
+criminal about using a pencil as well as inner thought; on the
+other hand there is no magical value in writing out for the
+teacher's inspection figures that the pupil does not need in
+order to attain, preserve, verify, or correct his result.</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><th><span class="smcap">Presentation of Initial Situation</span></th>
+<th><span class="smcap">Management of Intermediate Processes</span></th>
+<th><span class="smcap">Announcement of Final Response</span></th></tr>
+<tr><td align='left'>A. Printed or written</td><td align='left'>Written</td><td align='left'>Written</td></tr>
+<tr><td align='left'>B. &nbsp; &nbsp; "  &nbsp; &nbsp;  &nbsp; &nbsp;  "</td><td align='left'>Inner</td><td align='left'>Oral by one pupil, inner by the rest</td></tr>
+<tr><td align='left'>C. Oral (by teacher)</td><td align='left'>Written</td><td align='left'>Written</td></tr>
+<tr><td align='left'>D. &nbsp; &nbsp; "  &nbsp; &nbsp;  &nbsp; &nbsp;  "</td><td align='left'>Inner</td><td align='left'>Oral by one pupil, inner by the rest</td></tr>
+<tr><td align='left'>E. As in A or C</td><td align='left'>A mixture, the pupil writing what he needs</td><td align='left'>As in A or B or H</td></tr>
+<tr><td align='left'>F. The real situation itself, in part at least</td><td align='left'>As in E</td><td align='left'>As in A or B or H</td></tr>
+<tr><td align='left'>G. Both read by the pupil and put orally by the teacher</td><td align='left'>As in E</td><td align='left'>As in A or B or H</td></tr>
+<tr><td align='left'>H. As in A or C or G</td><td align='left'>As in E</td><td align='left'>Written by all pupils, announced orally by one pupil</td></tr>
+</table></div>
+
+
+
+
+
+<p><span class='pagenum'><a name="Page_265" id="Page_265">[Pg 265]</a></span></p>
+
+<p>The common practice of having the final responses of all
+<i>easy</i> tasks given orally has no sure justification. On the
+contrary, the great advantage of having all pupils really
+do the work should be secured in the easy work more than
+anywhere else. If the time cost of copying the figures is
+eliminated by the simple plan of having them printed, and
+if the supervision cost of examining the papers is eliminated
+by having the pupils correct each other's work in these easy
+tasks, written answers are often superior to oral except for
+the elements of sociability and 'go' and freedom from eyestrain
+of the oral exercise. Such written work provides the
+gifted pupils with from two to ten times as much practice
+as they would get in an oral drill on the same material, supposing
+them to give inner answers to every exercise done
+by the class as a whole; it makes sure that the dull pupils
+who would rarely get an inner answer at the rate demanded
+by the oral exercise, do as much as they are able to do.</p>
+
+<p>Two arguments often made for the oral statement of
+problems by the teacher are that problems so put are better
+understood, especially in the grades up through the fifth,
+and that the problems are more likely to be genuine and
+related to the life the pupils know. When these statements
+are true, the first is a still better argument for having
+the pupils read the problems <i>aided by the teacher's oral statement
+of them</i>. For the difficulty is largely that the pupils
+cannot read well enough; and it is better to help them to
+surmount the difficulty rather than simply evade it. The
+second is not an argument for oralness <i>versus</i> writtenness,
+but for good problems <i>versus</i> bad; the teacher who makes
+up such good problems should, in fact, take special care to
+write them down for later use, which may be by voice or by
+the blackboard or by printed sheet, as is best.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_266" id="Page_266">[Pg 266]</a></span></p>
+<h2><a name="CHAPTER_XIV" id="CHAPTER_XIV"></a>CHAPTER XIV</h2>
+
+<h3>THE CONDITIONS OF LEARNING: THE PROBLEM
+ATTITUDE</h3>
+
+
+<p>Dewey, and others following him, have emphasized the
+desirability of having pupils do their work as active seekers,
+conscious of problems whose solution satisfies some real
+need of their own natures. Other things being equal, it is
+unwise, they argue, for pupils to be led along blindfold as it
+were by the teacher and textbook, not knowing where they
+are going or why they are going there. They ought rather
+to have some living purpose, and be zealous for its attainment.</p>
+
+<p>This doctrine is in general sound, as we shall see, but it is
+often misused as a defense of practices which neglect the
+formation of fundamental habits, or as a recommendation
+to practices which are quite unworkable under ordinary
+classroom conditions. So it seems probable that its nature
+and limitations are not thoroughly known, even to its followers,
+and that a rather detailed treatment of it should be
+given here.</p>
+
+
+<h4>ILLUSTRATIVE CASES</h4>
+
+<p>Consider first some cases where time spent in making
+pupils understand the end to be attained before attacking
+the task by which it is attained, or care about attaining the
+end (well or ill understood) is well spent.<span class='pagenum'><a name="Page_267" id="Page_267">[Pg 267]</a></span></p>
+
+<p>It is well for a pupil who has learned (1) the meanings of
+the numbers one to ten, (2) how to count a collection of ten
+or less, and (3) how to measure in inches a magnitude of
+ten, nine, eight inches, etc., to be confronted with the
+problem of true adding without counting or measuring, as
+in 'hidden' addition and measurement by inference. For
+example, the teacher has three pencils counted and put
+under a book; has two more counted and put under the
+book; and asks, "How many pencils are there under the
+book?" Answers, when obtained, are verified or refuted
+by actual counting and measuring.</p>
+
+<p>The time here is well spent because the children can do the
+necessary thinking if the tasks are well chosen; because they
+are thereby prevented from beginning their study of addition
+by the bad habit of pseudo-adding by looking at the two
+groups of objects and counting their number instead of
+real adding, that is, thinking of the two numbers and inferring
+their sum; and further, because facing the problem of
+adding as a real problem is in the end more economical for
+learning arithmetic and for intellectual training in general
+than being enticed into adding by objective or other processes
+which conceal the difficulty while helping the pupil to master
+it.</p>
+
+<p>The manipulation of short multiplication may be introduced
+by confronting the pupils with such problems as,
+"How to tell how many Uneeda biscuit there are in four
+boxes, by opening only one box." Correct solutions by
+addition should be accepted. Correct solutions by multiplication,
+if any gifted children think of this way, should be
+accepted, even if the children cannot justify their procedure.
+(Inferring the manipulation from the place-values of numbers
+is beyond all save the most gifted and probably beyond
+them.) Correct solution by multiplication by some child<span class='pagenum'><a name="Page_268" id="Page_268">[Pg 268]</a></span>
+who happens to have learned it elsewhere should be accepted.
+Let the main proof of the trustworthiness of the
+manipulation be by measurement and by addition. Proof
+by the stock arguments from the place-values of numbers
+may also be used. If no child hits on the manipulation in
+question, the problem of finding the length <i>without</i> adding
+may be set. If they still fail, the problem may be made
+easier by being put as "4 times 22 gives the answer. Write
+down what you think 4 times 22 will be." Other reductions
+of the difficulty of the problem may be made, or the teacher
+may give the answer without very great harm being done.
+The important requirement is that the pupils should be
+aware of the problem and treat the manipulation as a solution
+of it, not as a form of educational ceremonial which
+they learn to satisfy the whims of parents and teachers.
+In the case of any particular class a situation that is more
+appealing to the pupils' practical interests than the situation
+used here can probably be devised.</p>
+
+<p>The time spent in this way is well spent (1) because all
+but the very dull pupils can solve the problem in some way,
+(2) because the significance of the manipulation as an
+economy over addition is worth bringing out, and (3) because
+there is no way of beginning training in short multiplication
+that is much better.</p>
+
+<p>In the same fashion multiplication by two-place numbers
+may be introduced by confronting pupils with the problem
+of the number of sheets of paper in 72 pads, or pieces of chalk
+in 24 boxes, or square inches in 35 square feet, or the number
+of days in 32 years, or whatever similar problem can be
+brought up so as to be felt as a problem.</p>
+
+<p>Suppose that it is the 35 square feet. Solutions by
+(5 &times; 144) &#43; (30 &times; 144), however arranged, or by (10 &times; 144) &#43;
+(10 &times; 144) &#43; (10 &times; 144) &#43; (5 &times; 144), or by 3500 &#43; (35 &times; 40) &#43;
+<span class='pagenum'><a name="Page_269" id="Page_269">[Pg 269]</a></span>
+(35 &times; 4), or by 7 &times; (5 &times; 144), however arranged, should all be
+listed for verification or rejection. The pupils need not be
+required to justify their procedures by a verbal statement.
+Answers like 432,720, or 720,432, or 1152, or 4220, or 3220
+should be listed for verification or rejection. Verification
+may be by a mixture of short multiplication and objective
+work, or by a mixture of short multiplication and addition,
+or by addition abbreviated by taking ten 144s as
+1440, or even (for very stupid pupils) by the authority
+of the teacher. Or the manipulation in cases like 53 &times; 9
+or 84 &times; 7 may be verified by the reverse short multiplication.
+The deductive proof of the correctness of the manipulation
+may be given in whole or in part in connection with
+exercises like</p>
+
+
+<div class='center'>
+<table border="0" cellpadding="2" cellspacing="5" width="60%" summary="">
+<tr><td align='left'>10 &times; 2 =</td><td align='left'>30 &times; 14 =</td></tr>
+<tr><td align='left'>10 &times; 3 =</td><td align='left'>&nbsp; 3 &times; 44 =</td></tr>
+<tr><td align='left'>10 &times; 4 =</td><td align='left'>30 &times; 44 =</td></tr>
+<tr><td align='left'>10 &times; 14 =</td><td align='left'>&nbsp; 3 &times; 144 =</td></tr>
+<tr><td align='left'>10 &times; 44 =</td><td align='left'>20 &times; 144 =</td></tr>
+<tr><td align='left'>10 &times; 144 =</td><td align='left'>40 &times; 144 =</td></tr>
+<tr><td align='left'>20 &times; 2 =</td><td align='left'>30 &times; 144 =</td></tr>
+<tr><td align='left'>20 &times; 3 =</td><td align='left'>&nbsp; 5 &times; 144 =</td></tr>
+<tr><td align='left'>30 &times; 3 =</td><td align='left'>35 = 30 &#43; ....</td></tr>
+<tr><td align='left'>30 &times; 4 =</td><td align='left'>30 &times; 144 added to 5 &times; 144 =</td></tr>
+<tr><td align='left'>&nbsp; 3 &times; 14 =</td></tr>
+</table></div>
+
+<p>Certain wrong answers may be shown to be wrong
+in many ways; <i>e.g.</i>, 432,720 is too big, for 35 times a
+thousand square inches is only 35,000; 1152 is too small,
+for 35 times a hundred square inches would be 3500, or
+more than 1152.</p>
+
+<p>The time spent in realizing the problem here is fairly well
+spent because (1) any successful original manipulation in
+<span class='pagenum'><a name="Page_270" id="Page_270">[Pg 270]</a></span>
+this case represents an excellent exercise of thought, because
+(2) failures show that it is useless to juggle the figures
+at random, and because (3) the previous experience with
+short multiplication makes it possible for the pupils to realize
+the problem in a very few minutes. It may, however, be
+still better to give the pupils the right method just as soon
+as the problem is realized, without having them spend more
+time in trying to solve it. Thus:&mdash;</p>
+
+<p>1 square foot has 144 square inches. How many square
+inches are there in 35 square feet (marked out in chalk on
+the floor as a piece 10 ft. &times; 3 ft. plus a piece 5 ft. &times; 1 ft.)?</p>
+
+<p>1 yard = 36 inches. How many inches long is this wall
+(found by measure to be 13 yards)?</p>
+
+<p>Here is a quick way to find the answers:&mdash;</p>
+
+<div class="blockquot">
+<p class="noidt">
+<span style="margin-left: 0.5em;">144</span><br />
+<span style="margin-left: 1em;">35</span><br />
+<span style="margin-left: 0em;">&mdash;&mdash;</span><br />
+<span style="margin-left: 0.5em;">720</span><br />
+432<br />
+&mdash;&mdash;<br />
+5040 sq. inches in 35 sq. ft.<br />
+<br />
+
+<br />
+<span style="margin-left: 0.5em;">36</span><br />
+<span style="margin-left: 0.5em;">13</span><br />
+&mdash;&mdash;<br />
+108<br />
+36<br />
+&mdash;&mdash;<br />
+468 inches in 13 yd.<br />
+</p>
+</div>
+
+<p>Consider now the following introduction to dividing by a
+decimal:&mdash;</p>
+
+<p class="center"><b>Dividing by a Decimal</b></p>
+
+<div class="pblockquot">
+<p><b>1.</b> How many minutes will it take a motorcycle, to go
+12.675 miles at the rate of .75 mi. per minute?</p>
+</div>
+
+<div class="iblockquot">
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='right'>16.9</td></tr>
+<tr><td align='right'>.75 |<span class="overline"> 12.675</span></td></tr>
+<tr><td align='right'><span class="u"> 7 5</span> &nbsp; &nbsp;</td></tr>
+<tr><td align='right'>5 17 &nbsp;</td></tr>
+<tr><td align='right'><span class="u"> 4 50</span> &nbsp;</td></tr>
+<tr><td align='right'>675</td></tr>
+<tr><td align='right'><span class="u">675</span></td></tr>
+</table></div>
+
+
+<p><span class='pagenum'><a name="Page_271" id="Page_271">[Pg 271]</a></span></p>
+
+<div class="pblockquot">
+<p><b>2.</b> Check by multiplying 16.9 by .75.</p>
+
+<p><b>3.</b> How do you know that the quotient cannot be as little as 1.69?</p>
+
+<p><b>4.</b> How do you know that the quotient cannot be as large as 169?</p>
+
+<p><b>5.</b> Find the quotient for 3.75 &divide; 1.5.</p>
+
+<p><b>6.</b> Check your result by multiplying the quotient by the divisor.</p>
+
+<p><b>7.</b> How do you know that the quotient cannot be .25 or 25 ?</p>
+
+<p><b>8.</b> Look at this problem.&nbsp; &nbsp;  .25<span class="overline">|7.5</span></p>
+
+<p>&nbsp; &nbsp; How do you know that 3.0 is wrong for the quotient?</p>
+
+<p>&nbsp; &nbsp; How do you know that 300 is wrong for the quotient?</p>
+
+<p>State which quotient is right for each of these:&mdash;</p>
+
+<p><b>9.</b> &nbsp; &nbsp; <small>.021 or .21 or 2.1 or 21 or 210</small><br />
+ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1.8<span class="overline">|3.78</span></p>
+
+<p><b>10.</b> &nbsp; <small>.021 or .21 or 21 or 210</small><br />
+ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1.8<span class="overline">|37.8</span></p>
+
+<p><b>11.</b> &nbsp; <small>.03 or .3 or 3 or 30 or 300</small><br />
+ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1.25<span class="overline">|37.5</span></p>
+
+<p><b>12.</b> &nbsp; <small>.03 or .3 or 3 or 30 or 300</small><br />
+ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 12.5<span class="overline">|37.5</span></p>
+
+<p><b>13.</b> &nbsp; <small>.05 or .5 or 5 or 50 or 500</small><br />
+ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1.25<span class="overline">|6.25</span></p>
+
+<p><b>14.</b> &nbsp; <small>.05 or .5 or 5 or 50 or 500</small><br />
+ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 12.5<span class="overline">|6.25</span></p>
+
+<p><b>15.</b> Is this rule true? If it is true, learn it.</p>
+
+<div class="blockquot">
+<p class="noidt"><b>In a correct result, the number of decimal places in
+the divisor and quotient together equals the number
+of decimal places in the dividend.</b></p>
+</div>
+</div>
+
+<p>These and similar exercises excite the problem attitude
+in children <i>who have a general interest in getting right answers</i>.
+Such a series carefully arranged is a desirable introduction
+to a statement of the rule for placing the decimal point in
+division with decimals. For it attracts attention to the
+general principle (divisor &times; quotient should equal dividend),
+which is more important than the rule for convenient location
+of the decimal point, and it gives training in placing
+the point by inspection of the divisor, quotient, and dividend,
+which suffices for nineteen out of twenty cases that the pupil
+will ever encounter outside of school. He is likely to remember
+this method by inspection long after he has forgotten
+the fixed rule.</p>
+
+<p>It is well for the pupil to be introduced to many arith<span class='pagenum'><a name="Page_272" id="Page_272">[Pg 272]</a></span>metical
+facts by way of problems about their common uses.
+The clockface, the railroad distance table in hundredths
+of a mile, the cyclometer and speedometer, the recipe, and
+the like offer problems which enlist his interest and energy
+and also connect the resulting arithmetical learning with
+the activities where it is needed. There is no time cost,
+but a time-saving, for the learning as a means to the solution
+of the problems is quicker than the mere learning of the
+arithmetical facts by themselves alone. A few samples of
+such procedure are shown below:&mdash;</p>
+
+<div class="blockquot">
+
+<div class="pblockquot">
+<p class="tabcap">GRADE 3</p>
+
+<p class="center"><b>To be Done at Home</b></p>
+
+<p>Look at a watch. Has it any hands besides the hour hand and
+the minute hand? Find out all that you can about how a watch
+tells seconds, how long a second is, and how many seconds make a
+minute.</p>
+
+<p class="tabcap">GRADE 5</p>
+
+<p class="center"><b>Measuring Rainfall</b></p>
+
+
+<div class='figleft'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='center' colspan='3'><b>Rainfall per Week</b><br />
+(<b>cu. in. per sq. in. of area</b>)</td></tr>
+<tr><td align='center'>June</td><td align='left'>&nbsp; 1-7</td><td align='center'>1.056</td></tr>
+<tr><td align='center'></td><td align='left'>&nbsp; 8-14</td><td align='center'>1.103</td></tr>
+<tr><td align='center'></td><td align='left'>15-21</td><td align='center'>1.040</td></tr>
+<tr><td align='center'></td><td align='left'>22-28</td><td align='center'>&nbsp; .960</td></tr>
+<tr><td align='center'></td><td align='left'>29-July 5</td><td align='center'>&nbsp; .915</td></tr>
+<tr><td align='center'>July</td><td align='left'>&nbsp; 6-12</td><td align='center'>&nbsp; .782</td></tr>
+<tr><td align='center'></td><td align='left'>13-19</td><td align='center'>&nbsp; .790</td></tr>
+<tr><td align='center'></td><td align='left'>20-26</td><td align='center'>&nbsp; .670</td></tr>
+<tr><td align='center'></td><td align='left'>27-Aug. 2</td><td align='center'>&nbsp; .503</td></tr>
+<tr><td align='center'>Aug.</td><td align='left'>&nbsp; 3-9</td><td align='center'>&nbsp; .512</td></tr>
+<tr><td align='center'></td><td align='left'>10-16</td><td align='center'>&nbsp; .240</td></tr>
+<tr><td align='center'></td><td align='left'>17-23</td><td align='center'>&nbsp; .215</td></tr>
+<tr><td align='center'></td><td align='left'>24-30</td><td align='center'>&nbsp; .811</td></tr>
+</table></div>
+<p><b>1.</b> In which weeks was the rainfall 1
+or more?</p>
+
+<p><b>2.</b> Which week of August had the
+largest rainfall for that month?</p>
+
+<p><b>3.</b> Which was the driest week of the
+summer? (Driest means with
+the least rainfall.)</p>
+
+<p><b>4.</b> Which week was the next to the
+driest?</p>
+
+<p><b>5.</b> In which weeks was the rainfall
+between .800 and 1.000?</p>
+
+<p><b>6.</b> Look down the table and estimate
+whether the average rainfall for
+one week was about .5, or about
+.6, or about .7, or about .8, or
+about .9.</p>
+<div style="clear: both;"></div>
+<p><span class='pagenum'><a name="Page_273" id="Page_273">[Pg 273]</a></span></p>
+
+<p class="center"><b>Dairy Records</b></p>
+
+
+<div class='figleft'>
+<table border="0" cellpadding="1" cellspacing="0" summary="">
+<tr><td align='center' colspan='3'><b>Record of Star Elsie</b></td></tr>
+<tr><th></th><th><small>Pounds of<br />Milk</small></th><th><small>Butter-Fat<br />per Pound<br />of Milk</small></th></tr>
+<tr><td align='left'>Jan.</td><td align='left'><b>1742</b></td><td align='left'><b>.0461</b></td></tr>
+<tr><td align='left'>Feb.</td><td align='left'><b>1690</b></td><td align='left'><b>.0485</b></td></tr>
+<tr><td align='left'>Mar.</td><td align='left'><b>1574</b></td><td align='left'><b>.0504</b></td></tr>
+<tr><td align='left'>Apr.</td><td align='left'><b>1226</b></td><td align='left'><b>.0490</b></td></tr>
+<tr><td align='left'>May</td><td align='left'><b>1202</b></td><td align='left'><b>.0466</b></td></tr>
+<tr><td align='left'>June</td><td align='left'><b>1251</b></td><td align='left'><b>.0481</b></td></tr>
+</table></div>
+
+<p>Read this record of the
+milk given by the cow Star
+Elsie. The first column tells
+the number of pounds of
+milk given by Star Elsie each
+month. The second column
+tells what fraction of a pound
+of butter-fat each pound of
+milk contained.</p>
+
+<p><b>1.</b> Read the first line, saying, "In January this cow gave 1742
+pounds of milk. There were 461 ten thousandths of a
+pound of butter-fat per pound of milk." Read the other
+lines in the same way.</p>
+
+<p><b>2.</b> How many pounds of butter-fat did the cow produce in Jan.?
+<b>3.</b> In Feb.? <b>4.</b> In Mar.? <b>5.</b> In Apr.? <b>6.</b> In May?
+<b>7.</b> In June?</p>
+<div style="clear: both;"></div>
+
+<p class="tabcap">GRADE 5 OR LATER</p>
+
+<p class="center"><b>Using Recipes to Make Larger or Smaller Quantities</b></p>
+
+<p>I. State how much you would use of each material in the
+following recipes: (<i>a</i>) To make double the quantity. (<i>b</i>) To
+make half the quantity. (<i>c</i>) To make 1&frac12; times the quantity.
+You may use pencil and paper when you cannot find the right
+amount mentally.</p>
+
+<div class='center sblockquot'>
+<table border="0" cellpadding="4" cellspacing="0" width="80%" summary="">
+<tr><td align='left'><b>1.</b> <span class="smcap">Peanut Penuche</span></td><td align='left'><b>2</b>. <span class="smcap">Molasses Candy</span></td></tr>
+<tr><td align='left'>1 tablespoon butter</td><td align='left'>&frac12; cup butter</td></tr>
+<tr><td align='left'>2 cups brown sugar</td><td align='left'>2 cups sugar</td></tr>
+<tr><td align='left'><sup>1</sup>&frasl;<sub>3</sub> cup milk or cream</td><td align='left'>1 cup molasses</td></tr>
+<tr><td align='left'>&frac34; cup chopped peanuts</td><td align='left'>1&frac12; cups boiling water</td></tr>
+<tr><td align='left'><sup>1</sup>&frasl;<sub>3</sub> teaspoon salt</td></tr>
+<tr><td colspan='2'>&nbsp;</td></tr>
+<tr><td align='left'><b>3.</b> <span class="smcap">Raisin Opera Caramels</span></td><td align='left'><b>4.</b> <span class="smcap">Walnut Molasses Squares</span></td></tr>
+<tr><td align='left'>2 cups light brown sugar</td><td align='left'>2 tablespoons butter</td></tr>
+<tr><td align='left'><sup>7</sup>&frasl;<sub>8</sub> cup thin cream</td><td align='left'>1 cup molasses</td></tr>
+<tr><td align='left'>&frac12; cup raisins</td><td align='left'><sup>1</sup>&frasl;<sub>3</sub> cup sugar</td></tr>
+<tr><td align='left'></td><td align='left'>&frac12; cup walnut meats</td></tr>
+<tr><td colspan='2'>&nbsp;<span class='pagenum'><a name="Page_274" id="Page_274">[Pg 274]</a></span></td></tr>
+<tr><td align='left'><b>5.</b> <span class="smcap">Reception Rolls</span></td><td align='left'><b>6.</b> <span class="smcap">Graham Raised Loaf</span></td></tr>
+<tr><td align='left'>1 cup scalded milk</td><td align='left'>2 cups milk</td></tr>
+<tr><td align='left'>1&frac12; tablespoons sugar</td><td align='left'>6 tablespoons molasses</td></tr>
+<tr><td align='left'>1 teaspoon salt</td><td align='left'>1&frac12; teaspoons salt</td></tr>
+<tr><td align='left'>&frac14; cup lard</td><td align='left'><sup>1</sup>&frasl;<sub>3</sub> yeast cake</td></tr>
+<tr><td align='left'>1 yeast cake</td><td align='left'>&frac14; cup lukewarm water</td></tr>
+<tr><td align='left'>&frac14; cup lukewarm water</td><td align='left'>2 cups sifted Graham flour</td></tr>
+<tr><td align='left'>White of 1 egg</td><td align='left'>&frac12; cup Graham bran</td></tr>
+<tr><td align='left'>3&frac12; cups flour</td><td align='left'>&frac34; cup flour (to knead)</td></tr>
+</table></div>
+
+
+<p>II. How much would you use of each material in the following
+recipes: (<i>a</i>) To make <sup>2</sup>&frasl;<sub>3</sub> as large a quantity? (<i>b</i>) To make 1<small><sup>1</sup>&frasl;<sub>3</sub></small>
+times as much? (<i>c</i>) To make 2&frac12; times as much?</p>
+
+<div class='center sblockquot'>
+<table border="0" cellpadding="4" cellspacing="0" width="80%" summary="">
+<tr><td align='left'><b>1.</b> <span class="smcap">English Dumplings</span></td><td align='left'><b>2.</b> <span class="smcap">White Mountain Angel Cake</span></td></tr>
+<tr><td align='left'>&frac12; pound beef suet</td><td align='left'>1&frac12; cups egg whites</td></tr>
+<tr><td align='left'>1&frac14; cups flour</td><td align='left'>1&frac12; cups sugar</td></tr>
+<tr><td align='left'>3 teaspoons baking powder</td><td align='left'>1 teaspoon cream of tartar</td></tr>
+<tr><td align='left'>1 teaspoon salt</td><td align='left'>1 cup bread flour</td></tr>
+<tr><td align='left'>&frac12; teaspoon pepper</td><td align='left'>&frac14; teaspoon salt</td></tr>
+<tr><td align='left'>1 teaspoon minced parsley</td><td align='left'>1 teaspoon vanilla</td></tr>
+<tr><td align='left'>&frac14; cup cold water</td></tr>
+</table></div>
+</div>
+</div>
+
+<p>In many cases arithmetical facts and principles can be
+well taught in connection with some problem or project
+which is not arithmetical, but which has special potency to
+arouse an intellectual activity in the pupil which by some
+ingenuity can be directed to arithmetical learning. Playing
+store is the most fundamental case. Planning for a
+party, seeing who wins a game of bean bag, understanding
+the calendar for a month, selecting Christmas presents,
+planning a picnic, arranging a garden, the clock, the watch
+with second hand, and drawing very simple maps are situations
+suggesting problems which may bring a living purpose
+into arithmetical learning in grade 2. These are all available
+under ordinary conditions of class instruction. A
+sample of such problems for a higher grade (6) is shown
+below.</p>
+
+<div class="pblockquot">
+<p class="center"><b>Estimating Areas</b></p>
+
+<p>The children in the geography class had a contest in estimating
+the areas of different surfaces. Each child wrote his estimates
+<span class='pagenum'><a name="Page_275" id="Page_275">[Pg 275]</a></span>
+for each of these maps, A, B, C, D, and E. (Only C and D are
+shown here.) In the arithmetic class they learned how to find
+the exact areas. Then they compared their estimates with the
+exact areas to find who came nearest.</p>
+
+<div class="figcenter" style="width: 512px;">
+<img src="images/img292.jpg" width="512" height="202" alt="Estimating Areas" title="Estimating Areas" />
+</div>
+
+<p>Write your estimates for A, B, C, D, and E. Then study the
+next 6 pages and learn how to find the exact areas.</p>
+
+<p>(The next 6 pages comprise training in the mensuration of
+parallelograms and triangles.)</p></div>
+
+<p>In some cases the affairs of individual pupils include problems
+which may be used to guide the individual in question
+to a zealous study of arithmetic as a means of achieving
+his purpose&mdash;of making a canoe, surveying an island, keeping
+the accounts of a Girls' Canning Club, or the like. It
+requires much time and very great skill to direct the work
+of thirty or more pupils each busy with a special type of his
+own, so as to make the work instructive for each, but in
+some cases the expense of time and skill is justified.</p>
+
+
+<h4>GENERAL PRINCIPLES</h4>
+
+<p>In general what should be meant when one says that it is
+desirable to have pupils in the problem-attitude when they
+are studying arithmetic is substantially as follows:&mdash;</p>
+
+<p><i>First.</i>&mdash;Information that comes as an answer to questions<span class='pagenum'><a name="Page_276" id="Page_276">[Pg 276]</a></span>
+is better attended to, understood, and remembered than
+information that just comes.</p>
+
+<p><i>Second.</i>&mdash;Similarly, movements that come as a step
+toward achieving an end that the pupil has in view are
+better connected with their appropriate situations, and such
+connections are longer retained, than is the case with movements
+that just happen.</p>
+
+<p><i>Third.</i>&mdash;The more the pupil is set toward getting the
+question answered or getting the end achieved, the greater
+is the satisfyingness attached to the bonds of knowledge
+or skill which mean progress thereto.</p>
+
+<p><i>Fourth.</i>&mdash;It is bad policy to rely exclusively on the purely
+intellectualistic problems of "How can I do this?" "How
+can I get the right answer?" "What is the reason for this?"
+"Is there a better way to do that?" and the like. It is bad
+policy to supplement these intellectualistic problems by only
+the remote problems of "How can I be fitted to earn a
+higher wage?" "How can I make sure of graduating?"
+"How can I please my parents?" and the like. The purely
+intellectualistic problems have too weak an appeal for many
+pupils; the remote problems are weak so long as they are
+remote and, what is worse, may be deprived of the strength
+that they would have in due time if we attempt to use them
+too soon. It is the extreme of bad policy to neglect those
+personal and practical problems furnished by life outside the
+class in arithmetic the solution of which can really be furthered
+by arithmetic then and there. It is good policy to
+spend time in establishing certain mental sets&mdash;stimulating,
+or even creating, certain needs&mdash;setting up problems
+themselves&mdash;when the time so spent brings a sufficient
+improvement in the quality and quantity of the pupils'
+interest in arithmetical learning.</p>
+
+<p><i>Fifth.</i>&mdash;It would be still worse policy to rely exclusively<span class='pagenum'><a name="Page_277" id="Page_277">[Pg 277]</a></span>
+on problems arising outside arithmetic. To learn arithmetic
+is itself a series of problems of intrinsic interest and worth
+to healthy-minded children. The need for ability to
+multiply with United States money or to add fractions or
+to compute percents may be as truly vital and engaging as
+the need for skill to make a party dress or for money to buy
+it or for time to play baseball. The intellectualistic needs
+and problems should be considered along with all others, and
+given whatever weight their educational value deserves.</p>
+
+
+<h4>DIFFICULTY AND SUCCESS AS STIMULI</h4>
+
+<p>There are certain misconceptions of the doctrine of the
+problem-attitude. The most noteworthy is that difficulty&mdash;temporary
+failure&mdash;an inadequacy of already existing bonds&mdash;is
+the essential and necessary stimulus to thinking and
+learning. Dewey himself does not, as I understand him,
+mean this, but he has been interpreted as meaning it by
+some of his followers.<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a></p>
+
+<p>Difficulty&mdash;temporary failure, inadequacy of existing
+bonds&mdash;on the contrary does nothing whatsoever in and
+of itself; and what is done by the annoying lack of success
+which sometimes accompanies difficulty sometimes hinders
+thinking and learning.</p>
+
+<p>Mere difficulty, mere failure, mere inadequacy of existing
+bonds, does nothing. It is hard for me to add three eight-place
+numbers at a glance; I have failed to find as effective
+illustrations for pages 276 to 277 as I wished; my existing
+sensori-motor connections are inadequate to playing a golf
+course in 65. But these events and conditions have done
+nothing to stimulate me in respect to the behavior in question.
+In the first of the three there is no annoying lack and
+no dynamic influence at all; in the second there was to some
+<span class='pagenum'><a name="Page_278" id="Page_278">[Pg 278]</a></span>
+degree an annoying lack&mdash;a slight irritation at not getting
+just what I wanted,&mdash;and this might have impelled me to
+further thinking (though it did not, and getting one tiptop
+illustration would as a rule stimulate me to hunt for others
+more than failing to get such). In the third case the lack
+of the 65 does not annoy me or have any noteworthy dynamic
+effect. The lack of 90 instead of 95-100 is annoying and is
+at times a stimulus to further learning, though not nearly
+so strong a stimulus as the attainment of the 90 would be!
+At other times this annoying lack is distinctly inhibitory&mdash;a
+stimulus to ceasing to learn. In the intellectual life the
+inhibitory effect seems far the commoner of the two. Not
+getting answers seems as a rule to make us stop trying to
+get them. The annoying lack of success with a theoretical
+problem most often makes us desert it for problems to whose
+solution the existing bonds promise to be more adequate.</p>
+
+<p>The real issue in all this concerns the relative strength,
+in the pupil's intellectual life, of the "negative reaction" of
+behavior in general. An animal whose life processes are
+interfered with so that an annoying state of affairs is set
+up, changes his behavior, making one after another responses
+as his instincts and learned tendencies prescribe, until the
+annoying state of affairs is terminated, or the animal dies,
+or suffers the annoyance as less than the alternatives which
+his responses have produced. When the annoying state of
+affairs is characterized by the failure of things as they are
+to minister to a craving&mdash;as in cases of hunger, loneliness,
+sex-pursuit, and the like,&mdash;we have stimulus to action by
+an annoying lack or need, with relief from action by the
+satisfaction of the need.</p>
+
+<p>Such is in some measure true of man's intellectual life.
+In recalling a forgotten name, in solving certain puzzles, or
+in simplifying an algebraic complex, there is an annoying<span class='pagenum'><a name="Page_279" id="Page_279">[Pg 279]</a></span>
+lack of the name, solution, or factor, a trial of one after
+another response, until the annoyance is relieved by success
+or made less potent by fatigue or distraction. Even here
+the <i>difficulty</i> does not do anything&mdash;but only the annoying
+interference with our intellectual peace by the problem.
+Further, although for the particular problem, the annoying
+lack stimulates, and the successful attainment stops thinking,
+the later and more important general effect on thinking is
+the reverse. Successful attainment stops our thinking <i>on
+that problem</i> but makes us more predisposed later to thinking
+<i>in general</i>.</p>
+
+<p>Overt negative reaction, however, plays a relatively small
+part in man's intellectual life. Filling intellectual voids or
+relieving intellectual strains in this way is much less frequent
+than being stimulated positively by things seen, words read,
+and past connections acting under modified circumstances.
+The notion of thinking as coming to a lack, filling it, meeting
+an obstacle, dodging it, being held up by a difficulty and
+overcoming it, is so one-sided as to verge on phantasy. The
+overt lacks, strains, and difficulties come perhaps once in
+five hours of smooth straightforward use and adaptation
+of existing connections, and they might as truly be called
+hindrances to thought&mdash;barriers which past successes help
+the thinker to surmount. Problems themselves come more
+often as cherished issues which new facts reveal, and whose
+contemplation the thinker enjoys, than as strains or lacks
+or 'problems which I need to solve.' It is just as true that
+the thinker gets many of his problems as results from, or
+bonuses along with, his information, as that he gets much of
+his information as results of his efforts to solve problems.</p>
+
+<p>As between difficulty and success, success is in the long
+run more productive of thinking. Necessity is not the
+mother of invention. Knowledge of previous inventions is<span class='pagenum'><a name="Page_280" id="Page_280">[Pg 280]</a></span>
+the mother; original ability is the father. The solutions
+of previous problems are more potent in producing both new
+problems and their solutions than is the mere awareness of
+problems and desire to have them solved.</p>
+
+<p>In the case of arithmetic, learning to cancel instead of
+getting the product of the dividends and the product of the
+divisors and dividing the former by the latter, is a clear
+case of very valuable learning, with ease emphasized rather
+than difficulty, with the adequacy of existing bonds (when
+slightly redirected) as the prime feature of the process
+rather than their inadequacy, and with no sense of failure
+or lack or conflict. It would be absurd to spend time in
+arousing in the pupil, before beginning cancellation, a sense
+of a difficulty&mdash;viz., that the full multiplying and dividing
+takes longer than one would like. A pupil in grade 4 or
+5 might well contemplate that difficulty for years to no advantage.
+He should at once begin to cancel and prove by
+checking that errorless cancellation always gives the right
+answer. To emphasize before teaching cancellation the
+inadequacy of the old full multiplying and dividing would,
+moreover, not only be uneconomical as a means to teaching
+cancellation; it would amount to casting needless slurs
+on valuable past acquisitions, and it would, scientifically,
+be false. For, until a pupil has learned to cancel, the old
+full multiplying is not inadequate; it is admirable in every
+respect. The issue of its inadequacy does not truly appear
+until the new method is found. It is the best way until
+the better way is mastered.</p>
+
+<p>In the same way it is unwise to spend time in making
+pupils aware of the annoying lacks to be supplied by the
+multiplication tables, the division tables, the casting out
+of nines, or the use of the product of the length and breadth
+of a rectangle as its area, the unit being changed to the<span class='pagenum'><a name="Page_281" id="Page_281">[Pg 281]</a></span>
+square erected on the linear unit as base. The annoying
+lack will be unproductive, while the learning takes place
+readily as a modification of existing habits, and is sufficiently
+appreciated as soon as it does take place. The
+multiplication tables come when instead of merely counting
+by 7s from 0 up saying "7, 14, 21," etc., the pupil counts
+by 7s from 0 up saying "Two sevens make 14, three sevens
+make 21, four sevens make 28," etc. The division tables
+come as easy selections from the known multiplications;
+the casting out of nines comes as an easy device. The
+computation of the area of a rectangle is best facilitated,
+not by awareness of the lack of a process for doing it, but by
+awareness of the success of the process as verified objectively.</p>
+
+<p>In all these cases, too, the pupil would be misled if we
+aroused first a sense of the inadequacy of counting, adding,
+and objective division, an awareness of the difficulties which
+the multiplication and division tables and nines device and
+area theorem relieve. The displaced processes are admirable
+and no unnecessary fault should be found with them, and they
+are <i>not</i> inadequate until the shorter ways have been learned.</p>
+
+
+<h4>FALSE INFERENCES</h4>
+
+<p>One false inference about the problem-attitude is that the
+pupil should always understand the aim or issue before beginning
+to form the bonds which give the method or process
+that provides the solution. On the contrary, he will often
+get the process more easily and value it more highly if he
+is taught what it is <i>for</i> gradually while he is learning it.
+The system of decimal notation, for example, may better
+be taken first as a mere fact, just as we teach a child to talk
+without trying first to have him understand the value of verbal
+intercourse, or to keep clean without trying first to have
+him understand the bacteriological consequences of filth.<span class='pagenum'><a name="Page_282" id="Page_282">[Pg 282]</a></span></p>
+
+<p>A second inference&mdash;that the pupil should always be
+taught to care about an issue and crave a process for managing
+it before beginning to learn the process&mdash;is equally
+false. On the contrary, the best way to become interested
+in certain issues and the ways of handling them is to learn
+the process&mdash;even to learn it by sheer habituation&mdash;and
+then note what it does for us. Such is the case with
+".1666<sup>2</sup>&frasl;<sub>3</sub> &times; = divide by 6," ".333<sup>1</sup>&frasl;<sub>3</sub> &times; = divide by 3," "multiply
+by .875 = divide the number by 8 and subtract the
+quotient from the number."</p>
+
+<p>A third unwise tendency is to degrade the mere giving of
+information&mdash;to belittle the value of facts acquired in any
+other way than in the course of deliberate effort by the pupil
+to relieve a problem or conflict or difficulty. As a protest
+against merely verbal knowledge, and merely memoriter
+knowledge, and neglect of the active, questioning search for
+knowledge, this tendency to belittle mere facts has been
+healthy, but as a general doctrine it is itself equally one-sided.
+Mere facts not got by the pupil's thinking are often
+of enormous value. They may stimulate to active thinking
+just as truly as that may stimulate to the reception of facts.
+In arithmetic, for example, the names of the numbers, the
+use of the fractional form to signify that the upper number is
+divided by the lower number, the early use of the decimal
+point in U. S. money to distinguish dollars from cents, and
+the meanings of "each," "whole," "part," "together,"
+"in all," "sum," "difference," "product," "quotient," and
+the like are self-justifying facts.</p>
+
+<p>A fourth false inference is that whatever teaching makes
+the pupil face a question and think out its answer is thereby
+justified. This is not necessarily so unless the question is a
+worthy one and the answer that is thought out an intrinsically
+valuable one and the process of thinking used one that is<span class='pagenum'><a name="Page_283" id="Page_283">[Pg 283]</a></span>
+appropriate for that pupil for that question. Merely to
+think may be of little value. To rely much on formal discipline
+is just as pernicious here as elsewhere. The tendency
+to emphasize the methods of learning arithmetic at the expense
+of what is learned is likely to lead to abuses different
+in nature but as bad in effect as that to which the emphasis
+on disciplinary rather than content value has led in the
+study of languages and grammar, or in the old puzzle problems
+of arithmetic.</p>
+
+<p>The last false inference that I shall discuss here is the
+inference that most of the problems by which arithmetical
+learning is stimulated had better be external to arithmetic
+itself&mdash;problems about Noah's Ark or Easter Flowers or
+the Merry Go Round or A Trip down the Rhine.</p>
+
+<p>Outside interests should be kept in mind, as has been
+abundantly illustrated in this volume, but it is folly to
+neglect the power, even for very young or for very stupid
+children, of the problem "How can I get the right answer?"
+Children do have intellectual interests. They do like
+dominoes, checkers, anagrams, and riddles as truly as playing
+tag, picking flowers, and baking cake. With carefully
+graded work that is within their powers they like to learn
+to add, subtract, multiply, and divide with integers, fractions,
+and decimals, and to work out quantitative relations.</p>
+
+<p>In some measure, learning arithmetic is like learning to
+typewrite. The learner of the latter has little desire to
+present attractive-looking excuses for being late, or to save
+expense for paper. He has no desire to hoard copies of such
+and such literary gems. He may gain zeal from the fact
+that a school party is to be given and invitations are to be
+sent out, but the problem "To typewrite better" is after
+all his main problem. Learning arithmetic is in some
+measure a game whose moves are motivated by the general<span class='pagenum'><a name="Page_284" id="Page_284">[Pg 284]</a></span>
+set of the mind toward victory&mdash;winning right answers. As
+a ball-player learns to throw the ball accurately to first-base,
+not primarily because of any particular problem concerning
+getting rid of the ball, or having the man at first-base
+possess it, or putting out an opponent against whom
+he has a grudge, but because that skill is required by the
+game as a whole, so the pupil, in some measure, learns the
+technique of arithmetic, not because of particular concrete
+problems whose solutions it furnishes, but because that
+technique is required by the game of arithmetic&mdash;a game
+that has intrinsic worth and many general recommendations.</p>
+
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_285" id="Page_285">[Pg 285]</a></span></p>
+<h2><a name="CHAPTER_XV" id="CHAPTER_XV"></a>CHAPTER XV</h2>
+
+<h3>INDIVIDUAL DIFFERENCES</h3>
+
+
+<p>The general facts concerning individual variations in
+abilities&mdash;that the variations are large, that they are continuous,
+and that for children of the same age they usually
+cluster around one typical or modal ability, becoming less
+and less frequent as we pass to very high or very low degrees
+of the ability&mdash;are all well illustrated by arithmetical
+abilities.</p>
+
+
+<h4>NATURE AND AMOUNT</h4>
+
+<p>The surfaces of frequency shown in Figs. 61, 62, and 63
+are samples. In these diagrams each space along the baseline
+represents a certain score or degree of ability, and the
+height of the surface above it represents the number of
+individuals obtaining that score. Thus in Fig. 61, 63 out of
+1000 soldiers had no correct answer, 36 out of 1000 had one
+correct answer, 49 had two, 55 had three, 67 had four, and
+so on, in a test with problems (stated in words).</p>
+
+<p>Figure 61 shows that these adults varied from no problems
+solved correctly to eighteen, around eight as a central
+tendency. Figure 62 shows that children of the same
+year-age (they were also from the same neighborhood and
+in the same school) varied from under 40 to over 200 figures
+correct. Figure 63 shows that even among children who
+have all reached the same school grade and so had rather
+<span class='pagenum'><a name="Page_286" id="Page_286">[Pg 286]</a></span>
+similar educational opportunities in arithmetic, the variation
+is still very great. It requires a range from 15 to over
+30 examples right to include even nine tenths of them.</p>
+
+
+<div class="figcenter" style="width: 1024px;">
+<img src="images/fig61.jpg" width="1024" height="297" alt="Fig. 61." title="Fig. 61." />
+<p class="nblockquot"><b><span class="smcap">Fig. 61.</span>&mdash;The scores of 1000 soldiers in the National Army born in English-speaking
+countries, in Test 2 of the Army Alpha. The score is the number of
+correct answers obtained in five minutes. Probably 10 to 15 percent of these
+men were unable to read or able to read only very easy sentences at a very slow
+rate. Data furnished by the Division of Psychology in the office of the Surgeon
+General.</b></p>
+</div>
+
+<p>It should, however, be noted that if each individual had
+been scored by the average of his work on eight or ten different
+days instead of by his work in just one test, the variability
+would have been somewhat less than appears in
+Figs. 61, 62, and 63.</p>
+
+
+<div class="figcenter" style="width: 1024px;">
+<img src="images/fig62.jpg" width="1024" height="257" alt="Fig. 62." title="Fig. 62." />
+<p class="nblockquot"><b><span class="smcap">Fig. 62.</span>&mdash;The scores of 100 11-year-old pupils in a test of computation. Estimated
+from the data given by Burt ['17, p. 68] for 10-, 11-, and 12-year-olds. The score
+equals the number of correct figures.</b></p>
+</div>
+
+<p>It is also the case that if each individual had been scored,
+not in problem-solving alone or division alone, but in an
+elaborate examination on the whole field of arithmetic, the
+variability would have been somewhat less than appears in
+Figs. 61, 62, and 63. On the other hand, if the officers and
+<span class='pagenum'><a name="Page_287" id="Page_287">[Pg 287]</a></span>
+the soldiers rejected for feeblemindedness had been included
+in Fig. 61, if the 11-year-olds in special classes for the very
+dull had been included in Fig. 62, and if all children who had
+been to school six years had been included in Fig. 63, no
+matter what grade they had reached, the effect would have
+been to <i>increase</i> the variability.</p>
+
+<div class="figcenter" style="width: 1024px;">
+<img src="images/fig63.jpg" width="1024" height="439" alt="Fig. 63." title="Fig. 63." />
+<p class="nblockquot"><b><span class="smcap">Fig. 63.</span>&mdash;The scores of pupils in grade 6 in city schools in the Woody Division
+Test A. The score is the number of correct answers obtained in 20 minutes.
+From Woody ['16, p. 61].</b></p>
+</div>
+
+<p>In spite of the effort by school officers to collect in any
+one school grade those somewhat equal in ability or in
+achievement or in a mixture of the two, the population of
+the same grades in the same school system shows a very
+wide range in any arithmetical ability. This is partly because
+promotion is on a more general basis than arithmetical
+ability so that some very able arithmeticians are deliberately
+held back on account of other deficiencies, and some very
+incompetent arithmeticians are advanced on account of
+other excellencies. It is partly because of general inaccuracy
+in classifying and promoting pupils.</p>
+
+<p>In a composite score made up of the sum of the scores in
+Woody tests,&mdash;Add. A, Subt. A, Mult. A, and Div. A, and
+two tests in problem-solving (ten and six graded problems,
+<span class='pagenum'><a name="Page_288" id="Page_288">[Pg 288]</a></span>
+with maximum attainable credits of 30 and 18), Kruse
+['18] found facts from which I compute those of Table 13,
+and Figs. 64 to 66, for pupils all having the training of the
+same city system, one which sought to grade its pupils very
+carefully.</p>
+
+<div class="figcenter" style="width: 768px;">
+<img src="images/figs64_66.jpg" width="768" height="797" alt="Figs. 64-66." title="Figs. 64-66." />
+<p class="nblockquot"><b><span class="smcap">Figs.</span> 64, 65, and 66.&mdash;The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65),
+and grade 8 (Fig. 66) in a composite of tests in computation and problem-solving.
+The time was about 120 minutes. The maximum score attainable was 196.</b></p>
+</div>
+
+<p><span class='pagenum'><a name="Page_289" id="Page_289">[Pg 289]</a></span></p>
+
+<p>The overlapping of grade upon grade should be noted.
+Of the pupils in grade 6 about 18 percent do better than the
+average pupil in grade 7, and about 7 percent do better than
+the average pupil in grade 8. Of the pupils in grade 8 about
+33 percent do worse than the average pupil in grade 7 and
+about 12 percent do worse than the average pupil in grade 6.</p>
+
+
+<p class="tabcap">TABLE 13</p>
+
+<p class="center"><span class="smcap">Relative Frequencies of Scores in an Extensive Team of
+Arithmetical Tests.<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> In Percents</span></p>
+
+
+<table border="1"  frame="hsides"  rules="cols" cellpadding="2" cellspacing="0" width="60%" summary="Table 13">
+<thead class="bb bt">
+<tr><th><span class="smcap">Score</span></th><th><span class="smcap">Grade 6</span></th><th><span class="smcap">Grade 7</span></th><th><span class="smcap">Grade 8</span></th></tr>
+</thead>
+<tbody>
+<tr><td align='center'>70 to 79</td><td align='center'> 1.3</td><td align='center'> .9</td><td align='center'> .4</td></tr>
+<tr><td align='center'>80 " 89</td><td align='center'> 5.5</td><td align='center'> 2.3</td><td align='center'> .4</td></tr>
+<tr><td align='center'>90 " 99</td><td align='center'> 10.6</td><td align='center'> 4.3</td><td align='center'> 2.9</td></tr>
+<tr><td align='center'>100 " 109</td><td align='center'> 19.4</td><td align='center'> 5.2</td><td align='center'> 4.4</td></tr>
+<tr><td align='center'>110 " 119</td><td align='center'> 19.8</td><td align='center'> 18.5</td><td align='center'> 5.8</td></tr>
+<tr><td align='center'>120 " 129</td><td align='center'> 23.5</td><td align='center'> 16.2</td><td align='center'> 16.8</td></tr>
+<tr><td align='center'>130 " 139</td><td align='center'> 12.6</td><td align='center'> 17.5</td><td align='center'> 16.8</td></tr>
+<tr><td align='center'>140 " 149</td><td align='center'> 4.6</td><td align='center'> 13.9</td><td align='center'> 22.9</td></tr>
+<tr><td align='center'>150 " 159</td><td align='center'> 1.7</td><td align='center'> 13.6</td><td align='center'> 17.1</td></tr>
+<tr><td align='center'>160 " 169</td><td align='center'> 1.2</td><td align='center'> 4.8</td><td align='center'> 9.4</td></tr>
+<tr><td align='center'>170 " 179</td><td align='center'></td><td align='center'> 2.5</td><td align='center'> 3.3</td></tr>
+</tbody>
+</table>
+
+<h4>DIFFERENCES WITHIN ONE CLASS</h4>
+
+<p>The variation within a single class for which a single
+teacher has to provide is great. Even when teaching is
+departmental and promotion is by subjects, and when also
+the school is a large one and classification within a grade is
+by ability&mdash;there may be a wide range for any given special
+component ability. Under ordinary circumstances the
+range is so great as to be one of the chief limiting conditions
+for the teaching of arithmetic. Many methods appropriate
+<span class='pagenum'><a name="Page_290" id="Page_290">[Pg 290]</a></span>
+to the top quarter of the class will be almost useless for the
+bottom quarter, and <i>vice versa</i>.</p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/fig67.jpg" width="600" height="766" alt="Fig. 67." title="Fig. 67." />
+<span class="caption"><span class="smcap">Fig. 67.</span></span>
+</div>
+
+<p>&nbsp;</p>
+<p><span class='pagenum'><a name="Page_291" id="Page_291">[Pg 291]</a></span></p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/fig68.jpg" width="600" height="735" alt="Fig. 68." title="Fig. 68." />
+<span class="caption"><span class="smcap">Fig. 68.</span><br />
+</span>
+<p class="nblockquot"><b><span class="smcap">Figs.</span> 67 and 68.&mdash;The scores of ten 6 B classes in a 12-minute test in computation
+with integers (the Courtis Test 7). The score is the number of units done.
+Certain long tasks are counted as two units.</b></p>
+
+</div>
+
+<p>Figures 67 and 68 show the scores of ten classes taken at
+random from ninety 6 B classes in one city by Courtis ['13,
+p. 64] in amount of computation done in 12 minutes. Observe
+the very wide variation present in the case of every
+<span class='pagenum'><a name="Page_292" id="Page_292">[Pg 292]</a></span>
+class. The variation within a class would be somewhat
+reduced if each pupil were measured by his average in eight
+or ten such tests given on different days. If a rather generous
+allowance is made for this we still have a variation in
+speed as great as that shown in Fig. 69, as the fact to be
+expected for a class of thirty-two 6 B pupils.</p>
+
+<div class="figcenter" style="width: 800px;">
+<img src="images/fig69.jpg" width="800" height="274" alt="Fig. 69." title="Fig. 69." />
+<p class="nblockquot"><b><span class="smcap">Fig. 69.</span>&mdash;A conservative estimate of the amount of variation to be expected
+within a single class of 32 pupils in grade 6, in the number of units done in
+Courtis Test 7 when all chance variations are eliminated.</b></p>
+</div>
+
+<p>The variations within a class in respect to what processes
+are understood so as to be done with only occasional errors
+may be illustrated further as follows:&mdash;A teacher in grade
+4 at or near the middle of the year in a city doing the customary
+work in arithmetic will probably find some pupil
+in her class who cannot do column addition even without
+carrying, or the easiest written subtraction</p>
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><td rowspan='2'><span class="sz30">(</span></td><td align='center'>8</td><td align='center'>9</td><td align='center'>&nbsp;</td><td align='center'>78</td><td rowspan='2'><span class="sz30">)</span>,</td></tr>
+<tr><td align='center'><span class="u">5</span></td><td align='center'><span class="u">3</span></td><td align='center'>or</td><td align='center'><span class="u">37</span></td></tr>
+</table></div>
+
+<p class="noidt">who does not know his multiplication tables or how to derive
+them, or understand the meanings of &#43; &minus; &times; and &divide;, or
+have any useful ideas whatever about division.</p>
+
+<p>There will probably be some child in the class who can do
+such work as that shown below, and with very few errors.<span class='pagenum'><a name="Page_293" id="Page_293">[Pg 293]</a></span></p>
+
+<p class="noidt">Add</p>
+<div class='center'>
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr><td align='center'></td>
+<td align='center'><sup>3</sup>&frasl;<sub>8</sub> &#43; <sup>5</sup>&frasl;<sub>8</sub> &#43; <sup>7</sup>&frasl;<sub>8</sub> &#43; <sup>1</sup>&frasl;<sub>8</sub>&nbsp; &nbsp; &nbsp; </td>
+<td align='center'>2&frac12;<br />6<small><sup>3</sup>&frasl;<sub>8</sub></small><br /><span class="u">3&frac34;</span></td>
+<td align='center'>&nbsp; &nbsp; &nbsp; <sup>1</sup>&frasl;<sub>6</sub> &#43; <sup>3</sup>&frasl;<sub>8</sub></td></tr>
+</table></div>
+
+
+<p class="noidt">Subtract</p>
+<div class='center'>
+<table border="0" cellpadding="10" cellspacing="0" summary="">
+<tr><td align='right'>10.00<br /><span class="u">3.49</span></td><td align='right'>4 yd. 1 ft. 6 in.<br /><span class="u">2 yd. 2 ft. 3 in.</span></td></tr>
+</table></div>
+
+<p class="noidt">Multiply</p>
+<div class='center'>
+<table border="0" cellpadding="0" cellspacing="0" summary="">
+<tr><td align='center'>1&frac14; &times; 8</td>
+<td align='center'>16<br />2<small><sup>5</sup>&frasl;<sub>8</sub></small></td><td align='center'>145<br />206</td></tr>
+<tr><td align='center'></td><td align='center'>&mdash;&mdash;</td><td align='center'>&mdash;&mdash;</td></tr>
+</table></div>
+
+
+<p class="noidt">Divide</p>
+<p class="center">2 )<span class="overline"> 13.50</span>         &nbsp; &nbsp; &nbsp;        25 )<span class="overline"> 9750</span>
+</p>
+
+<p>The invention of means of teaching thirty so different
+children at once with the maximum help and minimum
+hindrance from their different capacities and acquisitions
+is one of the great opportunities for applied science.</p>
+
+<p>Courtis, emphasizing the social demand for a certain
+moderate arithmetical attainment in the case of nearly all
+elementary school children of, say, grade 6, has urged that
+definite special means be taken to bring the deficient children
+up to certain standards, without causing undesirable 'overlearning'
+by the more gifted children. Certain experimental
+work to this end has been carried out by him and
+others, but probably much more must be done before an
+authoritative program for securing certain minimum standards
+for all or nearly all pupils can be arranged.</p>
+
+
+<h4>THE CAUSES OF INDIVIDUAL DIFFERENCES</h4>
+
+<p>The differences found among children of the same grade
+in the same city are due in large measure to inborn differences
+in their original natures. If, by a miracle, the children
+studied by Courtis, or by Woody, or by Kruse had all re<span class='pagenum'><a name="Page_294" id="Page_294">[Pg 294]</a></span>ceived
+exactly the same nurture from birth to date, they
+would still have varied greatly in arithmetical ability, perhaps
+almost as much as they now do vary.</p>
+
+<p>The evidence for this is the general evidence that variation
+in original nature is responsible for much of the eventual
+variation found in intellectual and moral traits, plus certain
+special evidence in the case of arithmetical abilities themselves.</p>
+
+<p>Thorndike found ['05] that in tests with addition and
+multiplication twins were very much more alike than
+siblings<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a> two or three years apart in age, though the resemblance
+in home and school training in arithmetic should
+be nearly as great for the latter as for the former. Also
+the young twins (9-11) showed as close a resemblance in
+addition and multiplication as the older twins (12-15),
+although the similarities of training in arithmetic have had
+twice as long to operate in the latter case.</p>
+
+<p>If the differences found, say among children in grade 6 in
+addition, were due to differences in the quantity and quality
+of training in addition which they have had, then by giving
+each of them 200 minutes of additional identical training
+the differences should be reduced. For the 200 minutes of
+identical training is a step toward equalizing training. It
+has been found in many investigations of the matter that
+when we make training in arithmetic more nearly equal for
+any group the variation within the group is not reduced.</p>
+
+<p>On the contrary, equalizing training seems rather to increase
+differences. The superior individual seems to have
+attained his superiority by his own superiority of nature
+rather than by superior past training, for, during a period
+of equal training for all, he increases his lead. For example,
+compare the gains of different individuals due to
+<span class='pagenum'><a name="Page_295" id="Page_295">[Pg 295]</a></span>
+about 300 minutes of practice in mental multiplication of a
+three-place number by a three-place number shown in
+Table 14 below, from data obtained by the author ['08].<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a></p>
+
+<p class="tabcap">TABLE 14</p>
+
+<p class="center"><span class="smcap">The Effect of Equal Amounts of Practice upon Individual Difference
+in the Multiplication Of Three-Place Numbers</span></p>
+
+<div class='center'>
+<table border="1" cellpadding="4" cellspacing="0" width="80%" summary="Table 14">
+<thead class="bb bt">
+<tr><th rowspan='2'>&nbsp;</th><th colspan='2'><span class="smcap">Amount</span></th><th colspan='2'><span class="smcap">Percentage of Correct Figures</span></th></tr>
+<tr><th>Initial Score</th><th>Gain</th><th>Initial Score</th><th>Gain</th></tr>
+</thead>
+<tbody>
+<tr><td align='center'>Initially highest five individuals</td><td align='center'> 85</td><td align='center'> 61</td><td align='center'> 70</td><td align='center'> 18</td></tr>
+<tr><td align='center'>next &nbsp;five &nbsp; &nbsp; "</td><td align='center'> 56</td><td align='center'> 51</td><td align='center'> 68</td><td align='center'> 10</td></tr>
+<tr><td align='center'>next &nbsp; six &nbsp; &nbsp; "</td><td align='center'> 46</td><td align='center'> 22</td><td align='center'> 74</td><td align='center'> 8</td></tr>
+<tr><td align='center'>next &nbsp; six &nbsp; &nbsp; "</td><td align='center'> 38</td><td align='center'> 8</td><td align='center'> 58</td><td align='center'> 12</td></tr>
+<tr><td align='center'>next &nbsp; six &nbsp; &nbsp; "</td><td align='center'> 29</td><td align='center'> 24</td><td align='center'> 56</td><td align='center'> 14</td></tr>
+</tbody>
+</table></div>
+
+<h4>THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES</h4>
+
+<p>Achievement in arithmetic depends upon a number of
+different abilities. For example, accuracy in copying numbers
+depends upon eyesight, ability to perceive visual details,
+and short-term memory for these. Long column
+addition depends chiefly upon great strength of the addition
+combinations especially in higher decades, 'carrying,' and
+keeping one's place in the column. The solution of problems
+framed in words requires understanding of language,
+the analysis of the situation described into its elements, the
+selection of the right elements for use at each step and their
+use in the right relations.</p>
+
+<p><span class='pagenum'><a name="Page_296" id="Page_296">[Pg 296]</a></span></p>
+<p>Since the abilities which together constitute arithmetic
+ability are thus specialized, the individual who is the best of
+a thousand of his age or grade in respect to, say, adding
+integers, may occupy different stations, perhaps from 1st
+to 600th, in multiplying with integers, placing the decimal
+point in division with decimals, solving novel problems,
+copying figures, etc., etc. Such specialization is in part
+due to his having had, relatively to the others in the thousand,
+more or better training in certain of these abilities than in
+others, and to various circumstances of life which have
+caused him to have, relatively to the others in the thousand,
+greater interest in certain of these achievements than in
+others. The specialization is not wholly due thereto, however.
+Certain inborn characteristics of an individual predispose
+him to different degrees of superiority or inferiority
+to other men in different features of arithmetic.</p>
+
+<p>We measure the extent to which ability of one sort goes
+with or fails to go with ability of some other sort by the
+coefficient of correlation between the two. If every individual
+keeps the same rank in the second ability&mdash;if
+the individual who is the best of the thousand in one is
+the best of the group in the other, and so on down the list&mdash;the
+correlation is 1.00. In proportion as the ranks of
+individuals vary in the two abilities the coefficient drops
+from 1.00, a coefficient of 0 meaning that the best individual
+in ability A is no more likely to be in first place in ability B
+than to be in any other rank.</p>
+
+<p>The meanings of coefficients of correlation of .90, .70, .50,
+and 0 are shown by Tables 15, 16, 17 and 18.<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a></p>
+
+<p><span class='pagenum'><a name="Page_297" id="Page_297">[Pg 297]</a></span></p>
+
+<p class="tabcap">TABLE 15</p>
+
+<p class="center"><span class="smcap">Distribution of Arrays in Successive Tenths of the Group
+When</span> <i>r</i> = .90</p>
+
+<div class='center'>
+<table border="1"  frame="hsides"  rules="cols" cellpadding="2" cellspacing="0" summary="Table 15">
+<thead class="bb bt">
+<tr><th>&nbsp;</th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr>
+</thead>
+<tbody>
+<tr><td align='left'>1st tenth</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> .1</td><td align='center'> .4</td><td align='center'> 1.8</td><td align='center'> 6.6</td><td align='center'>22.4</td><td align='center'>68.7</td></tr>
+<tr><td align='left'>2d tenth</td><td align='center'></td><td align='center'></td><td align='center'> .1</td><td align='center'> .4</td><td align='center'> 1.4</td><td align='center'> 4.7</td><td align='center'>11.5</td><td align='center'>23.5</td><td align='center'>36.0</td><td align='center'>22.4</td></tr>
+<tr><td align='left'>3d tenth</td><td align='center'></td><td align='center'> .1</td><td align='center'> .5</td><td align='center'> 2.1</td><td align='center'> 5.8</td><td align='center'>12.8</td><td align='center'>21.1</td><td align='center'>27.4</td><td align='center'>23.5</td><td align='center'> 6.6</td></tr>
+<tr><td align='left'>4th tenth</td><td align='center'></td><td align='center'> .4</td><td align='center'> 2.1</td><td align='center'> 6.4</td><td align='center'>12.8</td><td align='center'>20.1</td><td align='center'>23.8</td><td align='center'>21.2</td><td align='center'>11.5</td><td align='center'> 1.8</td></tr>
+<tr><td align='left'>5th tenth</td><td align='center'> .1</td><td align='center'> 1.4</td><td align='center'> 5.8</td><td align='center'>12.8</td><td align='center'>19.3</td><td align='center'>22.6</td><td align='center'>20.1</td><td align='center'>12.8</td><td align='center'> 4.7</td><td align='center'> .4</td></tr>
+<tr><td align='left'>6th tenth</td><td align='center'> .4</td><td align='center'> 4.7</td><td align='center'>12.8</td><td align='center'>20.1</td><td align='center'>22.6</td><td align='center'>19.3</td><td align='center'>12.8</td><td align='center'> 5.8</td><td align='center'> 1.4</td><td align='center'> .1</td></tr>
+<tr><td align='left'>7th tenth</td><td align='center'> 1.8</td><td align='center'>11.5</td><td align='center'>21.2</td><td align='center'>23.8</td><td align='center'>20.1</td><td align='center'>12.8</td><td align='center'> 6.4</td><td align='center'> 2.1</td><td align='center'> .4</td><td>&nbsp;</td></tr>
+<tr><td align='left'>8th tenth</td><td align='center'> 6.6</td><td align='center'>23.5</td><td align='center'>27.4</td><td align='center'>21.1</td><td align='center'>12.8</td><td align='center'> 5.8</td><td align='center'> 2.1</td><td align='center'> .5</td><td align='center'> .1</td><td>&nbsp;</td></tr>
+<tr><td align='left'>9th tenth</td><td align='center'>22.4</td><td align='center'>36.0</td><td align='center'>23.5</td><td align='center'>11.5</td><td align='center'> 4.7</td><td align='center'> 1.4</td><td align='center'> .4</td><td align='center'> .1</td><td align='center'></td><td>&nbsp;</td></tr>
+<tr><td align='left'>10th tenth</td><td align='center'>68.7</td><td align='center'>22.4</td><td align='center'> 6.6</td><td align='center'> 1.8</td><td align='center'> .4</td><td align='center'> .1</td><td align='center'></td><td align='center'></td><td align='center'>&nbsp;</td><td>&nbsp;</td></tr>
+</tbody>
+</table></div>
+
+<p>&nbsp;</p>
+
+<p class="tabcap">TABLE 16</p>
+
+<p class="center"><span class="smcap">Distribution of Arrays in Successive Tenths of the Group
+When</span> <i>r</i> = .70</p>
+
+<div class='center'>
+<table border="1"  frame="hsides"  rules="cols" cellpadding="2" cellspacing="0" summary="Table 16">
+<thead class="bb bt">
+<tr><th>&nbsp;</th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr>
+</thead>
+<tbody>
+<tr><td align='left'>1st tenth</td><td align='center'></td><td align='center'> .2</td><td align='center'> .7</td><td align='center'> 1.5</td><td align='center'> 2.8</td><td align='center'> 4.8</td><td align='center'> 8.0</td><td align='center'>13.0</td><td align='center'>22.3</td><td align='center'>46.7</td></tr>
+<tr><td align='left'>2d tenth</td><td align='center'> .2</td><td align='center'> 1.2</td><td align='center'> 2.6</td><td align='center'> 4.5</td><td align='center'> 7.0</td><td align='center'> 9.8</td><td align='center'>13.4</td><td align='center'>17.3</td><td align='center'>21.7</td><td align='center'>22.3</td></tr>
+<tr><td align='left'>3d tenth</td><td align='center'> .7</td><td align='center'> 2.6</td><td align='center'> 5.0</td><td align='center'> 7.3</td><td align='center'>10.0</td><td align='center'>12.5</td><td align='center'>14.9</td><td align='center'>16.7</td><td align='center'>17.3</td><td align='center'>13.0</td></tr>
+<tr><td align='left'>4th tenth</td><td align='center'> 1.5</td><td align='center'> 4.5</td><td align='center'> 7.3</td><td align='center'> 9.8</td><td align='center'>12.0</td><td align='center'>13.7</td><td align='center'>14.8</td><td align='center'>14.9</td><td align='center'>13.4</td><td align='center'> 8.0</td></tr>
+<tr><td align='left'>5th tenth</td><td align='center'> 2.8</td><td align='center'> 7.0</td><td align='center'>10.0</td><td align='center'>12.0</td><td align='center'>13.4</td><td align='center'>14.0</td><td align='center'>13.7</td><td align='center'>12.5</td><td align='center'> 9.8</td><td align='center'> 4.8</td></tr>
+<tr><td align='left'>6th tenth</td><td align='center'> 4.8</td><td align='center'> 9.8</td><td align='center'>12.5</td><td align='center'>13.7</td><td align='center'>14.0</td><td align='center'>13.4</td><td align='center'>12.0</td><td align='center'>10.0</td><td align='center'> 7.0</td><td align='center'> 2.8</td></tr>
+<tr><td align='left'>7th tenth</td><td align='center'> 8.0</td><td align='center'>13.4</td><td align='center'>14.9</td><td align='center'>14.8</td><td align='center'>13.7</td><td align='center'>12.0</td><td align='center'> 9.8</td><td align='center'> 7.3</td><td align='center'> 4.5</td><td align='center'> 1.5</td></tr>
+<tr><td align='left'>8th tenth</td><td align='center'>13.0</td><td align='center'>17.3</td><td align='center'>16.7</td><td align='center'>14.9</td><td align='center'>12.5</td><td align='center'>10.0</td><td align='center'> 7.3</td><td align='center'> 5.0</td><td align='center'> 2.6</td><td align='center'> .7</td></tr>
+<tr><td align='left'>9th tenth</td><td align='center'>22.3</td><td align='center'>21.7</td><td align='center'>17.3</td><td align='center'>13.4</td><td align='center'> 9.8</td><td align='center'> 7.0</td><td align='center'> 4.5</td><td align='center'> 2.6</td><td align='center'> 1.2</td><td align='center'> .2</td></tr>
+<tr><td align='left'>10th tenth</td><td align='center'>46.7</td><td align='center'>22.3</td><td align='center'>13.0</td><td align='center'> 8.0</td><td align='center'> 4.8</td><td align='center'> 2.8</td><td align='center'> 1.5</td><td align='center'> .7</td><td align='center'> .2</td><td>&nbsp;</td></tr>
+</tbody>
+</table></div>
+
+
+<p><span class='pagenum'><a name="Page_298" id="Page_298">[Pg 298]</a></span></p>
+
+<p>&nbsp;</p>
+
+<p class="tabcap">TABLE 17</p>
+
+<p class="center"><span class="smcap">Distribution of Arrays of Successive Tenths of the Group
+When</span> <i>r</i> = .50</p>
+
+
+<div class='center'>
+<table border="1"  frame="hsides"  rules="cols" cellpadding="2" cellspacing="0" summary="Table 17">
+<thead class="bb bt">
+<tr><th>&nbsp;</th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr>
+</thead>
+<tbody>
+<tr><td align='left'>1st tenth</td><td align='center'> .8</td><td align='center'> 2.0</td><td align='center'> 3.2</td><td align='center'> 4.6</td><td align='center'> 6.2</td><td align='center'> 8.1</td><td align='center'>10.5</td><td align='center'>13.9</td><td align='center'>18.0</td><td align='center'>31.8</td></tr>
+<tr><td align='left'>2d tenth</td><td align='center'> 2.0</td><td align='center'> 4.1</td><td align='center'> 5.7</td><td align='center'> 7.3</td><td align='center'> 8.8</td><td align='center'>10.5</td><td align='center'>12.2</td><td align='center'>14.1</td><td align='center'>16.4</td><td align='center'>18.9</td></tr>
+<tr><td align='left'>3d tenth</td><td align='center'> 3.2</td><td align='center'> 5.7</td><td align='center'> 7.4</td><td align='center'> 8.9</td><td align='center'>10.0</td><td align='center'>11.2</td><td align='center'>12.3</td><td align='center'>13.3</td><td align='center'>14.1</td><td align='center'>13.9</td></tr>
+<tr><td align='left'>4th tenth</td><td align='center'> 4.6</td><td align='center'> 7.3</td><td align='center'> 8.8</td><td align='center'> 9.9</td><td align='center'>10.8</td><td align='center'>11.6</td><td align='center'>12.0</td><td align='center'>12.3</td><td align='center'>12.2</td><td align='center'>10.5</td></tr>
+<tr><td align='left'>5th tenth</td><td align='center'> 6.2</td><td align='center'> 8.8</td><td align='center'>10.0</td><td align='center'>10.8</td><td align='center'>11.3</td><td align='center'>11.5</td><td align='center'>11.6</td><td align='center'>11.2</td><td align='center'>10.5</td><td align='center'> 8.1</td></tr>
+<tr><td align='left'>6th tenth</td><td align='center'> 8.1</td><td align='center'>10.5</td><td align='center'>11.2</td><td align='center'>11.6</td><td align='center'>11.5</td><td align='center'>11.3</td><td align='center'>10.8</td><td align='center'>10.0</td><td align='center'> 8.8</td><td align='center'> 6.2</td></tr>
+<tr><td align='left'>7th tenth</td><td align='center'>10.5</td><td align='center'>12.2</td><td align='center'>12.3</td><td align='center'>12.0</td><td align='center'>11.6</td><td align='center'>10.8</td><td align='center'> 9.9</td><td align='center'> 8.8</td><td align='center'> 7.5</td><td align='center'> 4.6</td></tr>
+<tr><td align='left'>8th tenth</td><td align='center'>13.9</td><td align='center'>14.1</td><td align='center'>13.3</td><td align='center'>12.3</td><td align='center'>11.2</td><td align='center'>10.0</td><td align='center'> 8.8</td><td align='center'> 7.4</td><td align='center'> 5.7</td><td align='center'> 3.2</td></tr>
+<tr><td align='left'>9th tenth</td><td align='center'>18.9</td><td align='center'>16.4</td><td align='center'>14.1</td><td align='center'>12.2</td><td align='center'>10.5</td><td align='center'> 8.8</td><td align='center'> 7.3</td><td align='center'> 5.7</td><td align='center'> 4.1</td><td align='center'> 2.0</td></tr>
+<tr><td align='left'>10th tenth</td><td align='center'>31.8</td><td align='center'>18.9</td><td align='center'>13.9</td><td align='center'>10.5</td><td align='center'> 8.1</td><td align='center'> 6.2</td><td align='center'> 4.6</td><td align='center'> 3.2</td><td align='center'> 2.0</td><td align='center'> .8</td></tr>
+</tbody>
+</table></div>
+
+<p>&nbsp;</p>
+
+<p class="tabcap">TABLE 18</p>
+
+<p class="center"><span class="smcap">Distribution of Arrays, in Successive Tenths of the Group
+When</span> <i>r</i> = .0</p>
+
+<div class='center'>
+<table border="1"  frame="hsides"  rules="cols" cellpadding="2" cellspacing="0" summary="Table 18">
+<thead class="bb bt">
+<tr><th>&nbsp;</th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr>
+</thead>
+<tbody>
+<tr><td align='left'>1st tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>2d tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>3d tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>4th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>5th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>6th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>7th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>8th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>9th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+<tr><td align='left'>10th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr>
+</tbody>
+</table></div>
+
+<p>&nbsp;</p>
+<p>The significance of any coefficient of correlation depends
+upon the group of individuals for which it is determined. A
+correlation of .40 between computation and problem-solving
+<span class='pagenum'><a name="Page_299" id="Page_299">[Pg 299]</a></span>
+in eighth-grade pupils of 14 years would mean a much closer
+real relation than a correlation of .40 in all 14-year-olds,
+and a very, very much closer relation than a correlation of
+.40 for all children 8 to 15.</p>
+
+<p>Unless the individuals concerned are very elaborately
+tested on several days, the correlations obtained are "attenuated"
+toward 0 by the "accidental" errors in the
+original measurements. This effect was not known until
+1904; consequently the correlations in the earlier studies of
+arithmetic are all too low.</p>
+
+<p>In general, the correlation between ability in any one
+important feature of computation and ability in any other
+important feature of computation is high. If we make
+enough tests to measure each individual exactly in:&mdash;</p>
+
+<p>(<i>A</i>) Subtraction with integers and decimals,</p>
+
+<p>(<i>B</i>) Multiplication with integers and decimals,</p>
+
+<p>(<i>C</i>) Division with integers and decimals,</p>
+
+<p>(<i>D</i>) Multiplication and division with common fractions,
+and</p>
+
+<p>(<i>E</i>) Computing with percents,</p>
+
+<p class="noidt">we shall probably find the intercorrelations for a thousand
+14-year-olds to be near .90. Addition of integers (<i>F</i>)
+will, however, correlate less closely with any of the
+above, being apparently dependent on simpler and more
+isolated abilities.</p>
+
+<p>The correlation between problem-solving (<i>G</i>) and computation
+will be very much less, probably not over .60.</p>
+
+<p>It should be noted that even when the correlation is as
+high as .90, there will be some individuals very high in one
+ability and very low in the other. Such disparities are to
+some extent, as Courtis ['13, pp. 67-75] and Cobb ['17] have
+argued, due to inborn characteristics of the individual in
+question which predispose him to very special sorts of
+<span class='pagenum'><a name="Page_300" id="Page_300">[Pg 300]</a></span>
+strength and weakness. They are often due, however, to
+defects in his learning whereby he has acquired more ability
+than he needs in one line of work or has failed to acquire
+some needed ability which was well within his capacity.</p>
+
+<p>In general, all correlations between an individual's divergence
+from the common type or average of his age for one
+arithmetical function, and his divergences from the average
+for any other arithmetical function, are positive. The
+correlation due to original capacity more than counterbalances
+the effects that robbing Peter to pay Paul may
+have.</p>
+
+<p>Speed and accuracy are thus positively correlated. The
+individuals who do the most work in ten minutes will be
+above the average in a test of accuracy. The common notion
+that speed is opposed to accuracy is correct when it means
+that the same person will tend to make more errors if he
+works at too rapid a rate; but it is entirely wrong when it
+means that the kind of person who works more rapidly
+than the average person is likely to be less accurate than the
+average person.</p>
+
+<p>Interest in arithmetic and ability at arithmetic are
+probably correlated positively in the sense that the pupil
+who has more interest than other pupils of his age tends in
+the long run to have more ability than they. They are
+certainly correlated in the sense that the pupil who 'likes'
+arithmetic better than geography or history tends to have
+relatively more ability in arithmetic, or, in other words,
+that the pupil who is more gifted at arithmetic than at
+drawing or English tends also to like it better than he likes
+these. These correlations are high.</p>
+
+<p>It is correct then to think of mathematical ability as, in
+a sense, a unitary ability of which any one individual may
+have much or little, most individuals possessing a moderate
+<span class='pagenum'><a name="Page_301" id="Page_301">[Pg 301]</a></span>
+amount of it. This is consistent, however, with the occasional
+appearance of individuals possessed of very great
+talents for this or that particular feature of mathematical
+ability and equally notable deficiencies in other features.</p>
+
+<p>Finally it may be noted that ability in arithmetic, though
+occasionally found in men otherwise very stupid, is usually
+associated with superior intelligence in dealing with ideas
+and symbols of all sorts, and is one of the best early indications
+thereof.</p>
+<p>&nbsp;</p>
+
+<div class="footnotes">
+<h3>FOOTNOTES</h3>
+<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The following and later problems are taken from actual textbooks or
+courses of study or state examinations; to avoid invidious comparisons, they
+are not exact quotations, but are equivalents in principle and form, as stated
+in the preface.</p></div>
+
+<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> The work of Mitchell has not been published, but the author has had
+the privilege of examining it.</p></div>
+
+<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> The form of Test 6 quoted here is that given by Courtis ['11-'12, p. 20].
+This differs a little from the other series of Test 6, shown on pages 43 and 44.</p></div>
+
+<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> Eight or ten times <i>in all</i>, not eight or ten times for each fact of the
+tables.</p></div>
+
+<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> The facts concerning the present inaccuracy of school work in arithmetic
+will be found on pages 102 to 105.</p></div>
+
+<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> McLellan and Ames, <i>Public School Arithmetic</i> [1900].</p></div>
+
+<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> These concern allowances for two errors occurring in the same example
+and for the same wrong answer being obtained in both original work and
+check work.</p></div>
+
+<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> The very early learning
+of 2 &times; 2, 2 &times; 3, 3 &times; 2, 2 &times; 4, 4 &times; 2, 3 &times; 3, and perhaps
+a few more multiplications is not considered here. It is advisable.
+The treatment of 0 &times; 0, 0 &times; 1, 1 &times; 0, etc., is not considered here. It is probably
+best to defer the '&times; 0' bonds until after all the others are formed and
+are being used in short multiplication, and to form them in close connection
+with their use in short multiplication. The '0 &times;' bonds may well be deferred
+until they are needed in 'long' multiplication, 0 &times; 0 coming last of all.</p></div>
+
+<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> See page 76.</p></div>
+
+<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> At the end of a volume or part, the count may be from as few as 5 or as
+many as 12 pages.</p></div>
+
+<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> Certain paragraphs in this and the following chapter are taken from the
+author's <i>Educational Psychology</i>, with slight modifications.</p></div>
+
+<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> It should be noted that just as concretes give rise to abstractions, so these
+in turn give rise to still more abstract abstractions. Thus fourness, fiveness,
+twentyness, and the like give rise to 'integral-number-ness.' Similarly just
+as individuals are grouped into general classes, so classes are grouped into still
+more general classes. Half, quarter, sixth, and tenth are general notions, but
+'one ...th' is more general; and 'fraction' is still more general.</p></div>
+
+<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> They may, of course, also result in a fusion or an alternation of responses,
+but only rarely.</p></div>
+
+<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> The more gifted children may be put to work using the principle after
+the first minute or two.</p></div>
+
+<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a>
+If desired this form may be used, with the appropriate difference in the form of the questions and statements.</p>
+
+<table border="0" cellpadding="4" cellspacing="0" summary="">
+<tr><td align='right'> 232<br />
+<span class="u"> &nbsp; 30</span><br />
+ 000<br />
+<span class="u"> 696&nbsp;&nbsp;</span><br />
+6960</td></tr>
+</table>
+</div>
+
+<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> Courtis finds in the case of addition that "of all the individuals making
+mistakes at any given time in a class, at least one third, and usually two
+thirds, will be making mistakes in carrying or copying."</p></div>
+
+<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> Facts concerning the conditions of learning in general will be found in
+the author's <i>Educational Psychology</i>, Vol. 2, Chapter 8, or in the <i>Educational
+Psychology, Briefer Course</i>, Chapter 15.</p></div>
+
+<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> See Thorndike ['00], King ['07], and Heck ['13].</p></div>
+
+<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> A special type could be constructed that would use a large type body,
+say 14 point, with integers in 10 or 12 point and fractions much larger than
+now.</p></div>
+
+<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> It will be still better if the 4 is replaced by an open-top 4.</p></div>
+
+<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> For an account in English of their main findings see Howell ['14], pp.
+149-251.</p></div>
+
+<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> In his <i>How We Think</i>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> Compiled from data on p. 89 of Kruse ['18].</p></div>
+
+<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> Siblings is used for children of the same parents.</p></div>
+
+<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> Similar results have been obtained in the case of arithmetical and other
+abilities by Thorndike ['08, '10, '15, '16], Whitley ['11], Starch ['11], Wells
+['12], Kirby ['13], Donovan and Thorndike ['13], Hahn and Thorndike ['14],
+and on a very large scale by Race in a study as yet unpublished.</p></div>
+
+<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> Unless he has a thorough understanding of the underlying theory, the
+student should be very cautious in making inferences from coefficients of
+correlation.</p></div>
+
+
+</div>
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_302" id="Page_302">[Pg 302]</a></span></p>
+<h2><a name="BIBLIOGRAPHY" id="BIBLIOGRAPHY"></a>BIBLIOGRAPHY OF REFERENCES MADE IN
+THE TEXT</h2>
+
+
+<div class='center'>
+<table border="0" cellpadding="4" cellspacing="20" summary="">
+<colgroup><col width="25%" /><col width="10%" /><col width="65%" /></colgroup>
+<tr><td align='left'>Ames, A. F., and McLellan, J. F.</td><td align='right'>'00</td><td align='left'>Public School Arithmetic.</td></tr>
+<tr><td align='left'>Ballou, F. W.</td><td align='right'>'16</td><td align='left'>Determining the Achievements of Pupils in the Addition of Fractions. School Document No. 3, 1916, Boston Public Schools.</td></tr>
+<tr><td align='left'>Brandell, G.</td><td align='right'>'13</td><td align='left'>Skolbarns intressen. Translated ['15] by W. Stern as, Das Interesse der Schulkinder an den Unterrichtsf&auml;chern.</td></tr>
+<tr><td align='left'>Brandford, B.</td><td align='right'>'08</td><td align='left'>A Study of Mathematical Education.</td></tr>
+<tr><td align='left'>Brown, J. C.</td><td align='right'>'11, '12</td><td align='left'>An Investigation on the Value of Drill Work in the Fundamental Operations in Arithmetic. Journal of Educational Psychology, vol. 2, pp. 81-88, vol. 3, pp. 485-492 and 561-570.</td></tr>
+<tr><td align='left'>Brown, J. C. and Coffman, L. D.</td><td align='right'>'14</td><td align='left'>How to Teach Arithmetic.</td></tr>
+<tr><td align='left'>Burgerstein, L.</td><td align='right'>'91</td><td align='left'>Die Arbeitscurve einer Schulstunde. Zeitschrift f&uuml;r Schulgesundheitspflege, vol. 4, pp. 543-562 and 607-627.<span class='pagenum'><a name="Page_303" id="Page_303">[Pg 303]</a></span></td></tr>
+<tr><td align='left'>Burnett, C. J.</td><td align='right'>'06</td><td align='left'>The Estimation of Number. Harvard Psychological Studies, vol. 2, pp. 349-404.</td></tr>
+<tr><td align='left'>Burt, C.</td><td align='right'>'17</td><td align='left'>The Distribution and Relations of Educational Abilities. Report of The London County Council, No. 1868.</td></tr>
+<tr><td align='left'>Chapman, J. C.</td><td align='right'>'14</td><td align='left'>Individual Differences in Ability and Improvement and Their Correlations. Teachers College Contributions to Education, No. 63.</td></tr>
+<tr><td align='left'>Chapman, J. C.</td><td align='right'>'17</td><td align='left'>The Scientific Measurement of Classroom Products. (With G. P. Rush.)</td></tr>
+<tr><td align='left'>Cobb, M. V.</td><td align='right'>'17</td><td align='left'>A Preliminary Study of the Inheritance of Arithmetical Abilities. Jour. of Educational Psychology, vol. 8, pp. 1-20. Jan., 1917.</td></tr>
+<tr><td align='left'>Coffman, L. D., and Brown, J. C.</td><td align='right'>'14</td><td align='left'>How to Teach Arithmetic.</td></tr>
+<tr><td align='left'>Coffman, L. D., and Jessup, W. A.</td><td align='right'>'16</td><td align='left'>The Supervision of Arithmetic.</td></tr>
+<tr><td align='left'>Courtis, S. A.</td><td align='right'>'09, '10, '11</td><td align='left'>Measurement of Growth and Efficiency in Arithmetic. Elementary School Teacher, vol. 10, pp. 58-74 and 177-199, vol. 11, pp. 171-185, 360-370, and 528-539.</td></tr>
+<tr><td align='left'>Courtis, S. A.</td><td align='right'>'11, '12</td><td align='left'>Report on Educational Aspects of the Public School System of the City of New York. Part II, Subdivision 1, Section D. Report on the Courtis Tests in Arithmetic.</td></tr>
+<tr><td align='left'>Courtis, S. A.</td><td align='right'>'13</td><td align='left'>Courtis Standard Tests. Second Annual Accounting.</td></tr>
+<tr><td align='left'>Courtis, S. A.</td><td align='right'>'14</td><td align='left'>Manual of Instructions for Giving and Scoring the Courtis Standard <span class='pagenum'><a name="Page_304" id="Page_304">[Pg 304]</a></span>Tests in the Three R's. Department of Comparative Research, 82 Elliot St., Detroit, Mich., 1914.</td></tr>
+<tr><td align='left'>Decroly, M., and Degand, J.</td><td align='right'>'12</td><td align='left'>L'evolution des notions de quantit&eacute;s continues et discontinues chez l'enfant. Archives de psychologie, vol. 12, pp. 81-121.</td></tr>
+<tr><td align='left'>Degand, J. <i>See</i> Decroly.</td></tr>
+<tr><td align='left'>De Voss, J. C. <i>See</i> Monroe, De Voss, and Kelly.</td></tr>
+<tr><td align='left'>Dewey, J.</td><td align='right'>'10</td><td align='left'>How We Think.</td></tr>
+<tr><td align='left'>Dewey, J., and McLellan, J. A.</td><td align='right'>'95</td><td align='left'>Psychology of Number and Its Applications to Methods of Teaching Arithmetic.</td></tr>
+<tr><td align='left'>Donovan, M. E., and Thorndike, E. L.</td><td align='right'>'13</td><td align='left'>Improvement in a Practice Experiment under School Conditions. American Journal of Psychology, vol. 24, pp. 426-428.</td></tr>
+<tr><td align='left'>Elliott, C. H.</td><td align='right'>'14</td><td align='left'>Variation in the Achievements of Pupils. Teachers College, Columbia University, Contributions to Education, No. 72.</td></tr>
+<tr><td align='left'>Flynn, F. J.</td><td align='right'>'12</td><td align='left'>Mathematical Games&mdash;Adaptations from Games Old and New. Teachers College Record, vol. 13, pp. 399-412.</td></tr>
+<tr><td align='left'>Freeman, F. N.</td><td align='right'>'10</td><td align='left'>Untersuchungen &uuml;ber den Aufmerksamkeitsumfang und die Zahlauffassung. P&auml;dagogische-Psychologische Arbeiten, I, 88-168.</td></tr>
+<tr><td align='left'>Friedrich, J.</td><td align='right'>'97</td><td align='left'>Untersuchungen &uuml;ber die Einfl&uuml;sse der Arbeitsdauer und die Arbeitspausen auf die geistige Leistungsf&auml;higkeit der Schulkinder. Zeitschrift f&uuml;r Psychologie, vol. 13, pp. 1-53.</td></tr>
+<tr><td align='left'>Gilbert, J. A.</td><td align='right'>'94</td><td align='left'>Researches on the Mental and<span class='pagenum'><a name="Page_305" id="Page_305">[Pg 305]</a></span> Physical Development of School Children. Studies from the Yale Psychological Laboratory, vol. 2, pp. 40-100.</td></tr>
+<tr><td align='left'>Greenleaf, B.</td><td align='right'>'73</td><td align='left'>Practical Arithmetic.</td></tr>
+<tr><td align='left'>Hahn, H. H., and Thorndike, E. L.</td><td align='right'>'14</td><td align='left'>Some Results of Practice in Addition under School Conditions. Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.</td></tr>
+<tr><td align='left'>Hall, G. S.</td><td align='right'>'83</td><td align='left'>The Contents of Children's Minds on Entering School. Princeton Review, vol. II, pp. 249-272, May, 1883. Reprinted in Aspects of Child Life and Education, 1907.</td></tr>
+<tr><td align='left'>Hartmann, B.</td><td align='right'>'90</td><td align='left'>Die Analyze des Kindlichen Gedanken-Kreises als die Naturgem&auml;ssedes Ersten Schulunterrichts, 1890.</td></tr>
+<tr><td align='left'>Heck, W. H.</td><td align='right'>'13</td><td align='left'>A Study of Mental Fatigue.</td></tr>
+<tr><td align='left'>Heck, W. H.</td><td align='right'>'13</td><td align='left'>A Second Study in Mental Fatigue in the Daily School Program. Psychological Clinic, vol. 7, pp. 29-34.</td></tr>
+<tr><td align='left'>Hoffmann, P.</td><td align='right'>'11</td><td align='left'>Das Interesse der Sch&uuml;ler an den Unterrichtsf&auml;chern. Zeitschrift f&uuml;r p&auml;dagogische Psychologie, XII, 458-470.</td></tr>
+<tr><td align='left'>Hoke, K. J., and Wilson, G. M.</td><td align='right'>'20</td><td align='left'>How to Measure.</td></tr>
+<tr><td align='left'>Holmes, M. E.</td><td align='right'>'95</td><td align='left'>The Fatigue of a School Hour. Pedagogical Seminary, vol. 3, pp. 213-234.</td></tr>
+<tr><td align='left'>Howell, H. B.</td><td align='right'>'14</td><td align='left'>A Foundation Study in the Pedagogy of Arithmetic.</td></tr>
+<tr><td align='left'>Hunt, C. W.</td><td align='right'>'12</td><td align='left'>Play and Recreation in Arithmetic. Teachers College Record, vol. 13, pp. 388-398.<span class='pagenum'><a name="Page_306" id="Page_306">[Pg 306]</a></span></td></tr>
+<tr><td align='left'>Jessup, W. A., and Coffman, L. D.</td><td align='right'>'16</td><td align='left'>The Supervision of Arithmetic.</td></tr>
+<tr><td align='left'>Kelly, F. J. <i>See</i> Monroe, De Voss and Kelly.</td></tr>
+<tr><td align='left'>King, A. C.</td><td align='right'>'07</td><td align='left'>The Daily Program in Elementary Schools. MSS.</td></tr>
+<tr><td align='left'>Kirby, T. J.</td><td align='right'>'13</td><td align='left'>Practice in the Case of School Children. Teachers College Contributions to Education, No. 58.</td></tr>
+<tr><td align='left'>Klapper, P.</td><td align='right'>'16</td><td align='left'>The Teaching of Arithmetic.</td></tr>
+<tr><td align='left'>Kruse, P. J.</td><td align='right'>'18</td><td align='left'>The Overlapping of Attainments in Certain Sixth, Seventh, and Eighth Grades. Teachers College, Columbia University, Contributions to Education, No. 92.</td></tr>
+<tr><td align='left'>Laser, H.</td><td align='right'>'94</td><td align='left'>Ueber geistige Erm&uuml;dung beim Schulunterricht. Zeitschrift f&uuml;r Schulgesundheitspflege, vol. 7, pp. 2-22.</td></tr>
+<tr><td align='left'>Lay, W. A.</td><td align='right'>'98</td><td align='left'>F&uuml;hrer durch den ersten Rechenunterricht.</td></tr>
+<tr><td align='left'>Lay, W. A.</td><td align='right'>'07</td><td align='left'>F&uuml;hrer durch den Rechenunterricht der Unterstufe.</td></tr>
+<tr><td align='left'>Lewis, E. O.</td><td align='right'>'13</td><td align='left'>Popular and Unpopular School-Subjects. The Journal of Experimental Pedagogy, vol. 2, pp. 89-98.</td></tr>
+<tr><td align='left'>Lobsien, M.</td><td align='right'>'03</td><td align='left'>Kinderideale. Zeitschrift f&uuml;r p&auml;dagogische Psychologie, V, 323-344 and 457-494.</td></tr>
+<tr><td align='left'>Lobsien, M.</td><td align='right'>'09</td><td align='left'>Beliebtheit und Unbeliebtheit der Unterrichtsf&auml;cher. P&auml;dagogisches Magazin, Heft 361.</td></tr>
+<tr><td align='left'>McCall, W. A.</td><td align='right'>'21</td><td align='left'>How to Measure in Education.</td></tr>
+<tr><td align='left'>McDougle, E. C.</td><td align='right'>'14</td><td align='left'>A Contribution to the Pedagogy of Arithmetic. Pedagogical Seminary, vol. 21, pp. 161-218.<span class='pagenum'><a name="Page_307" id="Page_307">[Pg 307]</a></span></td></tr>
+<tr><td align='left'>McKnight, J.A.</td><td align='right'>'07</td><td align='left'>Differentiation of the Curriculum in the Upper Grammar Grades. MSS. in the library of Teachers College, Columbia University.</td></tr>
+<tr><td align='left'>McLellan, J.A., and Dewey, J.</td><td align='right'>'95</td><td align='left'>Psychology of Number and Its Applications to Methods of Teaching.</td></tr>
+<tr><td align='left'>McLellan, J.A., and Ames, A.F.</td><td align='right'>'00</td><td align='left'>Public School Arithmetic.</td></tr>
+<tr><td align='left'>Messenger, J.F.</td><td align='right'>'03</td><td align='left'>The Perception of Number. Psychological Review, Monograph Supplement No. 22.</td></tr>
+<tr><td align='left'>Meumann, E.</td><td align='right'>'07</td><td align='left'>Vorlesungen zur Einf&uuml;hrung in die experimentelle P&auml;dagogik.</td></tr>
+<tr><td align='left'>Mitchell, H.E.</td><td align='right'>'20</td><td align='left'>Unpublished studies of the uses of arithmetic in factories, shops, farms, and the like.</td></tr>
+<tr><td align='left'>Monroe, W.S., De Voss, J.C., and Kelly, F.J.</td><td align='right'>'17</td><td align='left'>Educational Tests and Measurements.</td></tr>
+<tr><td align='left'>Nanu, H.A.</td><td align='right'>'04</td><td align='left'>Zur Psychologie der Zahl Auffassung.</td></tr>
+<tr><td align='left'>National Intelligence Tests</td><td align='right'>'20</td><td align='left'>Scale A, Form 1, Edition 1.</td></tr>
+<tr><td align='left'>Phillips, D.E.</td><td align='right'>'97</td><td align='left'>Number and Its Application Psychologically Considered. Pedagogical Seminary, vol. 5, pp. 221-281.</td></tr>
+<tr><td align='left'>Pommer, O.</td><td align='right'>'14</td><td align='left'>Die Erforschung der Beliebtheit der Unterrichtsf&auml;cher. Ihre psychologischen Grundlagen und ihre p&auml;dagog. Bedeutung. VII. Jahresber. des k.k. Ssaatsgymn. im XVIII Bez. v. Wien.</td></tr>
+<tr><td align='left'>Rice, J.M.</td><td align='right'>'02</td><td align='left'>Test in Arithmetic. Forum, vol. 34, pp. 281-297.</td></tr>
+<tr><td align='left'>Rice, J.M.</td><td align='right'>'03</td><td align='left'>Causes of Success and Failure in Arithmetic. Forum, vol. 34, pp. 437-452.</td></tr>
+<tr><td align='left'>Rush, G.P.</td><td align='right'>'17</td><td align='left'>The Scientific Measurement of Classroom Products. (With J. C. Chapman.)<span class='pagenum'><a name="Page_308" id="Page_308">[Pg 308]</a></span></td></tr>
+<tr><td align='left'>Seekel, E.</td><td align='right'>'14</td><td align='left'>Ueber die Beziehung zwischen der Beliebtheit und der Schwierigkeit der Schulf&auml;cher. Ergebnisse einer Erhebung. Zeitschrift f&uuml;r Angewandte Psychologie 9. S. 268-277.</td></tr>
+<tr><td align='left'>Selkin, F. B.</td><td align='right'>'12</td><td align='left'>Number Games Bordering on Arithmetic and Algebra. Teachers College Record, vol. 13, pp. 452-493.</td></tr>
+<tr><td align='left'>Smith, D. E.</td><td align='right'>'01</td><td align='left'>The Teaching of Elementary Mathematics.</td></tr>
+<tr><td align='left'>Smith, D. E.</td><td align='right'>'11</td><td align='left'>The Teaching of Arithmetic.</td></tr>
+<tr><td align='left'>Speer, W. W.</td><td align='right'>'97</td><td align='left'>Arithmetic: Elementary for Pupils.</td></tr>
+<tr><td align='left'>Starch, D.</td><td align='right'>'11</td><td align='left'>Transfer of Training in Arithmetical Operations. Journal of Educational Psychology, vol. 2, pp. 306-310.</td></tr>
+<tr><td align='left'>Starch, D.</td><td align='right'>'16</td><td align='left'>Educational Measurements.</td></tr>
+<tr><td align='left'>Stern, W.</td><td align='right'>'05</td><td align='left'>Ueber Beliebtheit und Unbeliebtheit der Schulf&auml;cher. Zeitschrift f&uuml;r p&auml;dagogische Psychologie, VII, 267-296.</td></tr>
+<tr><td align='left'>Stern, C., and Stern, W.</td><td align='right'>'13</td><td align='left'>Beliebtheit und Schwierigkeit der Schulf&auml;cher. (Freie Schulgemeinde Wickersdorf.) Auf Grund der von Herrn Luserke beschafften Materialien bearbeitet. In: "Die Ausstellung zur vergleichenden Jungendkunde der Geschlechter in Breslau." Arbeit 7 des Bundes f&uuml;r Schulreform. S. 24-26.</td></tr>
+<tr><td align='left'>Stern, W.</td><td align='right'>'14</td><td align='left'>Zur vergleichenden Jugendkunde der Geschlechter. Vortrag. III. Deutsch. Kongr. f. Jugendkunde <span class='pagenum'><a name="Page_309" id="Page_309">[Pg 309]</a></span>usw. Arbeiten 8 des Bundes f&uuml;r Schulreform. S. 17-38.</td></tr>
+<tr><td align='left'>Stone, C.W.</td><td align='right'>'08</td><td align='left'>Arithmetical Abilities and Some Factors Determining Them. Teachers College Contributions to Education, No. 19.</td></tr>
+<tr><td align='left'>Suzzallo, H.</td><td align='right'>'11</td><td align='left'>The Teaching of Primary Arithmetic.</td></tr>
+<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'00</td><td align='left'>Mental Fatigue. Psychological Review, vol. 7, pp. 466-482 and 547-579.</td></tr>
+<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'08</td><td align='left'>The Effect of Practice in the Case of a Purely Intellectual Function. American Journal of Psychology, vol. 10, pp. 374-384.</td></tr>
+<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'10</td><td align='left'>Practice in the case of Addition. American Journal of Psychology, vol. 21, pp. 483-486.</td></tr>
+<tr><td align='left'>Thorndike, E.L., and Donovan, M.E.</td><td align='right'>'13</td><td align='left'>Improvement in a Practice Experiment under School Conditions. American Journal of Psychology, vol. 24, pp. 426-428.</td></tr>
+<tr><td align='left'>Thorndike, E.L., and Donovan, M.E., and Hahn, H.H.</td><td align='right'>'14</td><td align='left'>Some Results of Practice in Addition under School Conditions. Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.</td></tr>
+<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'15</td><td align='left'>The Relation between Initial Ability and Improvement in a Substitution Test. School and Society, vol. 12, p. 429.</td></tr>
+<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'16</td><td align='left'>Notes on Practice, Improvability and the Curve of Work. American Journal of Psychology, vol 27, pp. 550-565.</td></tr>
+<tr><td align='left'>Walsh, J.H.</td><td align='right'>'06</td><td align='left'>Grammar School Arithmetic.</td></tr>
+<tr><td align='left'>Wells, F.L.</td><td align='right'>'12</td><td align='left'>The Relation of Practice to Individual Differences. <span class='pagenum'><a name="Page_310" id="Page_310">[Pg 310]</a></span>American Journal of Psychology, vol. 23, pp. 75-88.</td></tr>
+<tr><td align='left'>White, E. E.</td><td align='right'>'83</td><td align='left'>A New Elementary Arithmetic.</td></tr>
+<tr><td align='left'>Whitley, M. T.</td><td align='right'>'11</td><td align='left'>An Empirical Study of Certain Tests for Individual Differences. Archives of Psychology, No. 19.</td></tr>
+<tr><td align='left'>Wiederkehr, G.</td><td align='right'>'07</td><td align='left'>Statistiche Untersuchungen &uuml;ber die Art und den Grad des Interesses bei Kindern der Volksschule. Neue Bahnen, vol. 19, pp. 241-251, 289-299.</td></tr>
+<tr><td align='left'>Wilson, G. M.</td><td align='right'>'19</td><td align='left'>A Survey of the Social and Business Usage of Arithmetic. Teachers College Contributions to Education, No. 100.</td></tr>
+<tr><td align='left'>Wilson, G. M., and Hoke, K. J.</td><td align='right'>'20</td><td align='left'>How to Measure.</td></tr>
+<tr><td align='left'>Woody, C.</td><td align='right'>'10</td><td align='left'>Measurements of Some Achievements in Arithmetic. Teachers College Contributions to Education, No. 80.</td></tr>
+</table></div>
+
+
+<hr style="width: 100%;" />
+<p><span class='pagenum'><a name="Page_311" id="Page_311">[Pg 311]</a></span></p>
+<h2><a name="INDEX" id="INDEX"></a>INDEX</h2>
+
+
+<div class="pblockquot"><p class="noidt">
+Abilities, arithmetical, nature of, <a href="#Page_1">1 ff.</a>;<br />
+ &nbsp; &nbsp; measurement of, <a href="#Page_27">27 ff.</a>;<br />
+ &nbsp; &nbsp; constitution of, <a href="#Page_51">51 ff.</a>;<br />
+ &nbsp; &nbsp; organization of, <a href="#Page_137">137 ff.</a><br />
+<br />
+Abstract numbers, <a href="#Page_85">85 ff.</a><br />
+<br />
+Abstraction, <a href="#Page_169">169 ff.</a><br />
+<br />
+Accuracy, in relation to speed, <a href="#Page_31">31</a>;<br />
+ &nbsp; &nbsp; in fundamental operations, <a href="#Page_102">102 ff.</a><br />
+<br />
+Addition, measurement of, <a href="#Page_27">27 ff.</a>, <a href="#Page_34">34</a>;<br />
+ &nbsp; &nbsp; constitution of, <a href="#Page_52">52 f.</a>;<br />
+ &nbsp; &nbsp; habit in relation to, <a href="#Page_71">71 f.</a>;<br />
+ &nbsp; &nbsp; in the higher decades, <a href="#Page_75">75 f.</a>;<br />
+ &nbsp; &nbsp; accuracy in, <a href="#Page_108">108 f.</a>;<br />
+ &nbsp; &nbsp; amount of practice in, <a href="#Page_122">122 ff.</a>;<br />
+ &nbsp; &nbsp; interest in <a href="#Page_196">196 f.</a><br />
+<br />
+Aims of the teaching of arithmetic, <a href="#Page_23">23 f.</a><br />
+<br />
+<span class="smcap">Ames, A. F.</span>, <a href="#Page_89">89</a><br />
+<br />
+Analysis, learning by, <a href="#Page_169">169 ff.</a>;<br />
+ &nbsp; &nbsp; systematic and opportunistic stimuli to, <a href="#Page_178">178 f.</a>;<br />
+ &nbsp; &nbsp; gradual progress in, <a href="#Page_180">180 ff.</a><br />
+<br />
+Area, <a href="#Page_257">257 f.</a>, <a href="#Page_275">275</a><br />
+<br />
+Arithmetic, sociology of, <a href="#Page_24">24 ff.</a><br />
+<br />
+Arithmetical abilities. <i>See</i> Abilities.<br />
+<br />
+Arithmetical language, <a href="#Page_8">8 f.</a>, <a href="#Page_19">19</a>, <a href="#Page_89">89 ff.</a>, <a href="#Page_94">94 ff.</a><br />
+<br />
+Arithmetical learning, before school, <a href="#Page_199">199 ff.</a>;<br />
+ &nbsp; &nbsp; conditions of, <a href="#Page_227">227 ff.</a>;<br />
+ &nbsp; &nbsp; in relation to time of day, <a href="#Page_227">227 ff.</a>;<br />
+ &nbsp; &nbsp; in relation to time devoted to arithmetic, <a href="#Page_228">228 ff.</a><br />
+<br />
+Arithmetical reasoning. <i>See</i> Reasoning.<br />
+<br />
+Arithmetical terms, <a href="#Page_8">8</a>, <a href="#Page_19">19</a><br />
+<br />
+Averages, <a href="#Page_40">40 f.</a>; <a href="#Page_135">135 f.</a><br />
+<br />
+<br />
+<span class="smcap">Ballou, F. W.</span>, <a href="#Page_34">34</a>, <a href="#Page_38">38</a><br />
+<br />
+Banking, <a href="#Page_256">256 f.</a><br />
+<br />
+<span class="smcap">Binet, A.</span>, <a href="#Page_201">201</a><br />
+<br />
+Bonds, selection of, <a href="#Page_70">70 ff.</a>;<br />
+ &nbsp; &nbsp; strength of, <a href="#Page_102">102 ff.</a>;<br />
+ &nbsp; &nbsp; for temporary service, <a href="#Page_111">111 ff.</a>;<br />
+ &nbsp; &nbsp; order of formation of, <a href="#Page_141">141 ff.</a><br />
+ &nbsp; &nbsp; <i>See also</i> Habits.<br />
+<br />
+<span class="smcap">Brandell, G.</span>, <a href="#Page_211">211</a><br />
+<br />
+<span class="smcap">Brandford, B.</span>, <a href="#Page_198">198 f.</a><br />
+<br />
+<span class="smcap">Brown, J. C.</span>, <a href="#Page_xvi">xvi</a>, <a href="#Page_103">103</a><br />
+<br />
+<span class="smcap">Burgerstein, L.</span>, <a href="#Page_103">103</a><br />
+<br />
+<span class="smcap">Burnett, C. J.</span>, <a href="#Page_202">202</a><br />
+<br />
+<span class="smcap">Burt, C.</span>, <a href="#Page_286">286</a><br />
+<br />
+<br />
+Cardinal and ordinal numbers confused, <a href="#Page_206">206</a><br />
+<br />
+Catch problems, <a href="#Page_21">21 ff.</a><br />
+<br />
+<span class="smcap">Chapman, J. C.</span>, <a href="#Page_49">49</a><br />
+<br />
+Class, size of, in relation to arithmetical learning, <a href="#Page_228">228</a>;<br />
+ &nbsp; &nbsp; variation within a, <a href="#Page_289">289 ff.</a><br />
+<br />
+<span class="smcap">Cobb, M. V.</span>, <a href="#Page_299">299</a><br />
+<br />
+<span class="smcap">Coffman, L. D.</span>, <a href="#Page_xvi">xvi</a><br />
+<br />
+Collection meaning of numbers, <a href="#Page_3">3 ff.</a><br />
+<br />
+Computation, measurements of, <a href="#Page_33">33 ff.</a>;<br />
+ &nbsp; &nbsp; explanations of the processes in, <a href="#Page_60">60 ff.</a>;<br />
+ &nbsp; &nbsp; accuracy in, <a href="#Page_102">102 ff.</a><br />
+ &nbsp; &nbsp; <i>See also</i> Addition, Subtraction, Multiplication, Division, Fractions, Decimal numbers, Percents.<br />
+<br />
+Concomitants, law of varying, <a href="#Page_172">172 ff.</a>;<br />
+ &nbsp; &nbsp; law of contrasting, <a href="#Page_173">173 ff.</a><br />
+<br />
+Concrete numbers, <a href="#Page_85">85 ff.</a><br />
+<br />
+Concrete objects, use of, <a href="#Page_253">253 ff.</a><br />
+<br />
+Conditions of arithmetical learning, <a href="#Page_227">227 ff.</a><br />
+<br />
+Constitution of arithmetical abilities, <a href="#Page_51">51 ff.</a><br />
+<br />
+Copying of numbers, eyestrain due to, <a href="#Page_212">212 f.</a><br />
+<br />
+Correlations of arithmetical abilities, <a href="#Page_295">295 ff.</a><br />
+<br />
+Courses of study, <a href="#Page_232">232 f.</a><br />
+<br />
+<span class="smcap">Courtis, S. A.</span>, <a href="#Page_28">28 ff.</a>, <a href="#Page_43">43 ff.</a>, <a href="#Page_49">49</a>, <a href="#Page_103">103</a>, <a href="#Page_291">291</a>, <a href="#Page_293">293</a>, <a href="#Page_299">299</a><br />
+<br />
+Crutches, <a href="#Page_112">112 f.</a><br />
+<br />
+Culture-epoch theory, <a href="#Page_198">198 f.</a><br />
+<br />
+<br />
+Dairy records, <a href="#Page_273">273</a><br />
+<br />
+Decimal numbers, uses of, <a href="#Page_24">24 f.</a>;<br />
+ &nbsp; &nbsp; measurement of ability with, <a href="#Page_36">36 ff.</a>;<br />
+ &nbsp; &nbsp; learning, <a href="#Page_181">181 ff.</a>;<br />
+ &nbsp; &nbsp; division by, <a href="#Page_270">270 f.</a><br />
+<br />
+<span class="smcap">De Croly, M.</span>, <a href="#Page_205">205</a><br />
+<br />
+Deductive reasoning, <a href="#Page_60">60 ff.</a>, <a href="#Page_185">185 ff.</a><br />
+<br />
+<span class="smcap">Degand, J.</span>, <a href="#Page_205">205</a><br />
+<br />
+Denominate numbers, <a href="#Page_141">141 f.</a>, <a href="#Page_147">147 f.</a><br />
+<br />
+Described problems, <a href="#Page_10">10 ff.</a><br />
+<br />
+Development of knowledge of number, <a href="#Page_205">205 ff.</a><br />
+<span class='pagenum'><a name="Page_312" id="Page_312">[Pg 312]</a></span><br />
+<span class="smcap">De Voss, J. C.</span>, <a href="#Page_49">49</a><br />
+<br />
+<span class="smcap">Dewey, J.</span>, <a href="#Page_3">3</a>, <a href="#Page_83">83</a>, <a href="#Page_150">150</a>, <a href="#Page_205">205</a>, <a href="#Page_207">207</a>, <a href="#Page_208">208</a>, <a href="#Page_219">219</a>, <a href="#Page_266">266</a>, <a href="#Page_277">277</a><br />
+<br />
+Differences in arithmetical ability, <a href="#Page_285">285 ff.</a>;<br />
+ &nbsp; &nbsp; within a class, <a href="#Page_289">289 ff.</a><br />
+<br />
+Difficulty as a stimulus, <a href="#Page_277">277 ff.</a><br />
+<br />
+Drill, <a href="#Page_102">102 ff.</a><br />
+<br />
+Discipline, mental, <a href="#Page_20">20</a><br />
+<br />
+Distribution of practice, <a href="#Page_156">156 ff.</a><br />
+<br />
+Division, measurement of, <a href="#Page_35">35 f.</a>, <a href="#Page_37">37</a>;<br />
+ &nbsp; &nbsp; constitution of, <a href="#Page_57">57 ff.</a>;<br />
+ &nbsp; &nbsp; deductive explanations of, <a href="#Page_63">63</a>, <a href="#Page_64">64 f.</a>;<br />
+ &nbsp; &nbsp; inductive explanations of, <a href="#Page_63">63 f.</a>, <a href="#Page_65">65 f.</a>;<br />
+ &nbsp; &nbsp; habit in relation to, <a href="#Page_72">72</a>;<br />
+ &nbsp; &nbsp; with remainders, <a href="#Page_76">76</a>;<br />
+ &nbsp; &nbsp; with fractions, <a href="#Page_78">78 ff.</a>;<br />
+ &nbsp; &nbsp; amount of practice in, <a href="#Page_122">122 ff.</a>;<br />
+ &nbsp; &nbsp; distribution of practice in, <a href="#Page_167">167</a>;<br />
+ &nbsp; &nbsp; use of the problem attitude in teaching, <a href="#Page_270">270 f.</a><br />
+<br />
+<span class="smcap">Donovan, M. E.</span>, <a href="#Page_295">295</a><br />
+<br />
+<br />
+Elements, responses to, <a href="#Page_169">169 ff.</a><br />
+<br />
+Eleven, multiples of, <a href="#Page_85">85</a><br />
+<br />
+<span class="smcap">Elliott, C. H.</span>, <a href="#Page_228">228</a><br />
+<br />
+Equation form, importance of, <a href="#Page_77">77 f.</a><br />
+<br />
+Explanations of the processes of computation, <a href="#Page_60">60 ff.</a>;<br />
+ &nbsp; &nbsp; memory of, <a href="#Page_115">115 f.</a>;<br />
+ &nbsp; &nbsp; time for giving, <a href="#Page_154">154 ff.</a><br />
+<br />
+Eyestrain in arithmetical work, <a href="#Page_212">212 ff.</a><br />
+<br />
+<br />
+Facilitation, <a href="#Page_143">143 ff.</a><br />
+<br />
+Figures, printing of, <a href="#Page_235">235 ff.</a>;<br />
+ &nbsp; &nbsp; writing of, <a href="#Page_214">214 f.</a>, <a href="#Page_241">241</a><br />
+<br />
+<span class="smcap">Flynn, F. J.</span>, <a href="#Page_196">196</a><br />
+<br />
+Fractions, uses of, <a href="#Page_24">24 f.</a>;<br />
+ &nbsp; &nbsp; measurement of ability with, <a href="#Page_36">36 ff.</a>;<br />
+ &nbsp; &nbsp; knowledge of the meaning of, <a href="#Page_54">54 ff.</a><br />
+<br />
+<span class="smcap">Freeman, F. N.</span>, <a href="#Page_259">259</a>, <a href="#Page_261">261</a><br />
+<br />
+<span class="smcap">Friedrich, J.</span>, <a href="#Page_103">103</a><br />
+<br />
+<br />
+Generalization, <a href="#Page_169">169 ff.</a><br />
+<br />
+<span class="smcap">Gilbert, J. A.</span>, <a href="#Page_203">203</a><br />
+<br />
+Graded tests, <a href="#Page_28">28 ff.</a>, <a href="#Page_36">36 ff.</a><br />
+<br />
+Greatest common divisor, <a href="#Page_88">88 f.</a><br />
+<br />
+<br />
+Habits, importance of, in arithmetical learning, <a href="#Page_70">70 ff.</a>;<br />
+ &nbsp; &nbsp; now neglected, <a href="#Page_75">75 ff.</a>;<br />
+ &nbsp; &nbsp; harmful or wasteful, <a href="#Page_83">83 ff.</a>; <a href="#Page_91">91 ff.</a>;<br />
+ &nbsp; &nbsp; prop&aelig;deutic, <a href="#Page_117">117 ff.</a>;<br />
+ &nbsp; &nbsp; organization of, <a href="#Page_137">137 ff.</a>;<br />
+ &nbsp; &nbsp; arrangement of, <a href="#Page_141">141 ff.</a><br />
+<br />
+<span class="smcap">Hahn, H. H.</span>, <a href="#Page_295">295</a><br />
+<br />
+<span class="smcap">Hall, G. S.</span>, <a href="#Page_200">200 f.</a><br />
+<br />
+<span class="smcap">Hartmann, B.</span>, <a href="#Page_200">200 f.</a><br />
+<br />
+<span class="smcap">Heck, W. H.</span>, <a href="#Page_227">227</a><br />
+<br />
+Heredity in arithmetical abilities, <a href="#Page_293">293 ff.</a><br />
+<br />
+Highest common factor, <a href="#Page_88">88 f.</a><br />
+<br />
+<span class="smcap">Hoke, K. J.</span>, <a href="#Page_49">49</a><br />
+<br />
+<span class="smcap">Holmes, M. E.</span>, <a href="#Page_103">103</a><br />
+<br />
+<span class="smcap">Howell, H. B.</span>, <a href="#Page_259">259</a><br />
+<br />
+<span class="smcap">Hunt, C. W.</span>, <a href="#Page_196">196</a><br />
+<br />
+Hygiene of arithmetic, <a href="#Page_212">212 ff.</a>, <a href="#Page_234">234 ff.</a><br />
+<br />
+<br />
+Individual differences, <a href="#Page_285">285 ff.</a><br />
+<br />
+Inductive reasoning, <a href="#Page_60">60 ff.</a>, <a href="#Page_169">169 ff.</a><br />
+<br />
+Insurance, <a href="#Page_256">256</a><br />
+<br />
+Interest as a principle determining the order of topics, <a href="#Page_150">150 ff.</a><br />
+<br />
+Interests, instinctive <a href="#Page_195">195 ff.</a>;<br />
+ &nbsp; &nbsp; censuses of, <a href="#Page_209">209 ff.</a>;<br />
+ &nbsp; &nbsp; neglect of childish, <a href="#Page_226">226 ff.</a>;<br />
+ &nbsp; &nbsp; in self-management, <a href="#Page_223">223 f.</a>;<br />
+ &nbsp; &nbsp; intrinsic, <a href="#Page_224">224 ff.</a><br />
+<br />
+Interference, <a href="#Page_143">143 ff.</a><br />
+<br />
+Inventories of arithmetical knowledge and skill, <a href="#Page_199">199 ff.</a><br />
+<br />
+<br />
+<span class="smcap">Jessup, W. A.</span>, <a href="#Page_xvi">xvi</a><br />
+<br />
+<br />
+<span class="smcap">Kelly, F. J.</span>, <a href="#Page_49">49</a><br />
+<br />
+<span class="smcap">King, A. C.</span>, <a href="#Page_103">103</a>, <a href="#Page_227">227</a><br />
+<br />
+<span class="smcap">Kirby, T. J.</span>, <a href="#Page_76">76 f.</a>, <a href="#Page_104">104</a>, <a href="#Page_295">295</a><br />
+<br />
+<span class="smcap">Klapper, P.</span>, <a href="#Page_xvi">xvi</a><br />
+<br />
+<span class="smcap">Kruse, P. J.</span>, <a href="#Page_289">289</a>, <a href="#Page_293">293</a><br />
+<br />
+<br />
+Ladder tests, <a href="#Page_28">28 ff.</a>, <a href="#Page_36">36 ff.</a><br />
+<br />
+Language in arithmetic, <a href="#Page_8">8 f.</a>, <a href="#Page_19">19</a>, <a href="#Page_89">89 ff.</a>, <a href="#Page_94">94 ff.</a><br />
+<br />
+<span class="smcap">Laser, H.</span>, <a href="#Page_103">103</a><br />
+<br />
+<span class="smcap">Lay, W. A.</span>, <a href="#Page_259">259</a>, <a href="#Page_261">261</a><br />
+<br />
+Learning, nature of arithmetical, <a href="#Page_1">1 ff.</a><br />
+<br />
+Least common multiple, <a href="#Page_88">88 f.</a><br />
+<br />
+<span class="smcap">Lewis, E. O.</span>, <a href="#Page_210">210 f.</a><br />
+<br />
+<span class="smcap">Lobsien, M.</span>, <a href="#Page_209">209 f.</a><br />
+<br />
+<br />
+<span class="smcap">McCall, W. A.</span>, <a href="#Page_49">49</a><br />
+<br />
+<span class="smcap">McDougle, E. C.</span>, <a href="#Page_85">85 ff.</a><br />
+<br />
+<span class="smcap">McKnight, J. A.</span>, <a href="#Page_210">210</a><br />
+<br />
+<span class="smcap">McLellan, J. A.</span>, <a href="#Page_3">3</a>, <a href="#Page_83">83</a>, <a href="#Page_89">89</a>, <a href="#Page_205">205</a>, <a href="#Page_207">207</a><br />
+<br />
+Manipulation of numbers, <a href="#Page_60">60 ff.</a><br />
+<br />
+Meaning, of numbers, <a href="#Page_2">2 ff.</a>, <a href="#Page_171">171</a>;<br />
+ &nbsp; &nbsp; of a fraction, <a href="#Page_54">54 ff.</a>;<br />
+ &nbsp; &nbsp; of decimals, <a href="#Page_181">181 f.</a><br />
+<br />
+Measurement of arithmetical abilities, <a href="#Page_27">27 ff.</a><br />
+<br />
+Mental arithmetic, <a href="#Page_262">262 ff.</a><br />
+<br />
+<span class="smcap">Messenger, J. F.</span>, <a href="#Page_202">202</a><br />
+<br />
+Metric system, <a href="#Page_147">147</a><br />
+<br />
+<span class="smcap">Meumann, E.</span>, <a href="#Page_261">261</a><br />
+<br />
+<span class="smcap">Mitchell, H. E.</span>, <a href="#Page_24">24</a><br />
+<span class='pagenum'><a name="Page_313" id="Page_313">[Pg 313]</a></span><br />
+<span class="smcap">Monroe, W. S.</span>, <a href="#Page_49">49</a><br />
+<br />
+Multiplication, measurement of, <a href="#Page_35">35</a>, <a href="#Page_36">36</a>;<br />
+ &nbsp; &nbsp; constitution of, <a href="#Page_51">51</a>;<br />
+ &nbsp; &nbsp; deductive explanations of, <a href="#Page_61">61</a>;<br />
+ &nbsp; &nbsp; inductive explanations of, <a href="#Page_61">61 f.</a>;<br />
+ &nbsp; &nbsp; with fractions, <a href="#Page_78">78 ff.</a>;<br />
+ &nbsp; &nbsp; by eleven, <a href="#Page_85">85</a>;<br />
+ &nbsp; &nbsp; amount of practice in, <a href="#Page_122">122 ff.</a>;<br />
+ &nbsp; &nbsp; order of learning the elementary facts of, <a href="#Page_144">144 f.</a>;<br />
+ &nbsp; &nbsp; distribution of practice in, <a href="#Page_158">158 ff.</a>;<br />
+ &nbsp; &nbsp; use of the problem attitude in teaching, <a href="#Page_267">267 ff.</a><br />
+<br />
+<br />
+<span class="smcap">Nanu, H. A.</span>, <a href="#Page_202">202</a><br />
+<br />
+National Intelligence Tests, <a href="#Page_49">49 f.</a><br />
+<br />
+Negative reaction in intellectual life, <a href="#Page_278">278 f.</a><br />
+<br />
+Number pictures, <a href="#Page_259">259 ff.</a><br />
+<br />
+Numbers, meaning of, <a href="#Page_2">2</a>;<br />
+ &nbsp; &nbsp; as measures of continuous quantities, <a href="#Page_75">75</a>;<br />
+ &nbsp; &nbsp; abstract and concrete, <a href="#Page_85">85 ff.</a>;<br />
+ &nbsp; &nbsp; denominate, <a href="#Page_141">141 f.</a>, <a href="#Page_147">147 f.</a>;<br />
+ &nbsp; &nbsp; use of large, <a href="#Page_145">145 f.</a>;<br />
+ &nbsp; &nbsp; perception of, <a href="#Page_205">205 ff.</a>;<br />
+ &nbsp; &nbsp; early awareness of, <a href="#Page_205">205 ff.</a>;<br />
+ &nbsp; &nbsp; confusion of cardinal and ordinal, <a href="#Page_206">206</a>.<br />
+<i>See also</i> Decimal numbers <i>and</i> Fractions.<br />
+<br />
+<br />
+Objective aids, used for verification, <a href="#Page_154">154</a>;<br />
+ &nbsp; &nbsp; in general, <a href="#Page_243">243 ff.</a><br />
+<br />
+Oral arithmetic, <a href="#Page_262">262 ff.</a><br />
+<br />
+Order of topics, <a href="#Page_141">141 ff.</a><br />
+<br />
+Ordinal numbers, confused with cardinal, <a href="#Page_206">206</a><br />
+<br />
+Original tendencies and arithmetic, <a href="#Page_195">195 ff.</a><br />
+<br />
+Overlearning, <a href="#Page_134">134 ff.</a><br />
+<br />
+<br />
+Percents, <a href="#Page_80">80 f.</a><br />
+<br />
+Perception of number, <a href="#Page_202">202 ff.</a><br />
+<br />
+<span class="smcap">Phillips, D. E.</span>, <a href="#Page_3">3</a>, <a href="#Page_4">4</a>, <a href="#Page_205">205</a>, <a href="#Page_207">207</a><br />
+<br />
+Pictures, hygiene of, <a href="#Page_246">246 ff.</a>;<br />
+ &nbsp; &nbsp; number, <a href="#Page_259">259 ff.</a><br />
+<br />
+<span class="smcap">Pommer, O.</span>, <a href="#Page_212">212</a><br />
+<br />
+Practice, amount of, <a href="#Page_122">122 ff.</a>;<br />
+ &nbsp; &nbsp; distribution of, <a href="#Page_156">156 ff.</a><br />
+<br />
+Precision in fundamental operations, <a href="#Page_102">102 ff.</a><br />
+<br />
+Problem attitude, <a href="#Page_266">266 ff.</a><br />
+<br />
+Problems, <a href="#Page_9">9 ff.</a>;<br />
+ &nbsp; &nbsp; "catch," <a href="#Page_21">21 ff.</a>;<br />
+ &nbsp; &nbsp; measurement of ability with, <a href="#Page_42">42 ff.</a>;<br />
+ &nbsp; &nbsp; whose answer must be known in order to frame them, <a href="#Page_93">93 f.</a>;<br />
+ &nbsp; &nbsp; verbal form of, <a href="#Page_111">111 f.</a>;<br />
+ &nbsp; &nbsp; interest in, <a href="#Page_220">220 ff.</a>;<br />
+ &nbsp; &nbsp; as introductions to arithmetical learning, <a href="#Page_266">266 ff.</a><br />
+<br />
+Prop&aelig;deutic bonds, <a href="#Page_117">117 ff.</a><br />
+<br />
+Purposive thinking, <a href="#Page_193">193 ff.</a><br />
+<br />
+<br />
+Quantity, number and, <a href="#Page_85">85 ff.</a>;<br />
+ &nbsp; &nbsp; perception of, <a href="#Page_202">202 ff.</a><br />
+<br />
+<br />
+<span class="smcap">Race, H.</span>, <a href="#Page_295">295</a><br />
+<br />
+Rainfall, <a href="#Page_272">272</a><br />
+<br />
+Ratio, <a href="#Page_225">225 f.</a>;<br />
+ &nbsp; &nbsp; meaning of numbers, <a href="#Page_3">3 ff.</a><br />
+<br />
+Reaction, negative, <a href="#Page_278">278 f.</a><br />
+<br />
+Reality, in problems, <a href="#Page_9">9 ff.</a><br />
+<br />
+Reasoning, arithmetical, nature of, <a href="#Page_19">19 ff.</a>;<br />
+ &nbsp; &nbsp; measurement of ability in, <a href="#Page_42">42 ff.</a>;<br />
+ &nbsp; &nbsp; derivation of tables by, <a href="#Page_58">58 f.</a>;<br />
+ &nbsp; &nbsp; about the rationale of computations, <a href="#Page_60">60 ff.</a>;<br />
+ &nbsp; &nbsp; habit in relation to, <a href="#Page_73">73 f.</a>, <a href="#Page_190">190 ff.</a>;<br />
+ &nbsp; &nbsp; problems which provoke false, <a href="#Page_100">100 f.</a>;<br />
+ &nbsp; &nbsp; the essentials of arithmetical, <a href="#Page_185">185 ff.</a>;<br />
+ &nbsp; &nbsp; selection in, <a href="#Page_187">187 ff.</a>;<br />
+ &nbsp; &nbsp; as the co&ouml;peration of organized habits, <a href="#Page_190">190 ff.</a><br />
+<br />
+Recapitulation theory, <a href="#Page_198">198 f.</a><br />
+<br />
+Recipes, <a href="#Page_273">273 f.</a><br />
+<br />
+Rectangle, area of, <a href="#Page_257">257 f.</a><br />
+<br />
+<span class="smcap">Rice, J. M.</span>, <a href="#Page_228">228 ff.</a><br />
+<br />
+<span class="smcap">Rush, G. P.</span>, <a href="#Page_49">49</a><br />
+<br />
+<br />
+<span class="smcap">Seekel, E.</span>, <a href="#Page_212">212</a><br />
+<br />
+<span class="smcap">Selkin, F. B.</span>, <a href="#Page_196">196 f.</a><br />
+<br />
+Sequence of topics, <a href="#Page_141">141 ff.</a><br />
+<br />
+Series meaning of numbers, <a href="#Page_2">2 ff.</a><br />
+<br />
+Size of class in relation to arithmetical learning, <a href="#Page_228">228</a><br />
+<br />
+<span class="smcap">Smith, D. E.</span>, <a href="#Page_xvi">xvi</a>, <a href="#Page_224">224</a><br />
+<br />
+Social instincts, use of, <a href="#Page_195">195 f.</a><br />
+<br />
+Sociology of arithmetic, <a href="#Page_24">24 ff.</a><br />
+<br />
+Speed in relation to accuracy, <a href="#Page_31">31</a>, <a href="#Page_108">108</a><br />
+<br />
+<span class="smcap">Speer, W. W.</span>, <a href="#Page_3">3</a>, <a href="#Page_5">5</a>, <a href="#Page_83">83</a><br />
+<br />
+Spiral order, <a href="#Page_141">141</a>, <a href="#Page_145">145</a><br />
+<br />
+<span class="smcap">Starch, D.</span>, <a href="#Page_49">49</a>, <a href="#Page_295">295</a><br />
+<br />
+<span class="smcap">Stern, W.</span>, <a href="#Page_210">210</a>, <a href="#Page_212">212</a><br />
+<br />
+<span class="smcap">Stone, C. W.</span>, <a href="#Page_27">27 ff.</a>, <a href="#Page_42">42 ff.</a>, <a href="#Page_228">228 ff.</a><br />
+<br />
+Subtraction, measurement of, <a href="#Page_34">34 f.</a>;<br />
+ &nbsp; &nbsp; constitution of, <a href="#Page_57">57 f.</a>;<br />
+ &nbsp; &nbsp; amount of practice in, <a href="#Page_122">122 ff.</a><br />
+<br />
+Supervision, <a href="#Page_233">233 f.</a><br />
+<br />
+<span class="smcap">Suzzallo, H.</span>, <a href="#Page_xvi">xvi</a><br />
+<br />
+<br />
+Temporary bonds, <a href="#Page_111">111 ff.</a><br />
+<br />
+Terms, <a href="#Page_113">113 f.</a><br />
+<br />
+Tests of arithmetical abilities, <a href="#Page_27">27 ff.</a><br />
+<br />
+<span class="smcap">Thorndike, E. L.</span>, <a href="#Page_34">34</a>, <a href="#Page_38">38 ff.</a>, <a href="#Page_227">227</a>, <a href="#Page_294">294</a><br />
+<br />
+Time, devoted to arithmetic, <a href="#Page_228">228 ff.</a>;<br />
+ &nbsp; &nbsp; of day, in relation to arithmetical learning, <a href="#Page_227">227 f.</a><br />
+<span class='pagenum'><a name="Page_314" id="Page_314">[Pg 314]</a></span><br />
+Type, hygiene of, <a href="#Page_235">235 ff.</a><br />
+<br />
+<br />
+Underlearning, <a href="#Page_134">134 ff.</a><br />
+<br />
+United States money, <a href="#Page_148">148 ff.</a><br />
+<br />
+Units of measure, arbitrary, <a href="#Page_5">5</a>, <a href="#Page_83">83 f.</a><br />
+<br />
+<br />
+Variation, among individuals, <a href="#Page_285">285 ff.</a><br />
+<br />
+Variety, in teaching, <a href="#Page_153">153</a><br />
+<br />
+Verification, <a href="#Page_81">81 f.</a>;<br />
+ &nbsp; &nbsp; aided by greater strength of the fundamental bonds, <a href="#Page_107">107 ff.</a><br />
+<br />
+<br />
+<span class="smcap">Walsh, J. H.</span>, <a href="#Page_11">11</a><br />
+<br />
+<span class="smcap">Wells, F. L.</span>, <a href="#Page_295">295</a><br />
+<br />
+<span class="smcap">White, E. E.</span>, <a href="#Page_5">5</a><br />
+<br />
+<span class="smcap">Whitley, M. T.</span>, <a href="#Page_295">295</a><br />
+<br />
+<span class="smcap">Wiederkehr, G.</span>, <a href="#Page_212">212</a><br />
+<br />
+<span class="smcap">Wilson, G. M.</span>, <a href="#Page_24">24</a>, <a href="#Page_49">49</a><br />
+<br />
+<span class="smcap">Woody, C.</span>, <a href="#Page_29">29 ff.</a>, <a href="#Page_52">52</a>, <a href="#Page_287">287</a>, <a href="#Page_293">293</a><br />
+<br />
+Words. <i>See</i> Language <i>and</i> Terms.<br />
+<br />
+Written arithmetic, <a href="#Page_262">262 ff.</a><br />
+<br />
+<br />
+Zero in multiplication, <a href="#Page_179">179 f.</a><br />
+</p></div>
+
+<p>&nbsp;</p>
+<div class="notebox">
+<p class="noidt"><b>TRANSCRIBER'S NOTE: </b>
+Obvious errors in spelling and punctuation have been silently closed.
+Images have been moved from the middle of a paragraph to the closest paragraph break.
+Footnotes have been consolidated at the end of the HTML ebook.</p>
+</div>
+
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of the Project Gutenberg EBook of The Psychology of Arithmetic, by 
+Edward L. Thorndike
+
+*** END OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC ***
+
+***** This file should be named 39300-h.htm or 39300-h.zip *****
+This and all associated files of various formats will be found in:
+        http://www.gutenberg.org/3/9/3/0/39300/
+
+Produced by Jonathan Ingram and the Online Distributed
+Proofreading Team at http://www.pgdp.net
+
+
+Updated editions will replace the previous one--the old editions
+will be renamed.
+
+Creating the works from public domain print editions means that no
+one owns a United States copyright in these works, so the Foundation
+(and you!) can copy and distribute it in the United States without
+permission and without paying copyright royalties.  Special rules,
+set forth in the General Terms of Use part of this license, apply to
+copying and distributing Project Gutenberg-tm electronic works to
+protect the PROJECT GUTENBERG-tm concept and trademark.  Project
+Gutenberg is a registered trademark, and may not be used if you
+charge for the eBooks, unless you receive specific permission.  If you
+do not charge anything for copies of this eBook, complying with the
+rules is very easy.  You may use this eBook for nearly any purpose
+such as creation of derivative works, reports, performances and
+research.  They may be modified and printed and given away--you may do
+practically ANYTHING with public domain eBooks.  Redistribution is
+subject to the trademark license, especially commercial
+redistribution.
+
+
+
+*** START: FULL LICENSE ***
+
+THE FULL PROJECT GUTENBERG LICENSE
+PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
+
+To protect the Project Gutenberg-tm mission of promoting the free
+distribution of electronic works, by using or distributing this work
+(or any other work associated in any way with the phrase "Project
+Gutenberg"), you agree to comply with all the terms of the Full Project
+Gutenberg-tm License (available with this file or online at
+http://gutenberg.org/license).
+
+
+Section 1.  General Terms of Use and Redistributing Project Gutenberg-tm
+electronic works
+
+1.A.  By reading or using any part of this Project Gutenberg-tm
+electronic work, you indicate that you have read, understand, agree to
+and accept all the terms of this license and intellectual property
+(trademark/copyright) agreement.  If you do not agree to abide by all
+the terms of this agreement, you must cease using and return or destroy
+all copies of Project Gutenberg-tm electronic works in your possession.
+If you paid a fee for obtaining a copy of or access to a Project
+Gutenberg-tm electronic work and you do not agree to be bound by the
+terms of this agreement, you may obtain a refund from the person or
+entity to whom you paid the fee as set forth in paragraph 1.E.8.
+
+1.B.  "Project Gutenberg" is a registered trademark.  It may only be
+used on or associated in any way with an electronic work by people who
+agree to be bound by the terms of this agreement.  There are a few
+things that you can do with most Project Gutenberg-tm electronic works
+even without complying with the full terms of this agreement.  See
+paragraph 1.C below.  There are a lot of things you can do with Project
+Gutenberg-tm electronic works if you follow the terms of this agreement
+and help preserve free future access to Project Gutenberg-tm electronic
+works.  See paragraph 1.E below.
+
+1.C.  The Project Gutenberg Literary Archive Foundation ("the Foundation"
+or PGLAF), owns a compilation copyright in the collection of Project
+Gutenberg-tm electronic works.  Nearly all the individual works in the
+collection are in the public domain in the United States.  If an
+individual work is in the public domain in the United States and you are
+located in the United States, we do not claim a right to prevent you from
+copying, distributing, performing, displaying or creating derivative
+works based on the work as long as all references to Project Gutenberg
+are removed.  Of course, we hope that you will support the Project
+Gutenberg-tm mission of promoting free access to electronic works by
+freely sharing Project Gutenberg-tm works in compliance with the terms of
+this agreement for keeping the Project Gutenberg-tm name associated with
+the work.  You can easily comply with the terms of this agreement by
+keeping this work in the same format with its attached full Project
+Gutenberg-tm License when you share it without charge with others.
+
+1.D.  The copyright laws of the place where you are located also govern
+what you can do with this work.  Copyright laws in most countries are in
+a constant state of change.  If you are outside the United States, check
+the laws of your country in addition to the terms of this agreement
+before downloading, copying, displaying, performing, distributing or
+creating derivative works based on this work or any other Project
+Gutenberg-tm work.  The Foundation makes no representations concerning
+the copyright status of any work in any country outside the United
+States.
+
+1.E.  Unless you have removed all references to Project Gutenberg:
+
+1.E.1.  The following sentence, with active links to, or other immediate
+access to, the full Project Gutenberg-tm License must appear prominently
+whenever any copy of a Project Gutenberg-tm work (any work on which the
+phrase "Project Gutenberg" appears, or with which the phrase "Project
+Gutenberg" is associated) is accessed, displayed, performed, viewed,
+copied or distributed:
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+1.E.2.  If an individual Project Gutenberg-tm electronic work is derived
+from the public domain (does not contain a notice indicating that it is
+posted with permission of the copyright holder), the work can be copied
+and distributed to anyone in the United States without paying any fees
+or charges.  If you are redistributing or providing access to a work
+with the phrase "Project Gutenberg" associated with or appearing on the
+work, you must comply either with the requirements of paragraphs 1.E.1
+through 1.E.7 or obtain permission for the use of the work and the
+Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or
+1.E.9.
+
+1.E.3.  If an individual Project Gutenberg-tm electronic work is posted
+with the permission of the copyright holder, your use and distribution
+must comply with both paragraphs 1.E.1 through 1.E.7 and any additional
+terms imposed by the copyright holder.  Additional terms will be linked
+to the Project Gutenberg-tm License for all works posted with the
+permission of the copyright holder found at the beginning of this work.
+
+1.E.4.  Do not unlink or detach or remove the full Project Gutenberg-tm
+License terms from this work, or any files containing a part of this
+work or any other work associated with Project Gutenberg-tm.
+
+1.E.5.  Do not copy, display, perform, distribute or redistribute this
+electronic work, or any part of this electronic work, without
+prominently displaying the sentence set forth in paragraph 1.E.1 with
+active links or immediate access to the full terms of the Project
+Gutenberg-tm License.
+
+1.E.6.  You may convert to and distribute this work in any binary,
+compressed, marked up, nonproprietary or proprietary form, including any
+word processing or hypertext form.  However, if you provide access to or
+distribute copies of a Project Gutenberg-tm work in a format other than
+"Plain Vanilla ASCII" or other format used in the official version
+posted on the official Project Gutenberg-tm web site (www.gutenberg.org),
+you must, at no additional cost, fee or expense to the user, provide a
+copy, a means of exporting a copy, or a means of obtaining a copy upon
+request, of the work in its original "Plain Vanilla ASCII" or other
+form.  Any alternate format must include the full Project Gutenberg-tm
+License as specified in paragraph 1.E.1.
+
+1.E.7.  Do not charge a fee for access to, viewing, displaying,
+performing, copying or distributing any Project Gutenberg-tm works
+unless you comply with paragraph 1.E.8 or 1.E.9.
+
+1.E.8.  You may charge a reasonable fee for copies of or providing
+access to or distributing Project Gutenberg-tm electronic works provided
+that
+
+- You pay a royalty fee of 20% of the gross profits you derive from
+     the use of Project Gutenberg-tm works calculated using the method
+     you already use to calculate your applicable taxes.  The fee is
+     owed to the owner of the Project Gutenberg-tm trademark, but he
+     has agreed to donate royalties under this paragraph to the
+     Project Gutenberg Literary Archive Foundation.  Royalty payments
+     must be paid within 60 days following each date on which you
+     prepare (or are legally required to prepare) your periodic tax
+     returns.  Royalty payments should be clearly marked as such and
+     sent to the Project Gutenberg Literary Archive Foundation at the
+     address specified in Section 4, "Information about donations to
+     the Project Gutenberg Literary Archive Foundation."
+
+- You provide a full refund of any money paid by a user who notifies
+     you in writing (or by e-mail) within 30 days of receipt that s/he
+     does not agree to the terms of the full Project Gutenberg-tm
+     License.  You must require such a user to return or
+     destroy all copies of the works possessed in a physical medium
+     and discontinue all use of and all access to other copies of
+     Project Gutenberg-tm works.
+
+- You provide, in accordance with paragraph 1.F.3, a full refund of any
+     money paid for a work or a replacement copy, if a defect in the
+     electronic work is discovered and reported to you within 90 days
+     of receipt of the work.
+
+- You comply with all other terms of this agreement for free
+     distribution of Project Gutenberg-tm works.
+
+1.E.9.  If you wish to charge a fee or distribute a Project Gutenberg-tm
+electronic work or group of works on different terms than are set
+forth in this agreement, you must obtain permission in writing from
+both the Project Gutenberg Literary Archive Foundation and Michael
+Hart, the owner of the Project Gutenberg-tm trademark.  Contact the
+Foundation as set forth in Section 3 below.
+
+1.F.
+
+1.F.1.  Project Gutenberg volunteers and employees expend considerable
+effort to identify, do copyright research on, transcribe and proofread
+public domain works in creating the Project Gutenberg-tm
+collection.  Despite these efforts, Project Gutenberg-tm electronic
+works, and the medium on which they may be stored, may contain
+"Defects," such as, but not limited to, incomplete, inaccurate or
+corrupt data, transcription errors, a copyright or other intellectual
+property infringement, a defective or damaged disk or other medium, a
+computer virus, or computer codes that damage or cannot be read by
+your equipment.
+
+1.F.2.  LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
+of Replacement or Refund" described in paragraph 1.F.3, the Project
+Gutenberg Literary Archive Foundation, the owner of the Project
+Gutenberg-tm trademark, and any other party distributing a Project
+Gutenberg-tm electronic work under this agreement, disclaim all
+liability to you for damages, costs and expenses, including legal
+fees.  YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
+LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
+PROVIDED IN PARAGRAPH 1.F.3.  YOU AGREE THAT THE FOUNDATION, THE
+TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
+LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
+INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
+DAMAGE.
+
+1.F.3.  LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
+defect in this electronic work within 90 days of receiving it, you can
+receive a refund of the money (if any) you paid for it by sending a
+written explanation to the person you received the work from.  If you
+received the work on a physical medium, you must return the medium with
+your written explanation.  The person or entity that provided you with
+the defective work may elect to provide a replacement copy in lieu of a
+refund.  If you received the work electronically, the person or entity
+providing it to you may choose to give you a second opportunity to
+receive the work electronically in lieu of a refund.  If the second copy
+is also defective, you may demand a refund in writing without further
+opportunities to fix the problem.
+
+1.F.4.  Except for the limited right of replacement or refund set forth
+in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER
+WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
+WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.
+
+1.F.5.  Some states do not allow disclaimers of certain implied
+warranties or the exclusion or limitation of certain types of damages.
+If any disclaimer or limitation set forth in this agreement violates the
+law of the state applicable to this agreement, the agreement shall be
+interpreted to make the maximum disclaimer or limitation permitted by
+the applicable state law.  The invalidity or unenforceability of any
+provision of this agreement shall not void the remaining provisions.
+
+1.F.6.  INDEMNITY - You agree to indemnify and hold the Foundation, the
+trademark owner, any agent or employee of the Foundation, anyone
+providing copies of Project Gutenberg-tm electronic works in accordance
+with this agreement, and any volunteers associated with the production,
+promotion and distribution of Project Gutenberg-tm electronic works,
+harmless from all liability, costs and expenses, including legal fees,
+that arise directly or indirectly from any of the following which you do
+or cause to occur: (a) distribution of this or any Project Gutenberg-tm
+work, (b) alteration, modification, or additions or deletions to any
+Project Gutenberg-tm work, and (c) any Defect you cause.
+
+
+Section  2.  Information about the Mission of Project Gutenberg-tm
+
+Project Gutenberg-tm is synonymous with the free distribution of
+electronic works in formats readable by the widest variety of computers
+including obsolete, old, middle-aged and new computers.  It exists
+because of the efforts of hundreds of volunteers and donations from
+people in all walks of life.
+
+Volunteers and financial support to provide volunteers with the
+assistance they need, are critical to reaching Project Gutenberg-tm's
+goals and ensuring that the Project Gutenberg-tm collection will
+remain freely available for generations to come.  In 2001, the Project
+Gutenberg Literary Archive Foundation was created to provide a secure
+and permanent future for Project Gutenberg-tm and future generations.
+To learn more about the Project Gutenberg Literary Archive Foundation
+and how your efforts and donations can help, see Sections 3 and 4
+and the Foundation web page at http://www.pglaf.org.
+
+
+Section 3.  Information about the Project Gutenberg Literary Archive
+Foundation
+
+The Project Gutenberg Literary Archive Foundation is a non profit
+501(c)(3) educational corporation organized under the laws of the
+state of Mississippi and granted tax exempt status by the Internal
+Revenue Service.  The Foundation's EIN or federal tax identification
+number is 64-6221541.  Its 501(c)(3) letter is posted at
+http://pglaf.org/fundraising.  Contributions to the Project Gutenberg
+Literary Archive Foundation are tax deductible to the full extent
+permitted by U.S. federal laws and your state's laws.
+
+The Foundation's principal office is located at 4557 Melan Dr. S.
+Fairbanks, AK, 99712., but its volunteers and employees are scattered
+throughout numerous locations.  Its business office is located at
+809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email
+business@pglaf.org.  Email contact links and up to date contact
+information can be found at the Foundation's web site and official
+page at http://pglaf.org
+
+For additional contact information:
+     Dr. Gregory B. Newby
+     Chief Executive and Director
+     gbnewby@pglaf.org
+
+
+Section 4.  Information about Donations to the Project Gutenberg
+Literary Archive Foundation
+
+Project Gutenberg-tm depends upon and cannot survive without wide
+spread public support and donations to carry out its mission of
+increasing the number of public domain and licensed works that can be
+freely distributed in machine readable form accessible by the widest
+array of equipment including outdated equipment.  Many small donations
+($1 to $5,000) are particularly important to maintaining tax exempt
+status with the IRS.
+
+The Foundation is committed to complying with the laws regulating
+charities and charitable donations in all 50 states of the United
+States.  Compliance requirements are not uniform and it takes a
+considerable effort, much paperwork and many fees to meet and keep up
+with these requirements.  We do not solicit donations in locations
+where we have not received written confirmation of compliance.  To
+SEND DONATIONS or determine the status of compliance for any
+particular state visit http://pglaf.org
+
+While we cannot and do not solicit contributions from states where we
+have not met the solicitation requirements, we know of no prohibition
+against accepting unsolicited donations from donors in such states who
+approach us with offers to donate.
+
+International donations are gratefully accepted, but we cannot make
+any statements concerning tax treatment of donations received from
+outside the United States.  U.S. laws alone swamp our small staff.
+
+Please check the Project Gutenberg Web pages for current donation
+methods and addresses.  Donations are accepted in a number of other
+ways including checks, online payments and credit card donations.
+To donate, please visit: http://pglaf.org/donate
+
+
+Section 5.  General Information About Project Gutenberg-tm electronic
+works.
+
+Professor Michael S. Hart is the originator of the Project Gutenberg-tm
+concept of a library of electronic works that could be freely shared
+with anyone.  For thirty years, he produced and distributed Project
+Gutenberg-tm eBooks with only a loose network of volunteer support.
+
+
+Project Gutenberg-tm eBooks are often created from several printed
+editions, all of which are confirmed as Public Domain in the U.S.
+unless a copyright notice is included.  Thus, we do not necessarily
+keep eBooks in compliance with any particular paper edition.
+
+
+Most people start at our Web site which has the main PG search facility:
+
+     http://www.gutenberg.org
+
+This Web site includes information about Project Gutenberg-tm,
+including how to make donations to the Project Gutenberg Literary
+Archive Foundation, how to help produce our new eBooks, and how to
+subscribe to our email newsletter to hear about new eBooks.
+
+
+</pre>
+
+</body>
+</html>
diff --git a/39300-h/images/fig07.jpg b/39300-h/images/fig07.jpg
new file mode 100644
index 0000000..2925e35
--- /dev/null
+++ b/39300-h/images/fig07.jpg
diff --git a/39300-h/images/fig08.jpg b/39300-h/images/fig08.jpg
new file mode 100644
index 0000000..b78d07a
--- /dev/null
+++ b/39300-h/images/fig08.jpg
diff --git a/39300-h/images/fig09.jpg b/39300-h/images/fig09.jpg
new file mode 100644
index 0000000..58d097c
--- /dev/null
+++ b/39300-h/images/fig09.jpg
diff --git a/39300-h/images/fig1.jpg b/39300-h/images/fig1.jpg
new file mode 100644
index 0000000..250f843
--- /dev/null
+++ b/39300-h/images/fig1.jpg
diff --git a/39300-h/images/fig10.jpg b/39300-h/images/fig10.jpg
new file mode 100644
index 0000000..b2100ed
--- /dev/null
+++ b/39300-h/images/fig10.jpg
diff --git a/39300-h/images/fig11.jpg b/39300-h/images/fig11.jpg
new file mode 100644
index 0000000..efd0e4d
--- /dev/null
+++ b/39300-h/images/fig11.jpg
diff --git a/39300-h/images/fig12.jpg b/39300-h/images/fig12.jpg
new file mode 100644
index 0000000..65cff0c
--- /dev/null
+++ b/39300-h/images/fig12.jpg
diff --git a/39300-h/images/fig13.jpg b/39300-h/images/fig13.jpg
new file mode 100644
index 0000000..3f931d3
--- /dev/null
+++ b/39300-h/images/fig13.jpg
diff --git a/39300-h/images/fig14.jpg b/39300-h/images/fig14.jpg
new file mode 100644
index 0000000..bdc58a5
--- /dev/null
+++ b/39300-h/images/fig14.jpg
diff --git a/39300-h/images/fig15.jpg b/39300-h/images/fig15.jpg
new file mode 100644
index 0000000..26f8056
--- /dev/null
+++ b/39300-h/images/fig15.jpg
diff --git a/39300-h/images/fig16.jpg b/39300-h/images/fig16.jpg
new file mode 100644
index 0000000..585be65
--- /dev/null
+++ b/39300-h/images/fig16.jpg
diff --git a/39300-h/images/fig17.jpg b/39300-h/images/fig17.jpg
new file mode 100644
index 0000000..eec4277
--- /dev/null
+++ b/39300-h/images/fig17.jpg
diff --git a/39300-h/images/fig18.jpg b/39300-h/images/fig18.jpg
new file mode 100644
index 0000000..c827cc6
--- /dev/null
+++ b/39300-h/images/fig18.jpg
diff --git a/39300-h/images/fig19.jpg b/39300-h/images/fig19.jpg
new file mode 100644
index 0000000..ddf55b9
--- /dev/null
+++ b/39300-h/images/fig19.jpg
diff --git a/39300-h/images/fig2.jpg b/39300-h/images/fig2.jpg
new file mode 100644
index 0000000..e8e7929
--- /dev/null
+++ b/39300-h/images/fig2.jpg
diff --git a/39300-h/images/fig20.jpg b/39300-h/images/fig20.jpg
new file mode 100644
index 0000000..36e3b6d
--- /dev/null
+++ b/39300-h/images/fig20.jpg
diff --git a/39300-h/images/fig21.jpg b/39300-h/images/fig21.jpg
new file mode 100644
index 0000000..0721fa6
--- /dev/null
+++ b/39300-h/images/fig21.jpg
diff --git a/39300-h/images/fig22.jpg b/39300-h/images/fig22.jpg
new file mode 100644
index 0000000..ddb67c1
--- /dev/null
+++ b/39300-h/images/fig22.jpg
diff --git a/39300-h/images/fig23.jpg b/39300-h/images/fig23.jpg
new file mode 100644
index 0000000..0f579f6
--- /dev/null
+++ b/39300-h/images/fig23.jpg
diff --git a/39300-h/images/fig24.jpg b/39300-h/images/fig24.jpg
new file mode 100644
index 0000000..cfeba82
--- /dev/null
+++ b/39300-h/images/fig24.jpg
diff --git a/39300-h/images/fig25a.jpg b/39300-h/images/fig25a.jpg
new file mode 100644
index 0000000..b6b37aa
--- /dev/null
+++ b/39300-h/images/fig25a.jpg
diff --git a/39300-h/images/fig25b.jpg b/39300-h/images/fig25b.jpg
new file mode 100644
index 0000000..84d6112
--- /dev/null
+++ b/39300-h/images/fig25b.jpg
diff --git a/39300-h/images/fig26.jpg b/39300-h/images/fig26.jpg
new file mode 100644
index 0000000..77f1a9b
--- /dev/null
+++ b/39300-h/images/fig26.jpg
diff --git a/39300-h/images/fig27.jpg b/39300-h/images/fig27.jpg
new file mode 100644
index 0000000..2997c60
--- /dev/null
+++ b/39300-h/images/fig27.jpg
diff --git a/39300-h/images/fig28.jpg b/39300-h/images/fig28.jpg
new file mode 100644
index 0000000..017c7c8
--- /dev/null
+++ b/39300-h/images/fig28.jpg
diff --git a/39300-h/images/fig29.png b/39300-h/images/fig29.png
new file mode 100644
index 0000000..c81af39
--- /dev/null
+++ b/39300-h/images/fig29.png
diff --git a/39300-h/images/fig3.jpg b/39300-h/images/fig3.jpg
new file mode 100644
index 0000000..cf6aa44
--- /dev/null
+++ b/39300-h/images/fig3.jpg
diff --git a/39300-h/images/fig30.png b/39300-h/images/fig30.png
new file mode 100644
index 0000000..7bd5e1c
--- /dev/null
+++ b/39300-h/images/fig30.png
diff --git a/39300-h/images/fig31.jpg b/39300-h/images/fig31.jpg
new file mode 100644
index 0000000..077f511
--- /dev/null
+++ b/39300-h/images/fig31.jpg
diff --git a/39300-h/images/fig32.png b/39300-h/images/fig32.png
new file mode 100644
index 0000000..29202a0
--- /dev/null
+++ b/39300-h/images/fig32.png
diff --git a/39300-h/images/fig33.jpg b/39300-h/images/fig33.jpg
new file mode 100644
index 0000000..1b93ef9
--- /dev/null
+++ b/39300-h/images/fig33.jpg
diff --git a/39300-h/images/fig34.png b/39300-h/images/fig34.png
new file mode 100644
index 0000000..929184c
--- /dev/null
+++ b/39300-h/images/fig34.png
diff --git a/39300-h/images/fig35.jpg b/39300-h/images/fig35.jpg
new file mode 100644
index 0000000..90c1a29
--- /dev/null
+++ b/39300-h/images/fig35.jpg
diff --git a/39300-h/images/fig4.jpg b/39300-h/images/fig4.jpg
new file mode 100644
index 0000000..7d81cee
--- /dev/null
+++ b/39300-h/images/fig4.jpg
diff --git a/39300-h/images/fig40.png b/39300-h/images/fig40.png
new file mode 100644
index 0000000..75e1ae4
--- /dev/null
+++ b/39300-h/images/fig40.png
diff --git a/39300-h/images/fig41.png b/39300-h/images/fig41.png
new file mode 100644
index 0000000..1c3f45d
--- /dev/null
+++ b/39300-h/images/fig41.png
diff --git a/39300-h/images/fig42.jpg b/39300-h/images/fig42.jpg
new file mode 100644
index 0000000..022cc71
--- /dev/null
+++ b/39300-h/images/fig42.jpg
diff --git a/39300-h/images/fig43.jpg b/39300-h/images/fig43.jpg
new file mode 100644
index 0000000..ca86218
--- /dev/null
+++ b/39300-h/images/fig43.jpg
diff --git a/39300-h/images/fig44.jpg b/39300-h/images/fig44.jpg
new file mode 100644
index 0000000..e037bac
--- /dev/null
+++ b/39300-h/images/fig44.jpg
diff --git a/39300-h/images/fig45.png b/39300-h/images/fig45.png
new file mode 100644
index 0000000..7af6ef1
--- /dev/null
+++ b/39300-h/images/fig45.png
diff --git a/39300-h/images/fig46.png b/39300-h/images/fig46.png
new file mode 100644
index 0000000..b475d32
--- /dev/null
+++ b/39300-h/images/fig46.png
diff --git a/39300-h/images/fig47.png b/39300-h/images/fig47.png
new file mode 100644
index 0000000..f046e5b
--- /dev/null
+++ b/39300-h/images/fig47.png
diff --git a/39300-h/images/fig48.jpg b/39300-h/images/fig48.jpg
new file mode 100644
index 0000000..d1c1bd0
--- /dev/null
+++ b/39300-h/images/fig48.jpg
diff --git a/39300-h/images/fig49.png b/39300-h/images/fig49.png
new file mode 100644
index 0000000..4382215
--- /dev/null
+++ b/39300-h/images/fig49.png
diff --git a/39300-h/images/fig5.jpg b/39300-h/images/fig5.jpg
new file mode 100644
index 0000000..4402112
--- /dev/null
+++ b/39300-h/images/fig5.jpg
diff --git a/39300-h/images/fig50.png b/39300-h/images/fig50.png
new file mode 100644
index 0000000..5d36e0b
--- /dev/null
+++ b/39300-h/images/fig50.png
diff --git a/39300-h/images/fig51.png b/39300-h/images/fig51.png
new file mode 100644
index 0000000..b9a6f30
--- /dev/null
+++ b/39300-h/images/fig51.png
diff --git a/39300-h/images/fig52.png b/39300-h/images/fig52.png
new file mode 100644
index 0000000..aa98f8c
--- /dev/null
+++ b/39300-h/images/fig52.png
diff --git a/39300-h/images/fig53.png b/39300-h/images/fig53.png
new file mode 100644
index 0000000..34f3e57
--- /dev/null
+++ b/39300-h/images/fig53.png
diff --git a/39300-h/images/fig54.png b/39300-h/images/fig54.png
new file mode 100644
index 0000000..bef5739
--- /dev/null
+++ b/39300-h/images/fig54.png
diff --git a/39300-h/images/fig55.png b/39300-h/images/fig55.png
new file mode 100644
index 0000000..d1014e8
--- /dev/null
+++ b/39300-h/images/fig55.png
diff --git a/39300-h/images/fig56.png b/39300-h/images/fig56.png
new file mode 100644
index 0000000..75ea28a
--- /dev/null
+++ b/39300-h/images/fig56.png
diff --git a/39300-h/images/fig57.png b/39300-h/images/fig57.png
new file mode 100644
index 0000000..cb1efa9
--- /dev/null
+++ b/39300-h/images/fig57.png
diff --git a/39300-h/images/fig58.png b/39300-h/images/fig58.png
new file mode 100644
index 0000000..87f8b1b
--- /dev/null
+++ b/39300-h/images/fig58.png
diff --git a/39300-h/images/fig59.png b/39300-h/images/fig59.png
new file mode 100644
index 0000000..6c612c3
--- /dev/null
+++ b/39300-h/images/fig59.png
diff --git a/39300-h/images/fig6.jpg b/39300-h/images/fig6.jpg
new file mode 100644
index 0000000..798239a
--- /dev/null
+++ b/39300-h/images/fig6.jpg
diff --git a/39300-h/images/fig60.png b/39300-h/images/fig60.png
new file mode 100644
index 0000000..160376c
--- /dev/null
+++ b/39300-h/images/fig60.png
diff --git a/39300-h/images/fig61.jpg b/39300-h/images/fig61.jpg
new file mode 100644
index 0000000..9c66998
--- /dev/null
+++ b/39300-h/images/fig61.jpg
diff --git a/39300-h/images/fig62.jpg b/39300-h/images/fig62.jpg
new file mode 100644
index 0000000..c758ad8
--- /dev/null
+++ b/39300-h/images/fig62.jpg
diff --git a/39300-h/images/fig63.jpg b/39300-h/images/fig63.jpg
new file mode 100644
index 0000000..4b98b50
--- /dev/null
+++ b/39300-h/images/fig63.jpg
diff --git a/39300-h/images/fig67.jpg b/39300-h/images/fig67.jpg
new file mode 100644
index 0000000..0dec9ec
--- /dev/null
+++ b/39300-h/images/fig67.jpg
diff --git a/39300-h/images/fig68.jpg b/39300-h/images/fig68.jpg
new file mode 100644
index 0000000..e22e199
--- /dev/null
+++ b/39300-h/images/fig68.jpg
diff --git a/39300-h/images/fig69.jpg b/39300-h/images/fig69.jpg
new file mode 100644
index 0000000..620dd44
--- /dev/null
+++ b/39300-h/images/fig69.jpg
diff --git a/39300-h/images/figs64_66.jpg b/39300-h/images/figs64_66.jpg
new file mode 100644
index 0000000..b009e9a
--- /dev/null
+++ b/39300-h/images/figs64_66.jpg
diff --git a/39300-h/images/image04.jpg b/39300-h/images/image04.jpg
new file mode 100644
index 0000000..489d5f6
--- /dev/null
+++ b/39300-h/images/image04.jpg
diff --git a/39300-h/images/img220.jpg b/39300-h/images/img220.jpg
new file mode 100644
index 0000000..54dee51
--- /dev/null
+++ b/39300-h/images/img220.jpg
diff --git a/39300-h/images/img221.jpg b/39300-h/images/img221.jpg
new file mode 100644
index 0000000..2296098
--- /dev/null
+++ b/39300-h/images/img221.jpg
diff --git a/39300-h/images/img292.jpg b/39300-h/images/img292.jpg
new file mode 100644
index 0000000..c2021e3
--- /dev/null
+++ b/39300-h/images/img292.jpg
diff --git a/39300-h/images/p241.png b/39300-h/images/p241.png
new file mode 100644
index 0000000..06c3315
--- /dev/null
+++ b/39300-h/images/p241.png
diff --git a/39300-h/images/p242.png b/39300-h/images/p242.png
new file mode 100644
index 0000000..dc2fb36
--- /dev/null
+++ b/39300-h/images/p242.png
diff --git a/39300-h/images/publogo.png b/39300-h/images/publogo.png
new file mode 100644
index 0000000..057faf1
--- /dev/null
+++ b/39300-h/images/publogo.png
diff --git a/39300.txt b/39300.txt
new file mode 100644
index 0000000..9b26d82
--- /dev/null
+++ b/39300.txt
@@ -0,0 +1,11725 @@
+Project Gutenberg's The Psychology of Arithmetic, by Edward L. Thorndike
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: The Psychology of Arithmetic
+
+Author: Edward L. Thorndike
+
+Release Date: March 29, 2012 [EBook #39300]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC ***
+
+
+
+
+Produced by Jonathan Ingram and the Online Distributed
+Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+
+
+
+                         THE MACMILLAN COMPANY
+               NEW YORK  .  BOSTON  .  CHICAGO  .  DALLAS
+                       ATLANTA  .  SAN FRANCISCO
+
+                        MACMILLAN & CO., LIMITED
+                     LONDON  .  BOMBAY  .  CALCUTTA
+                               MELBOURNE
+
+                         THE MACMILLAN COMPANY
+                           OF CANADA, LIMITED
+                                TORONTO
+
+
+
+                           THE PSYCHOLOGY OF
+                               ARITHMETIC
+
+
+                                   BY
+                          EDWARD L. THORNDIKE
+
+                       TEACHERS COLLEGE, COLUMBIA
+                               UNIVERSITY
+
+
+                                New York
+                         THE MACMILLAN COMPANY
+                                  1929
+
+                         _All rights reserved_
+
+
+
+                            COPYRIGHT, 1922,
+                       BY THE MACMILLAN COMPANY.
+
+      Set up and electrotyped. Published January, 1922. Reprinted
+         October, 1924; May, 1926; August, 1927; October, 1929.
+
+
+              . PRINTED IN THE UNITED STATES OF AMERICA .
+
+
+
+
+PREFACE
+
+
+Within recent years there have been three lines of advance in psychology
+which are of notable significance for teaching. The first is the new
+point of view concerning the general process of learning. We now
+understand that learning is essentially the formation of connections or
+bonds between situations and responses, that the satisfyingness of the
+result is the chief force that forms them, and that habit rules in the
+realm of thought as truly and as fully as in the realm of action.
+
+The second is the great increase in knowledge of the amount, rate, and
+conditions of improvement in those organized groups or hierarchies of
+habits which we call abilities, such as ability to add or ability to
+read. Practice and improvement are no longer vague generalities, but
+concern changes which are definable and measurable by standard tests and
+scales.
+
+The third is the better understanding of the so-called "higher
+processes" of analysis, abstraction, the formation of general notions,
+and reasoning. The older view of a mental chemistry whereby sensations
+were compounded into percepts, percepts were duplicated by images,
+percepts and images were amalgamated into abstractions and concepts, and
+these were manipulated by reasoning, has given way to the understanding
+of the laws of response to elements or aspects of situations and to many
+situations or elements thereof in combination. James' view of reasoning
+as "selection of essentials" and "thinking things together" in a
+revised and clarified form has important applications in the teaching of
+all the school subjects.
+
+This book presents the applications of this newer dynamic psychology to
+the teaching of arithmetic. Its contents are substantially what have
+been included in a course of lectures on the psychology of the
+elementary school subjects given by the author for some years to
+students of elementary education at Teachers College. Many of these
+former students, now in supervisory charge of elementary schools, have
+urged that these lectures be made available to teachers in general. So
+they are now published in spite of the author's desire to clarify and
+reinforce certain matters by further researches.
+
+A word of explanation is necessary concerning the exercises and problems
+cited to illustrate various matters, especially erroneous pedagogy.
+These are all genuine, having their source in actual textbooks, courses
+of study, state examinations, and the like. To avoid any possibility of
+invidious comparisons they are not quotations, but equivalent problems
+such as represent accurately the spirit and intent of the originals.
+
+I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn
+and Company, D. C. Heath and Company, The Macmillan Company, The Oxford
+University Press, Rand, McNally and Company, Dr. C. W. Stone, The
+Teachers College Bureau of Publications, and The World Book Company, in
+permitting various quotations.
+
+                                                EDWARD L. THORNDIKE.
+
+   TEACHERS COLLEGE
+  COLUMBIA UNIVERSITY
+     April 1, 1920
+
+
+
+
+CONTENTS
+
+
+ CHAPTER                                                          PAGE
+
+    INTRODUCTION: THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS  xi
+
+ I. THE NATURE OF ARITHMETICAL ABILITIES                             1
+
+        Knowledge of the Meanings of Numbers
+        Arithmetical Language
+        Problem Solving
+        Arithmetical Reasoning
+        Summary
+        The Sociology of Arithmetic
+
+ II. THE MEASUREMENT OF ARITHMETICAL ABILITIES                      27
+
+        A Sample Measurement of an Arithmetical Ability
+        Ability to Add Integers
+        Measurements of Ability in Computation
+        Measurements of Ability in Applied Arithmetic:
+          the Solution of Problems
+
+ III. THE CONSTITUTION OF ARITHMETICAL ABILITIES                    51
+
+        The Elementary Functions of Arithmetical Learning
+        Knowledge of the Meaning of a Fraction
+        Learning the Processes of Computation
+
+ IV. THE CONSTITUTION OF ARITHMETICAL ABILITIES (_continued_)       70
+
+        The Selection of the Bonds to Be Formed
+        The Importance of Habit Formation
+        Desirable Bonds Now Often Neglected
+        Wasteful and Harmful Bonds
+        Guiding Principles
+
+ V. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS   102
+
+        The Need of Stronger Elementary Bonds
+        Early Mastery
+        The Strength of Bonds for Temporary Service
+        The Strength of Bonds with Technical Facts and Terms
+        The Strength of Bonds Concerning the Reasons for
+          Arithmetical Processes
+        Propaedeutic Bonds
+
+ VI. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF
+          PRACTICE AND THE ORGANIZATION OF ABILITIES               122
+
+        The Amount of Practice
+        Under-learning and Over-learning
+        The Organization of Abilities
+
+ VII. THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS      141
+
+        Conventional _versus_ Effective Orders
+        Decreasing Interference and Increasing Facilitation
+        Interest
+        General Principles
+
+ VIII. THE DISTRIBUTION OF PRACTICE                                156
+
+        The Problem
+        Sample Distributions
+        Possible Improvements
+
+ IX. THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL
+          NOTIONS IN ARITHMETIC                                    169
+
+        Responses to Elements and Classes
+        Facilitating the Analysis of Elements
+        Systematic and Opportunistic Stimuli to Analysis
+        Adaptations to Elementary-school Pupils
+
+ X. THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC            185
+
+        The Essentials of Arithmetical Reasoning
+        Reasoning as the Cooeperation of Organized Habits
+
+ XI. ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL            195
+
+        The Utilization of Instinctive Interests
+        The Order of Development of Original Tendencies
+        Inventories of Arithmetical Knowledge and Skill
+        The Perception of Number and Quantity
+        The Early Awareness of Number
+
+ XII. INTEREST IN ARITHMETIC                                       209
+
+        Censuses of Pupils' Interests
+        Relieving Eye Strain
+        Significance for Related Activities
+        Intrinsic Interest in Arithmetical Learning
+
+ XIII. THE CONDITIONS OF LEARNING                                  227
+
+        External Conditions
+        The Hygiene of the Eyes in Arithmetic
+        The Use of Concrete Objects in Arithmetic
+        Oral, Mental, and Written Arithmetic
+
+ XIV. THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE             266
+
+        Illustrative Cases
+        General Principles
+        Difficulty and Success as Stimuli
+        False Inferences
+
+ XV. INDIVIDUAL DIFFERENCES                                        285
+
+        Nature and Amount
+        Differences within One Class
+        The Causes of Individual Differences
+        The Interrelations of Individual Differences
+
+    BIBLIOGRAPHY OF REFERENCES                                     302
+
+    INDEX                                                          311
+
+
+
+
+GENERAL INTRODUCTION
+
+THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS
+
+
+The psychology of the elementary school subjects is concerned with the
+connections whereby a child is able to respond to the sight of printed
+words by thoughts of their meanings, to the thought of "six and eight"
+by thinking "fourteen," to certain sorts of stories, poems, songs, and
+pictures by appreciation thereof, to certain situations by acts of
+skill, to certain others by acts of courtesy and justice, and so on and
+on through the series of situations and responses which are provided by
+the systematic training of the school subjects and the less systematic
+training of school life during their study. The aims of elementary
+education, when fully defined, will be found to be the production of
+changes in human nature represented by an almost countless list of
+connections or bonds whereby the pupil thinks or feels or acts in
+certain ways in response to the situations the school has organized and
+is influenced to think and feel and act similarly to similar situations
+when life outside of school confronts him with them.
+
+We are not at present able to define the work of the elementary school
+in detail as the formation of such and such bonds between certain
+detached situations and certain specified responses. As elsewhere in
+human learning, we are at present forced to think somewhat vaguely in
+terms of mental functions, like "ability to read the vernacular,"
+"ability to spell common words," "ability to add, subtract, multiply,
+and divide with integers," "knowledge of the history of the United
+States," "honesty in examinations," and "appreciation of good music,"
+defined by some general results obtained rather than by the elementary
+bonds which constitute them.
+
+The psychology of the school subjects begins where our common sense
+knowledge of these functions leaves off and tries to define the
+knowledge, interest, power, skill, or ideal in question more adequately,
+to measure improvement in it, to analyze it into its constituent bonds,
+to decide what bonds need to be formed and in what order as means to the
+most economical attainment of the desired improvement, to survey the
+original tendencies and the tendencies already acquired before entrance
+to school which help or hinder progress in the elementary school
+subjects, to examine the motives that are or may be used to make the
+desired connections satisfying, to examine any other special conditions
+of improvement, and to note any facts concerning individual differences
+that are of special importance to the conduct of elementary school work.
+
+Put in terms of problems, the task of the psychology of the elementary
+school subjects is, in each case:--
+
+(1) _What is the function?_ For example, just what is "ability to read"?
+Just what does "the understanding of decimal notation" mean? Just what
+are "the moral effects to be sought from the teaching of literature"?
+
+(2) _How are degrees of ability or attainment, and degrees of progress
+or improvement in the function or a part of the function measured?_ For
+example, how can we determine how well a pupil should write, or how hard
+words we expect him to spell, or what good taste we expect him to show?
+How can we define to ourselves what knowledge of the meaning of a
+fraction we shall try to secure in grade 4?
+
+(3) _What can be done toward reducing the function to terms of
+particular situation-response connections, whose formation can be more
+surely and easily controlled?_ For example, how far does ability to
+spell involve the formation one by one of bonds between the thought of
+almost every word in the language and the thought of that word's letters
+in their correct order; and how far does, say, the bond leading from the
+situation of the sound of _ceive_ in _receive_ and _deceive_ to their
+correct spelling insure the correct spelling of that part of _perceive_?
+Does "ability to add" involve special bonds leading from "27 and 4" to
+"31," from "27 and 5" to "32," and "27 and 6" to "33"; or will the bonds
+leading from "7 and 4" to "11," "7 and 5" to "12" and "7 and 6" to "13"
+(each plus a simple inference) serve as well? What are the situations
+and responses that represent in actual behavior the quality that we call
+school patriotism?
+
+(4) _In almost every case a certain desired change of knowledge or skill
+or power can be attained by any one of several sets of bonds. Which of
+them is the best? What are the advantages of each?_ For example,
+learning to add may include the bonds "0 and 0 are 0," "0 and 1 are 1,"
+"0 and 2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these may be
+all left unformed, the pupil being taught the habits of entering 0 as
+the sum of a column that is composed of zeros and otherwise neglecting 0
+in addition. Are the rules of usage worth teaching as a means toward
+correct speech, or is the time better spent in detailed practice in
+correct speech itself?
+
+(5) _A bond to be formed may be formed in any one of many degrees of
+strength. Which of these is, at any given stage of learning the subject,
+the most desirable, all things considered?_ For example, shall the dates
+of all the early settlements of North America be learned so that the
+exact year will be remembered for ten years, or so that the exact date
+will be remembered for ten minutes and the date with an error plus or
+minus of ten years will be remembered for a year or two? Shall the
+tables of inches, feet, and yards, and pints, quarts, and gallons be
+learned at their first appearance so as to be remembered for a year, or
+shall they be learned only well enough to be usable in the work of that
+week, which in turn fixes them to last for a month or so? Should a pupil
+in the first year of study of French have such perfect connections
+between the sounds of French words and their meanings that he can
+understand simple sentences containing them spoken at an ordinary rate
+of speaking? Or is slow speech permissible, and even imperative, on the
+part of the teacher, with gradual increase of rate?
+
+(6) _In almost every case, any set of bonds may produce the desired
+change when presented in any one of several orders. Which is the best
+order? What are the advantages of each?_ Certain systems for teaching
+handwriting perfect the elementary movements one at a time and then
+teach their combination in words and sentences. Others begin and
+continue with the complex movement-series that actual words require.
+What do the latter lose and gain? The bonds constituting knowledge of
+the metric system are now formed late in the pupil's course. Would it be
+better if they were formed early as a means of facilitating knowledge of
+decimal fractions?
+
+(7) _What are the original tendencies and pre-school acquisitions upon
+which the connection-forming of the elementary school may be based or
+which it has to counteract?_ For example, if a pupil knows the meaning
+of a heard word, he may read it understandingly from getting its sound,
+as by phonic reconstruction. What words does the average beginner so
+know? What are the individual differences in this respect? What do the
+instincts of gregariousness, attention-getting, approval, and
+helpfulness recommend concerning group-work _versus_ individual-work,
+and concerning the size of a group that is most desirable? The original
+tendency of the eyes is certainly not to move along a line from left to
+right of a page, then back in one sweep and along the next line. What is
+their original tendency when confronted with the printed page, and what
+must we do with it in teaching reading?
+
+(8) _What armament of satisfiers and annoyers, of positive and negative
+interests and motives, stands ready for use in the formation of the
+intrinsically uninteresting connections between black marks and
+meanings, numerical exercises and their answers, words and their
+spelling, and the like?_ School practice has tried, more or less at
+random, incentives and deterrents from quasi-physical pain to the most
+sentimental fondling, from sheer cajolery to philosophical argument,
+from appeals to assumed savage and primitive traits to appeals to the
+interest in automobiles, flying-machines, and wireless telegraphy. Can
+not psychology give some rules for guidance, or at least limit
+experimentation to its more hopeful fields?
+
+(9) _The general conditions of efficient learning are described in
+manuals of educational psychology. How do these apply in the case of
+each task of the elementary school?_ For example, the arrangement of
+school drills in addition and in short division in the form of practice
+experiments has been found very effective in producing interest in the
+work and in improvement at it. In what other arithmetical functions may
+we expect the same?
+
+(10) _Beside the general principles concerning the nature and causation
+of individual differences, there must obviously be, in existence or
+obtainable as a possible result of proper investigation, a great fund
+of knowledge of special differences relevant to the learning of reading,
+spelling, geography, arithmetic, and the like. What are the facts as far
+as known? What are the means of learning more of them?_ Courtis finds
+that a child may be specially strong in addition and yet be specially
+weak in subtraction in comparison with others of his age and grade. It
+even seems that such subtle and intricate tendencies are inherited. How
+far is such specialization the rule? Is it, for example, the case that a
+child may have a special gift for spelling certain sorts of words, for
+drawing faces rather than flowers, for learning ancient history rather
+than modern?
+
+       *       *       *       *       *
+
+Such are our problems: this volume discusses them in the case of
+arithmetic. The student who wishes to relate the discussion to the
+general pedagogy of arithmetic may profitably read, in connection with
+this volume: The Teaching of Elementary Mathematics, by D. E. Smith
+['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to
+Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching
+of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic,
+by the author ['21].
+
+
+
+
+THE PSYCHOLOGY OF ARITHMETIC
+
+
+
+
+CHAPTER I
+
+THE NATURE OF ARITHMETICAL ABILITIES
+
+
+According to common sense, the task of the elementary school is to
+teach:--(1) the meanings of numbers, (2) the nature of our system
+of decimal notation, (3) the meanings of addition, subtraction,
+multiplication, and division, and (4) the nature and relations of
+certain common measures; to secure (5) the ability to add, subtract,
+multiply, and divide with integers, common and decimal fractions, and
+denominate numbers, (6) the ability to apply the knowledge and power
+represented by (1) to (5) in solving problems, and (7) certain specific
+abilities to solve problems concerning percentage, interest, and other
+common occurrences in business life.
+
+This statement of the functions to be developed and improved is sound
+and useful so far as it goes, but it does not go far enough to make the
+task entirely clear. If teachers had nothing but the statement above as
+a guide to what changes they were to make in their pupils, they would
+often leave out important features of arithmetical training, and put in
+forms of training that a wise educational plan would not tolerate. It is
+also the case that different leaders in arithmetical teaching, though
+they might all subscribe to the general statement of the previous
+paragraph, certainly do not in practice have identical notions of what
+arithmetic should be for the elementary school pupil.
+
+The ordinary view of the nature of arithmetical learning is obscure or
+inadequate in four respects. It does not define what 'knowledge of the
+meanings of numbers' is; it does not take account of the very large
+amount of teaching of _language_ which is done and should be done as a
+part of the teaching of arithmetic; it does not distinguish between the
+ability to meet certain quantitative problems as life offers them and
+the ability to meet the problems provided by textbooks and courses of
+study; it leaves 'the ability to apply arithmetical knowledge and power'
+as a rather mystical general faculty to be improved by some educational
+magic. The four necessary amendments may be discussed briefly.
+
+
+KNOWLEDGE OF THE MEANINGS OF NUMBERS
+
+Knowledge of the meanings of the numbers from one to ten may mean
+knowledge that 'one' means a single thing of the sort named, that two
+means one more than one, that three means one more than two, and so on.
+This we may call the _series_ meaning. To know the meaning of 'six' in
+this sense is to know that it is one more than five and one less than
+seven--that it is between five and seven in the number series. Or we may
+mean by knowledge of the meanings of numbers, knowledge that two fits a
+collection of two units, that three fits a collection of three units,
+and so on, each number being a name for a certain sized collection of
+discrete things, such as apples, pennies, boys, balls, fingers, and the
+other customary objects of enumeration in the primary school. This we
+may call the _collection_ meaning. To know the meaning of six in this
+sense is to be able to name correctly any collection of six separate,
+easily distinguishable individual objects. In the third place, knowledge
+of the numbers from one to ten may mean knowledge that two is twice
+whatever is called one, that three is three times whatever is one, and
+so on. This is, of course, the _ratio_ meaning. To know the meaning of
+six in this sense is to know that if ___________ is one, a line half a
+foot long is six, that if [___] is one, [____________] is about six,
+while if [__] is one, [______] is about six, and the like. In the fourth
+place, the meaning of a number may be a smaller or larger fraction of
+its _implications_--its numerical relations, facts about it. To know six
+in this sense is to know that it is more than five or four, less than
+seven or eight, twice three, three times two, the sum of five and one,
+or of four and two, or of three and three, two less than eight--that
+with four it makes ten, that it is half of twelve, and the like. This we
+may call the '_nucleus of facts_' or _relational_ meaning of a number.
+
+Ordinary school practice has commonly accepted the second meaning as
+that which it is the task of the school to teach beginners, but each of
+the other meanings has been alleged to be the essential one--the series
+idea by Phillips ['97], the ratio idea by McLellan and Dewey ['95] and
+Speer ['97], and the relational idea by Grube and his followers.
+
+This diversity of views concerning what the function is that is to be
+improved in the case of learning the meanings of the numbers one to ten
+is not a trifling matter of definition, but produces very great
+differences in school practice. Consider, for example, the predominant
+value assigned to counting by Phillips in the passage quoted below, and
+the samples of the sort of work at which children were kept employed
+for months by too ardent followers of Speer and Grube.
+
+THE SERIES IDEA OVEREMPHASIZED
+
+  "This is essentially the counting period, and any words that can be
+  arranged into a series furnish all that is necessary. Counting is
+  fundamental, and counting that is spontaneous, free from sensible
+  observation, and from the strain of reason. A study of these original
+  methods shows that multiplication was developed out of counting, and
+  not from addition as nearly all textbooks treat it. Multiplication is
+  counting. When children count by 4's, etc., they accent the same as
+  counting gymnastics or music. When a child now counts on its fingers
+  it simply reproduces a stage in the growth of the civilization of all
+  nations.
+
+  I would emphasize again that during the counting period there is a
+  somewhat spontaneous development of the number series-idea which
+  Preyer has discussed in his Arithmogenesis; that an immense momentum
+  is given by a systematic series of names; and that these names are
+  generally first learned and applied to objects later. A lady teacher
+  told me that the Superintendent did not wish the teachers to allow the
+  children to count on their fingers, but she failed to see why counting
+  with horse-chestnuts was any better. Her children could hardly avoid
+  using their fingers in counting other objects yet they followed the
+  series to 100 without hesitation or reference to their fingers. This
+  spontaneous counting period, or naming and following the series,
+  should precede its application to objects." [D.E. Phillips, '97,
+  p. 238.]
+
+THE RATIO IDEA OVEREMPHASIZED
+
+    [Illustration: FIG. 1.]
+
+  "Ratios.--1. Select solids having the relation, or ratio, of _a_, _b_,
+     _c_, _d_, _o_, _e_.
+
+  2. Name the solids, _a_, _b_, _c_, _d_, _o_, _e_.
+
+     The means of expressing must be as freely supplied as the means of
+     discovery. The pupil is not expected to invent terms.
+
+  3. Tell all you can about the relation of these units.
+
+  4. Unite units and tell what the sum equals.
+
+  5. Make statements like this: _o_ less _e_ equals _b_.
+
+  6. _c_ can be separated into how many _d_'s? into how many _b_'s?
+
+  7. _c_ can be separated into how many _b_'s? What is the name of the
+     largest unit that can be found in both _c_ and _d_ an exact number
+     of times?
+
+  8. Each of the other units equals what part of _c_?
+
+  9. If _b_ is 1, what is each of the other units?
+
+ 10. If _a_ is 1, what is each of the other units?
+
+ 11. If _b_ is 1, how many 1's are there in each of the other units?
+
+ 12. If _d_ is 1, how many 1's and parts of 1 in each of the other
+     units?
+
+ 13. 2 is the relation of what units?
+
+ 14. 3 is the relation of what units?
+
+ 15. 1/2 is the relation of what units?
+
+ 16. 2/3 is the relation of what units?
+
+ 17. Which units have the relation 3/2?
+
+ 18. Which unit is 3 times as large as 1/2 of _b_?
+
+ 19. _c_ equals 6 times 1/3 of what unit?
+
+ 20. 1/3 of what unit equals 1/6 of _c_?
+
+ 21. What equals 1/2 of _c_? _d_ equals how many sixths of _c_?
+
+ 22. _o_ equals 5 times 1/3 of what unit?
+
+ 23. 1/3 of what unit equals 1/5 of _o_?
+
+ 24. 2/3 of _d_ equals what unit? _b_ equals how many thirds of _d_?
+
+ 25. 2 is the ratio of _d_ to 1/3 of what unit? 3 is the ratio of _d_
+     to 1/2 of what unit?
+
+ 26. _d_ equals 3/4 of what unit? 3/4 is the ratio of what units?"
+     [Speer, '97, p. 9f.]
+
+THE RELATIONAL IDEA OVEREMPHASIZED
+
+  An inspection of books of the eighties which followed the "Grube
+  method" (for example, the _New Elementary Arithmetic_ by E.E. White
+  ['83]) will show undue emphasis on the relational ideas. There will be
+  over a hundred and fifty successive tasks all, or nearly all, on +7
+  and -7. There will be much written work of the sort shown below:
+
+  _Add:_
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    4    4
+           4    1    2
+          --   --   --
+
+  which must have sorely tried the eyes of all concerned. Pupils are
+  taught to "give the analysis and synthesis of each of the nine
+  digits." Yet the author states that he does not carry the principle
+  of the Grube method "to the extreme of useless repetition and
+  mechanism."
+
+It should be obvious that all four meanings have claims upon the
+attention of the elementary school. Four is the thing between three and
+five in the number series; it is the name for a certain sized collection
+of discrete objects; it is also the name for a continuous magnitude
+equal to four units--for four quarts of milk in a gallon pail as truly
+as for four separate quart-pails of milk; it is also, if we know it
+well, the thing got by adding one to three or subtracting six from ten
+or taking two two's or half of eight. To know the meaning of a number
+means to know somewhat about it in all of these respects. The difficulty
+has been the narrow vision of the extremists. A child must not be left
+interminably counting; in fact the one-more-ness of the number series
+can almost be had as a by-product. A child must not be restricted to
+exercises with collections objectified as in Fig. 2 or stated in words
+as so many apples, oranges, hats, pens, etc., when work with measurement
+of continuous quantities with varying units--inches, feet, yards,
+glassfuls, pints, quarts, seconds, minutes, hours, and the like--is
+so easy and so significant. On the other hand, the elaboration of
+artificial problems with fictitious units of measure just to have
+relative magnitudes as in the exercises on page 5 is a wasteful
+sacrifice. Similarly, special drills emphasizing the fact that eighteen
+is eleven and seven, twelve and six, three less than twenty-one, and the
+like, are simply idolatrous; these facts about eighteen, so far as they
+are needed, are better learned in the course of actual column-addition
+and -subtraction.
+
+    [Illustration: FIG. 2.]
+
+
+ARITHMETICAL LANGUAGE
+
+The second improvement to be made in the ordinary notion of what the
+functions to be improved are in the case of arithmetic is to include
+among these functions the knowledge of certain words. The understanding
+of such words as _both_, _all_, _in all_, _together_, _less_,
+_difference_, _sum_, _whole_, _part_, _equal_, _buy_, _sell_, _have
+left_, _measure_, _is contained in_, and the like, is necessary in
+arithmetic as truly as is the understanding of numbers themselves. It
+must be provided for by the school; for pre-school and extra-school
+training does not furnish it, or furnishes it too late. It can be
+provided for much better in connection with the teaching of arithmetic
+than in connection with the teaching of English.
+
+It has not been provided for. An examination of the first fifty pages of
+eight recent textbooks for beginners in arithmetic reveals very slight
+attention to this matter at the best and no attention at all in some
+cases. Three of the books do not even use the word _sum_, and one uses
+it only once in the fifty pages. In all the four hundred pages the word
+_difference_ occurs only twenty times. When the words are used, no great
+ingenuity or care appears in the means of making sure that their
+meanings are understood.
+
+The chief reason why it has not been provided for is precisely that the
+common notion of what the functions are that arithmetic is to develop
+has left out of account this function of intelligent response to
+quantitative terms, other than the names of the numbers and processes.
+
+Knowledge of language over a much wider range is a necessary element in
+arithmetical ability in so far as the latter includes ability to solve
+verbally stated problems. As arithmetic is now taught, it does include
+that ability, and a large part of the time of wise teaching is given to
+improving the function 'knowing what a problem states and what it asks
+for.' Since, however, this understanding of verbally stated problems may
+not be an absolutely necessary element of arithmetic, it is best to
+defer its consideration until we have seen what the general function of
+problem-solving is.
+
+
+PROBLEM-SOLVING
+
+The third respect in which the function, 'ability in arithmetic,' needs
+clearer definition, is this 'problem-solving.' The aim of the elementary
+school is to provide for correct and economical response to genuine
+problems, such as knowing the total due for certain real quantities at
+certain real prices, knowing the correct change to give or get, keeping
+household accounts, calculating wages due, computing areas, percentages,
+and discounts, estimating quantities needed of certain materials to make
+certain household or shop products, and the like. Life brings these
+problems usually either with a real situation (as when one buys and
+counts the cost and his change), or with a situation that one imagines
+or describes to himself (as when one figures out how much money he must
+save per week to be able to buy a forty-dollar bicycle before a certain
+date). Sometimes, however, the problem is described in words to the
+person who must solve it by another person (as when a life insurance
+agent says, 'You pay only 25 cents a week from now till--and you get
+$250 then'; or when an employer says, 'Your wages would be 9 dollars a
+week, with luncheon furnished and bonuses of such and such amounts').
+Sometimes also the problem is described in printed or written words to
+the person who must solve it (as in an advertisement or in the letter of
+a customer asking for an estimate on this or that). The problem may be
+in part real, in part imagined or described to oneself, and in part
+described to one orally or in printed or written words (as when the
+proposed articles for purchase lie before one, the amount of money one
+has in the bank is imagined, the shopkeeper offers 10 percent discount,
+and the printed price list is there to be read).
+
+To fit pupils to solve these real, personally imagined, or
+self-described problems, and 'described-by-another' problems, schools
+have relied almost exclusively on training with problems of the last
+sort only. The following page taken almost at random from one of the
+best recent textbooks could be paralleled by thousands of others; and
+the oral problems put by teachers have, as a rule, no real situation
+supporting them.
+
+  1. At 70 cents per 100 pounds, what will be the amount of duty on an
+     invoice of 3622 steel rails, each rail being 27 feet long and
+     weighing 60 pounds to the yard?
+
+  2. A man had property valued at $6500. What will be his taxes at the
+     rate of $10.80 per $1000?
+
+  3. Multiply seventy thousand fourteen hundred-thousandths by one
+     hundred nine millionths, and divide the product by five hundred
+     forty-five.
+
+  4. What number multiplied by 43-3/4 will produce 265-5/8?
+
+  5. What decimal of a bushel is 3 quarts?
+
+  6. A man sells 5/8 of an acre of land for $93.75. What would be the
+     value of his farm of 150-3/4 acres at the same rate?
+
+  7. A coal dealer buys 375 tons coal at $4.25 per ton of 2240 pounds.
+     He sells it at $4.50 per ton of 2000 pounds. What is his profit?
+
+  8. Bought 60 yards of cloth at the rate of 2 yards for $5, and 80
+     yards more at the rate of 4 yards for $9. I immediately sold the
+     whole of it at the rate of 5 yards for $12. How much did I gain?
+
+  9. A man purchased 40 bushels of apples at $1.50 per bushel.
+     Twenty-five hundredths of them were damaged, and he sold them
+     at 20 cents per peck. He sold the remainder at 50 cents per
+     peck. How much did he gain or lose?
+
+ 10. If oranges are 37-1/2 cents per dozen, how many boxes, each
+     containing 480, can be bought for $60?
+
+ 11. A man can do a piece of work in 18-3/4 days. What part of it can
+     he do in 6-2/3 days?
+
+ 12. How old to-day is a boy that was born Oct. 29, 1896?
+     [Walsh, '06, Part I, p. 165.]
+
+As a result, teachers and textbook writers have come to think of the
+functions of solving arithmetical problems as identical with the
+function of solving the described problems which they give in school in
+books, examination papers, and the like. If they do not think explicitly
+that this is so, they still act in training and in testing pupils as if
+it were so.
+
+It is not. Problems should be solved in school to the end that pupils
+may solve the problems which life offers. To know what change one should
+receive after a given real purchase, to keep one's accounts accurately,
+to adapt a recipe for six so as to make enough of the article for four
+persons, to estimate the amount of seed required for a plot of a given
+size from the statement of the amount required per acre, to make with
+surety the applications that the household, small stores, and ordinary
+trades require--such is the ability that the elementary school should
+develop. Other things being equal, the school should set problems in
+arithmetic which life then and later will set, should favor the
+situations which life itself offers and the responses which life itself
+demands.
+
+Other things are not always equal. The same amount of time and effort
+will often be more productive toward the final end if directed during
+school to 'made-up' problems. The keeping of personal financial accounts
+as a school exercise is usually impracticable, partly because some of
+the children have no earnings or allowance--no accounts to keep, and
+partly because the task of supervising work when each child has a
+different problem is too great for the teacher. The use of real
+household and shop problems will be easy only when the school program
+includes the household arts and industrial education, and when these
+subjects themselves are taught so as to improve the functions used by
+real life. Very often the most efficient course is to make sure that
+arithmetical procedures are applied to the real and personally initiated
+problems which they fit, by having a certain number of such problems
+arise and be solved; then to make sure that the similarity between these
+real problems and certain described problems of the textbook or
+teacher's giving is appreciated; and then to give the needed drill work
+with described problems. In many cases the school practice is fairly
+well justified in assuming that solving described problems will prepare
+the pupil to solve the corresponding real problems actually much better
+than the same amount of time spent on the real problems themselves.
+
+All this is true, yet the general principle remains that, other things
+being equal, the school should favor real situations, should present
+issues as life will present them.
+
+Where other things make the use of verbally described problems of the
+ordinary type desirable, these should be chosen so as to give a maximum
+of preparation for the real applications of arithmetic in life. We
+should not, for example, carelessly use any problem that comes to mind
+in applying a certain principle, but should stop to consider just what
+the situations of life really require and show clearly the application
+of that principle. For example, contrast these two problems applying
+cancellation:--
+
+  A. A man sold 24 lambs at $18 apiece on each of six days, and
+     bought 8 pounds of metal with the proceeds. How much did he
+     pay per ounce for the metal?
+
+  B. How tall must a rectangular tank 16" long by 8" wide be to
+     hold as much as a rectangular tank 24" by 18" by 6"?
+
+The first problem not only presents a situation that would rarely or
+never occur, but also takes a way to find the answer that would not, in
+that situation, be taken since the price set by another would determine
+the amount.
+
+Much thought and ingenuity should in the future be expended in
+eliminating problems whose solution does not improve the real function
+to be improved by applied arithmetic, or improves it at too great cost,
+and in devising problems which prepare directly for life's demands and
+still can fit into a curriculum that can be administered by one teacher
+in charge of thirty or forty pupils, under the limitations of school
+life.
+
+The following illustrations will to some extent show concretely what the
+ability to apply the knowledge and power represented by abstract or pure
+arithmetic--the so-called fundamentals--in solving problems should mean
+and what it should not mean.
+
+  _Samples of Desirable Applications of Arithmetic in Problems where
+    the Situation is Actually Present to Sense in Whole or in Part_
+
+Keeping the scores and deciding which side beat and by how much in
+appropriate classroom games, spelling matches, and the like.
+
+Computing costs, making and inspecting change, taking inventories, and
+the like with a real or play store.
+
+Mapping the school garden, dividing it into allotments, planning for the
+purchase of seeds, and the like.
+
+Measuring one's own achievement and progress in tests of word-knowledge,
+spelling, addition, subtraction, speed of writing, and the like.
+Measuring the rate of improvement per hour of practice or per week of
+school life, and the like.
+
+Estimating costs of food cooked in the school kitchen, articles made in
+the school shops, and the like.
+
+Computing the cost of telegrams, postage, expressage, for a real message
+or package, from the published tariffs.
+
+Computing costs from mail order catalogues and the like.
+
+  _Samples of Desirable Applications of Arithmetic where the Situation
+    is Not Present to Sense_
+
+The samples given here all concern the subtraction of fractions.
+Samples concerning any other arithmetical principle may be found in the
+appropriate pages of any text which contains problem-material selected
+with consideration of life's needs.
+
+A
+
+  1. Dora is making jelly. The recipe calls for 24 cups of sugar and
+     she has only 21-1/2. She has no time to go to the store so she
+     has to borrow the sugar from a neighbor. How much must she get?
+
+      _Subtract_
+          24        _Think "1/2 and 1/2 = 1." Write 1/2._
+          21-1/2    _Think "2 and 2 = 4." Write the 2._
+        --------
+           2-1/2
+
+  2. A box full of soap weighs 29-1/2 lb. The empty box weighs 3-1/2 lb.
+     How much does the soap alone weigh?
+
+  3. On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt. On July
+     15 there were just 11-1/2 lb. left. How much had he used in the two
+     weeks?
+
+  4. Grace promised to pick 30 qt. blueberries for her mother. So far
+     she has picked 18-1/2 qt. How many more quarts must she pick?
+
+B
+
+  This table of numbers tells                Weight of Mary Adams
+  what Nell's baby sister Mary            When born        7-3/8 lb.
+  weighed every two months from           2 months old    11-1/4 lb.
+  the time she was born till she          4 months old    14-1/8 lb.
+  was a year old.                         6 months old    15-3/4 lb.
+                                          8 months old    17-5/8 lb.
+                                         10 months old    19-1/2 lb.
+                                         12 months old    21-3/8 lb.
+
+  1. How much did the Adams baby gain in the first two months?
+  2. How much did the Adams baby gain in the second two months?
+  3. In the third two months?
+  4. In the fourth two months?
+  5. From the time it was 8 months old till it was 10 months old?
+  6. In the last two months?
+  7. From the time it was born till it was 6 months old?
+
+C
+
+  1. Helen's exact average for December was 87-1/3. Kate's was 84-1/2.
+     How much higher was Helen's than Kate's?
+
+      87-1/3   How do you think of 1/2 and 1/3?
+      84-1/2   How do you think of 1-2/6?
+      ------
+               How do you change the 4?
+
+  2. Find the exact average for each girl in the following list. Write
+     the answers clearly so that you can see them easily. You will use
+     them in solving problems 3, 4, 5, 6, 7, and 8.
+
+                 Alice  Dora  Emma  Grace  Louise  Mary  Nell  Rebecca
+      Reading      91    87    83     81     79     77    76     73
+      Language     88    78    82     79     73     78    73     75
+      Arithmetic   89    85    79     75     84     87    89     80
+      Spelling     90    79    75     80     82     91    68     81
+      Geography    91    87    83     75     78     85    73     79
+      Writing      90    88    75     72     93     92    95     78
+
+  3. Which girl had the highest average?
+
+  4. How much higher was her average than the next highest?
+
+  5. How much difference was there between the highest and the lowest
+     girl?
+
+  6. Was Emma's average higher or lower than Louise's? How much?
+
+  7. How much difference was there between Alice's average and Dora's?
+
+  8. How much difference was there between Mary's average and Nell's?
+
+  9. Write five other problems about these averages, and solve each of
+     them.
+
+_Samples of Undesirable Applications of Arithmetic_[1]
+
+  Will has XXI marbles, XII jackstones, XXXVI pieces of string. How many
+  things had he?
+
+  George's kite rose CDXXXV feet and Tom's went LXIII feet higher. How
+  high did Tom's kite rise?
+
+  If from DCIV we take CCIV the result will be a number IV times as
+  large as the number of dollars Mr. Dane paid for his horse. How much
+  did he pay for his horse?
+
+  Hannah has 5/8 of a dollar, Susie 7/25, Nellie 3/4, Norah 13/16. How
+  much money have they all together?
+
+  A man saves 3-17/80 dollars a week. How much does he save in a year?
+
+  A tree fell and was broken into 4 pieces, 13-1/6 feet, 10-3/7 feet,
+  8-1/2 feet, and 7-16/21 feet long. How tall was the tree?
+
+  Annie's father gave her 20 apples to divide among her friends. She
+  gave each one 2-2/9 apples apiece. How many playmates had she?
+
+  John had 17-2/5 apples. He divided his whole apples into fifths. How
+  many pieces had he in all?
+
+  A landlady has 3-3/7 pies to be divided among her 8 boarders. How much
+  will each boarder receive?
+
+  There are twenty columns of spelling words in Mary's lesson and 16
+  words in each column. How many words are in her lesson?
+
+  There are 9 nuts in a pint. How many pints in a pile of 5,888,673
+  nuts?
+
+  The Adams school contains eight rooms; each room contains 48 pupils;
+  if each pupil has eight cents, how much have they together?
+
+  A pile of wood in the form of a cube contains 15-1/2 cords. What are
+  the dimensions to the nearest inch?
+
+  A man 6 ft. high weighs 175 lb. How tall is his wife who is of similar
+  build, and weighs 125 lb.?
+
+  A stick of timber is in the shape of the frustum of a square pyramid,
+  the lower base being 22 in. square and the upper 14 in. square. How
+  many cubic feet in the log, if it is 22 ft. long?
+
+  Mr. Ames, being asked his age, replied: "If you cube one half of my
+  age and add 41,472 to the result, the sum will be one half the cube of
+  my age. How old am I?"
+
+  [1] The following and later problems are taken from actual textbooks
+      or courses of study or state examinations; to avoid invidious
+      comparisons, they are not exact quotations, but are equivalents
+      in principle and form, as stated in the preface.
+
+These samples, just given, of the kind of problem-solving that should
+not be emphasized in school training refer in some cases to books of
+forty years back, but the following represent the results of a
+collection made in 1910 from books then in excellent repute. It required
+only about an hour to collect them; and I am confident that a thousand
+such problems describing situations that the pupil will never encounter
+in real life, or putting questions that he will never be asked in real
+life, could easily be found in any ten textbooks of the decade from 1900
+to 1910.
+
+  If there are 250 kernels of corn on one ear, how many are there on 24
+  ears of corn the same size?
+
+  Maud is four times as old as her sister, who is 4 years old. What is
+  the sum of their ages?
+
+  If the first century began with the year 1, with what year does it
+  end?
+
+  Every spider has 8 compound eyes. How many eyes have 21 spiders?
+
+  A nail 4 inches long is driven through a board so that it projects
+  1.695 inches on one side and 1.428 on the other. How thick is the
+  board?
+
+  Find the perimeter of an envelope 5 in. by 3-1/4 in.
+
+  How many minutes in 5/9 of 9/4 of an hour?
+
+  Mrs. Knox is 3/4 as old as Mr. Knox, who is 48 years old. Their son
+  Edward is 4/9 as old as his mother. How old is Edward?
+
+  Suppose a pie to be exactly round and 10-1/2 miles in diameter. If it
+  were cut into 6 equal pieces, how long would the curved edge of each
+  piece be?
+
+  8-1/3% of a class of 36 boys were absent on a rainy day. 33-1/3% of
+  those present went out of the room to the school yard. How many were
+  left in the room?
+
+  Just after a ton of hay was weighed in market, a horse ate one pound
+  of it. What was the ratio of what he ate to what was left?
+
+  If a fan having 15 rays opens out so that the outer rays form a
+  straight line, how many degrees are there between any two adjacent
+  rays?
+
+  One half of the distance between St. Louis and New Orleans is 280
+  miles more than 1/10 of the distance; what is the distance between
+  these places?
+
+  If the pressure of the atmosphere is 14.7 lb. per square inch what is
+  the pressure on the top of a table 1-1/4 yd. long and 2/3 yd. wide?
+
+  13/28 of the total acreage of barley in 1900 was 100,000 acres; what
+  was the total acreage?
+
+  What is the least number of bananas that a mother can exactly divide
+  between her 2 sons, or among her 4 daughters, or among all her
+  children?
+
+  If Alice were two years older than four times her actual age she would
+  be as old as her aunt, who is 38 years old. How old is Alice?
+
+  Three men walk around a circular island, the circumference of which is
+  360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a
+  day. If they start together and walk in the same direction, how many
+  days will elapse before they will be together again?
+
+With only thirty or forty dollars a year to spend on a pupil's
+education, of which perhaps eight dollars are spent on improving his
+arithmetical abilities, the immediate guidance of his responses to real
+situations and personally initiated problems has to be supplemented
+largely by guidance of his responses to problems described in words,
+diagrams, pictures, and the like. Of these latter, words will be used
+most often. As a consequence the understanding of the words used in
+these descriptions becomes a part of the ability required in arithmetic.
+Such word knowledge is also required in so far as the problems to be
+solved in real life are at times described, as in advertisements,
+business letters, and the like.
+
+This is recognized by everybody in the case of words like _remainder_,
+_profit_, _loss_, _gain_, _interest_, _cubic capacity_, _gross_, _net_,
+and _discount_, but holds equally of _let_, _suppose_, _balance_,
+_average_, _total_, _borrowed_, _retained_, and many such semi-technical
+words, and may hold also of hundreds of other words unless the textbook
+and teacher are careful to use only words and sentence structures which
+daily life and the class work in English have made well known to the
+pupils. To apply arithmetic to a problem a pupil must understand what
+the problem is; problem-solving depends on problem-reading. In actual
+school practice training in problem-reading will be less and less
+necessary as we get rid of problems to be solved simply for the sake of
+solving them, unnecessarily unreal problems, and clumsy descriptions,
+but it will remain to some extent as an important joint task for the
+'arithmetic' and 'reading' of the elementary school.
+
+
+ARITHMETICAL REASONING
+
+The last respect in which the nature of arithmetical abilities requires
+definition concerns arithmetical reasoning. An adequate treatment of the
+reasoning that may be expected of pupils in the elementary school and of
+the most efficient ways to encourage and improve it cannot be given
+until we have studied the formation of habits. For reasoning is
+essentially the organization and control of habits of thought. Certain
+matters may, however, be decided here. The first concerns the use of
+computation and problems merely for discipline,--that is, the emphasis
+on training in reasoning regardless of whether the problem is otherwise
+worth reasoning about. It used to be thought that the mind was a set of
+faculties or abilities or powers which grew strong and competent by
+being exercised in a certain way, no matter on what they were exercised.
+Problems that could not occur in life, and were entirely devoid of any
+worthy interest, save the intellectual interest in solving them, were
+supposed to be nearly or quite as useful in training the mind to reason
+as the genuine problems of the home, shop, or trade. Anything that gave
+the mind a chance to reason would do; and pupils labored to find when
+the minute hand and hour hand would be together, or how many sheep a
+shepherd had if half of what he had plus ten was one third of twice what
+he had!
+
+We now know that the training depends largely on the particular data
+used, so that efficient discipline in reasoning requires that the pupil
+reason about matters of real importance. There is no magic essence or
+faculty of reasoning that works in general and irrespective of the
+particular facts and relations reasoned about. So we should try to find
+problems which not only stimulate the pupil to reason, but also direct
+his reasoning in useful channels and reward it by results that are of
+real significance. We should replace the purely disciplinary problems by
+problems that are also valuable as special training for important
+particular situations of life. Reasoning sought for reasoning's sake
+alone is too wasteful an expenditure of time and is also likely to be
+inferior as reasoning.
+
+The second matter concerns the relative merits of 'catch' problems,
+where the pupil has to go against some customary habit of thinking, and
+what we may call 'routine' problems, where the regular ways of thinking
+that have served him in the past will, except for some blunder, guide
+him rightly.
+
+Consider, for example, these four problems:
+
+  1. "A man bought ten dozen eggs for $2.50 and sold them for 30 cents
+     a dozen. How many cents did he lose?"
+
+  2. "I went into Smith's store at 9 A.M. and remained until 10 A.M.
+     I bought six yards of gingham at 40 cents a yard and three yards
+     of muslin at 20 cents a yard and gave a $5.00 bill. How long was
+     I in the store?"
+
+  3. "What must you divide 48 by to get half of twice 6?"
+
+  4. "What must you add to 19 to get 30?"
+
+The 'catch' problem is now in disrepute, the wise teacher feeling by a
+sort of intuition that to willfully require a pupil to reason to a
+result sharply contrary to that to which previous habits lead him is
+risky. The four illustrations just given show, however, that mere
+'catchiness' or 'contra-previous-habit-ness' in a problem is not enough
+to condemn it. The fourth problem is a catch problem, but so useful a
+one that it has been adopted in many modern books as a routine drill!
+The first problem, on the contrary, all, save those who demand no higher
+criterion for a problem than that it make the pupil 'think,' would
+reject. It demands the reversal of fixed habits _to no valid purpose_;
+for in life the question in such case would never (or almost never) be
+'How many cents did he lose?' but 'What was the result?' or simply 'What
+of it?' This problem weakens without excuse the child's confidence in
+the training he has had. Problems like (2) are given by teachers of
+excellent reputation, but probably do more harm than good. If a pupil
+should interrupt his teacher during the recitation in arithmetic by
+saying, "I got up at 7 o'clock to multiply 9 by 2-3/4 and got 24-3/4 for
+my answer; was that the right time to get up?" the teacher would not
+thank fortune for the stimulus to thought but would think the child a
+fool. Such catch questions may be fairly useful as an object lesson on
+the value of search for the essential element in a situation if a great
+variety of them are given one after another with routine problems
+intermixed and with warning of the general nature of the exercise at the
+beginning. Even so, it should be remembered that reasoning should be
+chiefly a force organizing habits, not opposing them; and also that
+there are enough bad habits to be opposed to give all necessary
+training. Fabricated puzzle situations wherein a peculiar hidden element
+of the situation makes the good habits called up by the situation
+misleading are useful therefore rather as a relief and amusing variation
+in arithmetical work than as stimuli to thought.
+
+Problems like the third quoted above we might call puzzling rather than
+'catch' problems. They have value as drills in analysis of a situation
+into its elements that will amuse the gifted children, and as tests of
+certain abilities. They also require that of many confusing habits, the
+right one be chosen, rather than that ordinary habits be set aside by
+some hidden element in the situation. Not enough is known about their
+effect to enable us to decide whether or not the elementary school
+should include special facility with them as one of the arithmetical
+functions that it specially trains.
+
+The fourth 'catch' quoted above, which all would admit is a good
+problem, is good because it opposes a good habit for the sake of another
+good habit, forces the analysis of an element whose analysis life very
+much requires, and does it with no obvious waste. It is not safe to
+leave a child with the one habit of responding to 'add, 19, 30' by 49,
+for in life the 'have 19, must get .... to have 30' situation is very
+frequent and important.
+
+On the whole, the ordinary problems which ordinary life proffers seem to
+be the sort that should be reasoned out, though the elementary school
+may include the less noxious forms of pure mental gymnastics for those
+pupils who like them.
+
+
+SUMMARY
+
+These discussions of the meanings of numbers, the linguistic demands of
+arithmetic, the distinction between scholastic and real applications of
+arithmetic, and the possible restrictions of training in reasoning,--may
+serve as illustrations of the significance of the question, "What are
+the functions that the elementary school tries to improve in its
+teaching of arithmetic?" Other matters might well be considered in this
+connection, but the main outline of the work of the elementary school is
+now fairly clear. The arithmetical functions or abilities which it seeks
+to improve are, we may say:--
+
+(1) Working knowledge of the meanings of numbers as names for certain
+sized collections, for certain relative magnitudes, the magnitude of
+unity being known, and for certain centers or nuclei of relations to
+other numbers.
+
+(2) Working knowledge of the system of decimal notation.
+
+(3) Working knowledge of the meanings of addition, subtraction,
+multiplication, and division.
+
+(4) Working knowledge of the nature and relations of certain common
+measures.
+
+(5) Working ability to add, subtract, multiply, and divide with
+integers, common and decimal fractions, and denominate numbers, all
+being real positive numbers.
+
+(6) Working knowledge of words, symbols, diagrams, and the like as
+required by life's simpler arithmetical demands or by economical
+preparation therefor.
+
+(7) The ability to apply all the above as required by life's simpler
+arithmetical demands or by economical preparation therefor, including
+(7 _a_) certain specific abilities to solve problems concerning areas
+of rectangles, volumes of rectangular solids, percents, interest, and
+certain other common occurrences in household, factory, and business
+life.
+
+
+THE SOCIOLOGY OF ARITHMETIC
+
+  The phrase 'life's simpler arithmetical demands' is necessarily left
+  vague. Just what use is being made of arithmetic in this country in
+  1920 by each person therein, we know only very roughly. What may be
+  called a 'sociology' of arithmetic is very much needed to investigate
+  this matter. For rare or difficult demands the elementary school
+  should not prepare; there are too many other desirable abilities that
+  it should improve.
+
+  A most interesting beginning at such an inventory of the actual uses
+  of arithmetic has been made by Wilson ['19] and Mitchell.[2] Although
+  their studies need to be much extended and checked by other methods of
+  inquiry, two main facts seem fairly certain.
+
+  First, the great majority of people in the great majority of their
+  doings use only very elementary arithmetical processes. In 1737 cases
+  of addition reported by Wilson, seven eighths were of five numbers or
+  less. Over half of the multipliers reported were one-figure numbers.
+  Over 95 per cent of the fractions operated with were included in this
+  list: 1/2 1/4 3/4 1/3 2/3 1/8 3/8 1/5 2/5 4/5. Three fourths of all
+  the cases reported were simple one-step computations with integers or
+  United States money.
+
+  Second, they often use these very elementary processes, not because
+  such are the quickest and most convenient, but because they have lost,
+  or maybe never had, mastery of the more advanced processes which would
+  do the work better. The 5 and 10 cent stores, the counter with
+  "Anything on this counter for 25c," and the arrangements for payments
+  on the installment plan are familiar instances of human avoidance of
+  arithmetic. Wilson found very slight use of decimals; and Mitchell
+  found men computing with 49ths as common fractions when the use of
+  decimals would have been more efficient. If given 120 seconds to
+  do a test like that shown below, leading lawyers, physicians,
+  manufacturers, and business men and their wives will, according to my
+  experience, get only about half the work right. Many women, finding on
+  their meat bill "7-3/8 lb. roast beef $2.36," will spend time and
+  money to telephone the butcher asking how much roast beef was per
+  pound, because they have no sure power in dividing by a mixed number.
+
+  [2] The work of Mitchell has not been published, but the author has
+      had the privilege of examining it.
+
+  Test
+
+  Perform the operations indicated. Express all fractions in answers in
+  lowest terms.
+
+      _Add:_
+
+      3/4 + 1/6 + .25       4 yr. 6 mo.
+                            1 yr. 2 mo.
+                            6 yr. 9 mo.
+                            3 yr. 6 mo.
+                            4 yr. 5 mo.
+                            -----------
+
+      _Subtract:_
+
+      8.6 - 6.05007     7/8 - 2/3 =    5-7/16 - 2-3/16 =
+
+
+      _Multiply:_
+
+      29 ft. 6 in.     7 x 8 x 4-1/2 =
+             8
+      ------------
+
+    _Divide:_
+
+    4-1/2 / 7 =
+
+  It seems probable that the school training in arithmetic of the past
+  has not given enough attention to perfecting the more elementary
+  abilities. And we shall later find further evidence of this. On the
+  other hand, the fact that people in general do not at present use a
+  process may not mean that they ought not to use it.
+
+  Life's simpler arithmetical demands certainly do not include matters
+  like the rules for finding cube root or true discount, which no
+  sensible person uses. They should not include matters like computing
+  the lateral surface or volume of pyramids and cones, or knowing the
+  customs of plasterers and paper hangers, which are used only by highly
+  specialized trades. They should not include matters like interest on
+  call loans, usury, exact interest, and the rediscounting of notes,
+  which concern only brokers, bank clerks, and rich men. They should not
+  include the technique of customs which are vanishing from efficient
+  practice, such as simple interest on amount for times longer than a
+  year, days of grace, or extremes and means in proportions. They should
+  not include any elaborate practice with very large numbers, or
+  decimals beyond thousandths, or the addition and subtraction of
+  fractions which not one person in a hundred has to add or subtract
+  oftener than once a year.
+
+  When we have an adequate sociology of arithmetic, stating accurately
+  just who should use each arithmetical ability and how often, we shall
+  be able to define the task of the elementary school in this respect.
+  For the present, we may proceed by common sense, guided by two
+  limiting rules. The first is,--"It is no more desirable for the
+  elementary school to teach all the facts of arithmetic than to teach
+  all the words in the English language, or all the topography of the
+  globe, or all the details of human physiology." The second is,--"It is
+  not desirable to eliminate any element of arithmetical training until
+  you have something better to put in its place."
+
+
+
+
+CHAPTER II
+
+THE MEASUREMENT OF ARITHMETICAL ABILITIES
+
+
+One of the best ways to clear up notions of what the functions are which
+schools should develop and improve is to get measures of them. If any
+given knowledge or skill or power or ideal exists, it exists in some
+amount. A series of amounts of it, varying from less to more, defines
+the ability itself in a way that no general verbal description can do.
+Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to
+tell us what we mean by weight. By finding a series of words like
+_only_, _smoke_, _another_, _pretty_, _answer_, _tailor_, _circus_,
+_telephone_, _saucy_, and _beginning_, which are spelled correctly by
+known and decreasing percentages of children of the same age, or of the
+same school grade, we know better what we mean by 'spelling-difficulty.'
+Indeed, until we can measure the efficiency and improvement of a
+function, we are likely to be vague and loose in our ideas of what the
+function is.
+
+
+A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: THE ABILITY TO ADD
+INTEGERS
+
+Consider first, as a sample, the measurement of ability to add integers.
+
+The following were the examples used in the measurements made by Stone
+['08]:
+
+           596     4695
+           428      872
+  2375      94     7948
+  4052      75     6786
+  6354     304      567
+   260     645      858
+  5041     984     9447
+  1543     897     7499
+  ----     ---     ----
+
+The scoring was as follows: Credit of 1 for each column added correctly.
+Stone combined measures of other abilities with this in a total score
+for amount done correctly in 12 minutes. Stone also scored the
+correctness of the additions in certain work in multiplication.
+
+Courtis uses a sheet of twenty-four tasks or 'examples,' each consisting
+of the addition of nine three-place numbers as shown below. Eight
+minutes is allowed. He scores the amount done by the number of examples,
+and also scores the number of examples done correctly, but does not
+suggest any combination of these two into a general-efficiency score.
+
+  927
+  379
+  756
+  837
+  924
+  110
+  854
+  965
+  344
+  ---
+
+The author long ago proposed that pupils be measured also with series
+like _a_ to _g_ shown below, in which the difficulty increases step by
+step.
+
+  _a._           3  2  2  3  2  2  1  2
+                 2  3  1  2  4  5  5  1
+                 4  2  3  3  3  2  2  2
+                 -  -  -  -  -  -  -  -
+
+  _b._    21    32    12    24    34    34    22    12
+          23    12    52    31    33    12    23    13
+          24    25    15    14    32    23    43    61
+          --    --    --    --    --    --    --    --
+
+  _c._    22     3     4    35    32    83    22     3
+           3    31     3     2    33    11     3    21
+          38    45    52    52     2     4    33    64
+          --    --    --    --    --    --    --    --
+
+  _d._    30    20    10    22    10    20    52    12
+          20    50    40    43    30     4     6    22
+          40    17    24    13    40    23    30    44
+          --    --    --    --    --    --    --    --
+
+  _e._           4     5    20    12    12    20    10
+          20    30     3    40     4    11    20    20
+          10    30    20     4     1    23     7     2
+          20     2    40    23    40    11    10    30
+          20    20    10    11    20    22    30    25
+          --    --    --    --    --    --    --    --
+
+  _f._                19     9                 9
+          14     2    19    24     9     4    13
+           9    14    13    12    13    13     9    14
+          17    23    13    15    15    34    12    25
+          26    29    18    19    25    28    18    39
+          --    --    --    --    --    --    --    --
+
+  _g._                                  13
+                13           9    14    12     9
+                 9          13    12     9    14    24
+          23    19    19    29     9     9    13    21
+          28    26    26    14     8     8    29    23
+          29    16    15    19    17    19    19    22
+          --    --    --    --    --    --    --    --
+
+Woody ['16] has constructed his well-known tests on this principle,
+though he uses only one example at each step of difficulty instead of
+eight or ten as suggested above. His test, so far as addition of
+integers goes, is:--
+
+SERIES A. ADDITION SCALE (in part)
+
+By Clifford Woody
+
+  (1)   (2)    (3)    (4)   (5)   (6)   (7)       (8)           (9)
+   2     2     17     53    72    60   3 + 1 =   2 + 5 + 1 =    20
+   3     4      2     45    26    37                            10
+  --     3     --     --    --    --                             2
+        --                                                      30
+                                                                25
+                                                                --
+
+  (10) (11)   (12)   (13)    (14)        (15)   (16)  (17)    (18)
+   21   32     43     23    25 + 42 =    100      9   199     2563
+   33   59      1     25                  33     24   194     1387
+   35   17      2     16                  45     12   295     4954
+   --   --     13     --                 201     15   156     2065
+               --                         46     19   ---     ----
+                                         ---     --
+
+            (19)        (20)            (21)       (22)
+           $ .75       $12.50          $8.00        547
+            1.25        16.75           5.75        197
+             .49        15.75           2.33        685
+           -----       ------           4.16        678
+                                         .94        456
+                                        6.32        393
+                                       -----        525
+                                                    240
+                                                    152
+                                                    ---
+
+In his original report, Woody gives no scheme for scoring an individual,
+wisely assuming that, with so few samples at each degree of difficulty,
+a pupil's score would be too unreliable for individual diagnosis. The
+test is reliable for a class; and for a class Woody used the degree of
+difficulty such that a stated fraction of the class can do the work
+correctly, if twenty minutes is allowed for the thirty-eight examples of
+the entire test.
+
+The measurement of even so simple a matter as the efficiency of a
+pupil's responses to these tests in adding integers is really rather
+complex. There is first of all the problem of combining speed and
+accuracy into some single estimate. Stone gives no credit for a column
+unless it is correctly added. Courtis evades the difficulty by reporting
+both number done and number correct. The author's scheme, which gives
+specified weights to speed and accuracy at each step of the series,
+involves a rather intricate computation.
+
+This difficulty of equating speed and accuracy in adding means precisely
+that we have inadequate notions of what the ability is that the
+elementary school should improve. Until, for example, we have decided
+whether, for a given group of pupils, fifteen Courtis attempts with ten
+right, is or is not a better achievement than ten Courtis attempts with
+nine right, we have not decided just what the business of the teacher of
+addition is, in the case of that group of pupils.
+
+There is also the difficulty of comparing results when short and long
+columns are used. Correctness with a short column, say of five figures,
+testifies to knowledge of the process and to the power to do four
+successive single additions without error. Correctness with a long
+column, say of ten digits, testifies to knowledge of the process and to
+the power to do nine successive single additions without error. Now if a
+pupil's precision was such that on the average he made one mistake in
+eight single additions, he would get about half of his five-digit
+columns right and almost none of his ten-digit columns right. (He would
+do this, that is, if he added in the customary way. If he were taught to
+check results by repeated addition, by adding in half-columns and the
+like, his percentages of accurate answers might be greatly increased in
+both cases and be made approximately equal.) Length of column in a test
+of addition under ordinary conditions thus automatically overweights
+precision in the single additions as compared with knowledge of the
+process, and ability at carrying.
+
+Further, in the case of a column of whatever size, the result as
+ordinarily scored does not distinguish between one, two, three, or more
+(up to the limit) errors in the single additions. Yet, obviously, a
+pupil who, adding with ten-digit columns, has half of his answer-figures
+wrong, probably often makes two or more errors within a column, whereas
+a pupil who has only one column-answer in ten wrong, probably almost
+never makes more than one error within a column. A short-column test is
+then advisable as a means of interpreting the results of a long-column
+test.
+
+Finally, the choice of a short-column or of a long-column test is
+indicative of the measurer's notion of the kind of efficiency the world
+properly demands of the school. Twenty years ago the author would have
+been readier to accept a long-column test than he now is. In the world
+at large, long-column addition is being more and more done by machine,
+though it persists still in great frequency in the bookkeeping of weekly
+and monthly accounts in local groceries, butcher shops, and the like.
+
+The search for a measure of ability to add thus puts the problem of
+speed _versus_ precision, and of short-column _versus_ long-column
+additions clearly before us. The latter problem has hardly been
+realized at all by the ordinary definitions of ability to add.
+
+It may be said further that the measurement of ability to add gives the
+scientific student a shock by the lack of precision found everywhere in
+schools. Of what value is it to a graduate of the elementary school to
+be able to add with examples like those of the Courtis test, getting
+only eight out of ten right? Nobody would pay a computer for that
+ability. The pupil could not keep his own accounts with it. The supposed
+disciplinary value of habits of precision runs the risk of turning
+negative in such a case. It appears, at least to the author, imperative
+that checking should be taught and required until a pupil can add single
+columns of ten digits with not over one wrong answer in twenty columns.
+Speed is useful, especially indirectly as an indication of control of
+the separate higher-decade additions, but the social demand for addition
+below a certain standard of precision is _nil_, and its disciplinary
+value is _nil_ or negative. This will be made a matter of further study
+later.
+
+
+MEASUREMENTS OF ABILITIES IN COMPUTATION
+
+Measurements of these abilities may be of two sorts--(1) of the speed
+and accuracy shown in doing one same sort of task, as illustrated by the
+Courtis test for addition shown on page 28; and (2) of how hard a task
+can be done perfectly (or with some specified precision) within a
+certain assigned time or less, as illustrated by the author's rough test
+for addition shown on pages 28 and 29, and by the Woody tests, when
+extended to include alternative forms.
+
+The Courtis tests, originated as an improvement on the Stone tests and
+since elaborated by the persistent devotion of their author, are a
+standard instrument of the first sort for measuring the so-called
+'fundamental' arithmetical abilities with integers. They are shown on
+this and the following page.
+
+Tests of the second sort are the Woody tests, which include operations
+with integers, common and decimal fractions, and denominate numbers, the
+Ballou test for common fractions ['16], and the "Ladder" exercises of
+the Thorndike arithmetics. Some of these are shown on pages 36 to 41.
+
+
+Courtis Test
+
+Arithmetic. Test No. 1. Addition
+
+Series B
+
+  You will be given eight minutes to find the answers to as many
+  of these addition examples as possible. Write the answers on this
+  paper directly underneath the examples. You are not expected
+  to be able to do them all. You will be marked for both speed and
+  accuracy, but it is more important to have your answers right than
+  to try a great many examples.
+
+    927    297    136    486    384    176    277    837
+    379    925    340    765    477    783    445    882
+    756    473    988    524    881    697    682    959
+    837    983    386    140    266    200    594    603
+    924    315    353    812    679    366    481    118
+    110    661    904    466    241    851    778    781
+    854    794    547    355    796    535    849    756
+    965    177    192    834    850    323    157    222
+    344    124    439    567    733    229    953    525
+    ---    ---    ---    ---    ---    ---    ---    ---
+
+  and sixteen more addition examples of nine three-place numbers.
+
+
+Courtis Test
+
+Arithmetic. Test No. 2. Subtraction
+
+Series B
+
+  You will be given four minutes to find the answers to as many
+  of these subtraction examples as possible. Write the answers
+  on this paper directly underneath the examples. You are not
+  expected to be able to do them all. You will be marked for both
+  speed and accuracy, but it is more important to have your answers
+  right than to try a great many examples.
+
+    107795491      75088824      91500053      87939983
+     77197029      57406394      19901563      72207316
+    ---------      --------      --------      --------
+
+  and twenty more tasks of the same sort.
+
+
+Courtis Test
+
+Arithmetic. Test No. 3. Multiplication
+
+Series B
+
+  You will be given six minutes to work as many of these multiplication
+  examples as possible. You are not expected to be able to do them all.
+  Do your work directly on this paper; use no other. You will be marked
+  for both speed and accuracy, but it is more important to get correct
+  answers than to try a large number of examples.
+
+    8246      7843      4837      3478      6482
+      29       702        83        15        46
+    ----      ----      ----      ----      ----
+
+  and twenty more multiplication examples of the same sort.
+
+
+Courtis Test
+
+Arithmetic. Test No. 4. Division
+
+Series B
+
+  You will be given eight minutes to work as many of these division
+  examples as possible. You are not expected to be able to do them all.
+  Do your work directly on this paper; use no other. You will be marked
+  for both speed and accuracy, but it is more important to get correct
+  answers than to try a large number of examples.
+      _____        ______        _____        ______
+    25)6775      94)85352      37)9990      86)80066
+
+  and twenty more division examples of the same sort.
+
+
+SERIES B. MULTIPLICATION SCALE
+
+By Clifford Woody
+
+   (1)               (3)               (4)               (5)
+   3 x 7 =           2 x 3 =           4 x 8 =            23
+                                                           3
+                                                          --
+
+   (8)               (9)              (11)              (12)
+    50               254              1036              5096
+     3                 6                 8                 6
+    --               ---              ----              ----
+
+  (13)              (16)              (18)              (20)
+  8754              7898                24               287
+     8                 9               234               .05
+  ----              ----               ---               ---
+
+  (24)              (26)              (27)              (29)
+    16              9742              6.25             1/8 x 2 =
+     2-5/8            59               3.2
+    ---             ----              ----
+
+  (33)                (35)              (37)               (38)
+  2-1/2 x 3-1/2 =     987-3/4    2-1/4 x 4-1/2 x 1-1/2 =   .0963-1/8
+                       25                                  .084
+                     ----                                  -----
+
+
+SERIES B. DIVISION  SCALE
+
+By Clifford Woody
+
+  (1)               (2)              (7)               (8)
+   __               ___                                 ___
+  3)6              9)27             4 / 2 =            9)0
+
+  (11)              (14)              (15)              (17)
+   ___              _____
+  2)13             8)5856          1/4 of 128 =        50 / 7 =
+
+  (19)              (23)              (27)              (28)
+                    ____                                 ______
+  248 / 7 =       23)469          7/8 of 624 =       .003).0936
+
+      (30)                (34)                (36)
+                                            ______________
+   3/4 / 5 =         62.50 / 1-1/4 =       9)69 lbs. 9 oz.
+
+
+Ballou Test
+
+Addition of Fractions
+
+            _Test 1_                                _Test 2_
+  (1)  1/4            (2) 3/14             (1) 1/3            (2) 2/7
+       1/4                1/14                 1/6                3/14
+       ---                ----                 ---                ----
+
+
+           _Test 3_                                 _Test 4_
+  (1)  3/5            (2) 5/6              (1) 1/7            (2) 7/9
+      11/15               1/2                  9/10               1/4
+      -----               ---                  ----               ---
+
+
+           _Test 5_                                 _Test 6_
+  (1)  1/10           (2) 4/9              (1) 1/6            (2) 5/6
+       1/6                5/12                 9/10               3/8
+       ----               ----                 ----               ---
+
+
+An Addition Ladder [Thorndike, '17, III, 5]
+
+Begin at the bottom of the ladder. See if you can climb to the top
+without making a mistake. Be sure to copy the numbers correctly.
+
+  #Step 6.#
+    _a._ Add 1-1/3 yd., 7/8 yd., 1-1/4 yd., 3/4 yd., 7/8 yd.,
+          and 1-1/2 yd.
+    _b._ Add 62-1/2c, 66-2/3c, 56-1/4c, 60c, and 62-1/2c.
+    _c._ Add 1-5/16, 1-9/32, 1-3/8, 1-11/32, and 1-7/16.
+    _d._ Add 1-1/3 yd., 1-1/4 yd., 1-1/2 yd., 2 yd., 3/4 yd.,
+          and 2/3 yd.
+
+  #Step 5.#
+    _a._ Add 4 ft. 6-1/2 in., 53-1/4 in., 5 ft. 1/2 in., 56-3/4 in.,
+          and 5 ft.
+    _b._ Add 7 lb., 6 lb. 11 oz., 7-1/2 lb., 6 lb. 4-1/2 oz.,
+          and 8-1/2 lb.
+    _c._ Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min.,
+          and 55 min.
+    _d._ Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes,
+          and 19 nickels.
+
+  #Step 4.#
+    _a._ Add .05-1/2, .06, .04-3/4, .02-3/4, and .05-1/4.
+    _b._ Add .33-1/3, .12-1/2, .18, .16-2/3, .08-1/3 and .15.
+    _c._ Add .08-1/3, .06-1/4, .21, .03-3/4, and .16-2/3.
+    _d._ Add .62, .64-1/2, .66-2/3, .10-1/4, and .68.
+
+  #Step 3.#
+    _a._ Add 7-1/4, 6-1/2, 8-3/8, 5-3/4, 9-5/8 and 3-7/8.
+    _b._ Add 4-5/8, 12, 7-1/2, 8-3/4, 6 and 5-1/4.
+    _c._ Add 9-3/4, 5-7/8, 4-1/8, 6-1/2, 7, 3-5/8.
+    _d._ Add 12, 8-1/2, 7-1/3, 5, 6-2/3, and 9-1/2.
+
+  #Step 2.#
+    _a._ Add 12.04, .96, 4.7, 9.625, 3.25, and 20.
+    _b._ Add .58, 6.03, .079, 4.206, 2.75, and 10.4.
+    _c._ Add 52, 29.8, 41.07, 1.913, 2.6, and 110.
+    _d._ Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05.
+
+  #Step 1.#
+    _a._ Add 10-3/5, 11-1/5, 10-4/5, 11, 11-2/5, 10-3/5, and 11.
+    _b._ Add 7-3/8, 6-5/8, 8, 9-1/8, 7-7/8, 5-3/8, and 8-1/8.
+    _c._ Add 21-1/2, 18-3/4, 31-1/2, 19-1/4, 17-1/4, 22, and 16-1/2.
+    _d._ Add 14-5/12, 12-7/12, 9-11/12, 6-1/12, and 5.
+
+
+A Subtraction Ladder [Thorndike, '17, III, 11]
+
+  #Step 9.#
+    _a._  2.16 mi. - 1-3/4 mi.
+    _b._  5.72 ft. - 5 ft. 3 in.
+    _c._  2 min. 10-1/2 sec. - 93.4 sec.
+    _d._  30.28 A. - 10-1/5 A.
+    _e._  10 gal. 2-1/2 qt. - 4.623 gal.
+
+  #Step 8.#
+     _a_             _b_            _c_            _d_            _e_
+    25-7/12         10-1/4          9-5/16         5-7/16         4-2/3
+    12-3/4           7-1/3          6-3/8          2-3/4          1-3/4
+    -------         ------          ------         ------         -----
+
+  #Step 7.#
+     _a_             _b_            _c_            _d_            _e_
+    28-3/4          40-1/2         10-1/4         24-1/3         37-1/2
+    16-1/8          14-3/8          6-1/2         11-1/2         14-3/4
+    ------          ------         ------         ------         ------
+
+  #Step 6.#
+     _a_             _b_            _c_            _d_            _e_
+    10-1/3           7-1/4         15-1/8         12-1/5          4-1/16
+     4-2/3           2-3/4          6-3/8         11-4/5          2-7/16
+    ------           -----         ------         ------          ------
+
+  #Step 5.#
+     _a_             _b_            _c_            _d_            _e_
+    58-4/5          66-2/3         28-7/8         62-1/2          9-7/12
+    52-1/5          33-1/3          7-5/8         37-1/2          4-5/12
+    ------          ------         ------         ------          ------
+
+  #Step 4.#
+    _a._  4 hr. - 2 hr. 17 min.
+    _b._  4 lb. 7 oz. - 2 lb. 11 oz.
+    _c._  1 lb. 5 oz. - 13 oz.
+    _d._  7 ft. - 2 ft. 8 in.
+    _e._  1 bu. - 1 pk.
+
+  #Step 3.#
+      _a_             _b_         _c_        _d_            _e_
+    92    mi.       6735 mi.    $3 - 89c    28.4  mi.     $508.40
+    84.15 mi.       6689 mi.                18.04 mi.      208.62
+    ---------       --------    --------    ---------     -------
+
+  #Step 2.#
+      _a_        _b_        _c_        _d_            _e_
+    $25.00     $100.00    $750.00   6124 sq. mi.   7846 sq. mi.
+      9.36       71.28     736.50   2494 sq. mi.   2789 sq. mi.
+    ------     -------    -------   ------------   ------------
+
+  #Step 1.#
+      _a_        _b_         _c_        _d_         _e_
+    $18.64     $25.39      $56.70     819.4 mi.    67.55 mi.
+      7.40      13.37       45.60     209.2 mi.    36.14 mi.
+    ------     ------      ------     ---------    ---------
+
+
+An Average Ladder [Thorndike, '17, III, 132]
+
+Find the average of the quantities on each line. Begin with #Step 1#.
+Climb to the top without making a mistake. Be sure to copy the numbers
+correctly. Extend the division to two decimal places if necessary.
+
+  #Step 6.#
+    _a_. 2-2/3, 1-7/8, 2-3/4, 4-1/4, 3-5/8, 3-1/2
+    _b_. 62-1/2c, 66-2/3c, 40c, 83-1/3c, $1.75, $2.25
+    _c_. 3-11/16, 3-9/32, 3-3/8, 3-17/32, 3-7/16
+    _d_. .17, 19, .16-2/3, .15-1/2, .23-1/4, .18
+
+  #Step 5.#
+    _a_. 5 ft. 3-1/2 in., 61-1/4 in., 58-3/4 in., 4 ft. 11 in.
+    _b_. 6 lb. 9 oz., 6 lb. 11 oz., 7-1/4 lb., 7-3/8 lb.
+    _c_. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1-1/4 hr.
+    _d_. 2.8 miles, 3-1/2 miles, 2.72 miles
+
+  #Step 4.#
+    _a._ .03-1/2, .06, .04-3/4, .05-1/2, .05-1/4
+    _b._ .043, .045, .049, .047, .046, .045
+    _c._ 2.20, .87-1/2, 1.18, .93-3/4, 1.2925, .80
+    _d._ .14-1/2, .12-1/2, .33-1/3, .16-2/3, .15, .17
+
+  #Step 3.#
+    _a._ 5-1/4, 4-1/2, 8-3/8, 7-3/4, 6-5/8, 9-3/8
+    _b._ 9-5/8, 12, 8-1/2, 8-3/4, 6, 5-1/4, 9
+    _c._ 9-3/8, 5-3/4, 4-1/8, 7-1/2, 6
+    _d._ 11, 9-1/2, 10-1/3, 13, 16-2/3, 9-1/2
+
+  #Step 2.#
+    _a._ 13.05, .97, 4.8, 10.625, 3.37
+    _b._ 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4
+    _c._ 68, 71.4, 59.8, 112, 96.1, 79.8
+    _d._ 2.079, 3.908, 4.165, 2.74
+
+  #Step 1.#
+    _a._ 4, 9-1/2, 6, 5, 7-1/2, 8, 10, 9
+    _b._ 6, 5, 3.9, 7.1, 8
+    _c._ 1086, 1141, 1059, 1302, 1284
+    _d._ $100.82, $206.49, $317.25, $244.73
+
+As such tests are widened to cover the whole task of the elementary
+school in respect to arithmetic, and accepted by competent authorities
+as adequate measures of achievement in computing, they will give, as has
+been said, a working definition of the task. The reader will observe,
+for example, that work such as the following, though still found in many
+textbooks and classrooms, does not, in general, appear in the modern
+tests and scales.
+
+Reduce the following improper fractions to mixed numbers:--
+
+          19/13    43/21   176/25   198/14
+
+Reduce to integral or mixed numbers:--
+
+          61381/37   2134/67   413/413   697/225
+
+Simplify:--
+
+          3/4 of 8/9 of 3/5 of 15/22
+
+Reduce to lowest terms:--
+
+  357/527  264/312  492/779  418/874  854/1769  30/735  44/242
+
+  77/847 18/243  96/224
+
+Find differences:--
+
+     6-2/7         8-5/11        8-4/13        5-1/4          7-1/8
+     3-1/14        5-1/7         3-7/13        2-11/14        2-1/7
+     ------        ------        ------        -------        ------
+
+Square:--
+
+  2/3  4/5  5/7  6/9  10/11  12/13  2/7  15/16  19/20  17/18
+
+  25/30  41/53
+
+Multiply:--
+
+     2/11 x 33    32 x 3/14    39 x 2/13    60 x 11/28    77 x 4/11
+
+     63 x 2/27    54 x 8/45    65 x 3/13     344-16/21   432-2/7
+
+
+MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE SOLUTION OF PROBLEMS
+
+Stone ['08] measured achievement with the following problems, fifteen
+minutes being the time allowed.
+
+"Solve as many of the following problems as you have time for; work them
+in order as numbered:
+
+  1. If you buy 2 tablets at 7 cents each and a book for 65 cents, how
+     much change should you receive from a two-dollar bill?
+
+  2. John sold 4 Saturday Evening Posts at 5 cents each. He kept 1/2
+     the money and with the other 1/2 he bought Sunday papers at 2 cents
+     each. How many did he buy?
+
+  3. If James had 4 times as much money as George, he would have $16.
+     How much money has George?
+
+  4. How many pencils can you buy for 50 cents at the rate of 2 for 5
+     cents?
+
+  5. The uniforms for a baseball nine cost $2.50 each. The shoes cost
+     $2 a pair. What was the total cost of uniforms and shoes for the
+     nine?
+
+  6. In the schools of a certain city there are 2200 pupils; 1/2 are
+     in the primary grades, 1/4 in the grammar grades, 1/8 in the high
+     school, and the rest in the night school. How many pupils are there
+     in the night school?
+
+  7. If 3-1/2 tons of coal cost $21, what will 5-1/2 tons cost?
+
+  8. A news dealer bought some magazines for $1. He sold them for
+     $1.20, gaining 5 cents on each magazine. How many magazines were
+     there?
+
+  9. A girl spent 1/8 of her money for car fare, and three times as
+     much for clothes. Half of what she had left was 80 cents. How much
+     money did she have at first?
+
+ 10. Two girls receive $2.10 for making buttonholes. One makes 42,
+     the other 28. How shall they divide the money?
+
+ 11. Mr. Brown paid one third of the cost of a building; Mr. Johnson
+     paid 1/2 the cost. Mr. Johnson received $500 more annual rent than
+     Mr. Brown. How much did each receive?
+
+ 12. A freight train left Albany for New York at 6 o'clock. An
+     express left on the same track at 8 o'clock. It went at the rate of
+     40 miles an hour. At what time of day will it overtake the freight
+     train if the freight train stops after it has gone 56 miles?"
+
+The criteria he had in mind in selecting the problems were as follows:--
+
+"The main purpose of the reasoning test is the determination of the
+ability of VI A children to reason in arithmetic. To this end, the
+problems, as selected and arranged, are meant to embody the following
+conditions:--
+
+  1. Situations equally concrete to all VI A children.
+
+  2. Graduated difficulties.
+      _a._ As to arithmetical thinking.
+      _b._ As to familiarity with the situation presented.
+
+  3. The omission of
+      _a._ Large numbers.
+      _b._ Particular memory requirements.
+      _c._ Catch problems.
+      _d._ All subject matter except whole numbers, fractions, and
+             United States money.
+
+The test is purposely so long that only very rarely did any pupil fully
+complete it in the fifteen minute limit."
+
+Credits were given of 1, for each of the first five problems, 1.4, 1.2,
+and 1.6 respectively for problems 6, 7, and 8, and of 2 for each of the
+others.
+
+Courtis sought to improve the Stone test of problem-solving, replacing
+it by the two tests reproduced below.
+
+
+ARITHMETIC--Test No. 6. Speed Test--Reasoning
+
+#Do not work# the following examples. Read each example through, make
+up your mind what operation you would use if you were going to work it,
+then write the name of the operation selected in the blank space after
+the example. Use the following abbreviations:--"Add." for addition,
+"Sub." for subtraction, "Mul." for multiplication, and "Div." for
+division.
+
+                                                      +-----------+----+
+                                                      | OPERATION |    |
+                                                      |-----------+----|
+  1. A girl brought a collection of 37 colored postal |           |    |
+  cards to school one day, and gave away 19 cards to  |           |    |
+  her friends. How many cards did she have left to    |           |    |
+  take home?                                          |           |    |
+                                                      |-----------+----|
+  2. Five boys played marbles. When the game was      |           |    |
+  over, each boy had the same number of marbles. If   |           |    |
+  there were 45 marbles altogether, how many did each |           |    |
+  boy have?                                           |           |    |
+                                                      |-----------+----|
+  3. A girl, watching from a window, saw 27           |           |    |
+  automobiles pass the school the first hour, and     |           |    |
+  33 the second. How many autos passed by the         |           |    |
+  school in the two hours?                            |           |    |
+                                                      |-----------+----|
+  4. In a certain school there were eight rooms and   |           |    |
+  each room had seats for 50 children. When all the   |           |    |
+  places were taken, how many children were there in  |           |    |
+  the school?                                         |           |    |
+                                                      |-----------+----|
+  5. A club of boys sent their treasurer to buy       |           |    |
+  baseballs. They gave him $3.15 to spend. How many   |           |    |
+  balls did they expect him to buy, if the balls cost |           |    |
+  45c. apiece?                                        |           |    |
+                                                      |-----------+----|
+  6. A teacher weighed all the girls in a certain     |           |    |
+  grade. If one girl weighed 79 pounds and another    |           |    |
+  110 pounds, how many pounds heavier was one girl    |           |    |
+  than the other?                                     |           |    |
+                                                      |-----------+----|
+  7. A girl wanted to buy a 5-pound box of candy to   |           |    |
+  give as a present to a friend. She decided to get   |           |    |
+  the kind worth 35c. a pound. What did she pay for   |           |    |
+  the present?                                        |           |    |
+                                                      |-----------+----|
+  8. One day in vacation a boy went on a fishing trip |           |    |
+  and caught 12 fish in the morning, and 7 in the     |           |    |
+  afternoon. How many fish did he catch altogether?   |           |    |
+                                                      |-----------+----|
+  9. A boy lived 15 blocks east of a school; his chum |           |    |
+  lived on the same street, but 11 blocks west of the |           |    |
+  school. How many blocks apart were the two boys'    |           |    |
+  houses?                                             |           |    |
+                                                      |-----------+----|
+  10. A girl was 5 times as strong as her small       |           |    |
+  sister. If the little girl could lift a weight of   |           |    |
+  20 pounds, how large a weight could the older girl  |           |    |
+  lift?                                               |           |    |
+                                                      |-----------+----|
+  11. The children of a school gave a sleigh-ride     |           |    |
+  party. There were 270 children to go on the ride    |           |    |
+  and each sleigh held 30 children. How many sleighs  |           |    |
+  were needed?                                        |           |    |
+                                                      |-----------+----|
+  12. In September there were 43 children in the      |           |    |
+  eighth grade of a certain school; by June there     |           |    |
+  were 59. How many children entered the grade        |           |    |
+  during the year?                                    |           |    |
+                                                      |-----------+----|
+  13. A girl who lived 17 blocks away walked to       |           |    |
+  school and back twice a day. What was the total     |           |    |
+  number of blocks the girl walked each day in        |           |    |
+  going to and from school?                           |           |    |
+                                                      |-----------+----|
+  14. A boy who made 67c. a day carrying papers, was  |           |    |
+  hired to run on a long errand for which he received |           |    |
+  50c. What was the total amount the boy earned that  |           |    |
+  day?                                                |           |    |
+                                                      |-----------+----|
+                                          Total Right |           |    |
+                                                      +-----------+----+
+
+(Two more similar problems follow.)
+
+Test 6 and Test 8 are from the Courtis Standard Test. Used by permission
+of S. A. Courtis.
+
+
+ARITHMETIC--Test No. 8. Reasoning
+
+In the blank space below, work as many of the following examples as
+possible in the time allowed. Work them in order as numbered, entering
+each answer in the "answer" column before commencing a new example. Do
+not work on any other paper.
+
+                                                            +--------+-+
+                                                            | ANSWER | |
+                                                            |--------+-|
+  1. The children in a certain school gave a Christmas      |        | |
+  party. One of the presents was a box of candy. In filling |        | |
+  the boxes, one grade used 16 pounds of candy, another 17  |        | |
+  pounds, a third 12 pounds, and a fourth 13 pounds. What   |        | |
+  did the candy cost at 26c. a pound?                       |        | |
+                                                            |--------+-|
+  2. A school in a certain city used 2516 pieces of chalk   |        | |
+  in 37 school days. Three new rooms were opened, each      |        | |
+  room holding 50 children, and the school was then found   |        | |
+  to use 84 sticks of chalk per day. How many more sticks   |        | |
+  of chalk were used per day than at first?                 |        | |
+                                                            |--------+-|
+  3. Several boys went on a bicycle trip of 1500 miles.     |        | |
+  The first week they rode 374 miles, the second week 264   |        | |
+  miles, the third 423 miles, the fourth 401 miles. They    |        | |
+  finished the trip the next week. How many miles did they  |        | |
+  ride the last week?                                       |        | |
+                                                            |--------+-|
+  4. Forty-five boys were hired to pick apples from 15      |        | |
+  trees in an apple orchard. In 50 minutes each boy had     |        | |
+  picked 48 choice apples. If all the apples picked were    |        | |
+  packed away carefully in 8 boxes of equal size, how many  |        | |
+  apples were put in each box?                              |        | |
+                                                            |--------+-|
+  5. In a certain school 216 children gave a sleigh-ride    |        | |
+  party. They rented 7 sleighs at a cost of $30.00 and paid |        | |
+  $24.00 for the refreshments. The party travelled 15 miles |        | |
+  in 2-1/2 hours and had a very pleasant time. What was     |        | |
+  each child's share of the expense?                        |        | |
+                                                            |--------+-|
+  6. A girl found, by careful counting, that there were     |        | |
+  2400 letters on one page of her history, and only 2295    |        | |
+  letters on a page of her reader. How many more letters    |        | |
+  had she read in one book than in the other if she had     |        | |
+  read 47 pages in each of the books?                       |        | |
+                                                            |--------+-|
+  7. Each of 59 rooms in the schools of a certain city      |        | |
+  contributed 25 presents to a Christmas entertainment for  |        | |
+  poor children. The stores of the city gave 1986 other     |        | |
+  articles for presents. What was the total number of       |        | |
+  presents given away at the entertainment?                 |        | |
+                                                            |--------+-|
+  8. Forty-eight children from a certain school paid 10c.   |        | |
+  apiece to ride 7 miles on the cars to a woods. There in a |        | |
+  few hours they gathered 2765 nuts. 605 of these were bad, |        | |
+  but the rest were shared equally among the children. How  |        | |
+  many good nuts did each one get?                          |        | |
+                                                            |--------+-|
+                                                      Total |        | |
+                                                            +--------+-+
+
+These proposed measures of ability to apply arithmetic illustrate very
+nicely the differences of opinion concerning what applied arithmetic and
+arithmetical reasoning should be. The thinker who emphasizes the fact
+that in life out of school the situation demanding quantitative
+treatment is usually real rather than described, will condemn a test all
+of whose constituents are _described_ problems. Unless we are
+excessively hopeful concerning the transfer of ideas of method and
+procedure from one mental function to another we shall protest against
+the artificiality of No. 3 of the Stone series, and of the entire
+Courtis Test 8 except No. 4. The Courtis speed-reasoning test (No. 6) is
+a striking example of the mixture of ability to understand quantitative
+relations with the ability to understand words. Consider these five, for
+example, in comparison with the revised versions attached.[3]
+
+  [3] The form of Test 6 quoted here is that given by Courtis ['11-'12,
+      p. 20]. This differs a little from the other series of Test 6,
+      shown on pages 43 and 44.
+
+  1. The children of a school gave a sleigh-ride party. There were 9
+  sleighs, and each sleigh held 30 children. How many children were
+  there in the party?
+
+  REVISION. _If one sleigh holds 30 children, 9 sleighs hold ....
+  children._
+
+  2. Two school-girls played a number-game. The score of the girl
+  that lost was 57 points and she was beaten by 16 points. What was
+  the score of the girl that won?
+
+  REVISION. _Mary and Nell played a game. Mary had a score of 57.
+  Nell beat Mary by 16. Nell had a score of ...._
+
+  3. A girl counted the automobiles that passed a school. The total
+  was 60 in two hours. If the girl saw 27 pass the first hour how
+  many did she see the second?
+
+  REVISION. _In two hours a girl saw 60 automobiles. She saw 27 the
+  first hour. She saw .... the second hour._
+
+  4. On a playground there were five equal groups of children each
+  playing a different game. If there were 75 children all together,
+  how many were there in each group?
+
+  REVISION. _75 pounds of salt just filled five boxes. The boxes were
+  exactly alike. There were .... pounds in a box._
+
+  5. A teacher weighed all the children in a certain grade. One girl
+  weighed 70 pounds. Her older sister was 49 pounds heavier. How many
+  pounds did the sister weigh?
+
+  REVISION. _Mary weighs 70 lb. Jane weighs 49 pounds more than Mary.
+  Jane weighs .... pounds._
+
+The distinction between a problem described as clearly and simply as
+possible and the same problem put awkwardly or in ill-known words or
+willfully obscured should be regarded; and as a rule measurements of
+ability to apply arithmetic should eschew all needless obscurity or
+purely linguistic difficulty. For example,
+
+  _A boy bought a two-cent stamp. He gave the man in the store 10
+  cents. The right change was .... cents._
+
+is better as a test than
+
+  _If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in
+  payment, what change should he be expected to receive in return?_
+
+The distinction between the description of a _bona fide_ problem that a
+human being might be called on to solve out of school and the
+description of imaginary possibilities or puzzles should also be
+considered. Nos. 3 and 9 of Stone are bad because to frame the problems
+one must first know the answers, so that in reality there could never be
+any point in solving them. It is probably safe to say that nobody in the
+world ever did or ever will or ever should find the number of apples in
+a box by the task of No. 4 of the Courtis Test 8.
+
+This attaches no blame to Dr. Stone or to Mr. Courtis. Until very
+recently we were all so used to the artificial problems of the
+traditional sort that we did not expect anything better; and so blind to
+the language demands of described problems that we did not see their
+very great influence. Courtis himself has been active in reform and has
+pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.
+
+"Tests Nos. 6 and 8, the so-called reasoning tests, have proved the
+least satisfactory of the series. The judgments of various teachers and
+superintendents as to the inequalities of the units in any one test, and
+of the differences between the different editions of the same test, have
+proved the need of investigating these questions. Tests of adults in
+many lines of commercial work have yielded in many cases lower scores
+than those of the average eighth grade children. At the same time the
+scores of certain individuals of marked ability have been high, and
+there appears to be a general relation between ability in these tests
+and accuracy in the abstract work. The most significant facts, however,
+have been the difficulties experienced by teachers in attempting to
+remedy the defects in reasoning. It is certain that the tests measure
+abilities of value but the abilities are probably not what they seem to
+be. In an attempt to measure the value of different units, for instance,
+as many problems as possible were constructed based upon a single
+situation. Twenty-one varieties were secured by varying the relative
+form of the question and the relative position of the different phrases.
+One of these proved nineteen times as hard as another as measured by the
+number of mistakes made by the children; yet the cause of the difference
+was merely the changes in the phrasing. This and other facts of the same
+kind seem to show that Tests 6 and 8 measure mainly the ability to
+read."
+
+The scientific measurement of the abilities and achievements concerned
+with applied arithmetic or problem-solving is thus a matter for the
+future. In the case of described problems a beginning has been made in
+the series which form a part of the National Intelligence Tests ['20],
+one of which is shown on page 49 f. In the case of problems with real
+situations, nothing in systematic form is yet available.
+
+Systematic tests and scales, besides defining the abilities we are to
+establish and improve, are of very great service in measuring the status
+and improvement of individuals and of classes, and the effects of
+various methods of instruction and of study. They are thus helpful to
+pupils, teachers, supervisors, and scientific investigators; and are
+being more and more widely used every year. Information concerning the
+merits of the different tests, the procedure to follow in giving and
+scoring them, the age and grade standards to be used in interpreting
+results, and the like, is available in the manuals of Educational
+Measurement, such as Courtis, _Manual of Instructions for Giving and
+Scoring the Courtis Standard Tests in the Three R's_ ['14]; Starch,
+_Educational Measurements_ ['16]; Chapman and Rush, _Scientific
+Measurement of Classroom Products_ ['17]; Monroe, DeVoss, and Kelly,
+_Educational Tests and Measurements_ ['17]; Wilson and Hoke, _How to
+Measure_ ['20]; and McCall, _How to Measure in Education_ ['21].
+
+TEST 1
+
+  National Intelligence Tests.
+  Scale A. Form 1, Edition 1
+
+  Find the answers as quickly as you can.
+  Write the answers on the dotted lines.
+  Use the side of the page to figure on.
+
+  #Begin here#
+
+  1  Five cents make 1 nickel. How many nickels make a
+     dime?                                                _Answer_ .....
+
+  2  John paid 5 dollars for a watch and 3 dollars for a chain.
+     How many dollars did he pay for the watch and chain? _Answer_ .....
+
+  3  Nell is 13 years old. Mary is 9 years old. How much
+     younger is Mary than Nell?                           _Answer_ .....
+
+  4  One quart of ice cream is enough for 5 persons. How
+     many quarts of ice cream are needed for 25 persons?  _Answer_ .....
+
+  5  John's grandmother is 86 years old. If she lives, in
+     how many years will she be 100 years old?            _Answer_ .....
+
+  6  If a man gets $2.50 a day, what will he be paid for six
+     days' work?                                          _Answer_ .....
+
+  7  How many inches are there in a foot and a half?      _Answer_ .....
+
+  8  What is the cost of 12 cakes at 6 for 5 cents?       _Answer_ .....
+
+  9  The uniforms for a baseball team of nine boys cost $2.50
+     each. The shoes cost $2 a pair. What was the total
+     cost of uniforms and shoes for the nine?             _Answer_ .....
+
+  10 A train that usually arrives at half-past ten was 17
+     minutes late. When did it arrive?                    _Answer_ .....
+
+  11 At 10c a yard, what is the cost of a piece 10-1/2 ft. long?
+                                                          _Answer_ .....
+
+  12 A man earns $6 a day half the time, $4.50 a day one
+     fourth of the time, and nothing on the remaining days
+     for a total period of 40 days. What did he earn in all
+     in the 40 days?                                      _Answer_ .....
+
+  13 What per cent of $800 is 4% of $1000?                _Answer_ .....
+
+  14 If 60 men need 1500 lb. flour per month, what is the
+     requirement per man per day counting a month as 30
+     days?                                                _Answer_ .....
+
+  15 A car goes at the rate of a mile a minute. A truck goes
+     20 miles an hour. How many times as far will the car
+     go as the truck in 10 seconds?                       _Answer_ .....
+
+  16 The area of the base (inside measure) of a cylindrical
+     tank is 90 square feet. How tall must it be to hold
+     100 cubic yards?                                     _Answer_ .....
+
+  From National Intelligence Tests by National Research Council.
+
+  Copyright, 1920, by The World Book Company, Yonkers-on-Hudson,
+  New York.
+
+  Used by permission of the publishers.
+
+
+
+
+CHAPTER III
+
+THE CONSTITUTION OF ARITHMETICAL ABILITIES
+
+
+THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING
+
+It would be a useful work for some one to try to analyze arithmetical
+learning into the unitary abilities which compose it, showing just what,
+in detail, the mind has to do in order to be prepared to pass a thorough
+test on the whole of arithmetic. These unitary abilities would make a
+very long list. Examination of a well-planned textbook will show that
+such an ability as multiplication is treated as a composite of the
+following: knowledge of the multiplications up to 9 x 9; ability to
+multiply two (or more)-place numbers by 2, 3, and 4 when 'carrying' is
+not required and no zeros occur in the multiplicand; ability to multiply
+by 2, 3, ... 9, with carrying; the ability to handle zeros in the
+multiplicand; the ability to multiply with two-place numbers not ending
+in zero; the ability to handle zero in the multiplier as last number;
+the ability to multiply with three (or more)-place numbers not including
+a zero; the ability to multiply with three- and four-place numbers with
+zero in second or third, or second and third, as well as in last place;
+the ability to save time by annexing zeros; and so on and on through a
+long list of further abilities required to multiply with United States
+money, decimal fractions, common fractions, mixed numbers, and
+denominate numbers.
+
+The units or 'steps' thus recognized by careful teaching would make a
+long list, but it is probable that a still more careful study of
+arithmetical ability as a hierarchy of mental habits or connections
+would greatly increase the list. Consider, for example, ordinary column
+addition. The majority of teachers probably treat this as a simple
+application of the knowledge of the additions to 9 + 9, plus
+understanding of 'carrying.' On the contrary there are at least seven
+processes or minor functions involved in two-place column addition, each
+of which is psychologically distinct and requires distinct educational
+treatment.
+
+These are:--
+
+  A. Learning to keep one's place in the column as one adds.
+
+  B. Learning to keep in mind the result of each addition until the
+        next number is added to it.
+
+  C. Learning to add a seen to a thought-of number.
+
+  D. Learning to neglect an empty space in the columns.
+
+  E. Learning to neglect 0s in the columns.
+
+  F. Learning the application of the combinations to higher decades
+        may for the less gifted pupils involve as much time and labor
+        as learning all the original addition tables. And even for
+        the most gifted child the formation of the connection
+        '8 and 7 = 15' probably never quite insures the presence
+        of the connections '38 and 7 = 45' and '18 + 7 = 25.'
+
+  G. Learning to write the figure signifying units rather than the
+        total sum of a column. In particular, learning to write 0 in
+        the cases where the sum of the column is 10, 20, etc. Learning
+        to 'carry' also involves in itself at least two distinct
+        processes, by whatever way it is taught.
+
+We find evidence of such specialization of functions in the results with
+such tests as Woody's. For example, 2 + 5 + 1 = .... surely involves
+abilities in part different from
+
+    2
+    4
+    3
+    -
+
+because only 77 percent of children in grade 3 do the former correctly,
+whereas 95 percent of children in that grade do the latter correctly. In
+grade 2 the difference is even more marked. In the case of subtraction
+
+    4
+    4
+    -
+
+involves abilities different from those involved in
+
+    9
+    3,
+    -
+
+being much less often solved correctly in grades 2 and 4.
+
+    6
+    0
+    -
+
+is much harder than either of the above.
+
+   43
+    1                       21
+    2                       33
+   13  is much harder than  35.
+   --                       --
+
+It may be said that these differences in difficulty are due to different
+amounts of practice. This is probably not true, but if it were, it would
+not change the argument; if the two abilities were identical, the
+practice of one would improve the other equally.
+
+I shall not undertake here this task of listing and describing the
+elementary functions which constitute arithmetical learning, partly
+because what they are is not fully known, partly because in many cases a
+final ability may be constituted in several different ways whose
+descriptions become necessarily tedious, and partly because an adequate
+statement of what is known would far outrun the space limits of this
+chapter. Instead, I shall illustrate the results by some samples.
+
+
+KNOWLEDGE OF THE MEANING OF A FRACTION
+
+As a first sample, consider knowledge of the meaning of a fraction. Is
+the ability in question simply to understand that a fraction is a
+statement of the number of parts, each of a certain size, the upper
+number or numerator telling how many parts are taken and the lower
+number or denominator telling what fraction of unity each part is? And
+is the educational treatment required simply to describe and illustrate
+such a statement and have the pupils apply it to the recognition of
+fractions and the interpretation of each of them? And is the learning
+process (1) the formation of the notions of part, size of part, number
+of part, (2) relating the last two to the numbers in a fraction, and, as
+a necessary consequence, (3) applying these notions adequately whenever
+one encounters a fraction in operation?
+
+Precisely this was the notion a few generations ago. The nature of
+fractions was taught as one principle, in one step, and the habits of
+dealing with fractions were supposed to be deduced from the general law
+of a fraction's nature. As a result the subject of fractions had to be
+long delayed, was studied at great cost of time and effort, and, even
+so, remained a mystery to all save gifted pupils. These gifted pupils
+probably of their own accord built up the ability piecemeal out of
+constituent insights and habits.
+
+At all events, scientific teaching now does build up the total ability
+as a fusion or organization of lesser abilities. What these are will be
+seen best by examining the means taken to get them. (1) First comes the
+association of 1/2 of a pie, 1/2 of a cake, 1/2 of an apple, and such
+like with their concrete meanings so that a pupil can properly name a
+clearly designated half of an obvious unit like an orange, pear, or
+piece of chalk. The same degree of understanding of 1/4, 1/8, 1/3, 1/6,
+and 1/5 is secured. The pupil is taught that 1 pie = 2 1/2s, 3 1/3s, 4
+1/4s, 5 1/5s, 6 1/6s, and 8 1/8s; similarly for 1 cake, 1 apple, and the
+like.
+
+So far he understands 1/_x_ of _y_ in the sense of certain simple parts
+of obviously unitary _y_s.
+
+(2) Next comes the association with 1/2 of an inch, 1/2 of a foot, 1/2
+of a glassful and other cases where _y_ is not so obviously a unitary
+object whose pieces still show their derivation from it. Similarly for
+1/4, 1/3, etc.
+
+(3) Next comes the association with 1/2 of a collection of eight pieces
+of candy, 1/3 of a dozen eggs, 1/5 of a squad of ten soldiers, etc.,
+until 1/2, 1/3, 1/4, 1/5, 1/6, and 1/8 are understood as names of
+certain parts of a collection of objects.
+
+(4) Next comes the similar association when the nature of the collection
+is left undefined, the pupil responding to
+
+  1/2 of 6 is ..., 1/4 of 8 is ..., 2 is 1/5 of ...,
+  1/3 of 6 is ..., 1/3 of 9 is ..., 2 is 1/3 of ..., and the like.
+
+Each of these abilities is justified in teaching by its intrinsic
+merits, irrespective of its later service in helping to constitute the
+general understanding of the meaning of a fraction. The habits thus
+formed in grades 3 or 4 are of constant service then and thereafter in
+and out of school.
+
+(5) With these comes the use of 1/5 of 10, 15, 20, etc., 1/6 of 12, 18,
+42, etc., as a useful variety of drill on the division tables, valuable
+in itself, and a means of making the notion of a unit fraction more
+general by adding 1/7 and 1/9 to the scheme.
+
+(6) Next comes the connection of 3/4, 2/5, 3/5, 4/5, 2/3, 1/6, 5/6, 3/8,
+5/8, 7/8, 3/10, 7/10, and 9/10, each with its meaning as a certain part
+of some conveniently divisible unit, and, (7) and (8), connections
+between these fractions and their meanings as parts of certain
+magnitudes (7) and collections (8) of convenient size, and (9)
+connections between these fractions and their meanings when the nature
+of the magnitude or collection is unstated, as in 4/5 of 15 = ...,
+5/8 of 32 = ....
+
+(10) That the relation is general is shown by using it with
+numbers requiring written division and multiplication, such as
+7/8 of 1736 = ..., and with United States money.
+
+Elements (6) to (10) again are useful even if the pupil never goes
+farther in arithmetic. One of the commonest uses of fractions is in
+calculating the cost of fractions of yards of cloth, and fractions of
+pounds of meat, cheese, etc.
+
+The next step (11) is to understand to some extent the principle that
+the value of any of these fractions is unaltered by multiplying or
+dividing the numerator and denominator by the same number. The drills in
+expressing fractions in lower and higher terms which accomplish this are
+paralleled by (12) and (13) simple exercises in adding and subtracting
+fractions to show that fractions are quantities that can be operated on
+like any quantities, and by (14) simple work with mixed numbers
+(addition and subtraction and reductions), and (15) improper fractions.
+All that is done with improper fractions is (_a_) to have the pupil use
+a few of them as he would any fractions and (_b_) to note their
+equivalent mixed numbers. In (12), (13), and (14) only fractions of the
+same denominators are added or subtracted, and in (12) (13), (14), and
+(15) only fractions with 2, 3, 4, 5, 6, 8, or 10 in the denominator are
+used. As hitherto, the work of (11) to (15) is useful in and of itself.
+(16) Definitions are given of the following type:--
+
+Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called whole numbers.
+
+Numbers like 7/8, 1/5, 2/3, 3/4, 11/8, 7/6, 1/3, 4/3, 1/8, 1/6 are
+called fractions.
+
+Numbers like 5-1/4, 7-3/8, 9-1/2, 16-4/5, 315-7/8, 1-1/3, 1-2/3 are
+called mixed numbers.
+
+(17) The terms numerator and denominator are connected with the upper
+and lower numbers composing a fraction.
+
+Building this somewhat elaborate series of minor abilities seems to be a
+very roundabout way of getting knowledge of the meaning of a fraction,
+and is, if we take no account of what is got along with this knowledge.
+Taking account of the intrinsically useful habits that are built up, one
+might retort that the pupil gets his knowledge of the meaning of a
+fraction at zero cost.
+
+
+KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES
+
+Consider next the knowledge of the subtraction and division 'Tables.'
+The usual treatment presupposes that learning them consists of forming
+independently the bonds:--
+
+   3 - 1 = 2                4 / 2 = 2
+   3 - 2 = 1                6 / 2 = 3
+   4 - 1 = 3                6 / 3 = 2
+     .                        .
+     .                        .
+     .                        .
+     .                        .
+     .                        .
+     .                        .
+  18 - 9 = 9               81 / 9 = 9
+
+In fact, however, these 126 bonds are not formed independently. Except
+perhaps in the case of the dullest twentieth of pupils, they are
+somewhat facilitated by the already learned additions and
+multiplications. And by proper arrangement of the learning they may be
+enormously facilitated thereby. Indeed, we may replace the independent
+memorizing of these facts by a set of instructive exercises wherein the
+pupil derives the subtractions from the corresponding additions by
+simple acts of reasoning or selective thinking. As soon as the additions
+giving sums of 9 or less are learned, let the pupil attack an exercise
+like the following:--
+
+Write the missing numbers:--
+
+         A                 B                 C                 D
+  3 and ... are 5.  5 and ... are 8.  4 and ... are 5.  4 and ... are 8.
+  3 and ... are 9.  3 and ... are 6.  5 and ... are 6.  1 and ... are 7.
+  4 and ... are 7.  4 and ... are 9.  6 and ... are 9.  6 and ... are 7.
+  5 and ... are 7.  2 and ...  =  6.  1 and ... are 8.  8 and ... are 9.
+  6 and ... are 8.  5 and ...  =  9.  3 and ... are 7.  3  +  ... are 4.
+  4 and ... are 6.  2 and ...  =  7.  1  +  ... are 3.  7  +  ... are 8.
+  2 and ... are 5.  3 and ...  =  8.  1  +  ... are 5.  4  +  ... are 9.
+  2 and ...  =  8.  1 and ...  =  4.  4  +  ... are 8.  2  +  ... are 3.
+  3 and ...  =  6.  2 and ...  =  4.  7  +  ... are 9.  1  +  ... are 9.
+  6 and ...  =  9.  3 and ...  =  8.  2  +  ...  =  4.  3  +  ...  =  6.
+  4 and ...  =  6.  6 and ...  =  7.  3  +  ...  =  8.  5  +  ...  =  9.
+  4 and ...  =  7.  2 and ...  =  5.  4  +  ...  =  5.  1  +  ...  =  3.
+
+The task for reasoning is only to try, one after another, numbers that
+seem promising and to select the right one when found. With a little
+stimulus and direction children can thus derive the subtractions up to
+those with 9 as the larger number. Let them then be taught to do the
+same with the printed forms:--
+
+Subtract
+
+  9   7   8   5   8   6
+  3   5   6   2   2   4   etc.
+  -   -   -   -   -   -
+
+and 9 - 7 = ..., 9 - 5 = ..., 7 - 5 = ..., etc.
+
+In the case of the divisions, suppose that the pupil has learned his
+first table and gained surety in such exercises as:--
+
+  4 5s = ....    6 x 5 = ....    9 nickels = .... cents.
+  8 5s = ....    4 x 5 = ....    6    "    = ....   "
+  3 5s = ....    2 x 5 = ....    5    "    = ....   "
+  7 5s = ....    9 x 5 = ....    7    "    = ....   "
+
+  If one ball costs 5 cents,
+                two balls cost .... cents,
+                three balls cost .... cents, etc.
+
+He may then be set at once to work at the answers to exercises like the
+following:--
+
+Write the answers and the missing numbers:--
+
+       A            B                 C                 D
+  .... 5s = 15   40 = .... 5s   .... x 5 = 25   20 cents = .... nickels.
+  .... 5s = 20   20 = .... 5s   .... x 5 = 50   30 cents = .... nickels.
+  .... 5s = 40   15 = .... 5s   .... x 5 = 35   15 cents = .... nickels.
+  .... 5s = 25   45 = .... 5s   .... x 5 = 10   40 cents = .... nickels.
+  .... 5s = 30   50 = .... 5s   .... x 5 = 40
+  .... 5s = 35   25 = .... 5s   .... x 5 = 45
+
+                          E
+  For  5 cents you can buy 1 small loaf of bread.
+  For 10 cents you can buy 2 small loaves of bread.
+  For 25 cents you can buy .... small loaves of bread.
+  For 45 cents you can buy .... small loaves of bread.
+  For 35 cents you can buy .... small loaves of bread.
+
+               F
+   5 cents pays 1 car fare.
+  15 cents pays .... car fares.
+  10 cents pays .... car fares.
+  20 cents pays .... car fares.
+
+                           G
+  How many 5 cent balls can you buy with 30 cents? ....
+  How many 5 cent balls can you buy with 35 cents? ....
+  How many 5 cent balls can you buy with 25 cents? ....
+  How many 5 cent balls can you buy with 15 cents? ....
+
+In the case of the meaning of a fraction, the ability, and so the
+learning, is much more elaborate than common practice has assumed; in
+the case of the subtraction and division tables the learning is much
+less so. In neither case is the learning either mere memorizing of facts
+or the mere understanding of a principle _in abstracto_ followed by its
+application to concrete cases. It is (and this we shall find true of
+almost all efficient learning in arithmetic) the formation of
+connections and their use in such an order that each helps the others to
+the maximum degree, and so that each will do the maximum amount for
+arithmetical abilities other than the one specially concerned, and for
+the general competence of the learner.
+
+
+LEARNING THE PROCESSES OF COMPUTATION
+
+As another instructive topic in the constitution of arithmetical
+abilities, we may take the case of the reasoning involved in
+understanding the manipulations of figures in two (or more)-place
+addition and subtraction, multiplication and division involving a two
+(or more)-place number, and the manipulations of decimals in all four
+operations. The psychology of these is of special interest and
+importance. For there are two opposite explanations possible here,
+leading to two opposite theories of teaching.
+
+The common explanation is that these methods of manipulation, if
+understood at all, are understood as deductions from the properties of
+our system of decimal notation. The other is that they are understood
+partly as inductions from the experience that they always give the right
+answer. The first explanation leads to the common preliminary deductive
+explanations of the textbooks. The other leads to explanations by
+verification; _e.g._, of addition by counting, of subtraction by
+addition, of multiplication by addition, of division by multiplication.
+Samples of these two sorts of explanation are given below.
+
+
+SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION
+
+MULTIPLICATION is the process of taking one number as many times as
+there are units in another number.
+
+The PRODUCT is the result of the multiplication.
+
+The MULTIPLICAND is the number to be taken.
+
+The MULTIPLIER is the number denoting how many times the multiplicand is
+to be taken.
+
+The multiplier and multiplicand are the FACTORS.
+
+    Multiply 623 by 3
+
+    OPERATION
+
+      _Multiplicand_    623
+      _Multiplier_        3
+                       ----
+      _Product_        1869
+
+    EXPLANATION.--For convenience we write the multiplier under the
+    multiplicand, and begin with units to multiply. 3 times 3 units are
+    9 units. We write the nine units in units' place in the product. 3
+    times 2 tens are 6 tens. We write the 6 tens in tens' place in the
+    product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8
+    hundreds. The 1 thousand we write in thousands' place and the 8
+    hundreds in hundreds' place in the product. Therefore, the product
+    is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869.
+
+
+SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION
+
+  1. The children of the third grade are to have a picnic. 32 are going.
+  How many sandwiches will they need if each of the 32 children has four
+  sandwiches?
+
+      _Here is a quick way to find out_:--
+
+       32   _Think "4 x 2," write 8 under the 2 in the ones column._
+        4   _Think "4 x 3," write 12 under the 3 in the tens column._
+       --
+
+  2. How many bananas will they need if each of the 32 children has
+  two bananas? 32 x 2 or 2 x 32 will give the answer.
+
+  3. How many little cakes will they need if each child has three
+  cakes? 32 x 3 or 3 x 32 will give the answer.
+
+    32           3 x 2 = ....  _Where do you write the 6?_
+     3           3 x 3 = ....  _Where do you write the 9?_
+    --
+
+  4.  Prove that 128, 64, and 96 are right by adding four 32s, two 32s,
+  and three 32s.
+
+      32
+      32  32
+      32  32  32
+      32  32  32
+      --  --  --
+
+
+Multiplication
+
+  You #multiply# when you find the answers to questions like
+
+    How many are 9 x 3?
+    How many are 3 x 32?
+    How many are 8 x 5?
+    How many are 4 x 42?
+
+  1. Read these lines. Say the right numbers where the dots are:
+
+     If you #add# 3 to 32, you have .... 35 is the #sum#.
+     If you #subtract# 3 from 32, the result is .... 29 is the
+          #difference# or #remainder#.
+     If you #multiply# 3 by 32 or 32 by 3, you have .... 96 is the
+          #product#.
+
+  Find the products. Check your answers to the first line by adding.
+
+   2.    3.    4.    5.    6.    7.    8.    9.
+
+   41    33    42    44    53    43    34    24
+    3     2     4     2     3     2     2     2
+   --    --    --    --    --    --    --    --
+
+      10.   11.   12.   13.   14.   15.   16.
+
+      43    52    32    23    41    51    14
+       3     3     3     3     2     4     2
+      --    --    --    --    --    --    --
+
+  17.     213  _Write the 9 in the ones column._
+            3  _Write the 3 in the tens column._
+          ---  _Write the 6 in the hundreds column._
+
+  _Check your answer by adding._    Add
+                                    213
+                                    213
+                                    213
+                                    ---
+
+   18.    19.    20.    21.    22.    23.    24.
+
+   214    312    432    231    132    314    243
+     2      3      2      3      3      2      2
+   ---    ---    ---    ---    ---    ---    ---
+
+
+SHORT DIVISION: DEDUCTIVE EXPLANATION
+
+Divide 1825 by 4
+
+  Divisor  4 |1825    Dividend
+             --------
+             456-1/4
+             Quotient
+
+    EXPLANATION.--For convenience we write the divisor at the left of
+    the dividend, and the quotient below it, and begin at the left to
+    divide. 4 is not contained in 1 thousand any thousand times,
+    therefore the quotient contains no unit of any order higher than
+    hundreds. Consequently we find how many times 4 is contained in the
+    hundreds of the dividend. 1 thousand and 8 hundreds are 18
+    hundreds. 4 is contained in 18 hundreds 4 hundred times and 2
+    hundreds remaining. We write the 4 hundreds in the quotient. The 2
+    hundreds we consider as united with the 2 tens, making 22 tens. 4
+    is contained in 22 tens 5 tens times, and 2 tens remaining. We
+    write the 5 tens in the quotient, and the remaining 2 tens we
+    consider as united with the 5 units, making 25 units. 4 is
+    contained in 25 units 6 units times and 1 unit remaining. We write
+    the 6 units in the quotient and indicate the division of the
+    remainder, 1 unit, by the divisor 4.
+
+    Therefore the quotient of 1825 divided by 4 is 456-1/4, or 456 and
+    1 remainder.
+
+
+SHORT DIVISION: INDUCTIVE EXPLANATION
+
+Dividing Large Numbers
+
+  1. Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent
+  bag of marbles. There are 128 marbles in it. How many should each boy
+  have, if they divide the marbles equally among the four boys?
+
+      -----
+    4 |128
+
+    _Think "12 = three 4s." Write the 3 over the 2 in the tens column._
+    _Think "8 = two 4s." Write the 2 over the 8 in the ones column._
+    _32 is right, because 4 x 32 = 128._
+
+  2. Mary, Nell, and Alice are going to buy a book as a present for
+  their Sunday-school teacher. The present costs 69 cents. How much
+  should each girl pay, if they divide the cost equally among the
+  three girls?
+     ----
+    3|69
+
+    _Think "6 = .... 3s." Write the 2 over the 6 in the tens column._
+    _Think "9 = .... 3s." Write the 3 over the 9 in the ones column._
+    _23 is right, for 3 x 23 = 69._
+
+  3. Divide the cost of a 96-cent present equally among three girls. How
+  much should each girl pay?
+     ------
+    3|96
+
+  4. Divide the cost of an 84-cent present equally among 4 girls. How
+  much should each girl pay?
+
+  5. Learn this: (Read / as "_divided by_.")
+
+      12 + 4 = 16.   16 is the sum.
+      12 - 4 =  8.    8 is the difference or remainder.
+      12 x 4 = 48.   48 is the product.
+      12 / 4 =  3.    3 is the quotient.
+
+  6. Find the quotients. Check your answers by multiplying.
+     ----     ----     ----     -----     -----     -----
+    3|99     2|86     5|155    6|246     4|168     3|219
+
+[Uneven division is taught by the same general plan, extended.]
+
+
+LONG DIVISION: DEDUCTIVE EXPLANATION
+
+To Divide by Long Division
+
+1. Let it be required to divide 34531 by 15.
+
+              _Operation_
+
+               Divided
+  Divisor  15)34531(2302-1/15  Quotient
+              30
+              --
+               45
+               45
+               --
+                 31
+                 30
+                 --
+                  1 Remainder
+
+For convenience we write the divisor at the left and the quotient at the
+right of the dividend, and begin to divide as in Short Division.
+
+15 is contained in 3 ten-thousands 0 ten-thousands times; therefore,
+there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is
+contained in 34 thousands 2 thousands times; we write the 2 thousands in
+the quotient. 15 x 2 thousands = 30 thousands, which, subtracted from 34
+thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we
+have 45 hundreds.
+
+15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the
+quotient. 15 x 3 hundreds = 45 hundreds, which subtracted from 45
+hundreds, leaves nothing. Adding the 3 tens, we have 3 tens.
+
+15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to
+the three tens, which equal 30 units, the 1 unit, we have 31 units.
+
+15 in 31 units 2 units times; we write the 2 units in the quotient. 15 x
+2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a
+remainder. Indicating the division of the 1 unit, we annex the
+fractional expression, 1/15 unit, to the integral part of the quotient.
+
+Therefore, 34531 divided by 15 is equal to 2302-1/15.
+
+[B. Greenleaf, _Practical Arithmetic_, '73, p. 49.]
+
+
+LONG DIVISION: INDUCTIVE EXPLANATION
+
+Dividing by Large Numbers
+
+  1. Just before Christmas Frank's father sent 360 oranges to be divided
+     among the children in Frank's class. There are 29 children. How
+     many oranges should each child receive? How many oranges will be
+     left over?
+
+  _Here is the best way to find out:_
+
+       12    and 12    _Think how many 29s there are in 36. 1 is right._
+     ______ remainder  _Write 1 over the 6 of 36. Multiply 29 by 1._
+  29 )360              _Write the 29 under the 36. Subtract 29 from 36._
+       29              _Write the 0 of 360 after the 7._
+      ---              _Think how many 29s there are in 70. 2 is right._
+       70              _Write 2 over the 0 of 360. Multiply 29 by 2._
+       58              _Write the 58 under 70. Subtract 58 from 70._
+       --              _There is 12 remainder._
+       12              _Each child gets 12 oranges, and there are 12
+                         left over. This is right, for 12 multiplied
+                         by 29 = 348, and 348 + 12 = 360._
+
+       *       *       *       *       *
+
+  8.                  _In No. 8, keep on dividing by 31 until you have
+    ________            used the 5, the 8, and the 7, and have four
+  31)99,587             figures in the quotient._
+
+  9.         10.         11.         12.        13.
+    _____      _____       _____       ____       _______
+  22)253     22)2895     21)8891     22)290     32)16,368
+
+Check your results for 9, 10, 11, 12, and 13.
+
+  1. The boys and girls of the Welfare Club plan to earn money to buy
+     a victrola. There are 23 boys and girls. They can get a good
+     second-hand victrola for $5.75. How much must each earn if they
+     divide the cost equally?
+
+     _Here is the best way to find out_:
+
+      $.25     _Think how many 23s there are in 57. 2 is right._
+     -----     _Write 2 over the 7 of 57. Multiply 23 by 2._
+  23|$5.75     _Write 46 under 57 and subtract. Write the 5 of 575
+        46       after the 11._
+      ----     _Think how many 23s there are in 115. 5 is right._
+       115     _Write 5 over the 5 of 575. Multiply 23 by 5._
+       115     _Write the 115 under the 115 that is there and subtract._
+      ----     _There is no remainder._
+               _Put $ and the decimal point where they belong._
+               _Each child must earn 25 cents. This is right, for $.25
+                 multiplied by 23 = $5.75._
+
+  2. Divide $71.76 equally among 23 persons. How much is each person's
+     share?
+
+  3. Check your result for No. 2 by multiplying the quotient by the
+     divisor.
+
+  Find the quotients. Check each quotient by multiplying it by the
+  divisor.
+
+  4.           5.           6.            7.           8.
+    _______      _______      ________      _______      _______
+  23)$99.13    25)$18.50    21)$129.15    13)$29.25    32)$73.92
+
+    1 bushel = 32 qt.
+
+  9. How many bushels are there in 288 qt.?
+
+  10. In 192 qt.?
+
+  11. In 416 qt.?
+
+Crucial experiments are lacking, but there are several lines of
+well-attested evidence. First of all, there can be no doubt that the
+great majority of pupils learn these manipulations at the start from the
+placing of units under units, tens under tens, etc., in adding, to the
+placing of the decimal point in division with decimals, by imitation and
+blind following of specific instructions, and that a very large
+proportion of the pupils do not to the end, that is to the fifth
+school-year, understand them as necessary deductions from decimal
+notation. It also seems probable that this proportion would not be much
+reduced no matter how ingeniously and carefully the deductions were
+explained by textbooks and teachers. Evidence of this fact will appear
+abundantly to any one who will observe schoolroom life. It also appears
+in the fact that after the properties of the decimal notation have been
+thus used again and again; _e.g._, for deducing 'carrying' in addition,
+'borrowing' in subtraction, 'carrying' in multiplication, the value of
+the digits in the partial product, the value of each remainder in short
+division, the value of the quotient figures in division, the addition,
+subtraction, multiplication, and division of United States money, and
+the placing of the decimal point in multiplication, no competent teacher
+dares to rely upon the pupil, even though he now has four or more years'
+experience with decimal notation, to deduce the placing of the decimal
+point in division with decimals. It may be an illusion, but one seems to
+sense in the better textbooks a recognition of the futility of the
+attempt to secure deductive derivations of those manipulations. I refer
+to the brevity of the explanations and their insertion in such a form
+that they will influence the pupils' thinking as little as possible. At
+any rate the fact is sure that most pupils do not learn the
+manipulations by deductive reasoning, or understand them as necessary
+consequences of abstract principles.
+
+It is a common opinion that the only alternative is knowing them by
+rote. This, of course, is one common alternative, but the other
+explanation suggests that understanding the manipulations by inductive
+reasoning from their results is another and an important alternative.
+The manipulations of 'long' multiplication, for instance, learned by
+imitation or mechanical drill, are found to give for 25 x _A_ a result
+about twice as large as for 13 x _A_, for 38 or 39 x _A_ a result about
+three times as large; for 115 x _A_ a result about ten times as large as
+for 11 x _A_. With even the very dull pupils the procedure is verified
+at least to the extent that it gives a result which the scientific
+expert in the case--the teacher--calls right. With even the very bright
+pupils, who can appreciate the relation of the procedure to decimal
+notation, this relation may be used not as the sole deduction of the
+procedure beforehand, but as one partial means of verifying it
+afterward. Or there may be the condition of half-appreciation of the
+relation in which the pupil uses knowledge of the decimal notation to
+convince himself that the procedure _does_, but not that it _must_ give
+the right answer, the answer being 'right' because the teacher, the
+answer-list, and collateral evidence assure him of it.
+
+I have taken the manipulation of the partial products as an illustration
+because it is one of the least favored cases for the explanation I am
+presenting. If we take the first case where a manipulation may be
+deduced from decimal notation, known merely by rote, or verified
+inductively, namely, the addition of two-place numbers, it seems sure
+that the mental processes just described are almost the universal rule.
+
+Surely in our schools at present children add the 3 of 23 to the 3 of 53
+and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten
+because they see the teacher do so and are told to do so. They are
+protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but
+because they do not know how to add 8 and 5, because they have been
+taught the habit of adding figures that stand one above the other, or
+with a + between them; and because they are shown or told what they are
+to do. They are protected from adding 3 + 5 and 2 + 3, again, by no
+deductive reasoning but for the second and third reasons just given.
+In nine cases out of ten they do not even think of the possibility
+of adding in any other way than the '3 + 3, 2 + 5' way, much less
+do they select that way on account of the facts that 53 = 50 + 3
+and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that
+(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_)!
+
+Just as surely all but the very dullest twentieth or so of children come
+in the end to something more than rote knowledge,--to _understand_, to
+_know_ that the procedure in question is right.
+
+Whether they know _why_ 76 is right depends upon what is meant by
+_why_. If it means that 76 is the result which competent people
+agree upon, they do. If it means that 76 is the result which would
+come from accurate counting they perhaps know why as well as they
+would have, had they been given full explanations of the relation
+of the procedure in two-place addition to decimal notation.
+If _why_ means because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and
+(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_), they do not.
+Nor, I am tempted to add, would most of them by any sort of teaching
+whatever.
+
+I conclude, therefore, that school children may and do reason about and
+understand the manipulations of numbers in this inductive, verifying way
+without being able to, or at least without, under present conditions,
+finding it profitable to derive them deductively. I believe, in fact,
+that pure arithmetic _as it is learned and known_ is largely an
+_inductive science_. At one extreme is a minority to whom it is a series
+of deductions from principles; at the other extreme is a minority to
+whom it is a series of blind habits; between the two is the great
+majority, representing every gradation but centering about the type of
+the inductive thinker.
+
+
+
+
+CHAPTER IV
+
+THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED): THE SELECTION OF
+THE BONDS TO BE FORMED
+
+
+When the analysis of the mental functions involved in arithmetical
+learning is made thorough it turns into the question, 'What are the
+elementary bonds or connections that constitute these functions?' and
+when the problem of teaching arithmetic is regarded, as it should be in
+the light of present psychology, as a problem in the development of a
+hierarchy of intellectual habits, it becomes in large measure a problem
+of the choice of the bonds to be formed and of the discovery of the best
+order in which to form them and the best means of forming each in that
+order.
+
+
+THE IMPORTANCE OF HABIT-FORMATION
+
+The importance of habit-formation or connection-making has been grossly
+underestimated by the majority of teachers and writers of textbooks.
+For, in the first place, mastery by deductive reasoning of such matters
+as 'carrying' in addition, 'borrowing' in subtraction, the value of the
+digits in the partial products in multiplication, the manipulation of
+the figures in division, the placing of the decimal point after
+multiplication or division with decimals, or the manipulation of the
+figures in the multiplication and division of fractions, is impossible
+or extremely unlikely in the case of children of the ages and experience
+in question. They do not as a rule deduce the method of manipulation
+from their knowledge of decimal notation. Rather they learn about
+decimal notation by carrying, borrowing, writing the last figure of each
+partial product under the multiplier which gives that product, etc. They
+learn the method of manipulating numbers by seeing them employed, and by
+more or less blindly acquiring them as associative habits.
+
+In the second place, we, who have already formed and long used the right
+habits and are thereby protected against the casual misleadings of
+unfortunate mental connections, can hardly realize the force of mere
+association. When a child writes sixteen as 61, or finds 428 as the sum
+of
+
+  15
+  19
+  16
+  18
+  --
+
+or gives 642 as an answer to 27 x 36, or says that 4 divided by 1/4 = 1,
+we are tempted to consider him mentally perverse, forgetting or perhaps
+never having understood that he goes wrong for exactly the same general
+reason that we go right; namely, the general law of habit-formation. If
+we study the cases of 61 for 16, we shall find them occurring in the
+work of pupils who after having been drilled in writing 26, 36, 46, 62,
+63, and so on, in which the order of the six in writing is the same as
+it is in speech, return to writing the 'teen numbers. If our language
+said onety-one for eleven and onety-six for sixteen, we should probably
+never find such errors except as 'lapses' or as the results of
+misperception or lack of memory. They would then be more frequent
+_before_ the 20s, 30s, etc., were learned.
+
+If pupils are given much drill on written single column addition
+involving the higher decades (each time writing the two-figure sum),
+they are forming a habit of writing 28 after the sum of 8, 6, 9, and 5
+is reached; and it should not surprise us if the pupil still
+occasionally writes the two-figure sum for the first column though a
+second column is to be added also. On the contrary, unless some counter
+force influences him, he is absolutely sure to make this mistake.
+
+The last mistake quoted (4 / 1/4 = 1) is interesting because here we
+have possibly one of the cases where deduction from psychology alone can
+give constructive aid to teaching. Multiplication and division by
+fractions have been notorious for their difficulty. The former is now
+alleviated by using _of_ instead of x until the new habit is fixed. The
+latter is still approached with elaborate caution and with various means
+of showing why one must 'invert and multiply' or 'multiply by the
+reciprocal.'
+
+But in the author's opinion it seems clear that the difficulty in
+multiplying and dividing by a fraction was not that children felt any
+logical objections to canceling or inverting. I fancy that the majority
+of them would cheerfully invert any fraction three times over or cancel
+numbers at random in a column if they were shown how to do so. But if
+you are a youngster inexperienced in numerical abstractions and if you
+have had _divide_ connected with 'make smaller' three thousand times and
+never once connected with 'make bigger,' you are sure to be somewhat
+impelled to make the number smaller the three thousand and first time
+you are asked to divide it. Some of my readers will probably confess
+that even now they feel a slight irritation or doubt in saying or
+writing that 16/1 / 1/8 = 128.
+
+The habits that have been confirmed by every multiplication and division
+by integers are, in this particular of '_the ratio of result to number
+operated upon_,' directly opposed to the formation of the habits
+required with fractions. And that is, I believe, the main cause of the
+difficulty. Its treatment then becomes easy, as will be shown later.
+
+These illustrations could be added to almost indefinitely, especially in
+the case of the responses made to the so-called 'catch' problems. The
+fact is that the learner rarely can, and almost never does, survey and
+analyze an arithmetical situation and justify what he is going to do by
+articulate deductions from principles. He usually feels the situation
+more or less vaguely and responds to it as he has responded to it or
+some situation like it in the past. Arithmetic is to him not a logical
+doctrine which he applies to various special instances, but a set of
+rather specialized habits of behavior toward certain sorts of quantities
+and relations. And in so far as he does come to know the doctrine it is
+chiefly by doing the will of the master. This is true even with the
+clearest expositions, the wisest use of objective aids, and full
+encouragement of originality on the pupil's part.
+
+Lest the last few paragraphs be misunderstood, I hasten to add that the
+psychologists of to-day do not wish to make the learning of arithmetic a
+mere matter of acquiring thousands of disconnected habits, nor to
+decrease by one jot the pupil's genuine comprehension of its general
+truths. They wish him to reason not less than he has in the past, but
+more. They find, however, that you do not secure reasoning in a pupil by
+demanding it, and that his learning of a general truth without the
+proper development of organized habits back of it is likely to be, not a
+rational learning of that general truth, but only a mechanical
+memorizing of a verbal statement of it. They have come to know that
+reasoning is not a magic force working in independence of ordinary
+habits of thought, but an organization and cooeperation of those very
+habits on a higher level.
+
+The older pedagogy of arithmetic stated a general law or truth or
+principle, ordered the pupil to learn it, and gave him tasks to do which
+he could not do profitably unless he understood the principle. It left
+him to build up himself the particular habits needed to give him
+understanding and mastery of the principle. The newer pedagogy is
+careful to help him build up these connections or bonds ahead of and
+along with the general truth or principle, so that he can understand it
+better. The older pedagogy commanded the pupil to reason and let him
+suffer the penalty of small profit from the work if he did not. The
+newer provides instructive experiences with numbers which will stimulate
+the pupil to reason so far as he has the capacity, but will still be
+profitable to him in concrete knowledge and skill, even if he lacks the
+ability to develop the experiences into a general understanding of the
+principles of numbers. The newer pedagogy secures more reasoning in
+reality by not pretending to secure so much.
+
+The newer pedagogy of arithmetic, then, scrutinizes every element of
+knowledge, every connection made in the mind of the learner, so as to
+choose those which provide the most instructive experiences, those which
+will grow together into an orderly, rational system of thinking about
+numbers and quantitative facts. It is not enough for a problem to be a
+test of understanding of a principle; it must also be helpful in and of
+itself. It is not enough for an example to be a case of some rule; it
+must help review and consolidate habits already acquired or lead up to
+and facilitate habits to be acquired. Every detail of the pupil's work
+must do the maximum service in arithmetical learning.
+
+
+DESIRABLE BONDS NOW OFTEN NEGLECTED
+
+As hitherto, I shall not try to list completely the elementary bonds
+that the course of study in arithmetic should provide for. The best
+means of preparing the student of this topic for sound criticism and
+helpful invention is to let him examine representative cases of bonds
+now often neglected which should be formed and representative cases of
+useless, or even harmful, bonds now often formed at considerable waste
+of time and effort.
+
+(1) _Numbers as measures of continuous quantities._--The numbers one,
+two, three, 1, 2, 3, etc., should be connected soon after the beginning
+of arithmetic each with the appropriate amount of some continuous
+quantity like length or volume or weight, as well as with the
+appropriate sized collection of apples, counters, blocks, and the like.
+Lines should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two
+inches, three inches, etc.; weights should be lifted and called one
+pound, two pounds, etc.; things should be measured in glassfuls,
+handfuls, pints, and quarts. Otherwise the pupil is likely to limit the
+meaning of, say, _four_ to four sensibly discrete things and to have
+difficulty in multiplication and division. Measuring, or counting by
+insensibly marked off repetitions of a unit, binds each number name to
+its meaning as ---- _times whatever 1 is_, more surely than mere
+counting of the units in a collection can, and should reenforce the
+latter.
+
+(2) _Additions in the higher decades._--In the case of all save the very
+gifted children, the additions with higher decades--that is, the bonds,
+16 + 7 = 23, 26 + 7 = 33, 36 + 7 = 43, 14 + 8 = 22, 24 + 8 = 32, and the
+like--need to be specifically practiced until the tendency becomes
+generalized. 'Counting' by 2s beginning with 1, and with 2, counting by
+3s beginning with 1, with 2, and with 3, counting by 4s beginning with
+1, with 2, with 3, and with 4, and so on, make easy beginnings in the
+formation of the decade connections. Practice with isolated bonds should
+soon be added to get freer use of the bonds. The work of column addition
+should be checked for accuracy so that a pupil will continually get
+beneficial practice rather than 'practice in error.'
+
+(3) _The uneven divisions._--The quotients with remainders for the
+divisions of every number to 19 by 2, every number to 29 by 3, every
+number to 39 by 4, and so on should be taught as well as the even
+divisions. A table like the following will be found a convenient means
+of making these connections:--
+
+  10 = .... 2s
+  10 = .... 3s and .... rem.
+  10 = .... 4s and .... rem.
+  10 = .... 5s
+  11 = .... 2s and .... rem.
+  11 = .... 3s and .... rem.
+  .
+  .
+  .
+  89 = .... 9s and .... rem.
+
+These bonds must be formed before short division can be efficient, are
+useful as a partial help toward selection of the proper quotient figures
+in long division, and are the chief instruments for one of the important
+problem series in applied arithmetic,--"How many _x_s can I buy for _y_
+cents at _z_ cents per _x_ and how much will I have left?" That these
+bonds are at present sadly neglected is shown by Kirby ['13], who found
+that pupils in the last half of grade 3 and the first half of grade 4
+could do only about four such examples per minute (in a ten-minute
+test), and even at that rate made far from perfect records, though they
+had been taught the regular division tables. Sixty minutes of practice
+resulted in a gain of nearly 75 percent in number done per minute, with
+an increase in accuracy as well.
+
+(4) _The equation form._--The equation form with an unknown quantity to
+be determined, or a missing number to be found, should be connected with
+its meaning and with the problem attitude long before a pupil begins
+algebra, and in the minds of pupils who never will study algebra.
+
+Children who have just barely learned to add and subtract learn easily
+to do such work as the following:--
+
+Write the missing numbers:--
+
+  4 + 8 = ....
+  5 + .... = 14
+  .... + 3 = 11
+  .... = 5 + 2
+  16 = 7 + ....
+  12 = .... + 5
+
+The equation form is the simplest uniform way yet devised to state a
+quantitative issue. It is capable of indefinite extension if certain
+easily understood conventions about parentheses and fraction signs are
+learned. It should be employed widely in accounting and the treatment of
+commercial problems, and would be except for outworn conventions. It is
+a leading contribution of algebra to business and industrial life.
+Arithmetic can make it nearly as well. It saves more time in the case of
+drills on reducing fractions to higher and lower terms alone than is
+required to learn its meaning and use. To rewrite a quantitative
+problem as an equation and then make the easy selection of the
+necessary technique to solve the equation is one of the most universally
+useful intellectual devices known to man. The words 'equals,' 'equal,'
+'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer
+equivalents should therefore early give way on many occasions to the '='
+which so far surpasses them in ultimate convenience and simplicity.
+
+(5) _Addition and subtraction facts in the case of fractions._--In the
+case of adding and subtracting fractions, certain specific
+bonds--between the situation of halves and thirds to be added and the
+responses of thinking of the numbers as equal to so many sixths, between
+the situation thirds and fourths to be added and thinking of them as so
+many twelfths, between fourths and eighths to be added and thinking of
+them as eighths, and the like--should be formed separately. The general
+rule of thinking of fractions as their equivalents with some convenient
+denominator should come as an organization and extension of such special
+habits, not as an edict from the textbook or teacher.
+
+(6) _Fractional equivalents._--Efficiency requires that in the end the
+much used reductions should be firmly connected with the situations
+where they are needed. They may as well, therefore, be so connected from
+the beginning, with the gain of making the general process far easier
+for the dull pupils to master. We shall see later that, for all save the
+very gifted pupils, the economical way to get an understanding of
+arithmetical principles is not, usually, to learn a rule and then apply
+it, but to perform instructive operations and, in the course of
+performing them, to get insight into the principles.
+
+(7) _Protective habits in multiplying and dividing with fractions._--In
+multiplying and dividing with fractions special bonds should be formed
+to counteract the now harmful influence of the 'multiply = get a larger
+number' and 'divide = get a smaller number' bonds which all work with
+integers has been reenforcing.
+
+For example, at the beginning of the systematic work with multiplication
+by a fraction, let the following be printed clearly at the top of every
+relevant page of the textbook and displayed on the blackboard:--
+
+_When you multiply a number by anything more than 1 the result is larger
+than the number._
+
+_When you multiply a number by 1 the result is the same as the number._
+
+_When you multiply a number by anything less than 1 the result is
+smaller than the number._
+
+Let the pupils establish the new habit by many such exercises as:--
+
+   18 x 4 = ....          9 x 2 = ....
+    4 x 4 = ....          6 x 2 = ....
+    2 x 4 = ....          3 x 2 = ....
+    1 x 4 = ....          1 x 2 = ....
+  1/2 x 4 = ....        1/3 x 2 = ....
+  1/4 x 4 = ....        1/6 x 2 = ....
+  1/8 x 4 = ....        1/9 x 2 = ....
+
+In the case of division by a fraction the old harmful habit should be
+counteracted and refined by similar rules and exercises as follows:--
+
+_When you divide a number by anything more than 1 the result is smaller
+than the number._
+
+_When you divide a number by 1 the result is the same as the number._
+
+_When you divide a number by anything less than 1 the result is larger
+than the number._
+
+State the missing numbers:--
+
+    8 = .... 4s      12 = .... 6s      9 = .... 9s
+    8 = .... 2s      12 = .... 4s      9 = .... 3s
+    8 = .... 1s      12 = .... 3s      9 = .... 1s
+    8 = .... 1/2s    12 = .... 2s      9 = .... 1/3s
+    8 = .... 1/4s    12 = .... 1s      9 = .... 1/9s
+    8 = .... 1/8s    12 = .... 1/2s
+                     12 = .... 1/3s
+                     12 = .... 1/4s
+
+  16 / 16  =         9 /  9  =      10 / 10   =    12 /  6  =
+  16 /  8  =         9 /  3  =      10 /  5   =    12 /  4  =
+  16 /  4  =         9 /  1  =      10 /  1   =    12 /  3  =
+  16 /  2  =         9 / 1/3 =      10 / 1/5  =    12 /  2  =
+  16 /  1  =         9 / 1/9 =      10 / 1/10 =    12 /  1  =
+  16 / 1/2 =                                       12 / 1/2 =
+  16 / 1/4 =                                       12 / 1/3 =
+  16 / 1/8 =                                       12 / 1/4 =
+                                                   12 / 1/6 =
+
+(8) _'% of' means 'hundredths times'._--In the case of percentage a
+series of bonds like the following should be formed:--
+
+   5 percent of = .05 times
+  20  "   "   " = .20   "
+   6  "   "   " = .06   "
+  25%         " = .25 x
+  12%         " = .12 x
+   3%         " = .03 x
+
+Four five-minute drills on such connections between '_x_ percent of' and
+'its decimal equivalent times' are worth an hour's study of verbal
+definitions of the meaning of percent as per hundred or the like. The
+only use of the study of such definitions is to facilitate the later
+formation of the bonds, and, with all save the brighter pupils, the
+bonds are more needed for an understanding of the definitions than the
+definitions are needed for the formation of the bonds.
+
+(9) _Habits of verifying results._--Bonds should early be formed between
+certain manipulations of numbers and certain means of checking, or
+verifying the correctness of, the manipulation in question. The
+additions to 9 + 9 and the subtractions to 18 - 9 should be verified by
+objective addition and subtraction and counting until the pupil has sure
+command; the multiplications to 9 x 9 should be verified by objective
+multiplication and counting of the result (in piles of tens and a pile
+of ones) eight or ten times,[4] and by addition eight or ten times;[4]
+the divisions to 81 / 9 should be verified by multiplication and
+occasionally objectively until the pupil has sure command; column
+addition should be checked by adding the columns separately and adding
+the sums so obtained, and by making two shorter tasks of the given task
+and adding the two sums; 'short' multiplication should be verified eight
+or ten times by addition; 'long' multiplication should be checked by
+reversing multiplier and multiplicand and in other ways; 'short' and
+'long' division should be verified by multiplication.
+
+  [4] Eight or ten times _in all_, not eight or ten times for each fact
+      of the tables.
+
+These habits of testing an obtained result are of threefold value. They
+enable the pupil to find his own errors, and to maintain a standard of
+accuracy by himself. They give him a sense of the relations of the
+processes and the reasons why the right ways of adding, subtracting,
+multiplying, and dividing are right, such as only the very bright pupils
+can get from verbal explanations. They put his acquisition of a certain
+power, say multiplication, to a real and intelligible use, in checking
+the results of his practice of a new power, and so instill a respect for
+arithmetical power and skill in general. The time spent in such
+verification produces these results at little cost; for the practice in
+adding to verify multiplications, in multiplying to verify divisions,
+and the like is nearly as good for general drill and review of the
+addition and multiplication themselves as practice devised for that
+special purpose.
+
+Early work in adding, subtracting, and reducing fractions should be
+verified by objective aids in the shape of lines and areas divided
+in suitable fractional parts. Early work with decimal fractions
+should be verified by the use of the equivalent common fractions
+for .25, .75, .125, .375, and the like. Multiplication and division
+with fractions, both common and decimal, should in the early stages
+be verified by objective aids. The placing of the decimal point in
+multiplication and division with decimal fractions should be verified
+by such exercises as:--
+
+           20  It cannot be 200; for 200 x 1.23 is much more than 24.6.
+       ______  It cannot be 2; for 2 x 1.23 is much less than 24.6.
+  1.23 )24.60
+        246
+       ----
+
+The establishment of habits of verifying results and their use is very
+greatly needed. The percentage of wrong answers in arithmetical work in
+schools is now so high that the pupils are often being practiced in
+error. In many cases they can feel no genuine and effective confidence
+in the processes, since their own use of the processes brings wrong
+answers as often as right. In solving problems they often cannot decide
+whether they have done the right thing or the wrong, since even if they
+have done the right thing, they may have done it inaccurately. A wrong
+answer to a problem is therefore too often ambiguous and uninstructive
+to them.[5]
+
+  [5] The facts concerning the present inaccuracy of school work in
+      arithmetic will be found on pages 102 to 105.
+
+These illustrations of the last few pages are samples of the procedures
+recommended by a consideration of all the bonds that one might form and
+of the contribution that each would make toward the abilities that the
+study of arithmetic should develop and improve. It is by doing more or
+less at haphazard what psychology teaches us to do deliberately and
+systematically in this respect that many of the past advances in the
+teaching of arithmetic have been made.
+
+
+WASTEFUL AND HARMFUL BONDS
+
+A scrutiny of the bonds now formed in the teaching of arithmetic with
+questions concerning the exact service of each, results in a list of
+bonds of small value or even no value, so far as a psychologist can
+determine. I present here samples of such psychologically unjustifiable
+bonds with some of the reasons for their deficiencies.
+
+(1) _Arbitrary units._--In drills intended to improve the ability to see
+and use the meanings of numbers as names for ratios or relative
+magnitudes, it is unwise to employ entirely arbitrary units. The
+procedure in II (on page 84) is better than that in I. Inches,
+half-inches, feet, and centimeters are better as units of length than
+arbitrary As. Square inches, square centimeters, and square feet are
+better for areas. Ounces and pounds should be lifted rather than
+arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and
+cubic inches are better for volume.
+
+All the real merit in the drills on relative magnitude advocated by
+Speer, McLellan and Dewey, and others can be secured without spending
+time in relating magnitudes for the sake of relative magnitude alone.
+The use of units of measure in drills which will never be used in _bona
+fide_ measuring is like the use of fractions like sevenths, elevenths,
+and thirteenths. A very little of it is perhaps desirable to test the
+appreciation of certain general principles, but for regular training it
+should give place to the use of units of practical significance.
+
+    [Illustration: FIG. 3.
+
+      A ----------
+      B ------------------------------
+      C --------------------
+      D ----------------------------------------
+
+    I. If _A_ is 1 which line is 2? Which line is 4? Which line is 3?
+    _A_ and _C_ together equal what line? _A_ and _B_ together equal
+    what line? How much longer is _B_ than _A_? How much longer is _B_
+    than _C_? How much longer is _D_ than _A_?]
+
+    [Illustration: FIG. 4.
+
+      A ----------
+      B ------------------------------
+      C --------------------
+      D ----------------------------------------
+
+    II. _A_ is 1 inch long. Which line is 2 inches long? Which line is
+    4 inches long? Which line is 3 inches long? _A_ and _C_ together
+    make ... inches? _A_ and _B_ together make ... inches? _B_ is ...
+    ... longer than _A_? _B_ is ... ... longer than _C_? _D_ is ...
+    ... longer than _A_?]
+
+(2) _Multiples of 11._--The multiplications of 2 to 12 by 11 and 12 as
+single connections should be left for the pupil to acquire by himself as
+he needs them. These connections interfere with the process of learning
+two-place multiplication. The manipulations of numbers there required
+can be learned much more easily if 11 and 12 are used as multipliers in
+just the same way that 78 or 96 would be. Later the 12 x 2, 12 x 3,
+etc., may be taught. There is less reason for knowing the multiples
+of 11 than for knowing the multiples of 15, 16, or 25.
+
+(3) _Abstract and concrete numbers._--The elaborate emphasis of the
+supposed fact that we cannot multiply 726 by 8 dollars and the still
+more elaborate explanations of why nevertheless we find the cost of 726
+articles at $8 each by multiplying 726 by 8 and calling the answer
+dollars are wasteful. The same holds of the corresponding pedantry about
+division. These imaginary difficulties should not be raised at all. The
+pupil should not think of multiplying or dividing men or dollars, but
+simply of the necessary equation and of the sort of thing that the
+missing number represents. "8 x 726 = .... Answer is dollars," or
+"8, 726, multiply. Answer is dollars," is all that he needs to think,
+and is in the best form for his thought. Concerning the distinction
+between abstract and concrete numbers, both logic and common sense as
+well as psychology support the contention of McDougle ['14, p. 206f.],
+who writes:--
+
+"The most elementary counting, even that stage when the counts were not
+carried in the mind, but merely in notches on a stick or by DeMorgan's
+stones in a pot, requires some thought; and the most advanced counting
+implies memory of things. The terms, therefore, abstract and concrete
+number, have long since ceased to be used by thinking people.
+
+"Recently the writer visited an arithmetic class in a State Normal
+School and saw a group of practically adult students confused about this
+very question concerning abstract and concrete numbers, according to
+their previous training in the conventionalities of the textbook. Their
+teacher diverted the work of the hour and she and the class spent almost
+the whole period in reestablishing the requirements 'that the product
+must always be the same kind of unit as the multiplicand,' and 'addends
+must all be alike to be added.' This is not an exceptional case.
+Throughout the whole range of teaching arithmetic in the public schools
+pupils are obfuscated by the philosophical encumbrances which have been
+imposed upon the simplest processes of numerical work. The time is
+surely ripe, now that we are readjusting our ideas of the subject of
+arithmetic, to revise some of these wasteful and disheartening
+practices. Algebra historically grew out of arithmetic, yet it has not
+been laden with this distinction. No pupil in algebra lets _x_ equal the
+horses; he lets _x_ equal the _number_ of horses, and proceeds to drop
+the idea of horses out of his consideration. He multiplies, divides, and
+extracts the root of the _number_, sometimes handling fractions in the
+process, and finally interprets the result according to the conditions
+of his problem. Of course, in the early number work there have been the
+sense-objects from which number has been perceived, but the mind
+retreats naturally from objectivity to the pure conception of number,
+and then to the number symbol. The following is taken from the appendix
+to Horn's thesis, where a seventh grade girl gets the population of the
+United States in 1820:--
+
+  7,862,166 whites
+    233,634 free negroes
+  1,538,022 slaves
+  ---------
+  9,633,822
+
+In this problem three different kinds of addends are combined, if we
+accept the usual distinction. Some may say that this is a mistake,--that
+the pupil transformed the 'whites,' 'free negroes,' and 'slaves' into a
+common unit, such as 'people' of 'population' and then added these
+common units. But this 'explanation' is entirely gratuitous, as one will
+find if he questions the pupil about the process. It will be found that
+the child simply added the figures as numbers only and then interpreted
+the result, according to the statement of the problem, without so much
+mental gymnastics. The writer has questioned hundreds of students in
+Normal School work on this point, and he believes that the ordinary
+mind-movement is correctly set forth here, no matter how well one may
+maintain as an academic proposition that this is not logical. Many
+classes in the Eastern Kentucky State Normal have been given this
+problem to solve, and they invariably get the same result:--
+
+'In a garden on the Summit are as many cabbage-heads as the total number
+of ladies and gentlemen in this class. How many cabbage-heads in the
+garden?'
+
+And the blackboard solution looks like this each time:--
+
+  29 ladies
+  15 gentlemen
+  --
+  44 cabbage-heads
+
+So, also, one may say: I have 6 times as many sheep as you have cows. If
+you have 5 cows, how many sheep have I? Here we would multiply the
+number of cows, which is 5, by 6 and call the result 30, which must be
+linked with the idea of sheep because the conditions imposed by the
+problem demand it. The mind naturally in this work separates the pure
+number from its situation, as in algebra, handles it according to the
+laws governing arithmetical combinations, and labels the result as the
+statement of the problem demands. This is expressed in the following,
+which is tacitly accepted in algebra, and should be accepted equally in
+arithmetic:
+
+'In all computations and operations in arithmetic, all numbers are
+essentially abstract and should be so treated. They are concrete only in
+the thought process that attends the operation and interprets the
+result.'"
+
+(4) _Least common multiple._--The whole set of bonds involved in
+learning 'least common multiple' should be left out. In adding and
+subtracting fractions the pupil should _not_ find the least common
+multiple of their denominators but should find any common multiple that
+he can find quickly and correctly. No intelligent person would ever
+waste time in searching for the least common multiple of sixths, thirds,
+and halves except for the unfortunate traditions of an oversystematized
+arithmetic, but would think of their equivalents in sixths or twelfths
+or twenty-fourths or _any other convenient common multiple_. The process
+of finding the least common multiple is of such exceedingly rare
+application in science or business or life generally that the textbooks
+have to resort to purely fantastic problems to give drill in its use.
+
+(5) _Greatest common divisor._--The whole set of bonds involved in
+learning 'greatest common divisor' should also be left out. In reducing
+fractions to lowest terms the pupil should divide by anything that he
+sees that he can divide by, favoring large divisors, and continue doing
+so until he gets the fraction in terms suitable for the purpose in hand.
+The reader probably never has had occasion to compute a greatest common
+divisor since he left school. If he has computed any, the chances are
+that he would have saved time by solving the problem in some other way!
+
+The following problems are taken at random from those given by one of
+the best of the textbooks that make the attempt to apply the facts of
+Greatest Common Divisor and Least Common Multiple to problems.[6] Most
+of these problems are fantastic. The others are trivial, or are better
+solved by trial and adaptation.
+
+  1. A certain school consists of 132 pupils in the high school, 154
+     in the grammar, and 198 in the primary grades. If each group is
+     divided into sections of the same number containing as many pupils
+     as possible, how many pupils will there be in each section?
+
+  2. A farmer has 240 bu. of wheat and 920 bu. of oats, which he
+     desires to put into the least number of boxes of the same capacity,
+     without mixing the two kinds of grain. Find how many bushels each
+     box must hold.
+
+  3. Four bells toll at intervals of 3, 7, 12, and 14 seconds
+     respectively, and begin to toll at the same instant. When will
+     they next toll together?
+
+  4. A, B, C, and D start together, and travel the same way around an
+     island which is 600 mi. in circuit. A goes 20 mi. per day, B 30,
+     C 25, and D 40. How long must their journeying continue, in order
+     that they may all come together again?
+
+  5. The periods of three planets which move uniformly in circular
+     orbits round the sun, are respectively 200, 250, and 300 da.
+     Supposing their positions relatively to each other and the sun
+     to be given at any moment, determine how many da. must elapse
+     before they again have exactly the same relative positions.
+
+  [6] McLellan and Ames, _Public School Arithmetic_ [1900].
+
+(6) _Rare and unimportant words._--The bonds between rare or unimportant
+words and their meanings should not be formed for the mere sake of
+verbal variety in the problems of the textbook. A pupil should not be
+expected to solve a problem that he cannot read. He should not be
+expected in grades 2 and 3, or even in grade 4, to read words that he
+has rarely or never seen before. He should not be given elaborate drill
+in reading during the time devoted to the treatment of quantitative
+facts and relations.
+
+All this is so obvious that it may seem needless to relate. It is not.
+With many textbooks it is now necessary to give definite drill in
+reading the words in the printed problems intended for grades 2, 3, and
+4, or to replace them by oral statements, or to leave the pupils in
+confusion concerning what the problems are that they are to solve. Many
+good teachers make a regular reading-lesson out of every page of
+problems before having them solved. There should be no such necessity.
+
+To define _rare_ and _unimportant_ concretely, I will say that for
+pupils up to the middle of grade 3, such words as the following are rare
+and unimportant (though each of them occurs in the very first fifty
+pages of some well-known beginner's book in arithmetic).
+
+  absentees
+  account
+  Adele
+  admitted
+  Agnes
+  agreed
+  Albany
+  Allen
+  allowed
+  alternate
+  Andrew
+  Arkansas
+  arrived
+  assembly
+  automobile
+  baking powder
+  balance
+  barley
+  beggar
+  Bertie
+  Bessie
+  bin
+  Boston
+  bouquet
+  bronze
+  buckwheat
+  Byron
+  camphor
+  Carl
+  Carrie
+  Cecil
+  Charlotte
+  charity
+  Chicago
+  cinnamon
+  Clara
+  clothespins
+  collect
+  comma
+  committee
+  concert
+  confectioner
+  cranberries
+  crane
+  currants
+  dairyman
+  Daniel
+  David
+  dealer
+  debt
+  delivered
+  Denver
+  department
+  deposited
+  dictation
+  discharged
+  discover
+  discovery
+  dish-water
+  drug
+  due
+  Edgar
+  Eddie
+  Edwin
+  election
+  electric
+  Ella
+  Emily
+  enrolled
+  entertainment
+  envelope
+  Esther
+  Ethel
+  exceeds
+  explanation
+  expression
+  generally
+  gentlemen
+  Gilbert
+  Grace
+  grading
+  Graham
+  grammar
+  Harold
+  hatchet
+  Heralds
+  hesitation
+  Horace Mann
+  impossible
+  income
+  indicated
+  inmost
+  inserts
+  installments
+  instantly
+  insurance
+  Iowa
+  Jack
+  Jennie
+  Johnny
+  Joseph
+  journey
+  Julia
+  Katherine
+  lettuce-plant
+  library
+  Lottie
+  Lula
+  margin
+  Martha
+  Matthew
+  Maud
+  meadow
+  mentally
+  mercury
+  mineral
+  Missouri
+  molasses
+  Morton
+  movements
+  muslin
+  Nellie
+  nieces
+  Oakland
+  observing
+  obtained
+  offered
+  office
+  onions
+  opposite
+  original
+  package
+  packet
+  palm
+  Patrick
+  Paul
+  payments
+  peep
+  Peter
+  perch
+  phaeton
+  photograph
+  piano
+  pigeons
+  Pilgrims
+  preserving
+  proprietor
+  purchased
+  Rachel
+  Ralph
+  rapidity
+  rather
+  readily
+  receipts
+  register
+  remanded
+  respectively
+  Robert
+  Roger
+  Ruth
+  rye
+  Samuel
+  San Francisco
+  seldom
+  sheared
+  shingles
+  skyrockets
+  sloop
+  solve
+  speckled
+  sponges
+  sprout
+  stack
+  Stephen
+  strap
+  successfully
+  suggested
+  sunny
+  supply
+  Susan
+  Susie's
+  syllable
+  talcum
+  term
+  test
+  thermometer
+  Thomas
+  torpedoes
+  trader
+  transaction
+  treasury
+  tricycle
+  tube
+  two-seated
+  united
+  usually
+  vacant
+  various
+  vase
+  velocipede
+  votes
+  walnuts
+  Walter
+  Washington
+  watched
+  whistle
+  woodland
+  worsted
+
+(7) _Misleading facts and procedures._--Bonds should not be formed
+between articles of commerce and grossly inaccurate prices therefor,
+between events and grossly improbable consequences, or causes or
+accompaniments thereof, nor between things, qualities, and events which
+have no important connections one with another in the real world. In
+general, things should not be put together in the pupil's mind that do
+not belong together.
+
+If the reader doubts the need of this warning let him examine problems 1
+to 5, all from reputable books that are in common use, or have been
+within a few years, and consider how addition, subtraction, and the
+habits belonging with each are confused by exercise 6.
+
+  1. If a duck flying 3/5 as fast as a hawk flies 90 miles in an hour,
+     how fast does the hawk fly?
+
+  2. At 5/8 of a cent apiece how many eggs can I buy for $60?
+
+  3. At $.68 a pair how many pairs of overshoes can you buy for $816?
+
+  4. At $.13 a dozen how many dozen bananas can you buy for $3.12?
+
+  5. How many pecks of beans can be put into a box that will hold just
+     21 bushels?
+
+  6. Write answers:
+
+     537    Beginning at the bottom say 11, 18, and 2 (writing it in
+     365    its place) are 20. 5, 11, 14, and 6 (writing it) are 20,
+       ?    5, 10. The number, omitted, is 62.
+      36
+    ----
+    1000
+
+  _a._  581     _b._  625     _c._  752     _d._  314     _e._    ?
+         97             ?           414           429           845
+        364            90           130             ?           223
+          ?           417             ?            76            95
+       ----          ----          ----          ----          ----
+       1758          2050          2460          1000          2367
+
+(8) _Trivialities and absurdities._--Bonds should not be formed between
+insignificant or foolish questions and the labor of answering them,
+nor between the general arithmetical work of the school and such
+insignificant or foolish questions. The following are samples from
+recent textbooks of excellent standing:--
+
+  On one side of George's slate there are 32 words, and on the other
+  side 26 words. If he erases 6 words from one side, and 8 from the
+  other, how many words remain on his slate?
+
+  A certain school has 14 rooms, and an average of 40 children in a
+  room. If every one in the school should make 500 straight marks on
+  each side of his slate, how many would be made in all?
+
+  8 times the number of stripes in our flag is the number of years
+  from 1800 until Roosevelt was elected President. In what year was
+  he elected President?
+
+  From the Declaration of Independence to the World's Fair in Chicago
+  was 9 times as many years as there are stripes in the flag. How
+  many years was it?
+
+(9) _Useless methods._--Bonds should not be formed between a described
+situation and a method of treating the situation which would not be a
+useful one to follow in the case of the real situation. For example, "If
+I set 96 trees in rows, sixteen trees in a row, how many rows will I
+have?" forms the habit of treating by division a problem that in reality
+would be solved by counting the rows. So also "I wish to give 25 cents
+to each of a group of boys and find that it will require $2.75. How many
+boys are in the group?" forms the habit of answering a question by
+division whose answer must already have been present to give the data of
+the problem.
+
+(10) _Problems whose answers would, in real life, be already
+known._--The custom of giving problems in textbooks which could not
+occur in reality because the answer has to be known to frame the problem
+is a natural result of the lazy author's tendency to work out a problem
+to fit a certain process and a certain answer. Such bogus problems are
+very, very common. In a random sampling of a dozen pages of "General
+Review" problems in one of the most widely used of recent textbooks, I
+find that about 6 percent of the problems are of this sort. Among the
+problems extemporized by teachers these bogus problems are probably
+still more frequent. Such are:--
+
+  A clerk in an office addressed letters according to a given list.
+  After she had addressed 2500, 4/9 of the names on the list had not
+  been used; how many names were in the entire list?
+
+  The Canadian power canal at Sault Ste. Marie furnished 20,000
+  horse power. The canal on the Michigan side furnished 2-1/2
+  times as much. How many horse power does the latter furnish?
+
+It may be asserted that the ideal of giving as described problems only
+problems that might occur and demand the same sort of process for
+solution with a real situation, is too exacting. If a problem is
+comprehensible and serves to illustrate a principle or give useful
+drill, that is enough, teachers may say. For really scientific teaching
+it is not enough. Moreover, if problems are given merely as tests of
+knowledge of a principle or as means to make some fact or principle
+clear or emphatic, and are not expected to be of direct service in the
+quantitative work of life, it is better to let the fact be known. For
+example, "I am thinking of a number. Half of this number is twice six.
+What is the number?" is better than "A man left his wife a certain sum
+of money. Half of what he left her was twice as much as he left to his
+son, who receives $6000. How much did he leave his wife?" The former is
+better because it makes no false pretenses.
+
+(11) _Needless linguistic difficulties._--It should be unnecessary to
+add that bonds should not be formed between the pupil's general attitude
+toward arithmetic and needless, useless difficulty in language or
+needless, useless, wrong reasoning. Our teaching is, however, still
+tainted by both of these unfortunate connections, which dispose the
+pupil to think of arithmetic as a mystery and folly.
+
+Consider, for example, the profitless linguistic difficulty of problems
+1-6, whose quantitative difficulties are simply those of:--
+
+  1.  5 + 8 + 3 + 7
+  2.  64 / 8, and knowledge that 1 peck = 8 quarts
+  3.  12 / 4
+  4.  6 / 2
+  5.  3 x 2
+  6.  4 x 4
+
+  1. What amount should you obtain by putting together 5 cents, 8
+     cents, 3 cents, and 7 cents? Did you find this result by adding or
+     multiplying?
+
+  2. How many times must you empty a peck measure to fill a basket
+     holding 64 quarts of beans?
+
+  3. If a girl commits to memory 4 pages of history in one day, in
+     how many days will she commit to memory 12 pages?
+
+  4. If Fred had 6 chickens how many times could he give away 2
+     chickens to his companions?
+
+  5. If a croquet-player drove a ball through 2 arches at each
+     stroke, through how many arches will he drive it by 3 strokes?
+
+  6. If mamma cut the pie into 4 pieces and gave each person a piece,
+     how many persons did she have for dinner if she used 4 whole pies
+     for dessert?
+
+Arithmetically this work belongs in the first or second years of
+learning. But children of grades 2 and 3, save a few, would be utterly
+at a loss to understand the language.
+
+We are not yet free from the follies illustrated in the lessons of pages
+96 to 99, which mystified our parents.
+
+LESSON I
+
+    [Illustration: FIG. 5.]
+
+  1. In this picture, how many girls are in the swing?
+
+  2. How many girls are pulling the swing?
+
+  3. If you count both girls together, how many are they?
+     _One_ girl and _one_ other girl are how many?
+
+  4. How many kittens do you see on the stump?
+
+  5. How many on the ground?
+
+  6. How many kittens are in the picture? One kitten and one other
+     kitten are how many?
+
+  7. If you should ask me how many girls are in the swing, or how
+     many kittens are on the stump, I could answer aloud, _One_; or I
+     could write _One_; or thus, _1_.
+
+  8. If I write _One_, this is called the _word One_.
+
+  9. This, _1_, is named a _figure One_, because it means the same as
+     the word _One_, and stands for _One_.
+
+ 10. Write 1. What is this named? Why?
+
+ 11. A figure 1 may stand for _one_ girl, _one_ kitten, or _one_
+     anything.
+
+ 12. When children first attend school, what do they begin to learn?
+     _Ans._ Letters and words.
+
+ 13. Could you read or write before you had learned either letters
+     or words?
+
+ 14. If we have all the _letters_ together, they are named the
+     Alphabet.
+
+ 15. If we write or speak _words_, they are named Language.
+
+ 16. You are commencing to study Arithmetic; and you can read and
+     write in Arithmetic only as you learn the Alphabet and Language
+     of Arithmetic. But little time will be required for this purpose.
+
+LESSON II
+
+    [Illustration: FIG. 6.]
+
+  1. If we speak or write words, what do we name them, when taken
+     together?
+
+  2. What are you commencing to study? _Ans._ Arithmetic.
+
+  3. What Language must you now learn?
+
+  4. What do we name this, 1? Why?
+
+  5. This figure, 1, is part of the Language of Arithmetic.
+
+  6. If I should write something to stand for _Two_--_two_ girls,
+     _two_ kittens, or _two_ things of any kind--what do you think we
+     would name it?
+
+  7. A _figure Two_ is written thus: _2._ Make a _figure two_.
+
+  8. Why do we name this a _figure two_?
+
+  9. This figure two (2) is part of the Language of Arithmetic.
+
+ 10. In this picture one boy is sitting, playing a flageolet. What
+     is the other boy doing? If the boy standing should sit down by the
+     other, how many boys would be sitting together? One boy and one
+     other boy are how many boys?
+
+ 11. You see a flageolet and a violin. They are musical instruments.
+     One musical instrument and one other musical instrument are how
+     many?
+
+ 12. I will write thus: 1 1 2. We say that 1 boy and 1 other boy,
+     counted together, are 2 boys; or are equal to 2 boys. We will now
+     write something to show that the first 1 and the other 1 are to be
+     counted together.
+
+ 13. We name a line drawn thus, -, a _horizontal line_. Draw such a
+     line. Name it.
+
+ 14. A line drawn thus, |, we name a _vertical line_. Draw such a
+     line. Name it.
+
+ 15. Now I will put two such lines together; thus, +. What kind of a
+     line do we name the first (-)? And what do we name the last? (|)?
+     Are these lines long or short? Where do they cross each other?
+
+ 16. Each of you write thus: -, |, +.
+
+ 17. This, +, is named _Plus_. _Plus_ means _more_; and + also means
+     _more_.
+
+ 18. I will write.
+
+        _One and One More Equal Two._
+
+ 19. Now I will write part of this in the Language of Arithmetic.
+     I write the first _One_ thus, 1; then the other _One_ thus, 1.
+     Afterward I write, for the word _More_, thus, +, placing
+     the + between 1 and 1, so that the whole stands thus: 1 + 1.
+     As I write, I say, _One and One more_.
+
+ 20. Each of you write 1 + 1. Read what you have written.
+
+ 21. This +, when written between the 1s, shows that they are to be
+     put together, or counted together, so as to make 2.
+
+ 22. Because + shows what is to be done, it is called a _Sign_. If
+     we take its name, _Plus_, and the word _Sign_, and put both words
+     together, we have _Sign Plus_, or _Plus Sign_. In speaking of this
+     we may call it _Sign Plus_, or _Plus Sign_, or _Plus_.
+
+ 23. 1, 2, +, are part of the Language of Arithmetic.
+
+        _Write the following in the Language of Arithmetic_:
+
+ 24. One and one more.
+
+ 25. One and two more.
+
+ 26. Two and one more.
+
+(12) _Ambiguities and falsities._--Consider the ambiguities and false
+reasoning of these problems.
+
+  1. If you can earn 4 cents a day, how much can you earn in 6 weeks?
+     (Are Sundays counted? Should a child who earns 4 cents some day
+     expect to repeat the feat daily?)
+
+  2. How many lines must you make to draw ten triangles and five
+     squares? (I can do this with 8 lines, though the answer the book
+     requires is 50.)
+
+  3. A runner ran twice around an 1/8 mile track in two minutes. What
+     distance did he run in 2/3 of a minute? (I do not know, but I do
+     know that, save by chance, he did not run exactly 2/3 of 1/8 mile.)
+
+  4. John earned $4.35 in a week, and Henry earned $1.93. They put
+     their money together and bought a gun. What did it cost? (Maybe $5,
+     maybe $10. Did they pay for the whole of it? Did they use all their
+     earnings, or less, or more?)
+
+  5. Richard has 12 nickels in his purse. How much more than 50 cents
+     would you give him for them? (Would a wise child give 60 cents to a
+     boy who wanted to swap 12 nickels therefor, or would he suspect a
+     trick and hold on to his own coins?)
+
+  6. If a horse trots 10 miles in one hour how far will he travel in
+     9 hours?
+
+  7. If a girl can pick 3 quarts of berries in 1 hour how many quarts
+     can she pick in 3 hours?
+
+       (These last two, with a teacher insisting on the 90 and 9, might
+       well deprive a matter-of-fact boy of respect for arithmetic for
+       weeks thereafter.)
+
+     The economics and physics of the next four problems speak for
+     themselves.
+
+  8. I lost $15 by selling a horse for $85. What was the value of the
+     horse?
+
+  9. If floating ice has 7 times as much of it under the surface of
+     the water as above it, what part is above water? If an iceberg is
+     50 ft. above water, what is the entire height of the iceberg? How
+     high above water would an iceberg 300 ft. high have to be?
+
+ 10. A man's salary is $1000 a year and his expenses $625. How many
+     years will elapse before he is worth $10,000 if he is worth $2500
+     at the present time?
+
+ 11. Sound travels 1120 ft. a second. How long after a cannon is
+     fired in New York will the report be heard in Philadelphia, a
+     distance of 90 miles?
+
+
+GUIDING PRINCIPLES
+
+The reader may be wearied of these special details concerning bonds now
+neglected that should be formed and useless or harmful bonds formed for
+no valid reason. Any one of them by itself is perhaps a minor matter,
+but when we have cured all our faults in this respect and found all the
+possibilities for wiser selection of bonds, we shall have enormously
+improved the teaching of arithmetic. The ideal is such choice of bonds
+(and, as will be shown later, such arrangement of them) as will most
+improve the functions in question at the least cost of time and effort.
+The guiding principles may be kept in mind in the form of seven simple
+but golden rules:--
+
+1. Consider the situation the pupil faces.
+
+2. Consider the response you wish to connect with it.
+
+3. Form the bond; do not expect it to come by a miracle.
+
+4. Other things being equal, form no bond that will have to be broken.
+
+5. Other things being equal, do not form two or three bonds when one
+will serve.
+
+6. Other things being equal, form bonds in the way that they are
+required later to act.
+
+7. Favor, therefore, the situations which life itself will offer, and
+the responses which life itself will demand.
+
+
+
+
+CHAPTER V
+
+THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS
+
+
+An inventory of the bonds to be formed in learning arithmetic should be
+accompanied by a statement of how strong each bond is to be made and
+kept year by year. Since, however, the inventory itself has been
+presented here only in samples, the detailed statement of desired
+strength for each bond cannot be made. Only certain general facts will
+be noted here.
+
+
+THE NEED OF STRONGER ELEMENTARY BONDS
+
+The constituent bonds involved in the fundamental operations with
+numbers need to be much stronger than they now are. Inaccuracy in these
+operations means weakness of the constituent bonds. Inaccuracy exists,
+and to a degree that deprives the subject of much of its possible
+disciplinary value, makes the pupil's achievements of slight value for
+use in business or industry, and prevents the pupil from verifying his
+work with new processes by some previously acquired process.
+
+The inaccuracy that exists may be seen in the measurements made by the
+many investigators who have used arithmetical tasks as tests of fatigue,
+practice, individual differences and the like, and in the special
+studies of arithmetical achievements for their own sake made by Courtis
+and others.
+
+Burgerstein ['91], using such examples as
+
+    28704516938276546397
+  + 35869427359163827263
+  ----------------------
+
+and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5
+or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred
+answer-figures, or 3/5 of an error per example. The children were 9-1/2
+to 15 years old. Laser ['94], using the same sort of addition and
+multiplication, found somewhat over 3 errors per hundred answer-figures
+in the case of boys and girls averaging 11-1/2 years, during the period
+of their most accurate work. Holmes ['95], using addition of the sort
+just described, found 346 errors in 23,713 answer-figures or about 1-1/2
+per hundred. The children were from all grades from the third to the
+eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained
+per minute. Friedrich ['97] with similar examples, giving the very long
+time of 20 minutes for obtaining about 200 answer-figures, found from 1
+to 2 per hundred wrong. King ['07] had children in grade 5 do sums, each
+consisting of 5 two-place numbers. In the most accurate work-period,
+they made 1 error per 20 columns. In multiplying a four-place by a
+four-place number they had less than one total answer right out of
+three. In New York City Courtis found ['11-'12] with his Test 7 that in
+12 minutes the average achievement of fourth-grade children is 8.8 units
+attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with
+5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15
+attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right.
+These results are near enough to those obtained from the country at
+large to serve as a text here.
+
+The following were set as official standards, in an excellent school
+system, Courtis Series B being used:--
+
+
+                            SPEED        PERCENT OF
+                  GRADE.    ATTEMPTS.    CORRECT ANSWERS.
+  Addition          8         12           80
+                    7         11           80
+                    6         10           70
+                    5          9           70
+                    4          8           70
+
+  Subtraction       8         12           90
+                    7         11           90
+                    6         10           90
+                    5          9           80
+                    4          7           80
+
+  Multiplication    8         11           80
+                    7         10           80
+                    6          9           80
+                    5          7           70
+                    4          6           60
+
+  Division          8         11           90
+                    7         10           90
+                    6          8           80
+                    5          6           70
+                    4          4           60
+
+Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like
+those printed below, children in grade 4 got on the average less than 80
+percent of correct answers. Their average speed was about 2 columns per
+minute. In doing division of the sort printed below children of grades 3
+_B_ and 4 _A_ got less than 95 percent of correct answers, the average
+speed being 4 divisions per minute. In both cases the slower computers
+were no more accurate than the faster ones. Practice improved the speed
+very rapidly, but the accuracy remained substantially unchanged. Brown
+['11 and '12] found a similar low status of ability and notable
+improvement from a moderate amount of special practice.
+
+   3    5    6    2    3    8    9    7    4    9
+   7    9    6    5    5    6    4    5    8    2
+   3    4    7    8    7    3    7    9    3    7
+   8    8    4    8    2    6    8    2    9    8
+   2    2    4    7    6    9    8    5    6    2
+   6    9    5    7    8    5    2    3    2    4
+   9    6    4    2    7    2    9    4    4    5
+   3    3    7    9    9    9    2    8    9    7
+   6    8    9    6    4    7    7    9    2    4
+   8    4    6    9    9    2    6    9    8    9
+  --   --   --   --   --   --   --   --   --   --
+
+  20 = .... 5s
+  56 = .... 9s and .... _r_.
+  30 = .... 7s and .... _r_.
+  89 = .... 9s and .... _r_.
+  20 = .... 8s and .... _r_.
+  56 = .... 6s and .... _r_.
+  31 = .... 4s and .... _r_.
+  86 = .... 9s and .... _r_.
+
+It is clear that numerical work as inaccurate as this has little or no
+commercial or industrial value. If clerks got only six answers out of
+ten right as in the Courtis tests, one would need to have at least four
+clerks make each computation and would even then have to check many of
+their discrepancies by the work of still other clerks, if he wanted his
+accounts to show less than one error per hundred accounting units of the
+Courtis size.
+
+It is also clear that the "habits of ... absolute accuracy, and
+satisfaction in truth as a result" which arithmetic is supposed to
+further must be largely mythical in pupils who get right answers only
+from three to nine times out of ten!
+
+
+EARLY MASTERY
+
+The bonds in question clearly must be made far stronger than they now
+are. They should in fact be strong enough to abolish errors in
+computation, except for those due to temporary lapses. It is much
+better for a child to know half of the multiplication tables, and to
+know that he does not know the rest, than to half-know them all; and
+this holds good of all the elementary bonds required for computation.
+Any bond should be made to work perfectly, though slowly, very soon
+after its formation is begun. Speed can easily be added by proper
+practice.
+
+The chief reasons why this is not done now seem to be the following:
+(1) Certain important bonds (like the additions with higher decades)
+are not given enough attention when they are first used. (2) The special
+training necessary when a bond is used in a different connection (as
+when the multiplications to 9 x 9 are used in examples like
+
+  729
+    8
+  ---
+
+where the pupil has also to choose the right number to multiply, keep in
+mind what is carried, use it properly, and write the right figure in the
+right place, and carry a figure, or remember that he carries none) is
+neglected. (3) The pupil is not taught to check his work. (4) He is not
+made responsible for substantially accurate results. Furthermore, the
+requirement of (4) without the training of (1), (2), and (3) will
+involve either a fruitless failure on the part of many pupils, or an
+utterly unjust requirement of time. The common error of supposing that
+the task of computation with integers consists merely in learning the
+additions to 9 + 9, the subtractions to 18 - 9, the multiplications to
+8 x 9, and the divisions to 81 / 9, and in applying this knowledge in
+connection with the principles of decimal notation, has had a large
+share in permitting the gross inaccuracy of arithmetical work. The bonds
+involved in 'knowing the tables' do not make up one fourth of the bonds
+involved in real adding, subtracting, multiplying, and dividing (with
+integers alone).
+
+It should be noted that if the training mentioned in (1) and (2) is
+well cared for, the checking of results as recommended in (3) becomes
+enormously more valuable than it is under present conditions, though
+even now it is one of our soundest practices. If a child knows the
+additions to higher decades so that he can add a seen one-place number
+to a thought-of two-place number in three seconds or less with a correct
+answer 199 times out of 200, there is only an infinitesimal chance that
+a ten-figure column twice added (once up, once down) a few minutes apart
+with identical answers will be wrong. Suppose that, in long
+multiplication, a pupil can multiply to 9 x 9 while keeping his place
+and keeping track of what he is 'carrying' and of where to write the
+figure he writes, and can add what he carries without losing track of
+what he is to add it to, where he is to write the unit figure, what he
+is to multiply next and by what, and what he will then have to carry, in
+each case to a surety of 99 percent of correct responses. Then two
+identical answers got by multiplying one three-place number by another a
+few minutes apart, and with reversal of the numbers, will not be wrong
+more than twice in his entire school career. Checks approach proofs when
+the constituent bonds are strong.
+
+If, on the contrary, the fundamental bonds are so weak that they do not
+work accurately, checking becomes much less trustworthy and also very
+much more laborious. In fact, it is possible to show that below a
+certain point of strength of the fundamental bonds, the time required
+for checking is so great that part of it might better be spent in
+improving the fundamental bonds.
+
+For example, suppose that a pupil has to find the sum of five numbers
+like $2.49, $5.25, $6.50, $7.89, and $3.75. Counting each act of
+holding in mind the number to be carried and each writing of a column's
+result as equivalent in difficulty to one addition, such a sum equals
+nineteen single additions. On this basis and with certain additional
+estimates[7] we can compute the practical consequences for a pupil's use
+of addition in life according to the mastery of it that he has gained in
+school.
+
+  [7] These concern allowances for two errors occurring in the same
+      example and for the same wrong answer being obtained in both
+      original work and check work.
+
+I have so computed the amount of checking a pupil will have to do to
+reach two agreeing numbers (out of two, or three, or four, or five, or
+whatever the number before he gets two that are alike), according to his
+mastery of the elementary processes. The facts appear in Table 1.
+
+It is obvious that a pupil whose mastery of the elements is that denoted
+by getting them right 96 times out of 100 will require so much time for
+checking that, even if he were never to use this ability for anything
+save a few thousand sums in addition, he would do well to improve this
+ability before he tried to do the sums. An ability of 199 out of 200, or
+995 out of 1000, seems likely to save much more time than would be taken
+to acquire it, and a reasonable defense could be made for requiring 996
+or 997 out of 1000.
+
+A precision of from 995 to 997 out of 1000 being required, and ordinary
+sagacity being used in the teaching, speed will substantially take care
+of itself. Counting on the fingers or in words will not give that
+precision. Slow recourse to memory of serial addition tables will not
+give that precision. Nothing save sure memory of the facts operating
+under the conditions of actual examples will give it. And such memories
+will operate with sufficient speed.
+
+TABLE 1
+
+THE EFFECT OF MASTERY OF THE ELEMENTARY FACTS OF ADDITION UPON THE LABOR
+REQUIRED TO SECURE TWO AGREEING ANSWERS WHEN ADDING FIVE THREE-FIGURE
+NUMBERS
+
+  ======================================================================
+  MASTERY OF   |APPROXIMATE  |APPROXIMATE  |APPROXIMATE  |APPROXIMATE
+  THE          |NUMBER OF    |NUMBER OF    |NUMBER OF    |NUMBER OF
+  ELEMENTARY   |WRONG ANSWERS|AGREEING     |AGREEING     |CHECKINGS
+  ADDITIONS    |IN SUMS OF 5 |ANSWERS,     |ANSWERS,     |REQUIRED (OVER
+  TIMES RIGHT  |THREE-PLACE  |AFTER ONE    |AFTER A      |AND ABOVE THE
+  IN 1000      |NUMBERS PER  |CHECKING,    |CHECKING OF  |FIRST GENERAL
+               |1000         |PER 1000     |THE FIRST    |CHECKING OF
+               |             |             |DISCREPANCIES|THE 1000 SUMS)
+               |             |             |             |TO SECURE TWO
+               |             |             |             |AGREEING
+               |             |             |             |RESULTS
+  -------------+-------------+-------------+-------------+--------------
+       960     |     700     |      90     |     216     |     4500
+       980     |     380     |     384     |     676     |     1200
+       990     |     190     |     656     |     906     |      470
+       995     |      95     |     819     |     975     |      210
+       996     |      76     |     854     |     984     |      165
+       997     |      54     |     895     |     992     |      115
+       998     |      38     |     925     |     996     |       80
+       999     |      19     |     962     |     999     |       40
+  -------------+-------------+-------------+-------------+--------------
+
+There is one intelligent objection to the special practice necessary to
+establish arithmetical connections so fully as to give the accuracy
+which both utilitarian and disciplinary aims require. It may be said
+that the pupils in grades 3, 4, and 5 cannot appreciate the need and
+that consequently the work will be dull, barren, and alien, without
+close personal appropriation by the pupil's nature. It is true that no
+vehement life-purpose is directly involved by the problem of perfecting
+one's power to add 7 to 28 in grade 2, or by the problem of multiplying
+253 by 8 accurately in grade 3 or by precise subtraction in long
+division in grade 4. It is also true, however, that the most humanly
+interesting of problems--one that the pupil attacks most
+whole-heartedly--will not be solved correctly unless the pupil has the
+necessary associative mechanisms in order; and the surer he is of them,
+the freer he is to think out the problem as such. Further, computation
+is not dull if the pupil can compute. He does not himself object to its
+barrenness of vital meaning, so long as the barrenness of failure is
+prevented. We must not forget that pupils like to learn. In teaching
+excessively dull individuals, who has not often observed the great
+interest which they display in anything that they are enabled to master?
+There is pathos in their joy in learning to recognize parts of speech,
+perform algebraic simplifications, or translate Latin sentences, and in
+other accomplishments equally meaningless to all their interests save
+the universal human interest in success and recognition. Still further,
+it is not very hard to show to pupils the imperative need of accuracy in
+scoring games, in the shop, in the store, and in the office. Finally,
+the argument that accurate work of this sort is alien to the pupil in
+these grades is still stronger against _inaccurate_ work of the same
+sort. If we are to teach computation with two- and three- and four-place
+numbers at all, it should be taught as a reliable instrument, not as a
+combination of vague memories and faith. The author is ready to cut
+computation with numbers above 10 out of the curriculum of grades 1-6 as
+soon as more valuable educational instruments are offered in its place,
+but he is convinced that nothing in child-nature makes a large variety
+of inaccurate computing more interesting or educative or germane to felt
+needs, than a smaller variety of accurate computing!
+
+
+THE STRENGTH OF BONDS FOR TEMPORARY SERVICE
+
+The second general fact is that certain bonds are of service for only a
+limited time and so need to be formed only to a limited and slight
+degree of strength. The data of problems set to illustrate a principle
+or improve some habit of computation are, of course, the clearest cases.
+The pupil needs to remember that John bought 3 loaves of bread and that
+they were 5-cent loaves and that he gave 25 cents to the baker only long
+enough to use the data to decide what change John should receive. The
+connections between the total described situation and the answer
+obtained, supposing some considerable computation to intervene, is a
+bond that we let expire almost as soon as it is born.
+
+It is sometimes assumed that the bond between a certain group of
+features which make a problem a 'Buy _a_ things at _b_ per thing, find
+total cost' problem or a 'Buy _a_ things at _b_ per thing, what change
+from _c_' problem or a 'What gain on buying for _a_ and selling for _b_'
+problem or a 'How many things at _a_ each can I buy for _b_ cents'
+problem--it is assumed that the bond between these essential defining
+features and the operation or operations required for solution is as
+temporary as the bonds with the name of the buyer or the price of the
+thing. It is assumed that all problems are and should be solved by some
+pure act of reasoning without help or hindrance from bonds with the
+particular verbal structure and vocabulary of the problems. Whether or
+not they _should_ be, they _are not_. Every time that a pupil solves a
+'bought-sold' problem by subtraction he strengthens the tendency to
+respond to any problem whatsoever that contains the words 'bought for'
+and 'sold for' by subtraction; and he will by no means surely stop and
+survey every such problem in all its elements to make sure that no
+other feature makes inapplicable the tendency to subtract which the
+'bought sold' evokes.
+
+To prevent pupils from responding to the form of statement rather than
+the essential facts, we should then not teach them to forget the form of
+statement, but rather give them all the common forms of statement to
+which the response in question is an appropriate response, and only
+such. If a certain form of statement does in life always signify a
+certain arithmetical procedure, the bond between it and that procedure
+may properly be made very strong.
+
+Another case of the formation of bonds to only a slight degree
+of strength concerns the use of so-called 'crutches' such as
+writing +, -, and x in copying problems like those below:--
+
+  Add       Subtract       Multiply
+  23           79             32
+  61           24              3
+  --           --             --
+
+or altering the figures when 'borrowing' in subtraction, and the like.
+Since it is undesirable that the pupil should regard the 'crutch'
+response as essential to the total procedure, or become so used to
+having it that he will be disturbed by its absence later, it is supposed
+that the bond between the situation and the crutch should not be fully
+formed. There is a better way out of the difficulty, in case crutches
+are used at all. This is to associate the crutch with a special 'set,'
+and its non-use with the general set which is to be the permanent one.
+For example, children may be taught from the start never to write
+the crutch sign or crutch figure unless the work is accompanied by
+"Write ... to help you to...."
+
+  Write - to help you to        Find the differences:--
+  remember that you must        39     67     78     56     45
+  subtract in this row.         23     44     36     26     24
+                                --     --     --     --     --
+
+  Remember that you must        Find the differences:--
+  subtract in this row.         85     27     96     38     78
+                                63     14     51     45     32
+                                --      --    --     --     --
+
+The bond evoking the use of the crutch may then be formed thoroughly
+enough so that there is no hesitation, insecurity, or error, without
+interfering to any harmful extent with the more general bond from the
+situation to work without the crutch.
+
+
+THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS
+
+Another instructive case concerns the bonds between certain words and
+their meanings, and between certain situations of commerce, industry, or
+agriculture and useful facts about these situations. Illustrations of
+the former are the bonds between _cube root_, _hectare_, _brokerage_,
+_commission_, _indorsement_, _vertex_, _adjacent_, _nonagon_, _sector_,
+_draft_, _bill of exchange_, and their meanings. Illustrations of the
+latter are the bonds from "Money being lent 'with interest' at no
+specified rate, what rate is charged?" to "The legal rate of the state,"
+from "$_X_ per M as a rate for lumber" to "Means $_X_ per thousand board
+feet, a board foot being 1 ft. by 1 ft. by 1 in."
+
+It is argued by many that such bonds are valuable for a short time;
+namely, while arithmetical procedures in connection with which they
+serve are learned, but that their value is only to serve as a means for
+learning these procedures and that thereafter they may be forgotten.
+"They are formed only as accessory means to certain more purely
+arithmetical knowledge or discipline; after this is acquired they may
+be forgotten. Everybody does in fact forget them, relearning them later
+if life requires." So runs the argument.
+
+In some cases learning such words and facts only to use them in solving
+a certain sort of problems and then forget them may be profitable. The
+practice is, however, exceedingly risky. It is true that everybody does
+in fact forget many such meanings and facts, but this commonly means
+either that they should not have been learned at all at the time that
+they were learned, or that they should have been learned more
+permanently, or that details should have been learned with the
+expectation that they themselves would be forgotten but that a general
+fact or attitude would remain. For example, duodecagon should not be
+learned at all in the elementary school; indorsement should either not
+be learned at all there, or be learned for permanence of a year or more;
+the details of the metric system should be so taught as to leave for
+several years at least knowledge of the facts that there is a system so
+named that is important, whose tables go by tens, hundreds, or
+thousands, and a tendency (not necessarily strong) to connect meter,
+kilogram, and liter with measurement by the metric system and with
+approximate estimates of their several magnitudes.
+
+If an arithmetical procedure seems to require accessory bonds which are
+to be forgotten, once the procedure is mastered, we should be suspicious
+of the value of the procedure itself. If pupils forget what compound
+interest is, we may be sure that they will usually also have forgotten
+how to compute it. Surely there is waste if they have learned what it is
+only to learn how to compute it only to forget how to compute it!
+
+
+THE STRENGTH OF BONDS CONCERNING THE REASONS FOR ARITHMETICAL PROCESSES
+
+The next case of the formation of bonds to slight strength is the
+problematic one of forming the bonds involved in understanding the
+reasons for certain processes only to forget them after the process has
+become a habit. Should a pupil, that is, learn why he inverts and
+multiplies, only to forget it as soon as he can be trusted to divide by
+a fraction? Should he learn why he puts the units figure of each partial
+product in multiplication under the figure that he multiplies by, only
+to forget the reason as soon as he has command of the process? Should he
+learn why he gets the number of square inches in a rectangle by
+multiplying the length by the width, both being expressed in linear
+inches, and forget why as soon as he is competent to make computations
+of the areas of rectangles?
+
+On general psychological grounds we should be suspicious of forming
+bonds only to let them die of starvation later, and tend to expect that
+elaborate explanations learned only to be forgotten either should not be
+learned at all, or should be learned at such a time and in such a way
+that they would not be forgotten. Especially we should expect that the
+general principles of arithmetic, the whys and wherefores of its
+fundamental ways of manipulating numbers, ought to be the last bonds of
+all to be forgotten. Details of _how_ you arranged numbers to multiply
+might vanish, but the general reasons for the placing would be expected
+to persist and enable one to invent the detailed manipulations that had
+been forgotten.
+
+This suspicion is, I think, justified by facts. The doctrine that the
+customary deductive explanations of why we invert and multiply, or place
+the partial products as we do before adding, may be allowed to be
+forgotten once the actual habits are in working order, has a suspicious
+source. It arose to meet the criticism that so much time and effort were
+required to keep these deductive explanations in memory. The fact was
+that the pupil learned to compute correctly _irrespective of_ the
+deductive explanations. They were only an added burden. His inductive
+learning that the procedure gave the right answer really taught him. So
+he wisely shuffled off the extra burden of facts about the consequences
+of the nature of a fraction or the place values of our decimal notation.
+The bonds weakened because they were not used. They were not used
+because they were not useful in the shape and at the time that they were
+formed, or because the pupil was unable to understand the explanations
+so as to form them at all.
+
+The criticism was valid and should have been met in part by replacing
+the deductive explanations by inductive verifications, and in part by
+using the deductive reasoning as a check after the process itself is
+mastered. The very same discussions of place-value which are futile as
+proof that you must do a certain thing before you have done it, often
+become instructive as an explanation of why the thing that you have
+learned to do and are familiar with and have verified by other tests
+works as well as it does. The general deductive theory of arithmetic
+should not be learned only to be forgotten. Much of it should, by most
+pupils, not be learned at all. What is learned should be learned much
+later than now, as a synthesis and rationale of habits, not as their
+creator. What is learned of such deductive theory should rank among the
+most rather than least permanent of a pupil's stock of arithmetical
+knowledge and power. There are bonds which are formed only to be lost,
+and bonds formed only to be lost _in their first form_, being used in a
+new organization as material for bonds of a higher order; but the bonds
+involved in deductive explanations of why certain processes are right
+are not such: they are not to be formed just to be forgotten, nor as
+mere propaedeutics to routine manipulations.
+
+
+PROPAEDEUTIC BONDS
+
+The formation of bonds to a limited strength because they are to be lost
+in their first form, being worked over in different ways in other bonds
+to which they are propaedeutic or contributing is the most important case
+of low strength, or rather low permanence, in bonds.
+
+The bond between four 5s in a column to be added and the response of
+thinking '10, 15, 20' is worth forming, but it is displaced later by the
+multiplication bond or direct connection of 'four 5s to be added' with
+'20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc.,
+forms serial bonds which as series might well be left to disappear.
+Their separate steps are kept as permanent bonds for use in column
+addition, but their serial nature is changed from 2 (and 2) 4, (and 2)
+6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.;
+after playing their part in producing the bonds whereby any multiple of
+2 by 2 to 9, can be got, the original serial bonds are, as series,
+needed no longer. The verbal response of saying 'and' in adding, after
+helping to establish the bonds whereby the general set of the mind
+toward adding cooeperates with the numbers seen or thought of to produce
+their sum, should disappear; or remain so slurred in inner speech as to
+offer no bar to speed.
+
+The rule for such bonds is, of course, to form them strongly enough so
+that they work quickly and accurately for the time being and facilitate
+the bonds that are to replace them, but not to overlearn them. There is
+a difference between learning something to be held for a short time, and
+the same amount of energy spent in learning for long retention. The
+former sort of learning is, of course, appropriate with many of these
+propaedeutic bonds.
+
+The bonds mentioned as illustrations are not _purely_ propaedeutic, nor
+formed _only_ to be transmuted into something else. Even the saying of
+'and' in addition has some genuine, intrinsic value in distinguishing
+the process of addition, and may perhaps be usefully reviewed for a
+brief space during the first steps in adding common fractions. Some such
+propaedeutic bonds may be worth while apart from their value in preparing
+for other bonds. Consider, for example, exercises like those shown below
+which are propaedeutic to long division, giving the pupil some basis in
+experience for his selection of the quotient figures. These
+multiplications are intrinsically worth doing, especially the 12s and
+25s. Whatever the pupil remembers of them will be to his advantage.
+
+  1. Count by 11s to 132, beginning 11, 22, 33.
+
+  2. Count by 12s to 144, beginning 12, 24, 36.
+
+  3. Count by 25s to 300, beginning 25, 50, 75.
+
+  4. State the missing numbers:--
+
+          A.          B.             C.                D.
+       3 11s =     5 11s =     8 ft. = .... in.     2 dozen =
+       4 12s =     3 12s =    10 ft. = .... in.     4 dozen =
+       5 12s =     6 12s =     7 ft. = .... in.    10 dozen =
+       6 11s =    12 11s =     4 ft. = .... in.     5 dozen =
+       9 11s =     2 12s =     6 ft. = .... in.     7 dozen =
+       7 12s =     9 12s =     9 ft. = .... in.    12 dozen =
+       8 12s =     7 11s =    11 ft. = .... in.     9 dozen =
+      11 11s =    12 12s =     5 ft. = .... in.     6 dozen =
+
+  5. Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents,
+     one dollar," and so on.
+
+  6. Count by 15s to $1.50.
+
+  7. Find the products. Do not use pencil. Think what they are.
+
+         A.        B.         C.          D.          E.
+       2 x 25    3 x 15     2 x 12      4 x 11      6 x 25
+       3 x 25   10 x 15     2 x 15      4 x 15      6 x 15
+       5 x 25    4 x 15     2 x 25      4 x 12      6 x 12
+      10 x 25    2 x 15     2 x 11      4 x 25      6 x 11
+       4 x 25    7 x 15     3 x 25      5 x 11      7 x 12
+       6 x 25    9 x 15     3 x 15      5 x 12      7 x 15
+       8 x 25    5 x 15     3 x 11      5 x 15      7 x 25
+       7 x 25    8 x 15     3 x 12      5 x 25      7 x 11
+       9 x 25    6 x 15     8 x 12      9 x 12      8 x 25
+
+  State the missing numbers:--
+
+    A.  36 = .... 12s    B.  44 = .... 11s    C.  50 = .... 25s
+        60 = .... 12s        88 = .... 11s       125 = .... 25s
+        24 = .... 12s        77 = .... 11s        75 = .... 25s
+        48 = .... 12s        55 = .... 11s       200 = .... 25s
+       144 = .... 12s        99 = .... 11s       250 = .... 25s
+       108 = .... 12s       110 = .... 11s       175 = .... 25s
+        72 = .... 12s        33 = .... 11s       225 = .... 25s
+        96 = .... 12s        66 = .... 11s       150 = .... 25s
+        84 = .... 12s        22 = .... 11s       100 = .... 25s
+
+  Find the quotients and remainders. If you need to use paper and pencil
+  to find them, you may. But find as many as you can without pencil and
+  paper. Do Row A first. Then do Row B. Then Row C, etc.
+             __      __      __      __      __      __
+  Row A.  11|45   12|45   25|45   15|45   21|45   22|45
+             __      __      __      __      __      __
+  Row B.  25|55   11|55   12|55   15|55   22|55   30|55
+             __      __      __      __      __      __
+  Row C.  12|60   25|60   15|60   11|60   30|60   21|60
+             __      __      __      __      __      __
+  Row D.  12|75   11|75   15|75   25|75   30|75   35|75
+             ___     ___     ___     ___     ___     ___
+  Row E.  11|100  12|100  25|100  15|100  30|100  22|100
+             __      __      __      __      __      __
+  Row F.  11|96   12|96   25|96   15|96   30|96   22|96
+             ___     ___     ___     ___     ___     ___
+  Row G.  25|105  11|105  15|105  12|105  22|105  35|105
+             __      __      __      __      __      __
+  Row H.  12|64   15|64   25|64   11|64   22|64   21|64
+             __      __      __      __      __      __
+  Row I.  11|80   12|80   15|80   25|80   35|80   21|80
+             ___     ___     ___     ___     ___     ___
+  Row J.  25|200  30|200  75|200  63|200  65|200  66|200
+
+  Do this section again. Do all the first column first. Then do
+  the second column, then the third, and so on.
+
+Consider, from the same point of view, exercises like (3 x 4) + 2,
+(7 x 6) + 5, (9 x 4) + 6, given as a preparation for written
+multiplication. The work of
+
+  48   68   47
+   3    7    9
+  --   --   --
+
+and the like is facilitated if the pupil has easy control of the process
+of getting a product, and keeping it in mind while he adds a one-place
+number to it. The practice with (3 x 4) + 2 and the like is also good
+practice intrinsically. So some teachers provide systematic preparatory
+drills of this type just before or along with the beginning of short
+multiplication.
+
+In some cases the bonds are purely propaedeutic or are formed _only_ for
+later reconstruction. They then differ little from 'crutches.' The
+typical crutch forms a habit which has actually to be broken, whereas
+the purely propaedeutic bond forms a habit which is left to rust out from
+disuse.
+
+For example, as an introduction to long division, a pupil may be given
+exercises using one-figure divisors in the long form, as:--
+
+     773 and 5 remainder
+   ______
+  7)5416
+    49
+    --
+     51
+     49
+     --
+      26
+      21
+      --
+       5
+
+The important recommendation concerning these purely propaedeutic bonds,
+and bonds formed only for later reconstruction, is to be very critical
+of them, and not indulge in them when, by the exercise of enough
+ingenuity, some bond worthy of a permanent place in the individual's
+equipment can be devised which will do the work as well. Arithmetical
+teaching has done very well in this respect, tending to err by leaving
+out really valuable preparatory drills rather than by inserting
+uneconomical ones. It is in the teaching of reading that we find the
+formation of propaedeutic bonds of dubious value (with letters,
+phonograms, diacritical marks, and the like) often carried to
+demonstrably wasteful extremes.
+
+
+
+
+CHAPTER VI
+
+THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE
+ORGANIZATION OF ABILITIES
+
+
+THE AMOUNT OF PRACTICE
+
+It will be instructive if the reader will perform the following
+experiment as an introduction to the discussion of this chapter, before
+reading any of the discussion.
+
+Suppose that a pupil does all the work, oral and written, computation
+and problem-solving, presented for grades 1 to 6 inclusive (that is, in
+the first two books of a three-book series) in the average textbook now
+used in the elementary school. How many times will he have exercised
+each of the various bonds involved in the four operations with integers
+shown below? That is, how many times will he have thought, "1 and 1
+are 2," "1 and 2 are 3," etc.? Every case of the action of each bond
+is to be counted.
+
+
+THE FUNDAMENTAL BONDS
+
+  1 + 1                             2 - 1        1 x 1      2 / 1
+  1 + 2                             2 - 2        2 x 1      2 / 2
+  1 + 3                                          3 x 1
+  1 + 4                                          4 x 1
+  1 + 5                             3 - 1        5 x 1      3 / 1
+  1 + 6                             3 - 2        6 x 1      3 / 2
+  1 + 7                             3 - 3        7 x 1      3 / 3
+  1 + 8                                          8 x 1
+  1 + 9                                          9 x 1
+
+                                    4 - 1                   4 / 1
+                                    4 - 2                   4 / 2
+  11 (or 21 or 31, etc.) + 1        4 - 3        1 x 2      4 / 3
+  11        "            + 2        4 - 4        2 x 2      4 / 4
+  11        "            + 3                     3 x 2
+  11        "            + 4                     4 x 2
+  11        "            + 5        5 - 1        5 x 2      5 / 1
+  11        "            + 6        5 - 2        6 x 2      5 / 2
+  11        "            + 7        5 - 3        7 x 2      5 / 3
+  11        "            + 8        5 - 4        8 x 2      5 / 4
+  11        "            + 9        5 - 5        9 x 2      5 / 5
+
+                                    6 - 1        1 x 3      6 / 1
+  2 + 1                             6 - 2        2 x 3      6 / 2
+  2 + 2                             6 - 3        3 x 3      6 / 3
+  2 + 3                             6 - 4        4 x 3      6 / 4
+  2 + 4                             6 - 5        5 x 3      6 / 5
+  2 + 5                             6 - 6        6 x 3      6 / 6
+  2 + 6                                          7 x 3
+  2 + 7                                          8 x 3
+  2 + 8                             7 - 1        9 x 3      7 / 1
+  2 + 9                             7 - 2                   7 / 2
+                                    7 - 3                   7 / 3
+                                    7 - 4        1 x 4      7 / 4
+  12 (or 22 or 32, etc.) + 1        7 - 5        2 x 4      7 / 5
+  12        "            + 2        7 - 6        and so on  7 / 6
+                                    7 - 7        to 9 x 9   7 / 7
+  and so on to                      and so on               and so on to
+  9 + 9                             to 18 - 9               82 / 9
+  19 (or 29 or 39, etc.) + 9                                83 / 9, etc.
+
+If estimating for the entire series is too long a task, it will be
+sufficient to use eight or ten from each, say:--
+
+   3 + 2    13, 23, etc. + 2     7 + 2    17, 27, etc. + 2
+     " 3           "       3       " 3            "      3
+     " 4           "       4       " 4            "      4
+     " 5           "       5       " 5            "      5
+     " 6           "       6       " 6            "      6
+     " 7           "       7       " 7            "      7
+     " 8           "       8       " 8            "      8
+     " 9           "       9       " 9            "      9
+
+   3 - 3     7 - 7     9 x 7    63 / 9
+   4   "     8   "     7 x 9    64   "
+   5   "     9   "     8 x 6    65   "
+   6   "    10   "     6 x 8    66   "
+   7   "    11   "              67   "
+   8   "    12   "              68   "
+   9   "    13   "              69   "
+  10   "    14   "              70   "
+  11   "    15   "              71   "
+  12   "    16   "
+
+
+TABLE 2
+
+ESTIMATES OF THE AMOUNT OF PRACTICE PROVIDED IN BOOKS I AND II OF THE
+AVERAGE THREE-BOOK TEXT IN ARITHMETIC; BY 50 EXPERIENCED TEACHERS
+
+  ======================================================================
+                         | LOWEST | MEDIAN | HIGHEST |RANGE REQUIRED TO
+    ARITHMETICAL FACT    |ESTIMATE|ESTIMATE|ESTIMATE | INCLUDE HALF OF
+                         |        |        |         |  THE ESTIMATES
+  -----------------------+--------+--------+---------+------------------
+  3 or 13 or 23, etc. + 2|   25   |  1500  |1,000,000|    800-5000
+       "         "      3|   24   |  1450  |   80,000|    475-5000
+       "         "      4|   23   |  1150  |   50,000|    750-5000
+       "         "      5|   22   |  1400  |   44,000|    700-5000
+       "         "      6|   21   |  1350  |   41,000|    700-4500
+       "         "      7|   21   |  1500  |   37,000|    600-4000
+       "         "      8|   20   |  1400  |   33,000|    550-4100
+       "         "      9|   20   |  1150  |   28,000|    650-4500
+                         |        |        |         |
+  7 or 17 or 27, etc. + 2|   20   |  1250  |2,000,000|    600-5000
+       "         "      3|   19   |  1100  |1,000,000|    650-4900
+       "         "      4|   18   |  1000  |   80,000|    650-4900
+       "         "      5|   17   |  1300  |   80,000|    650-4400
+       "         "      6|   16   |  1100  |   29,000|    650-4500
+       "         "      7|   15   |  1100  |   25,000|    500-4500
+       "         "      8|   13   |  1100  |   21,000|    650-3800
+       "         "      9|   10   |  1275  |   17,000|    500-4000
+                         |        |        |         |
+          3 - 3          |   25   |  1000  |  100,000|    500-4000
+          4 - 3          |   20   |  1050  |  500,000|    525-3000
+          5 - 3          |   20   |  1100  |2,500,000|    650-4200
+          6 - 3          |   10   |  1050  |   21,000|    650-3250
+          7 - 3          |   22   |  1100  |   15,000|    550-3050
+          8 - 3          |   21   |  1075  |   15,000|    650-3000
+          9 - 3          |   21   |  1000  |   15,000|    700-2600
+         10 - 3          |   20   |  1000  |   20,000|    600-2500
+         11 - 3          |   20   |  1000  |   15,000|    465-2550
+         12 - 3          |   18   |  1000  |   15,000|    650-2100
+                         |        |        |         |
+          7 - 7          |   10   |  1000  |   18,000|    425-3000
+          8 - 7          |   15   |  1000  |   18,000|    413-3100
+          9 - 7          |   15   |   950  |   18,000|    550-3000
+         10 - 7          |   15   |   950  |   18,000|    600-3950
+         11 - 7          |   10   |   900  |   18,000|    550-3000
+         12 - 7          |   10   |   925  |   18,000|    525-3100
+         13 - 7          |   10   |   900  |   18,000|    500-2600
+         14 - 7          |   10   |   900  |   18,000|    500-3100
+         15 - 7          |   10   |   925  |   18,000|    500-3000
+         16 - 7          |   10   |   875  |   18,000|    500-2500
+                         |        |        |         |
+          9 x 7          |   10   |   700  |   20,000|    500-2000
+          7 x 9          |   10   |   700  |   20,000|    500-1750
+          8 x 6          |   10   |   750  |   20,000|    500-2500
+          6 x 8          |    9   |   700  |   20,000|    500-2500
+                         |        |        |         |
+         63 / 9          |    9   |   500  |    4,500|    300-2500
+         64 / 9          |    9   |   200  |    4,000|    100- 700
+         65 / 9          |    8   |   200  |    4,000|    100- 600
+         66 / 9          |    7   |   200  |    4,000|    100- 550
+         67 / 9          |    7   |   200  |    4,000|     75- 450
+         68 / 9          |    6   |   200  |    4,000|     87- 575
+         69 / 9          |    6   |   200  |    4,000|     87- 450
+         70 / 9          |    5   |   200  |    4,000|     75- 575
+         71 / 9          |    5   |   200  |    4,000|     75- 700
+                         |        |        |         |
+           _XX_          |   40   |   550  |1,000,000|    300-2000
+           _XO_          |   20   |   500  |   11,500|    150-2000
+          _XXX_          |   15   |   450  |   12,000|    100-1000
+          _XXO_          |   25   |   400  |   15,000|    150-1000
+          _XOO_          |   15   |   400  |    5,000|    100-1000
+          _XOX_          |   10   |   400  |   10,000|    100- 975
+  ======================================================================
+
+Having made his estimates the reader should compare them first with
+similar estimates made by experienced teachers (shown on page 124 f.),
+and then with the results of actual counts for representative textbooks
+in arithmetic (shown on pages 126 to 132).
+
+It will be observed in Table 2 that even experienced teachers vary
+enormously in their estimates of the amount of practice given by an
+average textbook in arithmetic, and that most of them are in serious
+error by overestimating the amount of practice. In general it is the
+fact that we use textbooks in arithmetic with very vague and erroneous
+ideas of what is in them, and think they give much more practice than
+they do.
+
+The authors of the textbooks as a rule also probably had only very vague
+and erroneous ideas of what was in them. If they had known, they would
+almost certainly have revised their books. Surely no author would
+intentionally provide nearly four times as much practice on 2 + 2 as on
+8 + 8, or eight times as much practice on 2 x 2 as on 9 x 8, or eleven
+times as much practice on 2 - 2 as on 17 - 8, or over forty times as
+much practice on 2 / 2 as on 75 / 8 and 75 / 9, both together. Surely
+no author would have provided intentionally only twenty to thirty
+occurrences each of 16 - 7, 16 - 8, 16 - 9, 17 - 8, 17 - 9, and 18 - 9
+for the entire course through grade 6; or have left the practice on
+60 / 7, 60 / 8, 60 / 9, 61 / 7, 61 / 8, 61 / 9, and the like to occur
+only about once a year!
+
+
+TABLE 3
+
+AMOUNT OF PRACTICE: ADDITION BONDS IN A RECENT TEXTBOOK (A) OF EXCELLENT
+REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY
+MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION
+
+The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times,
+22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.
+
+  ======================================================================
+                  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9 | TOTAL
+  ----------------+-----+-----+-----+-----+-----+-----+-----+----+------
+    2             | 226 | 154 | 162 | 150 |  97 |  87 |  66 |  45|
+   12             |  74 |  53 |  76 |  46 |  51 |  37 |  36 |  33|
+   22, etc.       |  50 |  60 |  68 |  63 |  42 |  50 |  38 |  26|
+                  |     |     |     |     |     |     |     |    |
+    3             | 216 | 141 | 127 |  89 |  82 |  54 |  58 |  40|
+   13             |  43 |  43 |  60 |  70 |  52 |  30 |  22 |  18|
+   23, etc.       |  15 |  30 |  51 |  50 |  42 |  32 |  29 |  30|
+                  |     |     |     |     |     |     |     |    |
+    7             |  85 |  90 | 103 | 103 |  84 |  81 |  61 |  47|
+   17             |  35 |  25 |  42 |  32 |  35 |  21 |  29 |  16|
+   27, etc.       |  30 |  23 |  32 |  29 |  24 |  23 |  25 |  28|
+                  |     |     |     |     |     |     |     |    |
+    8             | 185 | 112 | 146 |  99 |  75 |  71 |  73 |  61|
+   18             |  28 |  35 |  52 |  46 |  28 |  29 |  24 |  14|
+   28, etc.       |  53 |  36 |  34 |  38 |  23 |  36 |  27 |  27|
+                  |     |     |     |     |     |     |     |    |
+    9             | 104 |  81 | 112 |  96 |  63 |  74 |  58 |  57|
+   19             |  13 |  11 |  31 |  38 |  25 |  14 |  22 |  11|
+   29, etc.       |  19 |  17 |  27 |  20 |  32 |  32 |  19 |  18|
+                  |     |     |     |     |     |     |     |    |
+  2, 12, 22, etc. | 350 | 277 | 306 | 260 | 190 | 174 | 140 | 104| 1801
+  3, 13, 23, etc. | 274 | 214 | 230 | 209 | 176 | 116 | 109 |  88| 1406
+                  |     |     |     |     |     |     |     |    |
+  7, 17, 27, etc. | 148 | 138 | 187 | 164 | 141 | 125 | 115 |  91| 1109
+  8, 18, 28, etc. | 266 | 183 | 232 | 185 | 126 | 136 | 124 | 102| 1354
+  9, 19, 29, etc. | 136 | 109 | 170 | 154 | 120 | 120 |  99 |  86|  994
+                  |     |     |     |     |     |     |     |    |
+  Totals          |1164 | 921 |1125 | 972 | 753 | 671 | 687 | 471|
+  ======================================================================
+
+
+TABLE  4
+
+AMOUNT OF PRACTICE: SUBTRACTION BONDS IN A RECENT TEXTBOOK (A)
+OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF
+SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION
+
+  ================================================================
+            |                     SUBTRAHENDS
+   MINUENDS |-----------------------------------------------------
+            |  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----
+    1       | 372 |     |     |     |     |     |     |     |
+    2       | 214 | 311 |     |     |     |     |     |     |
+    3       | 136 | 149 | 189 |     |     |     |     |     |
+    4       | 146 | 142 | 103 | 205 |     |     |     |     |
+    5       | 171 |  91 |  92 | 164 | 136 |     |     |     |
+    6       |  80 |  59 |  69 |  71 |  81 | 192 |     |     |
+    7       | 106 |  57 |  55 |  67 |  59 | 156 |  80 |     |
+    8       |  73 |  50 |  50 |  75 |  50 |  62 |  48 | 152 |
+    9       |  71 |  75 |  54 |  74 |  48 |  55 |  55 | 124 | 133
+   10       | 261 |  84 |  63 | 100 | 193 |  83 |  57 | 124 |  91
+            |     |     |     |     |     |     |     |     |
+   11       |     |  48 |  31 |  50 |  36 |  41 |  32 |  46 |  35
+   12       |     |     |  48 |  77 |  57 |  51 |  35 |  80 |  30
+   13       |     |     |     |  35 |  22 |  40 |  29 |  35 |  28
+   14       |     |     |     |     |  25 |  37 |  36 |  49 |  32
+   15       |     |     |     |     |     |  33 |  19 |  48 |  20
+            |     |     |     |     |     |     |     |     |
+   16       |     |     |     |     |     |     |  16 |  36 |  26
+   17       |     |     |     |     |     |     |     |  27 |  20
+   18       |     |     |     |     |     |     |     |     |  19
+            |     |     |     |     |     |     |     |     |
+  Total     |     |     |     |     |     |     |     |     |
+  excluding |     |     |     |     |     |     |     |     |
+  1-1, 2-2, |     |     |     |     |     |     |     |     |
+  etc.      |1258 | 755 | 565 | 713 | 571 | 558 | 327 | 569 | 301
+  ================================================================
+
+
+TABLE 5
+
+FREQUENCIES OF SUBTRACTIONS NOT INCLUDED IN TABLE 4
+
+These are cases where the pupil would, by reason of his stage of
+advancement, probably operate 35-30, 46-46, etc., as one bond.
+
+  ======================================================================
+                      |                    SUBTRAHENDS
+                      |----+----+----+----+----+----+----+----+----+----
+                      |   1|  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |
+        MINUENDS      |  11| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |  10
+                      |  21| 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |  20
+                      |etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.
+  --------------------+----+----+----+----+----+----+----+----+----+----
+  10, 20, 30, 40, etc.| 11 | 29 | 16 | 52 | 32 | 51 |  7 | 30 | 22 | 60
+  11, 21, 31, 41, etc.| 42 | 14 | 22 | 32 | 12 | 26 | 19 | 52 | 17 | 10
+  12, 22, 32, 42, etc.| 47 | 97 |  5 | 13 |  9 | 21 | 11 | 24 | 19 | 17
+  13, 23, 33, 43, etc.|  7 | 40 |  7 | 14 | 15 | 13 | 19 | 19 | 22 |  3
+  14, 24, 34, 44, etc.|  8 | 28 | 14 | 58 | 13 | 16 | 14 | 26 | 19 |  7
+  15, 25, 35, 45, etc.| 21 | 28 | 29 | 54 | 51 | 15 | 21 | 12 | 24 |  8
+  16, 26, 36, 46, etc.|  5 | 18 | 12 | 27 | 35 | 69 | 13 | 17 | 19 |  2
+  17, 27, 37, 47, etc.|  5 |  9 | 12 | 40 | 32 | 54 | 24 | 12 | 12 |  1
+  18, 28, 38, 48, etc.|  2 | 16 | 10 | 23 | 22 | 36 | 18 | 47 | 16 |  0
+  19, 29, 39, etc.    |  5 |  7 |  7 | 10 | 13 | 28 | 14 | 23 | 16 |  0
+                      |    |    |    |    |    |    |    |    |    |
+  Totals              |153 |286 |134 |323 |234 |329 |160 |262 |186 |108
+  =====================================================================
+
+
+TABLE 6
+
+AMOUNT OF PRACTICE: MULTIPLICATION BONDS IN ANOTHER RECENT TEXTBOOK (B)
+OF EXCELLENT REPUTE. BOOKS I AND II
+
+  ======================================================================
+              |                     MULTIPLICANDS
+  MULTIPLIERS |---------------------------------------------------------
+              |  0 |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |Totals
+  ------------+----+----+----+----+----+----+----+----+----+----+-------
+       1      | 299| 534| 472| 271| 310| 293| 261| 178| 195|  99| 2912
+       2      | 350| 644| 668| 480| 458| 377| 332| 238| 239| 155| 3941
+       3      | 280| 487| 509| 388| 318| 302| 247| 199| 227| 152| 3109
+       4      | 186| 375| 398| 242| 203| 265| 197| 163| 159|  93| 2281
+       5      | 268| 359| 393| 234| 263| 243| 217| 192| 197| 114| 2480
+       6      | 180| 284| 265| 199| 196| 191| 168| 169| 165| 106| 1923
+       7      | 135| 283| 277| 176| 187| 158| 155| 121| 145| 118| 1755
+       8      | 137| 272| 292| 175| 192| 164| 158| 157| 126| 126| 1799
+       9      |  71| 173| 140| 122|  97| 102| 101| 100|  82| 110| 1098
+              |    |    |    |    |    |    |    |    |    |    |
+  Totals      |1906|3411|3414|2287|2224|2095|1836|1517|1535|1073|
+  ======================================================================
+
+
+TABLE  7
+
+AMOUNT OF PRACTICE: DIVISIONS WITHOUT REMAINDER IN TEXTBOOK B,
+PARTS I AND II
+
+  ======================================================================
+                         |                   DIVISORS
+        DIVIDENDS        |----------------------------------------------
+                         |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |Totals
+  -----------------------+----+----+----+----+----+----+----+----+------
+  Integral               | 397| 224| 250| 130|  93|  44|  98|  23| 1259
+  multiples              |    |    |    |    |    |    |    |    |
+  of 2 to 9              | 256| 124| 152|  79|  28|  43|  61|  25|  768
+  in sequence;           |    |    |    |    |    |    |    |    |
+  _i.e._, 4 / 2          | 318| 123| 130|  65|  50|  19|  39|  19|  763
+  occurred               |    |    |    |    |    |    |    |    |
+  397 times,             | 258|  98|  86| 105|  25|  24|  34|  20|  650
+  6 / 2 occurred         |    |    |    |    |    |    |    |    |
+  256 times,             | 198|  49|  76|  27|  22|  30|  33|  16|  451
+  6 / 3, 224 times,      |    |    |    |    |    |    |    |    |
+  9 / 3, 124 times.      |  77|  54|  36|  31|  28|  27|  16|   9|  278
+                         | 180|  91|  50|  38|  17|  13|  22|  16|  427
+                         |  69|  46|  37|  24|  12|  17|  16|  15|  236
+                         |    |    |    |    |    |    |    |    |
+  Totals                 |1753| 809| 817| 499| 275| 217| 319| 142|
+  ======================================================================
+
+
+TABLE 8
+
+DIVISION BONDS, WITH AND WITHOUT REMAINDERS. BOOK B
+
+All work through grade 6, except estimates of quotient figures in long
+division.
+
+  Dividend       2       3           4               5
+  Divisor        1   2   1   2   3   1   2   3   4   1   2   3   4   5
+  Number of
+  Occurrences   41 386  27 189 240  26 397  66 185  23 136  43  53 135
+
+  Dividend       6                       7
+  Divisor        1   2   3   4   5   6   1   2   3   4   5   6   7
+  Number of
+  Occurrences   21 256 224  68  43  83  23  72  55  38  46  32  54
+
+  Dividend       8                            9
+  Divisor        1   2   3   4   5  6  7  8   1  2   3  4  5  6  7  8  9
+  Number of
+  Occurrences   17 318  30 250  22 28 39 91  19 50 124 49 25 15 18 30 38
+
+  Dividend      10                         11
+  Divisor        2  3  4   5  6 7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences  258 38 46 120 19 9 24 24    32 21 16  3  7 11 14  3
+
+  Dividend      12                         13
+  Divisor        2   3   4  5  6 7  8 9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences  198 123 152 29 93 9 16 7    45 16 15 11  7  4  5  3
+
+  Dividend      14                         15
+  Divisor        2  3  4  5  6  7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences   77 20 13  5  8 44  8  6    69 98 16 79  8  8  4  6
+
+  Dividend      16                         17
+  Divisor        2  3   4  5 6  7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences  180 19 130 14 6  9 98  3    61  9 15 14  6  6 12  3
+
+  Dividend      18                         19
+  Divisor        2  3  4  5  6  7  8  9     2  3  4  5  6  7  8  9
+  Number of
+  Occurrences   69 49 13  6 28  7  7 23    21  6 10  5  3  4 10  4
+
+  Dividend      20                      21
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences   24 86 65 11  3 23  5    54 12  8  5 43 10  5
+
+  Dividend      22                      23
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences   17 16 15  8 13  6 15     7  8 11  8  6  3  2
+
+  Dividend      24                      25
+  Divisor        3  4  5  6  7  8  9     3  4   5 6  7  8  9
+  Number of
+  Occurrences   91 76 18 50  5 61  1    11 13 105 5  6  5  3
+
+  Dividend      26                      27
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences    5  6  3  3  4  6  3    46  8 10  4  2  6 25
+
+  Dividend      28                       29
+  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
+  Number of
+  Occurrences    4 36  8  3 19  3  7     6  8  0  5 11  2  3
+
+  Dividend      30                 31                 32
+  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
+  Number of
+  Occurrences   21 27 25  6  7 13   4  3  1  1  4  2  50 11  3  6 39  5
+
+  Dividend      33                 34                 35
+  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
+  Number of
+  Occurrences    8  7  7  2  6  1   8  3  5  2  1  1  10 31  5 24  5  3
+
+  Dividend      36                 37                 38
+  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
+  Number of
+  Occurrences   37 16 22  2  6 19   12 8  7  5  3  9   7  8  7  1  1  5
+
+  Dividend      39             40            41            42
+  Divisor        4 5 6 7 8 9    5 6 7  8 9    5 6 7 8 9     5  6  7  8 9
+  Number of
+  Occurrences    4 3 7 4 3 1   38 9 2 34 2    6 6 3 7 5     7 28 30 10 3
+
+  Dividend      43             44            45            46
+  Divisor        5 6  7 8 9     5 6 7 8 9     5 6 7  8  9   5 6 7 8 9
+  Number of
+  Occurrences    7 5 10 3 2     7 6 4 5 0    24 6 7 10 20   3 3 2 2 2
+
+  Dividend      47             48            49            50
+  Divisor        5 6 7 8 9      5  6 7  8 9   5 6  7 8 9    6 7 8 9
+  Number of
+  Occurrences    6 2 2 0 3      7 17 4 33 2   4 7 27 9 2    4 6 3 8
+
+  Dividend      51         52           53         54
+  Divisor        6 7 8 9    6 7 8 9      6 7 8 9    6 7 8  9
+  Number of
+  Occurrences    2 3 1 2    5 5 5 3      4 3 2 2    12 5 1 16
+
+  Dividend      55         56           57         58         59
+  Divisor        6 7 8 9    6  7  8 9    6 7 8 9    6 7 8 9    6 7 8 9
+  Number of
+  Occurrences    5 3 4 2    0 13 16 8    0 3 1 3    2 2 3 1    2 3 0 3
+
+  Dividend      60       61       62       63       64         65
+  Divisor        7 8 9    7 8 9    7 8 9    7 8 9    7  8  9    7  8 9
+  Number of
+  Occurrences    3 9 1    1 2 5    4 6 1   17 5 9    5 22  0    1 10 1
+
+  Dividend      66       67       68       69       70     71
+  Divisor        7 8 9    7 8 9    7 8 9    7 8 9    8 9    8 9
+  Number of
+  Occurrences    2 1 4    0 1 1    1 3 2    0 6 1    6 2    1 0
+
+  Dividend      72      73     74     75     76     77     78     79
+  Divisor        8  9    8 9    8 9    8 9    8 9    8 9    8 9    8 9
+  Number of
+  Occurrences   16 10    7 5    3 3    5 3    3 2    3 0    4 1    0 2
+
+  Dividend      80   81   82   83   84   85   86   87   88   89
+  Divisor        9    9    9    9    9    9    9    9    9    9
+  Number of
+  Occurrences    4   15    2    4    1    2    0    3    2    7
+
+Tables 3 to 8 show that even gifted authors make instruments for
+instruction in arithmetic which contain much less practice on certain
+elementary facts than teachers suppose; and which contain relatively
+much more practice on the more easily learned facts than on those which
+are harder to learn.
+
+How much practice should be given in arithmetic? How should it be
+divided among the different bonds to be formed? Below a certain amount
+there is waste because, as has been shown in Chapter VI, the pupil will
+need more time to detect and correct his errors than would have been
+required to give him mastery. Above a certain amount there is waste
+because of unproductive overlearning. If 668 is just enough for 2 x 2,
+82 is not enough for 9 x 8. If 82 is just enough for 9 x 8, 668 is too
+much for 2 x 2.
+
+It is possible to find the answers to these questions for the pupil of
+median ability (or any stated ability) by suitable experiments. The
+amount of practice will, of course, vary according to the ability of
+the pupil. It will also vary according to the interest aroused in him
+and the satisfaction he feels in progress and mastery. It will also vary
+according to the amount of practice of other related bonds; 7 + 7 = 14
+and 60 / 7 = 8 and 4 remainder will help the formation of 7 + 8 = 15
+and 61 / 7 = 8 and 5 remainder. It will also, of course, vary with the
+general difficulty of the bond, 17 - 8 = 9 being under ordinary
+conditions of teaching harder to form than 7 - 2 = 5.
+
+Until suitable experiments are at hand we may estimate for the
+fundamental bonds as follows, assuming that by the end of grade 6 a
+strength of 199 correct out of 200 is to be had, and that the teaching
+is by an intelligent person working in accord with psychological
+principles as to both ability and interest.
+
+For one of the easier bonds, most facilitated by other bonds (such
+as 2 x 5 = 10, or 10 - 2 = 8, or the double bond 7 = two 3s and 1
+remainder) in the case of the median or average pupil, twelve practices
+in the week of first learning, supported by twenty-five practices during
+the two months following, and maintained by thirty practices well spread
+over the later periods should be enough. For the more gifted pupils
+lesser amounts down to six, twelve, and fifteen may suffice. For the
+less gifted pupils more may be required up to thirty, fifty, and a
+hundred. It is to be doubted, however, whether pupils requiring nearly
+two hundred repetitions of each of these easy bonds should be taught
+arithmetic beyond a few matters of practical necessity.
+
+For bonds of ordinary difficulty, with average facilitation from other
+bonds (such as 11 - 3, 4 x 7, or 48 / 8 = 6) in the case of the median
+or average pupil, we may estimate twenty practices in the week of first
+learning, supported by thirty, and maintained by fifty practices well
+spread over the later periods. Gifted pupils may gain and keep mastery
+with twelve, fifteen, and twenty practices respectively. Pupils dull at
+arithmetic may need up to twenty, sixty, and two hundred. Here, again,
+it is to be doubted whether a pupil for whom arithmetical facts, well
+taught and made interesting, are so hard to acquire as this, should
+learn many of them.
+
+For bonds of greater difficulty, less facilitated by other bonds (such
+as 17 - 9, 8 x 7, or 12-1/2% of = 1/8 of), the practice may be from ten
+to a hundred percent more than the above.
+
+
+UNDERLEARNING AND OVERLEARNING
+
+If we accept the above provisional estimates as reasonable, we may
+consider the harm done by giving less and by giving more than these
+reasonable amounts. Giving less is indefensible. The pupil's time is
+wasted in excessive checking to find his errors. He is in danger of
+being practiced in error. His attention is diverted from the learning of
+new facts and processes by the necessity of thinking out these
+supposedly mastered facts. All new bonds are harder to learn than they
+should be because the bonds which should facilitate them are not strong
+enough to do so. Giving more does harm to some extent by using up time
+that could be spent better for other purposes, and (though not
+necessarily) by detracting from the pupil's interest in arithmetic. In
+certain cases, however, such excess practice and overlearning are
+actually desirable. Three cases are of special importance.
+
+The first is the case of a bond operating under a changed mental set or
+adjustment. A pupil may know 7 x 8 adequately as a thing by itself, but
+need more practice to operate it in
+
+  285
+    7
+  ---
+
+where he has to remember that 3 is to be added to the 56 when he
+obtains it, and that only the 9 is to be written down, the 5 to be held
+in mind for later use. The practice required to operate the bond
+efficiently in this new set is desirable, even though it is excess from
+a narrower point of view, and causes the straightforward 'seven eights
+are fifty-six' to be overlearned. So also a pupil's work with 24, 34,
+44, etc., +9 may react to give what would be excess practice from the
+point of view of 4 + 9 alone; his work in estimating approximate
+quotient figures in long division may give excess practice on the
+division tables. There are many such cases. Even adding the 5 and 7 in
+5/12 + 7/12 is not quite the same task as adding 5 and 7 undisturbed by
+the fact that they are twelfths. We know far too little about the amount
+of practice needed to adapt arithmetical bonds to efficient operation in
+these more complicated conditions to estimate even approximately the
+allowances to be made. But some allowance, and often a rather large
+allowance, must be made.
+
+The second is the case where the computation in general should be made
+very easy and sure for the pupil except for some one new element that
+is being learned. For example, in teaching the meaning and uses of
+'Averages' and of uneven division, we may deliberately use 2, 3, and 4
+as divisors rather than 7 and 9, so as to let all the pupil's energy be
+spent in learning the new facts, and so that the fraction in the
+quotient may be something easily understood, real, and significant. In
+teaching the addition of mixed numbers, we may use, in the early steps,
+
+  11-1/2
+  13-1/2
+  24
+  ------
+
+rather than
+
+  79-1/2
+  98-1/2
+  67
+  ------
+
+so as to save attention for the new process itself. In cancellation, we
+may give excess practice to divisions by 2, 3, 4, and 5 in order to make
+the transfer to the new habits of considering two numbers together from
+the point of view of their divisibility by some number. In introducing
+trade discount, we may give excess practice on '5% of' and '10% of'
+deliberately, so that the meaning of discount may not be obscured by
+difficulties in the computation itself. Excess practice on, and
+overlearning of, certain bonds is thus very often justifiable.
+
+The third case concerns bonds whose importance for practical uses in
+life or as notable facilitators of other bonds is so great that they may
+profitably be brought to a greater strength than 199 correct out of 200
+at a speed of 2 sec. or less, or be brought to that degree of strength
+very early. Examples of bonds of such special practical use are the
+subtractions from 10, 1/2 + 1/2, 1/2 + 1/4, 1/2 of 60, 1/4 of 60,
+and the fractional parts of 12 and of $1.00. Examples of notable
+facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like
+2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 x 9.
+
+In consideration of these three modifying cases or principles, a volume
+could well be written concerning just how much practice to give to each
+bond, in each of the types of complex situations where it has to
+operate. There is evidently need for much experimentation to expose the
+facts, and for much sagacity and inventiveness in making sure of
+effective learning without wasteful overlearning.
+
+The facts of primary importance are:--
+
+  (1) The textbook or other instrument of instruction which is a
+  teacher's general guide may give far too little practice on
+  certain bonds.
+
+  (2) It may divide the practice given in ways that are apparently
+  unjustifiable.
+
+  (3) The teacher needs therefore to know how much practice it does
+  give, where to supplement it, and what to omit.
+
+  (4) The omissions, on grounds of apparent excess practice, should
+  be made only after careful consideration of the third principle
+  described above.
+
+  (5) The amount of practice should always be considered in the
+  light of its interest and appeal to the pupil's tendency to work
+  with full power and zeal. Mere repetition of bonds when the
+  learner does not care whether he is improving is rarely
+  justifiable on any grounds.
+
+  (6) Practice that is actually in excess is not a very grave defect
+  if it is enjoyed and improves the pupil's attitude toward
+  arithmetic. Not much time is lost; a hundred practices for each of
+  a thousand bonds after mastery to 199 in 200 at 2 seconds will use
+  up less than 60 hours, or 15 hours per year in grades 3 to 6.
+
+  (7) By the proper division of practice among bonds, the
+  arrangement of learning so that each bond helps the others, the
+  adroit shifting of practice of a bond to each new type of
+  situation requiring it to operate under changed conditions, and
+  the elimination of excess practice where nothing substantial is
+  gained, notable improvements over the past hit-and-miss customs
+  may be expected.
+
+  (8) Unless the material for practice is adequate, well balanced
+  and sufficiently motivated, the teacher must keep close account of
+  the learning of pupils. Otherwise disastrous underlearning of many
+  bonds is almost sure to occur and retard the pupil's development.
+
+
+THE ORGANIZATION OF ABILITIES
+
+There is danger that the need of brevity and simplicity which has made
+us speak so often of a bond or an ability, and of the amount of
+practice it requires, may mislead the reader into thinking that these
+bonds and abilities are to be formed each by itself alone and kept so.
+They should rarely be formed so and never kept so. This we have
+indicated from time to time by references to the importance of forming a
+bond in the way in which it is to be used, to the action of bonds in
+changed situations, to facilitation of one bond by others, to the
+cooeperation of abilities, and to their integration into a total
+arithmetical ability.
+
+As a matter of fact, only a small part of drill work in arithmetic
+should be the formation of isolated bonds. Even the very young
+pupil learning 5 and 3 are 8 should learn it with '5 and 5 = 10,'
+'5 and 2 = 7,' at the back of his mind, so to speak. Even so early,
+5 + 3 = 8 should be part of an organized, cooeperating system of bonds.
+Later 50 + 30 = 80 should become allied to it. Each bond should be
+considered, not simply as a separate tool to be put in a compartment
+until needed, but also as an improvement of one total tool or machine,
+arithmetical ability.
+
+There are differences of course. Knowledge of square root can be
+regarded somewhat as a separate tool to be sharpened, polished, and used
+by itself, whereas knowledge of the multiplication tables cannot. Yet
+even square root is probably best made more closely a part of the total
+ability, being taught as a special case of dividing where divisor is to
+be the same as quotient, the process being one of estimating and
+correcting.
+
+In general we do not wish the pupil to be a repository of separated
+abilities, each of which may operate only if you ask him the sort of
+questions which the teacher used to ask him, or otherwise indicate to
+him which particular arithmetical tool he is to use. Rather he is to be
+an effective organization of abilities, cooeperating in useful ways to
+meet the quantitative problems life offers. He should not as a rule
+have to think in such fashion as: "Is this interest or discount? Is it
+simple interest or compound interest? What did I do in compound
+interest? How do I multiply by 2 percent?" The situation that calls up
+interest should also call up the kind of interest that is appropriate,
+and the technique of operating with percents should be so welded
+together with interest in his mind that the right cooeperation will occur
+almost without supervision by him.
+
+As each new ability is acquired, then, we seek to have it take its place
+as an improvement of a thinking being, as a cooeperative member of a
+total organization, as a soldier fighting together with others, as an
+element in an educated personality. Such an organization of bonds will
+not form itself any more than any one bond will create itself. If the
+elements of arithmetical ability are to act together as a total
+organized unified force they must be made to act together in the course
+of learning. What we wish to have work together we must put together and
+give practice in teamwork.
+
+We can do much to secure such cooeperative action when and where and as
+it is needed by a very simple expedient; namely, to give practice with
+computation and problems such as life provides, instead of making up
+drills and problems merely to apply each fact or principle by itself.
+Though a pupil has solved scores of problems reading, "A triangle has a
+base of _a_ feet and an altitude of _b_ feet, what is its area?" he may
+still be practically helpless in finding the area of a triangular plot
+of ground; still more helpless in using the formula for a triangle which
+is one of two into which a trapezoid is divided. Though a pupil has
+learned to solve problems in trade discount, simple interest, compound
+interest, and bank discount one at a time, stated in a few set forms,
+he may be practically helpless before the actual series of problems
+confronting him in starting in business, and may take money out of the
+savings bank when he ought to borrow on a time loan, or delay payment on
+his bills when by paying cash he could save money as well as improve his
+standing with the wholesaler.
+
+Instead of making up problems to fit the abilities given by school
+instruction, we should preferably modify school instruction so that
+arithmetical abilities will be organized into an effective total ability
+to meet the problems that life will offer. Still more generally, _every
+bond formed should be formed with due consideration of every other bond
+that has been or will be formed; every ability should be practiced in
+the most effective possible relations with other abilities_.
+
+
+
+
+CHAPTER VII
+
+THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS
+
+
+The bonds to be formed having been chosen, the next step is to arrange
+for their most economical order of formation--to arrange to have each
+help the others as much as possible--to arrange for the maximum of
+facilitation and the minimum of inhibition.
+
+The principle is obvious enough and would probably be admitted in theory
+by any intelligent teacher, but in practice we are still wedded to
+conventional usages which arose long before the psychology of arithmetic
+was studied. For example, we inherit the convention of studying addition
+of integers thoroughly, and then subtraction, and then multiplication,
+and then division, and many of us follow it though nobody has ever given
+a proof that this is the best order for arithmetical learning. We
+inherit also the opposite convention of studying in a so-called "spiral"
+plan, a little addition, subtraction, multiplication, and division, and
+then some more of each, and then some more, and many of us follow this
+custom, with an unreasoned faith that changing about from one process to
+another is _per se_ helpful.
+
+Such conventions are very strong, illustrating our common tendency to
+cherish most those customs which we cannot justify! The reductions of
+denominate numbers ascending and descending were, until recently, in
+most courses of study, kept until grade 4 or grade 5 was reached,
+although this material is of far greater value for drills on the
+multiplication and division tables than the customary problems about
+apples, eggs, oranges, tablets, and penholders. By some historical
+accident or for good reasons the general treatment of denominate numbers
+was put late; by our naive notions of order and system we felt that any
+use of denominate numbers before this time was heretical; we thus became
+blind to the advantages of quarts and pints for the tables of 2s; yards
+and feet for the tables of 3s; gallons and quarts for the tables of 4s;
+nickels and cents for the 5s; weeks and days for the 7s; pecks and
+quarts for the 8s; and square yards and square feet for the 9s.
+Problems like 5 yards = __ feet or 15 feet = __ yards have not only the
+advantages of brevity, clearness, practical use, real reference, and
+ready variation, but also the very great advantage that part of the data
+have to be _thought of_ in a useful way instead of _read off_ from the
+page. In life, when a person has twenty cents with which to buy tablets
+of a certain sort, he _thinks of_ the price in making his purchase,
+asking it of the clerk only in case he does not know it, and in planning
+his purchases beforehand he _thinks of_ prices as a rule. In spite of
+these and other advantages, not one textbook in ten up to 1900 made
+early use of these exercises with denominate numbers. So strong is mere
+use and wont.
+
+Besides these conventional customs, there has been, in those responsible
+for arithmetical instruction, an admiration for an arrangement of topics
+that is easy for a person, after he knows the subject, to use in
+thinking of its constituent parts and their relations. Such arrangements
+are often called 'logical' arrangements of subject matter, though they
+are often far from logical in any useful sense. Now the easiest order in
+which to think of a hierarchy of habits after you have formed them all
+may be an extremely difficult order in which to form them. The criticism
+of other orders as 'scrappy,' or 'unsystematic,' valid enough if the
+course of study is thought of as an object of contemplation, may be
+foolish if the course of study is regarded as a working instrument for
+furthering arithmetical learning.
+
+We must remember that all our systematizing and labeling is largely
+without meaning to the pupils. They cannot at any point appreciate the
+system as a progression from that point toward this and that, since they
+have no knowledge of the 'this or that.' They do not as a rule think of
+their work in grade 4 as an outcome of their work in grade 3 with
+extensions of a to a_1, and additions of b_2 and b_3 to b and b_1, and
+refinements of c and d by c_4 and d_5. They could give only the vaguest
+account of what they did in grade 3, much less of why it should have
+been done then. They are not much disturbed by a lack of so-called
+'system' and 'logical' progression for the same reason that they are not
+much helped by their presence. What they need and can use is a
+_dynamically_ effective system or order, one that they can learn easily
+and retain long by, regardless of how it would look in a museum of
+arithmetical systems. Unless their actual arithmetical habits are
+usefully related it does no good to see the so-called logical relations;
+and if their habits are usefully related, it does not very much matter
+whether or not they do see these; finally, they can be brought to see
+them best by first acquiring the right habits in a dynamically effective
+order.
+
+
+DECREASING INTERFERENCE AND INCREASING FACILITATION
+
+Psychology offers no single, easy, royal road to discovering this
+dynamically best order. It can only survey the bonds, think what each
+demands as prerequisite and offers as future help, recommend certain
+orders for trial, and measure the efficiency of each order as a means of
+attaining the ends desired. The ingenious thought and careful
+experimentation of many able workers will be required for many years to
+come.
+
+Psychology can, however, even now, give solid constructive help in many
+instances, either by recommending orders that seem almost certainly
+better than those in vogue, or by proposing orders for trial which can
+be justified or rejected by crucial tests.
+
+Consider, for example, the situation, 'a column of one-place numbers to
+be added, whose sum is over 9,' and the response 'writing down the sum.'
+This bond is commonly firmly fixed before addition with two-place
+numbers is undertaken. As a result the pupil has fixed a habit that he
+has to break when he learns two-place addition. If _oral_ answers only
+are given with such single columns until two-place addition is well
+under way, the interference is avoided.
+
+In many courses of study the order of systematic formation of the
+multiplication table bonds is: 1 x 1, 2 x 1, etc., 1 x 2, 2 x 2, etc.,
+1 x 3, 2 x 3, etc., 1 x 9, 2 x 9, etc. This is probably wrong in two
+respects. There is abundant reason to believe that the x 5s should be
+learned first, since they are easier to learn than the 1s or the 2s, and
+give the idea of multiplying more emphatically and clearly. There is
+also abundant reason to believe that the 1 x 5, 1 x 2, 1 x 3, etc.,
+should be put very late--after at least three or four tables are
+learned, since the question "What is 1 times 2?" (or 3 or 5) is
+unnecessary until we come to multiplication of two- and three-place
+numbers, seems a foolish question until then, and obscures the notion of
+multiplication if put early. Also the facts are best learned once for
+all as the habits "1 times _k_ is the same as _k_," and "_k_ times 1 is
+the same as _k_."[8]
+
+  [8] The very early learning of 2 x 2, 2 x 3, 3 x 2, 2 x 4, 4 x 2,
+      3 x 3, and perhaps a few more multiplications is not considered
+      here. It is advisable. The treatment of 0 x 0, 0 x 1, 1 x 0, etc.,
+      is not considered here. It is probably best to defer the 'x 0'
+      bonds until after all the others are formed and are being used
+      in short multiplication, and to form them in close connection
+      with their use in short multiplication. The '0 x' bonds may well
+      be deferred until they are needed in 'long' multiplication,
+      0 x 0 coming last of all.
+
+In another connection it was recommended that the divisions to 81 / 9
+be learned by selective thinking or reasoning from the multiplications.
+This determines the order of bonds so far as to place the formation of
+the division bonds soon after the learning of the multiplications. For
+other reasons it is well to make the proximity close.
+
+One of the arbitrary systematizations of the order of formation of bonds
+restricts operations at first to the numbers 1 to 10, then to numbers
+under 100, then to numbers under 1000, then to numbers under 10,000.
+Apart from the avoidance of unreal and pedantic problems in applied
+arithmetic to which work with large numbers in low grades does somewhat
+predispose a teacher, there is little merit in this restriction of the
+order of formation of bonds. Its demerits are many. For example, when
+the pupil is learning to 'carry' in addition he can be given better
+practice by soon including tasks with sums above 100, and can get a
+valuable sense of the general use of the process by being given a few
+examples with three- and four-place numbers to be added. The same holds
+for subtraction. Indeed, there is something to be said in favor of using
+six- or seven-place numbers in subtraction, enforcing the 'borrowing'
+process by having it done again and again in the same example, and
+putting it under control by having the decision between 'borrowing' and
+'not borrowing' made again and again in the same example. When the
+multiplication tables are learned the most important use for them is not
+in tedious reviews or trivial problems with answers under 100, but in
+regular 'short' multiplication of two- and three- and even four-place
+numbers. Just as the addition combinations function mainly in the
+higher-decade modifications of them, so the multiplication combinations
+function chiefly in the cases where the bond has to operate while the
+added tasks of keeping one's place, adding what has been carried,
+writing down the right figure in the right place, and holding the right
+number for later addition, are also taken care of. It seems best to
+introduce such short multiplication as soon as the x 5s, x 2s, x 3s,
+and x 4s are learned and to put the x 6s, x 7s, and the rest to work
+in such short multiplication as soon as each is learned.
+
+Still surer is the need for four-, five-, and six-place numbers when
+two-place numbers are used in multiplying. When the process with a
+two-place multiplier is learned, multiplications by three-place numbers
+should soon follow. They are not more difficult then than later. On the
+contrary, if the pupil gets used to multiplying only as one does with
+two-place multipliers, he will suffer more by the resulting interference
+than he does from getting six- or seven-place answers whose meaning he
+cannot exactly realize. They teach the rationale and the manipulations
+of long multiplication with especial economy because the principles and
+the procedures are used two or three times over and the contrasts
+between the values which the partial products have in adding become
+three instead of one.
+
+The entire matter of long multiplication with integers and United States
+money should be treated as a teaching unit and the bonds formed in close
+organization, even though numbers as large as 900,000 are occasionally
+involved. The reason is not that it is more logical, or less scrappy,
+but that each of the bonds in question thus gets much help from, and
+gives much help to, the others.
+
+In sharp contrast to a topic like 'long multiplication' stands a topic
+like denominate numbers. It most certainly should not be treated as a
+large teaching unit, and all the bonds involved in adding, subtracting,
+multiplying, and dividing with all the ordinary sorts of measures should
+certainly not be formed in close sequence. The reductions ascending and
+descending for many of the measures should be taught as drills on the
+appropriate multiplication and division tables. The reduction of feet
+and inches to inches, yards and feet to yards, gallons and quarts to
+quarts, and the like are admirable exercises in connection with the
+(_a_ x _b_) + _c_ = .... problems,--the 'Bought 3 lbs. of sugar at
+7 cents and 5 cents worth of matches' problems. The reductions of
+inches to feet and inches and the like are admirable exercises in
+the _d_ = (.... x _b_) + _c_ or 'making change' problem, which in its
+small-number forms is an excellent preparatory step for short division.
+They are also of great service in early work with fractions. The
+feet-mile, square-foot-square-inch, and other simple relations give a
+genuine and intelligible demand for multiplication with large numbers.
+
+Knowledge of the metric system for linear and square measure would
+perhaps, as an introduction to decimal fractions, more than save the
+time spent to learn it. It would even perhaps be worth while to invent a
+measure (call it the _twoqua_) midway between the quart and gallon and
+teach carrying in addition and borrowing in subtraction by teaching
+first the addition and subtraction of 'gallon, twoqua, quart, and pint'
+series! Many of the bonds which a system-made tradition huddled together
+uselessly in a chapter on denominate numbers should thus be formed as
+helpful preparations for and applications of other bonds all the way
+from the first to the eighth half-year of instruction in arithmetic.
+
+The bonds involved in the ability to respond correctly to the series:--
+
+   5 = .... 2s and .... remainder
+   5 = .... 3s and .... remainder
+  88 = .... 9s and .... remainder
+
+should be formed before, not during, the training in short division.
+They are admirable at that point as practice on the division tables; are
+of practical service in the making-change problems of the small purchase
+and the like; and simplify the otherwise intricate task of keeping one's
+place, choosing the quotient figure, multiplying by it, subtracting and
+holding in mind the new number to be divided, which is composed half of
+the remainder and half of a figure in the written dividend. This change
+of order is a good illustration of the nearly general rule that "_When
+the practice or review required to perfect or hold certain bonds can, by
+an inexpensive modification, be turned into a useful preparation for new
+bonds, that modification should be made._"
+
+The bonds involved in the four operations with United States money
+should be formed in grades 3 and 4 along with or very soon after the
+corresponding bonds with three-place and four-place integers. This
+statement would have seemed preposterous to the pedagogues of fifty
+years ago. "United States money," they would have said, "is an
+application of decimals. How can it be learned until the essentials of
+decimal fractions are known? How will the child understand when
+multiplying $.75 by 3 that 3 times 5 cents is 1 dime and 5 cents, or
+that 3 times 70 cents is 2 dollars and 1 dime? Why perplex the young
+pupils with the difficulties of placing the decimal point? Why disturb
+the learning of the four operations with integers by adding at each step
+a second 'procedure with United States money'?"
+
+The case illustrates very well the error of the older oversystematic
+treatment of the order of topics and the still more important error of
+confusing the logic of proof with the psychology of learning. To prove
+that 3 x $.75 = $2.25 to the satisfaction of certain arithmeticians, you
+may need to know the theory of decimal fractions; but to do such
+multiplication all a child needs is to do just what he has been doing
+with integers and then "Put a $ before the answer to show that it means
+dollars and cents, and put a decimal point in the answer to show which
+figures mean dollars and which figures mean cents." And this is general.
+The ability to operate with integers plus the two habits of prefixing $
+and separating dollars from cents in the result will enable him to
+operate with United States money.
+
+Consequently good practice came to use United States money not as a
+consequence of decimal fractions, learned by their aid, but as an
+introduction to decimal fractions which aids the pupil to learn them. So
+it has gradually pushed work with United States money further and
+further back, though somewhat timidly.
+
+We need not be timid. The pupil will have no difficulty in adding,
+subtracting, multiplying, and dividing with United States money--unless
+we create it by our explanations! If we simply form the two bonds
+described above and show by proper verification that the procedure
+always gives the right answer, the early teaching of the four operations
+with United States money will in fact actually show a learning profit!
+It will save more time in the work with integers than was spent in
+teaching it! For, in the first place, it will help to make work with
+four-place and five-place numbers more intelligible and vital. A pupil
+can understand $16.75 or $28.79 more easily than 1675 or 2879. The
+former may be the prices of a suit or sewing machine or bicycle. In the
+second place, it permits the use of a large stock of genuine problems
+about spending, saving, sharing, and the like with advertisements and
+catalogues and school enterprises. In the third place, it permits the
+use of common-sense checks. A boy may find one fourth of 3000 as 7050 or
+75 and not be disturbed, but he will much more easily realize that one
+fourth of $30.00 is not over $70 or less than $1. Even the decimal point
+of which we used to be so afraid may actually help the eye to keep its
+place in adding.
+
+
+INTEREST
+
+So far, the illustrations of improvements in the order of bonds so as to
+get less interference and more facilitation than the customary orders
+secure have sought chiefly to improve the mechanical organization of the
+bonds. Any gain in interest which the changes described effected would
+be largely due to the greater achievement itself. Dewey and others have
+emphasized a very different principle of improving the order of
+formation of bonds--the principle of determination of the bonds to be
+formed by some vital, engaging problem which arouses interest enough to
+lighten the labor and which goes beyond or even against cut-and-dried
+plans for sequences in order to get effective problems. For example, the
+work of the first month in grade 2B might sacrifice facilitations of the
+mechanical sort in order to put arithmetic to use in deciding what
+dimensions a rabbit's cage should have to give him 12 square feet of
+floor space, how much bread he should have per meal to get 6 ounces a
+day, how long a ten-cent loaf would last, how many loaves should be
+bought per week, how much it costs to feed the rabbit, how much he has
+gained in weight since he was brought to the school, and so on.
+
+Such sacrifices of the optimal order if interest were equal, in order to
+get greater interest or a healthier interest, are justifiable. Vital
+problems as nuclei around which to organize arithmetical learning are of
+prime importance. It is even safe probably to insist that some genuine
+problem-situation requiring a new process, such as addition with
+carrying, multiplication by two-place numbers, or division with
+decimals, be provided in every case as a part of the introduction to
+that process. The sacrifice should not be too great, however; the search
+for vital problems that fit an economical order of subject matter is as
+much needed as the amendment of that order to fit known interests; and
+the assurance that a problem helps the pupil to learn arithmetic is as
+important as the assurance that arithmetic is used to help the pupil
+solve his personal problems.
+
+Much ingenuity and experimentation will be required to find the order
+that is satisfactory in both quality and quantity of interest or motive
+and helpfulness of the bonds one to another. The difficulty of
+organizing arithmetic around attractive problems is much increased by
+the fact of class instruction. For any one pupil vital, personal
+problems or projects could be found to provide for many arithmetical
+abilities; and any necessary knowledge and technique which these
+projects did not develop could be somehow fitted in along with them. But
+thirty children, half boys and half girls, varying by five years in age,
+coming from different homes, with different native capacities, will not,
+in September, 1920, unanimously feel a vital need to solve any one
+problem, and then conveniently feel another on, say, October 15! In the
+mechanical laws of learning children are much alike, and the gain we
+may hope to make from reducing inhibitions and increasing facilitations
+is, for ordinary class-teaching, probably greater than that to be made
+from the discovery of attractive central problems. We should, however,
+get as much as possible of both.
+
+
+GENERAL PRINCIPLES
+
+The reader may by now feel rather helpless before the problem of the
+arrangement of arithmetical subject matter. "Sometimes you complete a
+topic, sometimes you take it piecemeal months or years apart, often you
+make queer twists and shifts to get a strategic advantage over the
+enemy," he may think, "but are there no guiding principles, no general
+rules?" There is only one that is absolutely general, to _take the order
+that works best for arithmetical learning_. There are particular rules,
+but there are so many and they are so limited by an 'other things being
+equal' clause, that probably a general eagerness to think out the _pros_
+and _cons_ for any given proposal is better than a stiff attempt to
+adhere to these rules. I will state and illustrate some of them, and let
+the reader judge.
+
+_Other things being equal, one new sort of bonds should not be started
+until the previous set is fairly established, and two different sets
+should not be started at once._ Thus, multiplication of two- and
+three-place numbers by 2, 3, 4, and 5 will first use numbers such that
+no carrying is required, and no zero difficulties are encountered, then
+introduce carrying, then introduce multiplicands like 206 and 320.
+If other things were equal, the carrying would be split into two
+steps--first drills with (4 x 6) + 2, (3 x 7) + 3, (5 x 4) + 1, and the
+like, and second the actual use of these habits in the multiplication.
+The objection to this separation of the double habit is that the first
+part of it in isolation is too artificial--that it may be better to
+suffer the extra difficulty of forming the two together than to teach so
+rarely used habits as the (_a_ x _b_) + _c_ series. Experimental tests
+are needed to decide this point.
+
+_Other things being equal, bonds should be formed in such order that
+none will have to be broken later._ For example, there is a strong
+argument for teaching long division first, or very early, with
+remainders, letting the case of zero remainder come in as one of many.
+If the pupils have been familiarized with the remainder notion by the
+drills recommended as preparation for short division,[9] the use of
+remainders in long division will offer little difficulty. The exclusive
+use of examples without remainders may form the habit of not being exact
+in computation, of trusting to 'coming out even' as a sole check, and
+even of writing down a number to fit the final number to be divided
+instead of obtaining it by honest multiplication.
+
+  [9] See page 76.
+
+For similar reasons additions with 2 and 3 as well as 1 to be 'carried'
+have much to recommend them in the very first stages of column addition
+with carrying. There is here the added advantage that a pupil will be
+more likely to remember to carry if he has to think _what_ to carry. The
+present common practice of using small numbers for ease in the addition
+itself teaches many children to think of carrying as adding one.
+
+_Other things being equal, arrange to have variety._ Thus it is
+probably, though not surely, wise to interrupt the monotony of learning
+the multiplication and division tables, by teaching the fundamentals of
+'short' multiplication and perhaps of division after the 5s, 2s, 3s, and
+4s are learned. This makes a break of several weeks. The facts for the
+6s, 7s, 8s, and 9s can then be put to varied use as fast as learned. It
+is almost certainly wise to interrupt the first half-year's work with
+addition and subtraction, by teaching 2 x 2, 2 x 3, 3 x 2, 2 x 4, 4 x 2,
+2 x 5, later by 2 x 10, 3 x 10, 4 x 10, 5 x 10, later by 1/2 + 1/2,
+1-1/2 + 1/2, 1/2 of 2, 1/2 of 4, 1/2 of 6, and at some time by certain
+profitable exercises wherein a pupil tells all he knows about certain
+numbers which may be made nuclei of important facts (say, 5, 8, 10, 12,
+15, and 20).
+
+_Other things being equal, use objective aids to verify an arithmetical
+process or inference after it is made, as well as to provoke it._ It is
+well at times to let pupils do everything that they can with relations
+abstractly conceived, testing their results by objective counting,
+measuring, adding, and the like. For example, an early step in adding
+should be to show three things, put them under a book, show two more,
+put these under the book, and then ask how many there are under the
+book, letting the objective counting come later as the test of the
+correctness of the addition.
+
+_Other things being equal, reserve all explanations of why a process
+must be right until the pupils can use the process accurately, and have
+verified the fact that it is right._ Except for the very gifted pupils,
+the ordinary preliminary deductive explanations of what must be done are
+probably useless as means of teaching the pupils what to do. They use up
+much time and are of so little permanent effect that, as we have seen,
+the very arithmeticians who advocate making them, admit that after a
+pupil has mastered the process he may be allowed to forget the reasons
+for it. I am not sure that the deductive proofs of why we place the
+decimal point as we do in division by a decimal, or invert and multiply
+in dividing by a fraction, and the like, are worth teaching at all. If
+they are to be taught at all, the time to teach them is (except for the
+very gifted) after the pupil has mastered the process and has confidence
+in it. He then at least knows what process he is to prove is right, and
+that it is right, and has had some chance of seeing _why_ it is right
+from his experience with it.
+
+One more principle may be mentioned without illustration. _Arrange the
+order of bonds with due regard for the aims of the other studies of the
+curriculum and the practical needs of the pupil outside of school._
+Arithmetic is not a book or a closed system of exercises. It is the
+quantitative work of the pupils in the elementary school. No narrower
+view of it is adequate.
+
+
+
+
+CHAPTER VIII
+
+THE DISTRIBUTION OF PRACTICE
+
+
+THE PROBLEM
+
+The same amount of practice may be distributed in various ways. Figures
+7 to 10, for example, show 200 practices with division by a fraction
+distributed over three and a half years of 10 months in four different
+ways. In Fig. 7, practice is somewhat equally distributed over the whole
+period. In Fig. 8 the practice is distributed at haphazard. In Fig. 9
+there is a first main learning period, a review after about ten weeks, a
+review at the beginning of the seventh grade, another review at the
+beginning of the eighth grade, and some casual practice rather at
+random. In Fig. 10 there is a main learning period, with reviews
+diminishing in length and separated by wider and wider intervals, with
+occasional practice thereafter to keep the ability alive and healthy.
+
+Plans I and II are obviously inferior to Plans III and IV; and Plan IV
+gives promise of being more effective than Plan III, since there seems
+danger that the pupil working by Plan III might in the ten weeks lose
+too much of what he had gained in the initial practice, and so again in
+the next ten weeks.
+
+It is not wise, however, to try now to make close decisions in the case
+of practice with division by a fraction; or to determine what the best
+distribution of practice is for that or any other ability to be
+improved. The facts of psychology are as yet not adequate for very close
+decisions, nor are the types of distribution of practice that are best
+adapted to different abilities even approximately worked out.
+
+    [Illustration: FIG. 7.--Plan I. 200 practices distributed somewhat
+    evenly over 3-1/2 years of 10 months. In Figs. 7, 8, 9, and 10,
+    each tenth of an inch along the base line represents one month.
+    Each hundredth of a square inch represents four practices, a
+    little square 1/20 of an inch wide and 1/20 inch high representing
+    one practice.]
+
+    [Illustration: FIG. 8.--Plan II. 200 practices distributed
+    haphazard over 3-1/2 years of 10 months.]
+
+    [Illustration: FIG. 9.--Plan III. A learning period, three reviews,
+    and incidental practice.]
+
+    [Illustration: FIG. 10.--Plan IV. A learning period with reviews
+    of decreasing length at increasing intervals.]
+
+
+SAMPLE DISTRIBUTIONS
+
+Let us rather examine some actual cases of distribution of practice
+found in school work and consider, not the attainment of the best
+possible distribution, but simply the avoidance of gross blunders and
+the attainment of reasonable, defensible procedures in this regard.
+
+Figures 11 to 18 show the distribution of examples in multiplication
+with multipliers of various sorts. _X_ stands for any digit except zero.
+_O_ stands for 0. _XXO_ thus means a multiplier like 350 or 270 or 160;
+_XOX_ means multipliers like 407, 905, or 206; _XX_ means multipliers
+like 25, 17, 38. Each of these diagrams covers approximately 3-1/2 years
+of school work, or from about the middle of grade 3 to the end of grade
+6. They are made from counts of four textbooks (A, B, C, and D), the
+count being taken for each successive 8 pages.[10] Each tenth of an inch
+along the base line equals 8 pages of the text in question. Each .01 sq.
+in. equals one example. The books, it will be observed, differ in the
+amount of practice given, as well as in the way in which it is
+distributed.
+
+  [10] At the end of a volume or part, the count may be from as
+       few as 5 or as many as 12 pages.
+
+These distributions are worthy of careful study; we shall note only a
+few salient facts about them here. Of the distributions of
+multiplications with multipliers of the _XX_ type, that of book D (Fig.
+14) is perhaps the best. A (Fig. 11) has too much of the practice too
+late; B (Fig. 12) gives too little practice in the first learning; C
+(Fig. 13) gives too much in the first learning and in grade 6. Among the
+distributions of multiplication with multipliers of the _XOX_ type, that
+of book D (Fig. 18) is again probably the best. A, B, and C (Figs. 15,
+16, and 17) have too much practice early and too long intervals between
+reviews. Book C (Fig. 17) by a careless oversight has one case of this
+very difficult process, without any explanation, weeks before the
+process is taught!
+
+    [Illustration: FIG. 11.--Distribution of practise with multipliers
+    of the _XX_ type in the first two books of the three-book text A.]
+
+    [Illustration: FIG. 12.--Same as Fig. 11, but for text B. Following
+    this period come certain pages of computation to be used by the
+    teacher at her discretion, containing 24 _XX_ multiplications.]
+
+    [Illustration: FIG. 13.--Same as Fig. 11, but for text C.]
+
+    [Illustration: FIG. 14.--Same as Fig. 11, but for text D.]
+
+    [Illustration: FIG. 15.--Distribution of practice with multipliers
+    of the _XOX_ type in the first two books of the three-book text
+    A.]
+
+    [Illustration: FIG. 16.--Same as Fig. 15, but for text B. Following
+    this period come certain pages of computation to be used by the
+    teacher at her discretion, containing 17 _XOX_ multiplications.]
+
+    [Illustration: FIG. 17.--Same as Fig. 16, but for text C.]
+
+    [Illustration: FIG. 18.--Same as Fig. 16, but for text D.]
+
+Figures 19, 20, 21, 22, and 23 all concern the first two books of the
+three-book text E.
+
+Figure 19 shows the distribution of practice on 5 x 5 in the first two
+books of text E. The plan is the same as in Figs. 11 to 18, except that
+each tenth of an inch along the base line represents ten pages. Figure
+20 shows the distribution of practice on 7 x 7; Fig. 21 shows it for
+6 x 7 and 7 x 6 together. In Figs. 20 and 21 also, 0.1 inch along the
+base line equals ten pages.
+
+Figures 22 and 23 show the distribution of practice on the divisions of
+72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or 9, and on the
+divisions of 81, 82 ... 89 by 9. Each tenth of an inch along the base
+line represents ten pages here also.
+
+Figures 19 to 23 show no consistent plan for distributing practice.
+With 5 x 5 (Fig. 19) the amount of practice increases from the first
+treatment in grade 3 to the end of grade 6, so that the distribution
+would be better if the pupil began at the end and went backward! With
+7 x 7 (Fig. 20) the practice is distributed rather evenly and in small
+doses. With 6 x 7 and 7 x 6 (Fig. 21) much of it is in very large doses.
+With the divisions (Figs. 22 and 23) the practice is distributed more
+suitably, though in Fig. 23 there is too much of it given at one time
+in the middle of the period.
+
+    [Illustration: FIG. 19.--Distribution of practice with 5 x 5 in
+    the first two books of the three-book text E.]
+
+    [Illustration: FIG. 20.--Distribution of practice with 7 x 7 in
+    the first two books of text E.]
+
+    [Illustration: FIG. 21.--Distribution of practice with 6 x 7
+    or 7 x 6 in the first two books of text E.]
+
+    [Illustration: FIG. 22.--Distribution of practice with
+    72, 73 ... 79 / 8 or 9 in the first two books of text E.]
+
+    [Illustration: FIG. 23.--Distribution of practice with
+    81, 82 ... 89 / 9 in the first two books of text E.]
+
+
+POSSIBLE IMPROVEMENTS
+
+Even if we knew what the best distribution of practice was for each
+ability of the many to be inculcated by arithmetical instruction, we
+could perhaps not provide it for all of them. For, in the first place,
+the allotments for some of them might interfere with those for others.
+In the second place, there are many other considerations of importance
+in the ordering of topics besides giving the optimal distribution of
+practice to each ability. Such are considerations of interest, of
+welding separate abilities into an integrated total ability, and of the
+limitations due to the school schedule with its Saturdays, Sundays,
+holidays, and vacations.
+
+Improvement can, however, be made over present practice in many
+respects. A scientific examination of the teaching of almost any class
+for a year, or of many of our standard instruments of instruction, will
+reveal opportunities for improving the distribution of practice with no
+sacrifice of interest, and with an actual gain in integrated functioning
+arithmetical power. In particular it will reveal cases where an ability
+is given practice and then, never being used again, left to die of
+inactivity. It will reveal cases where an ability is given practice and
+then left so long without practice that the first effect is nearly lost.
+There will be cases where practice is given and reviews are given, but
+all in such isolation from everything else in arithmetic that the
+ability, though existent, does not become a part of the pupil's general
+working equipment. There will be cases where more practice is given in
+the late than the earlier periods for no apparent extrinsic advantage;
+and cases where the practice is put where it is for no reason that is
+observable save that the teacher or author in question has decided to
+have some drill work at that time!
+
+Each ability has its peculiar needs in this matter, and no set rules are
+at present of much value. It will be enough for the present if we are
+aroused to the problem of distribution, avoid obvious follies like those
+just noted, and exercise what ingenuity we have.
+
+
+
+
+CHAPTER IX
+
+THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL NOTIONS IN
+ARITHMETIC[11]
+
+  [11] Certain paragraphs in this and the following chapter are
+       taken from the author's _Educational Psychology_, with
+       slight modifications.
+
+
+RESPONSES TO ELEMENTS AND CLASSES
+
+The plate which you see, the egg before you at the breakfast table, and
+this page are concrete things, but whiteness, whether of plate, egg, or
+paper, is, we say, an abstract quality. To be able to think of whiteness
+irrespective of any concrete white object is to be able to have an
+abstract idea or notion of white; to be able to respond to whiteness,
+irrespective of whether it is a part of china, eggshell, paper or
+whatever object, is to be able to respond to the abstract element of
+whiteness.
+
+Learning arithmetic involves the formation of very many such ideas, the
+acquisition of very many such powers of response to elements regardless
+of the gross total situations in which they appear. To appreciate the
+fiveness of five boys, five pencils, five inches, five rings of a bell;
+to understand the division into eight equal parts of 40 cents, 32 feet,
+64 minutes, or 16 ones; to respond correctly to the fraction relation in
+2/3, 5/6, 3/4, 7/12, 1/8, or any other; to be sensitive to the common
+element of 9 = 3 x 3, 16 = 4 x 4, 625 = 25 x 25, .04 = .2 x .2, 1/4 =
+1/2 x 1/2,--these are obvious illustrations. All the numbers which the
+pupil learns to understand and manipulate are in fact abstractions; all
+the operations are abstractions; percent, discount, interest, height,
+length, area, volume, are abstractions; sum, difference, product,
+quotient, remainder, average, are facts that concern elements or aspects
+which may appear with countless different concrete surroundings or
+concomitants.
+
+Towser is a particular dog; your house lot on Elm Street is a particular
+rectangle; Mr. and Mrs. I.S. Peterson and their daughter Louise are a
+particular family of three. In contrast to these particulars, we mean
+by a dog, a rectangle, and a family of three, _any_ specimens of these
+classes of facts. The idea of a dog, of rectangles in general, of any
+family of three is a general notion, a concept or idea of a class or
+species. The ability to respond to any dog, or rectangle, or family of
+three, regardless of which particular one it may be, is the general
+notion in action.
+
+Learning arithmetic involves the formation of very many such general
+notions, such powers of response to any member of a certain class. Thus
+a hundred different sized lots may all be responded to as rectangles;
+9/18, 12/27, 15/24, and 27/36 may all be responded to as members of the
+class, 'both members divisible by 3.' The same fact may be responded to
+in different ways according to the class to which it is assigned. Thus 4
+in 3/4, 4/5, 45, 54, and 405 is classed respectively as 'a certain sized
+part of unity,' 'a certain number of parts of the size shown by the 5,'
+'a certain number of tens,' 'a certain number of ones,' and 'a certain
+number of hundreds.' Each abstract quality may become the basis of a
+class of facts. So fourness as a quality corresponds to the class
+'things four in number or size'; the fractional quality or relation
+corresponds to the class 'fractions.' The bonds formed with classes of
+facts and with elements or features by which one whole class of facts is
+distinguished from another, are in fact, a chief concern of arithmetical
+learning.[12]
+
+  [12] It should be noted that just as concretes give rise to
+       abstractions, so these in turn give rise to still more
+       abstract abstractions. Thus fourness, fiveness, twentyness,
+       and the like give rise to 'integral-number-ness.' Similarly
+       just as individuals are grouped into general classes, so
+       classes are grouped into still more general classes.
+       Half, quarter, sixth, and tenth are general notions, but
+       'one ...th' is more general; and 'fraction' is still more
+       general.
+
+
+FACILITATING THE ANALYSIS OF ELEMENTS
+
+Abstractions and generalizations then depend upon analysis and upon
+bonds formed with more or less subtle elements rather than with gross
+total concrete situations. The process involved is most easily
+understood by considering the means employed to facilitate it.
+
+The first of these is having the learner respond to the total situations
+containing the element in question with the attitude of piecemeal
+examination, and with attentiveness to one element after another,
+especially to so near an approximation to the element in question as he
+can already select for attentive examination. This attentiveness to one
+element after another serves to emphasize whatever appropriate minor
+bonds from the element in question the learner already possesses. Thus,
+in teaching children to respond to the 'fiveness' of various
+collections, we show five boys or five girls or five pencils, and say,
+"See how many boys are standing up. Is Jack the only boy that is
+standing here? Are there more than two boys standing? Name the boys
+while I point at them and count them. (Jack) is one, and (Fred) is one
+more, and (Henry) is one more. Jack and Fred make (two) boys. Jack and
+Fred and Henry make (three) boys." (And so on with the attentive
+counting.) The mental set or attitude is directed toward favoring the
+partial and predominant activity of 'how-many-ness' as far as may be;
+and the useful bonds that the 'fiveness,' the 'one and one and one and
+one and one-ness,' already have, are emphasized as far as may be.
+
+The second of the means used to facilitate analysis is having the
+learner respond to many situations each containing the element in
+question (call it A), but with varying concomitants (call these V. C.)
+his response being so directed as, so far as may be, to separate each
+total response into an element bound to the A and an element bound to
+the V. C.
+
+Thus the child is led to associate the responses--'Five boys,' 'Five
+girls,' 'Five pencils,' 'Five inches,' 'Five feet,' 'Five books,' 'He
+walked five steps,' 'I hit my desk five times,' and the like--each with
+its appropriate situation. The 'Five' element of the response is thus
+bound over and over again to the 'fiveness' element of the situation,
+the mental set being 'How many?,' but is bound only once to any one of
+the concomitants. These concomitants are also such as have preferred
+minor bonds of their own (the sight of a row of boys _per se_ tends
+strongly to call up the 'Boys' element of the response). The other
+elements of the responses (boys, girls, pencils, etc.) have each only a
+slight connection with the 'fiveness' element of the situations. These
+slight connections also in large part[13] counteract each other, leaving
+the field clear for whatever uninhibited bond the 'fiveness' has.
+
+  [13] They may, of course, also result in a fusion or an alternation
+       of responses, but only rarely.
+
+The third means used to facilitate analysis is having the learner
+respond to situations which, pair by pair, present the element in a
+certain context and present that same context with _the opposite of the
+element in question_, or with something at least very unlike the
+element. Thus, a child who is being taught to respond to 'one fifth' is
+not only led to respond to 'one fifth of a cake,' 'one fifth of a pie,'
+'one fifth of an apple,' 'one fifth of ten inches,' 'one fifth of an
+army of twenty soldiers,' and the like; he is also led to respond to
+each of these _in contrast with_ 'five cakes,' 'five pies,' 'five
+apples,' 'five times ten inches,' 'five armies of twenty soldiers.'
+Similarly the 'place values' of tenths, hundredths, and the rest are
+taught by contrast with the tens, hundreds, and thousands.
+
+These means utilize the laws of connection-forming to disengage a
+response element from gross total responses and attach it to some
+situation element. The forces of use, disuse, satisfaction, and
+discomfort are so maneuvered that an element which never exists by
+itself in nature can influence man almost as if it did so exist, bonds
+being formed with it that act almost or quite irrespective of the gross
+total situation in which it inheres. What happens can be most
+conveniently put in a general statement by using symbols.
+
+Denote by _a_ + _b_, _a_ + _g_, _a_ + _l_, _a_ + _q_, _a_ + _v_, and
+_a_ + _B_ certain situations alike in the element _a_ and different in
+all else. Suppose that, by original nature or training, a child responds
+to these situations respectively by r_{1} + r_{2}, r_{1} + r_{7},
+r_{1} + r_{12}, r_{1} + r_{17}, r_{1} + r_{22}, r_{1} + r_{27}. Suppose
+that man's neurones are capable of such action that r_{1}, r_{2}, r_{7},
+r_{12}, r_{22}, and r_{27}, can each be made singly.
+
+
+Case I. Varying Concomitants
+
+Suppose that _a_ + _b_, _a_ + _g_, _a_ + _l_, etc., occur once each.
+
+  We have _a_ + _b_ responded to by r_{1} + r_{2},
+          _a_ + _g_      "       "  r_{1} + r_{7},
+          _a_ + _l_      "       "  r_{1} + r_{12},
+          _a_ + _q_      "       "  r_{1} + r_{17},
+          _a_ + _v_      "       "  r_{1} + r_{22}, and
+          _a_ + _B_      "       "  r_{1} + r_{27}, as shown in
+                                      Scheme I.
+
+Scheme I
+
+              _a_   _b_   _g_   _l_   _q_   _v_   _B_
+  r_{1}        6     1     1     1     1     1     1
+  r_{2}        1     1
+  r_{7}        1           1
+  r_{12}       1                 1
+  r_{17}       1                       1
+  r_{22}       1                             1
+  r_{27}       1                                   1
+
+_a_ is thus responded to by r_{1} (that is, connected with r_{1}) each
+time, or six in all, but only once each with _b_, _g_, _l_, _q_, _v_,
+and _B_. _b_, _g_, _l_, _q_, _v_, and _B_ are connected once each with
+r_{1} and once respectively with r_{2}, r_{7}, r_{12}, etc. The bond
+from _a_ to r_{1}, has had six times as much exercise as the bond from
+_a_ to r_{2}, or from _a_ to r_{7}, etc. In any new gross situation, _a_
+0, _a_ will be more predominant in determining response than it would
+otherwise have been; and r_{1} will be more likely to be made than
+r_{2}, r_{7}, r_{12}, etc., the other previous associates in the
+response to a situation containing _a_. That is, the bond from the
+element _a_ to the response r_{1} has been notably strengthened.
+
+
+Case II. Contrasting Concomitants
+
+Now suppose that _b_ and _g_ are very dissimilar elements (_e.g._, white
+and black), that _l_ and _q_ are very dissimilar (_e.g._, long and
+short), and that _v_ and _B_ are also very dissimilar. To be very
+dissimilar means to be responded to very differently, so that r_{7}, the
+response to _g_, will be very unlike r_{2}, the response to _b_. So
+r_{7} may be thought of as r_{not 2} or r_{-2}. In the same way r_{12}
+may be thought of as r_{not 12} or r_{-12}, and r_{27} may be called
+r_{not 22} or r_{-22}.
+
+Then, if the situations _a_ _b_, _a _g_, _a _l_, _a _q_, _a _v_, and
+_a_ _B_ are responded to, each once, we have:--
+
+  _a_ + _b_ responded to by r_{1} + r_{2},
+  _a_ + _g_     "        "  r_{1} + r_{not 2},
+  _a_ + _l_     "        "  r_{1} + r_{12},
+  _a_ + _q_     "        "  r_{1} + r_{not 12},
+  _a_ + _v_     "        "  r_{1} + r_{22}, and
+  _a_ + _B_     "        "  r_{1} + r_{not 22}, as shown in Scheme II.
+
+Scheme II
+
+              _a_  _b_     _g_       _l_     _q_       _v_     _B_
+                       (opp. of _b_)     (opp. of _l_)     (opp. of _v_)
+  r_{1}        6    1       1         1       1         1       1
+  r_{not 1}
+  r_{2}        1    1
+  r_{not 2}    1            1
+  r_{12}       1                      1
+  r_{not 12}   1                               1
+  r_{22}       1                                        1
+  r_{not 22}   1                                                1
+
+r_{1} is connected to _a_ by 6 repetitions. r_{2} and r_{not 2} are each
+connected to _a_ by 1 repetition, but since they interfere, canceling
+each other so to speak, the net result is for _a_ to have zero tendency
+to call up r_{2} or r_{not 2}. r_{12} and r_{not 12} are each connected
+to _a_ by 1 repetition, but they interfere with or cancel each other
+with the net result that _a_ has zero tendency to call up r_{12} or
+r_{not 12}. So with r_{22} and r_{not 22}. Here then the net result of
+the six connections of _a_ _b_, _a_ _g_, _a_ _l_, _a_ _q_, _a_ _v_, and
+_a_ _B_ is to connect _a_ with _r_, and with nothing else.
+
+
+Case III. Contrasting Concomitants and Contrasting Element
+
+Suppose now that the facts are as in Case II, but with the addition of
+six experiences where a certain element which is the opposite of, or
+very dissimilar to, _a_ is connected with the response r_{not 1}, or
+r_{-1} which is opposite to, or very dissimilar to r_{1}. Call this
+opposite of _a_, - _a_.
+
+That is, we have not only
+
+  _a_ + _b_ responded to by r_{1} + r_{2},
+  _a_ + _g_     "       "   r_{1} + r_{not 2},
+  _a_ + _l_     "       "   r_{1} + r_{12},
+  _a_ + _q_     "       "   r_{1} + r_{not 12},
+  _a_ + _v_     "       "   r_{1} + r_{22}, and
+  _a_ + _B_     "       "   r_{1} + r_{not 22},
+
+but also
+
+  - _a_ + _b_ responded to by r_{not 1} + r_{2},
+  - _a_ + _g_     "       "   r_{not 1} + r_{not 2},
+  - _a_ + _l_     "       "   r_{not 1} + r_{12},
+  - _a_ + _q_     "       "   r_{not 1} + r_{not 12},
+  - _a_ + _v_     "       "   r_{not 1} + r_{22}, and
+  - _a_ + _B_     "       "   r_{not 1} + r_{not 22}, as shown in
+                                    Scheme III.
+
+Scheme III
+
+            _a_    opp.     _b_     _g_     _l_     _q_     _v_     _B_
+                  of _a_   (opp. of _b_)   (opp. of _l_)   (opp. of _v_)
+  r_{1}       6              1       1       1       1       1       1
+  r_{not 1}         6        1       1       1       1       1       1
+  r_{2}       1     1        2
+  r_{not 2}   1     1                2
+  r_{12}      1     1                        2
+  r_{not 12}  1     1                                2
+  r_{22}      1     1                                        2
+  r_{not 22}  1     1                                                2
+
+In this series of twelve experiences _a_ connects with r_{1} six times
+and the opposite of _a_ connects with r_{not 1} six times. _a_ connects
+equally often with three pairs of mutual destructives r_{2} and
+r_{not 2}, r_{12} and r_{not 12}, r_{22} and r_{not 22}, and so has zero
+tendency to call them up. - _a_ has also zero tendency to call up any of
+these responses except its opposite, r_{not 1}. _b_, _g_, _l_, _q_, _v_,
+and _B_ are made to connect equally often with r_{1} and r_{not 1}. So,
+of these elements, _a_ is the only one left with a tendency to call up
+r_{1}.
+
+Thus, by the mere action of frequency of connection, r_{1} is connected
+with _a_; the bonds from _a_ to anything except r_{1} are being
+counteracted, and the slight bonds from anything except _a_ to r_{1} are
+being counteracted. The element _a_ becomes predominant in situations
+containing it; and its bond toward r_{1} becomes relatively enormously
+strengthened and freed from competition.
+
+These three processes occur in a similar, but more complicated,
+form if the situations _a_ + _b_, _a_ + _g_, etc., are replaced by
+_a_ + _b_ + _c_ + _d_ + _e_ + _f_, _a_ + _g_ + _h_ + _i_ + _j_ + _k_,
+etc., and the responses r_{1} + r_{2}, r_{1} + r_{7}, r_{1} + r_{12},
+etc., are replaced by r_{1} + r_{2} + r_{3} + r_{4} + r_{5} + r_{6},
+r_{1} + r_{7} + r_{8} + r_{9} + r_{10} + r_{11}, etc.--_provided the_
+r_{1}, r_{2}, r_{3}, r_{4}, etc., _can be made singly_. In so far as any
+one of the responses is necessarily co-active with any one of the others
+(so that, for example, r_{13} always brings r_{26} with it and _vice
+versa_), the exact relations of the numbers recorded in schemes like
+schemes I, II, and III on pages 172 to 174 will change; but, unless
+r_{1} has such an inevitable co-actor, the general results of schemes I,
+II, and III will hold good. If r_{1} does have such an inseparable
+co-actor, say r_{2}, then, of course, _a_ can never acquire bonds with
+r_{1} alone, but everywhere that r_{1} or r_{2} appears in the preceding
+schemes the other element must appear also. r_{1} r_{2} would then have
+to be used as a unit in analysis.
+
+The '_a_ + _b_,' '_a_ + _g_,' '_a_ + _l_,' ... '_a_ + _B_' situations
+may occur unequal numbers of times, altering the exact numerical
+relations of the connections formed and presented in schemes I, II,
+and III; but the process in general remains the same.
+
+So much for the effect of use and disuse in attaching appropriate
+response elements to certain subtle elements of situations. There are
+three main series of effects of satisfaction and discomfort. They
+serve, first, to emphasize, from the start, the desired bonds leading to
+the responses r_{1} + r_{2}, r_{1} + r_{7}, etc., to the total
+situations, and to weed out the undesirable ones. They also act to
+emphasize, in such comparisons and contrasts as have been described,
+every action of the bond from _a_ to r_{1}; and to eliminate every
+tendency of _a_ to connect with aught save r_{1}, and of aught save _a_
+to connect with r_{1}. Their third service is to strengthen the bonds
+produced of appropriate responses to _a_ wherever it occurs, whether or
+not any formal comparisons and contrasts take place.
+
+The process of learning to respond to the difference of pitch in tones
+from whatever instrument, to the 'square-root-ness' of whatever number,
+to triangularity in whatever size or combination of lines, to equality
+of whatever pairs, or to honesty in whatever person or instance, is thus
+a consequence of associative learning, requiring no other forces than
+those of use, disuse, satisfaction, and discomfort. "What happens in
+such cases is that the response, by being connected with many situations
+alike in the presence of the element in question and different in other
+respects, is bound firmly to that element and loosely to each of its
+concomitants. Conversely any element is bound firmly to any one response
+that is made to all situations containing it and very, very loosely to
+each of those responses that are made to only a few of the situations
+containing it. The element of triangularity, for example, is bound
+firmly to the response of saying or thinking 'triangle' but only very
+loosely to the response of saying or thinking white, red, blue, large,
+small, iron, steel, wood, paper, and the like. A situation thus acquires
+bonds not only with some response to it as a gross total, but also with
+responses to any of its elements that have appeared in any other gross
+totals. Appropriate response to an element regardless of its
+concomitants is a necessary consequence of the laws of exercise and
+effect if an animal learns to make that response to the gross total
+situations that contain the element and not to make it to those that do
+not. Such prepotent determination of the response by one or another
+element of the situation is no transcendental mystery, but, given the
+circumstances, a general rule of all learning." Such are at bottom only
+extreme cases of the same learning as a cat exhibits that depresses a
+platform in a certain box whether it faces north or south, whether the
+temperature is 50 or 80 degrees, whether one or two persons are in
+sight, whether she is exceedingly or moderately hungry, whether fish or
+milk is outside the box. All learning is analytic, representing the
+activity of elements within a total situation. In man, by virtue of
+certain instincts and the course of his training, very subtle elements
+of situations can so operate.
+
+       *       *       *       *       *
+
+Learning by analysis does not often proceed in the carefully organized
+way represented by the most ingenious marshaling of comparing and
+contrasting activities. The associations with gross totals, whereby in
+the end an element is elevated to independent power to determine
+response, may come in a haphazard order over a long interval of time.
+Thus a gifted three-year-old boy will have the response element of
+'saying or thinking _two_,' bound to the 'two-ness' element of very many
+situations in connection with the 'how-many' mental set; and he will
+have made this analysis without any formal, systematic training. An
+imperfect and inadequate analysis already made is indeed usually the
+starting point for whatever systematic abstraction the schools direct.
+Thus the kindergarten exercises in analyzing out number, color, size,
+and shape commonly assume that 'one-ness' _versus_ 'more-than-one-ness,'
+black and white, big and little, round and not round are, at least
+vaguely, active as elements responded to in some independence of their
+contexts. Moreover, the tests of actual trial and success in further
+undirected exercises usually cooeperate to confirm and extend and refine
+what the systematic drills have given. Thus the ordinary child in school
+is left, by the drills on decimal notation, with only imperfect power
+of response to the 'place-values.' He continues to learn to respond
+properly to them by finding that 4 x 40 = 160, 4 x 400 = 1600,
+800 - 80 = 720, 800 - 8 = 792, 800-800 = 0, 42 x 48 = 2016,
+24 x 48 = 1152, and the like, are satisfying; while 4 x 40 = 16,
+23 x 48 = 832, 800 - 8 = 0, and the like, are not. The process of
+analysis is the same in such casual, unsystematized formation of
+connections with elements as in the deliberately managed, piecemeal
+inspection, comparison, and contrast described above.
+
+
+SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS
+
+The arrangement of a pupil's experiences so as to direct his attention
+to an element, vary its concomitants instructively, stimulate
+comparison, and throw the element into relief by contrast may be by
+fixed, formal, systematic exercises. Or it may be by much less formal
+exercises, spread over a longer time, and done more or less incidentally
+in other connections. We may call these two extremes the 'systematic'
+and 'opportunistic,' since the chief feature of the former is that it
+systematically provides experiences designed to build up the power of
+correct response to the element, whereas the chief feature of the latter
+is that it uses especially such opportunities as occur by reason of the
+pupil's activities and interests.
+
+Each method has its advantages and disadvantages. The systematic method
+chooses experiences that are specially designed to stimulate the
+analysis; it provides these at a certain fixed time so that they may
+work together; it can then and there test the pupils to ascertain
+whether they really have the power to respond to the element or aspect
+or feature in question. Its disadvantages are, first, that many of the
+pupils will feel no need for and attach no interest or motive to these
+formal exercises; second, that some of the pupils may memorize the
+answers as a verbal task instead of acquiring insight into the facts;
+third, that the ability to respond to the element may remain restricted
+to the special cases devised for the systematic training, and not be
+available for the genuine uses of arithmetic.
+
+The opportunistic method is strong just where the systematic is weak.
+Since it seizes upon opportunities created by the pupil's abilities and
+interests, it has the attitude of interest more often. Since it builds
+up the experiences less formally and over a wider space of time, the
+pupils are less likely to learn verbal answers. Since its material comes
+more from the genuine uses of life, the power acquired is more likely to
+be applicable to life.
+
+Its disadvantage is that it is harder to manage. More thought and
+experimentation are required to find the best experiences; greater care
+is required to keep track of the development of an abstraction which is
+taught not in two days, but over two months; and one may forget to test
+the pupils at the end. In so far as the textbook and teacher are able to
+overcome these disadvantages by ingenuity and care, the opportunistic
+method is better.
+
+
+ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS
+
+We may expect much improvement in the formation of abstract and general
+ideas in arithmetic from the application of three principles in addition
+to those already described. They are: (1) Provide enough actual
+experiences before asking the pupil to understand and use an abstract or
+general idea. (2) Develop such ideas gradually, not attempting to give
+complete and perfect ideas all at once. (3) Develop such ideas so far as
+possible from experiences which will be valuable to the pupil in and of
+themselves, quite apart from their merit as aids in developing the
+abstraction or general notion. Consider these three principles in order.
+
+Children, especially the less gifted intellectually, need more
+experiences as a basis for and as applications of an arithmetical
+abstraction or concept than are usually given them. For example, in
+paving the way for the principle, "Any number times 0 equals 0," it is
+not safe to say, "John worked 8 days for 0 minutes per day. How many
+minutes did he work?" and "How much is 0 times 4 cents?" It will be
+much better to spend ten or fifteen minutes as follows:[14] "What does
+zero mean? (Not any. No.) How many feet are there in eight yards?
+In 5 yards? In 3 yards? In 2 yards? In 1 yard? In 0 yard? How many
+inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... qt.
+5 pk. = .... qt. 0 pk. = .... qt. A boy receives 60 cents an hour
+when he works. How much does he receive when he works 3 hr.? 8 hr.?
+6 hr.? 0 hr.? A boy received 60 cents a day for 0 days. How much did he
+receive? How much is 0 times $600? How much is 0 times $5000? How much
+is 0 times a million dollars? 0 times any number equals....
+
+   232  (At the blackboard.)  0 time 232 equals what?
+    30  I write 0 under the 0.[15]  3 times 232 equals what?
+  ----
+  6960  Continue at the blackboard with
+
+        734         321         312          41
+         20          40          30          60  etc."
+        ---         ---         ---          --
+
+  [14] The more gifted children may be put to work using the principle
+       after the first minute or two.
+
+  [15] 232
+        30   If desired this form may be used, with the appropriate
+       ---      difference in the form of the questions and statements.
+       000
+      696
+      ----
+      6960
+
+Pupils in the elementary school, except the most gifted, should not be
+expected to gain mastery over such concepts as _common fraction_,
+_decimal fraction_, _factor_, and _root_ quickly. They can learn a
+definition quickly and learn to use it in very easy cases, where even a
+vague and imperfect understanding of it will guide response correctly.
+But complete and exact understanding commonly requires them to take,
+not one intellectual step, but many; and mastery in use commonly comes
+only as a slow growth. For example, suppose that pupils are taught
+that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03,
+etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean
+1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal
+fractions. They may then respond correctly when asked to write a decimal
+fraction, or to state which of these,--1/4, .4, 3/8, .07, .002,
+5/6,--are common fractions and which are decimal fractions. They may
+be able, though by no means all of them will be, to write decimal
+fractions which equal 1/2 and 1/5, and the common fractions which
+equal .1 and .09. Most of them will not, however, be able to respond
+correctly to "Write a decimal mixed number"; or to state which of
+these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which
+are decimals; or to write the decimal fractions which equal 3/4 and 1/3.
+
+If now the teacher had given all at once the additional experiences
+needed to provide the ability to handle these more intricate and subtle
+features of decimal-fraction-ness, the result would have been confusion
+for most pupils. The general meaning of .32, .14, .99, and the like
+requires some understanding of .30, .10, .90, and .02, .04, .08; but it
+is not desirable to disturb the child with .30 while he is trying to
+master 2.3, 4.3, 6.3, and the like. Decimals in general require
+connection with place value and the contrasts of .41 with 41, 410, 4.1,
+and the like, but if the relation to place values in general is taught
+in the same lesson with the relation to /10s, /100s, /1000s, the mind
+will suffer from violent indigestion.
+
+A wise pedagogy in fact will break up the process of learning the
+meaning and use of decimal fractions into many teaching units, for
+example, as follows:--
+
+(1) Such familiarity with fractions with large denominators as is
+desirable for pupils to have, as by an exercise in reducing to lowest
+terms, 8/10, 36/64, 20/25, 18/24, 24/32, 21/30, 25/100, 40/100, and the
+like. This is good as a review of cancellation, and as an extension of
+the idea of a fraction.
+
+(2) Objective work, showing 1/10 sq. ft., 1/50 sq. ft., 1/100 sq. ft.,
+and 1/1000 sq. ft., and having these identified and the forms 1/10 sq.
+ft., 1/100 sq. ft., and 1/1000 sq. ft. learned. Finding how many
+feet = 1/10 mile and 1/100 mile.
+
+(3) Familiarity with /100s and /1000s by reductions of 750/1000, 50/100,
+etc., to lowest terms and by writing the missing numerators in
+500/1000 = /100 = /10 and the like, and by finding 1/10, 1/100, and
+1/1000 of 3000, 6000, 9000, etc.
+
+(4) Writing 1/10 as .1 and 1/100 as .01, 11/100, 12/100, 13/100, etc.,
+as .11, .12, .13. United States money is used as the introduction.
+Application is made to miles.
+
+(5) Mixed numbers with a first decimal place. The cyclometer or
+speedometer. Adding numbers like 9.1, 14.7, 11.4, etc.
+
+(6) Place value in general from thousands to hundredths.
+
+(7) Review of (1) to (6).
+
+(8) Tenths and hundredths of a mile, subtraction when both numbers
+extend to hundredths, using a railroad table of distances.
+
+(9) Thousandths. The names 'decimal fractions or decimals,' and 'decimal
+mixed numbers or decimals.' Drill in reading any number to thousandths.
+The work will continue with gradual extension and refinement of the
+understanding of decimals by learning how to operate with them in
+various ways.
+
+Such may seem a slow progress, but in fact it is not, and many of these
+exercises whereby the pupil acquires his mastery of decimals are useful
+as organizations and applications of other arithmetical facts.
+
+That, it will be remembered, was the third principle:--"Develop abstract
+and general ideas by experiences which will be intrinsically valuable."
+The reason is that, even with the best of teaching, some pupils will
+not, within any reasonable limits of time expended, acquire ideas that
+are fully complete, rigorous when they should be, flexible when they
+should be, and absolutely exact. Many children (and adults, for that
+matter) could not within any reasonable limits of time be so taught the
+nature of a fraction that they could decide unerringly in original
+exercises like:--
+
+Is 2.75/25 a common fraction?
+
+Is $.25 a decimal fraction?
+
+Is one _x_th of _y_ a fraction?
+
+Can the same words mean both a common fraction and a decimal fraction?
+
+Express 1 as a common fraction.
+
+Express 1 as a decimal fraction.
+
+These same children can, however, be taught to operate correctly with
+fractions in the ordinary uses thereof. And that is the chief value of
+arithmetic to them. They should not be deprived of it because they
+cannot master its subtler principles. So we seek to provide experiences
+that will teach all pupils something of value, while stimulating in
+those who have the ability the growth of abstract ideas and general
+principles.
+
+Finally, we should bear in mind that working with qualities and
+relations that are only partly understood or even misunderstood does
+under certain conditions give control over them. The general process of
+analytic learning in life is to respond as well as one can; to get a
+clearer idea thereby; to respond better the next time; and so on. For
+instance, one gets some sort of notion of what 1/5 means; he then
+answers such questions as 1/5 of 10 = ? 1/5 of 5 = ? 1/5 of 20 = ?;
+by being told when he is right and when he is wrong, he gets from
+these experiences a better idea of 1/5; again he does his best with
+1/5 = _/10, 1/5 = _/15, etc., and as before refines and enlarges his
+concept of 1/5. He adds 1/5 to 2/5, etc., 1/5 to 3/10, etc., 1/5 to 1/2,
+etc., and thereby gains still further, and so on.
+
+What begins as a blind habit of manipulation started by imitation may
+thus grow into the power of correct response to the essential element.
+The pupil who has at the start no notion at all of 'multiplying' may
+learn what multiplying is by his experience that '4 6 multiplying
+gives 24'; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing
+something with numbers and differentiates right results, he will often
+reach in the end the abstractions which he is supposed to need in the
+beginning. It may even be the case with some of the abstractions
+required in arithmetic that elaborate provision for comprehension
+beforehand is not so efficient as the same amount of energy devoted
+partly to provision for analysis itself beforehand and partly to
+practice in response to the element in question without full
+comprehension.
+
+It certainly is not the best psychology and not the best educational
+theory to think that the pupil first masters a principle and then merely
+applies it--first does some thinking and then computes by mere routine.
+On the contrary, the applications should help to establish, extend, and
+refine the principle--the work a pupil does with numbers should be a
+main means of increasing his understanding of the principles of
+arithmetic as a science.
+
+
+
+
+CHAPTER X
+
+THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC
+
+
+THE ESSENTIALS OF ARITHMETICAL REASONING
+
+We distinguish aimless reverie, as when a child dreams of a vacation
+trip, from purposive thinking, as when he tries to work out the answer
+to "How many weeks of vacation can a family have for $120 if the cost is
+$22 a week for board, $2.25 a week for laundry, and $1.75 a week for
+incidental expenses, and if the railroad fares for the round trip are
+$12?" We distinguish the process of response to familiar situations,
+such as five integral numbers to be added, from the process of response
+to novel situations, such as (for a child who has not been trained with
+similar problems):--"A man has four pieces of wire. The lengths are 120
+yd., 132 meters, 160 feet, and 1/8 mile. How much more does he need to
+have 1000 yd. in all?" We distinguish 'thinking things together,' as
+when a diagram or problem or proof is understood, from thinking of one
+thing after another as when a number of words are spelled or a poem in
+an unknown tongue is learned. In proportion as thinking is purposive,
+with selection from the ideas that come up, and in proportion as it
+deals with novel problems for which no ready-made habitual response is
+available, and in proportion as many bonds act together in an organized
+way to produce response, we call it reasoning.
+
+When the conclusion is reached as the effect of many particular
+experiences, the reasoning is called inductive. When some principle
+already established leads to another principle or to a conclusion about
+some particular fact, the reasoning is called deductive. In both cases
+the process involves the analysis of facts into their elements, the
+selection of the elements that are deemed significant for the question
+at hand, the attachment of a certain amount of importance or weight to
+each of them, and their use in the right relations. Thought may fail
+because it has not suitable facts, or does not select from them the
+right ones, or does not attach the right amount of weight to each, or
+does not put them together properly.
+
+In the world at large, many of our failures in thinking are due to not
+having suitable facts. Some of my readers, for example, cannot solve the
+problem--"What are the chances that in drawing a card from an ordinary
+pack of playing-cards four times in succession, the same card will be
+drawn each time?" And it will be probably because they do not know
+certain facts about the theory of probabilities. The good thinkers among
+such would look the matter up in a suitable book. Similarly, if a person
+did not happen to know that there were fifty-two cards in all and that
+no two were alike, he could not reason out the answer, no matter what
+his mastery of the theory of probabilities. If a competent thinker, he
+would first ask about the size and nature of the pack. In the actual
+practice of reasoning, that is, we have to survey our facts to see if we
+lack any that are necessary. If we do, the first task of reasoning is to
+acquire those facts.
+
+This is specially true of the reasoning about arithmetical facts in
+life. "Will 3-1/2 yards of this be enough for a dress?" Reason directs
+you to learn how wide it is, what style of dress you intend to make of
+it, how much material that style normally calls for, whether you are a
+careful or a wasteful cutter, and how big the person is for whom the
+dress is to be made. "How much cheaper as a diet is bread alone, than
+bread with butter added to the extent of 10% of the weight of the
+bread?" Reason directs you to learn the cost of bread, the cost of
+butter, the nutritive value of bread, and the nutritive value of butter.
+
+In the arithmetic of the school this feature of reasoning appears in
+cases where some fact about common measures must be brought to bear, or
+some table of prices or discounts must be consulted, or some business
+custom must be remembered or looked up.
+
+Thus "How many badges, each 9 inches long, can be made from 2-1/2 yd.
+ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches.
+"At Jones' prices, which costs more, 3-3/4 lb. butter or 6-1/2 lb. lard?
+How much more?" is a problem which directs the thinker to ascertain
+Jones' prices.
+
+It may be noted that such problems are, other things being equal,
+somewhat better training in thinking than problems where all the data
+are given in the problem itself (_e.g._, "Which costs more, 3-3/4 lb.
+butter at 48c per lb. or 6-1/2 lb. lard at 27c per lb.? How much
+more?"). At least it is unwise to have so many problems of the latter
+sort that the pupil may come to think of a problem in applied arithmetic
+as a problem where everything is given and he has only to manipulate the
+data. Life does not present its problems so.
+
+The process of selecting the right elements and attaching proper weight
+to them may be illustrated by the following problem:--"Which of these
+offers would you take, supposing that you wish a D.C.K. upright piano,
+have $50 saved, can save a little over $20 per month, and can borrow
+from your father at 6% interest?"
+
+  A
+
+  A Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down
+  and $21 a month for only a year and a half. _No interest_ to pay.
+  We ask you to pay only for the piano and allow you plenty of time.
+
+  B
+
+  We offer the well-known D.C.K. Piano for $390. $50 cash and $20 a
+  month thereafter. Regular interest at 6%. The interest soon is
+  reduced to less than $1 a month.
+
+  C
+
+  The D.C.K. Piano. Special Offer, $375, cash. Compare our prices
+  with those of any reliable firm.
+
+If you consider chiefly the "only," "No interest to pay," "only," and
+"plenty of time" in offer A, attaching much weight to them and little to
+the thought, "How much will $50 plus (18 x $21) be?", you will probably
+decide wrongly.
+
+The situations of life are often complicated by many elements of little
+or even of no relevance to the correct solution. The offerer of A may
+belong to your church; your dearest friend may urge you to accept offer
+B; you may dislike to talk with the dealer who makes offer C; you may
+have a prejudice against owing money to a relative; that prejudice may
+be wise or foolish; you may have a suspicion that the B piano is
+shopworn; that suspicion may be well-founded or groundless; the salesman
+for C says, "You don't want your friends to say that you bought on the
+installment plan. Only low-class persons do that," etc. The statement of
+arithmetical problems in school usually assists the pupil to the extent
+of ruling out all save definitely quantitative elements, and of ruling
+out all quantitative elements except those which should be considered.
+The first of the two simplifications is very beneficial, on the whole,
+since otherwise there might be different correct solutions to a problem
+according to the nature and circumstances of the persons involved. The
+second simplification is often desirable, since it will often produce
+greater improvement in the pupils, per hour of time spent, than would be
+produced by the problems requiring more selection. It should not,
+however, be a universal custom; for in that case the pupils are tempted
+to think that in every problem they must use all the quantities given,
+as one must use all the pieces in a puzzle picture.
+
+It is obvious that the elements selected must not only be right but also
+be in the right relations to one another. For example, in the problems
+below, the 6 must be thought of in relation to a dozen and as being half
+of a dozen, and also as being 6 times 1. 1 must be mentally tied to
+"each." The 6 as half of a dozen must be related to the $1.00, $1.60,
+etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc.
+
+  Buying in Quantity
+
+  These are a grocer's prices for certain things by the dozen and
+  for a single one. He sells a half dozen at half the price of a
+  dozen. Find out how much you save by buying 6 all at one time
+  instead of buying them one at a time.
+
+                              Doz.  Each
+  1. Evaporated Milk         $1.00  $.09
+  2. Puffed Rice              1.60   .14
+  3. Puffed Wheat             1.10   .10
+  4. Canned Soup              1.90   .17
+  5. Sardines                 1.80   .16
+  6. Beans (No. 2 cans)       1.50   .13
+  7. Pork and Beans           1.70   .15
+  8. Peas (No. 2 cans)        1.40   .12
+  9. Tomatoes (extra cans)    3.20   .28
+  10. Ripe olives (qt. cans)  7.20   .65
+
+It is obvious also that in such arithmetical work as we have been
+describing, the pupil, to be successful, must 'think things together.'
+Many bonds must cooeperate to determine his final response.
+
+As a preface to reasoning about a problem we often have the discovery of
+the problem and the classification of just what it is, and as a
+postscript we have the critical inspection of the answer obtained to
+make sure that it is verified by experiment or is consistent with known
+facts. During the process of searching for, selecting, and weighting
+facts, there may be similar inspection and validation, item by item.
+
+
+REASONING AS THE COOePERATION OF ORGANIZED HABITS
+
+The pedagogy of the past made two notable errors in practice based on
+two errors about the psychology of reasoning. It considered reasoning as
+a somewhat magical power or essence which acted to counteract and
+overrule the ordinary laws of habit in man; and it separated too sharply
+the 'understanding of principles' by reasoning from the 'mechanical'
+work of computation, reading problems, remembering facts and the like,
+done by 'mere' habit and memory.
+
+Reasoning or selective, inferential thinking is not at all opposed to,
+or independent of, the laws of habit, but really is their necessary
+result under the conditions imposed by man's nature and training. A
+closer examination of selective thinking will show that no principles
+beyond the laws of readiness, exercise, and effect are needed to explain
+it; that it is only an extreme case of what goes on in associative
+learning as described under the 'piecemeal' activity of situations; and
+that attributing certain features of learning to mysterious faculties of
+abstraction or reasoning gives no real help toward understanding or
+controlling them.
+
+It is true that man's behavior in meeting novel problems goes beyond, or
+even against, the habits represented by bonds leading from gross total
+situations and customarily abstracted elements thereof. One of the two
+reasons therefor, however, is simply that the finer, subtle,
+preferential bonds with subtler and less often abstracted elements go
+beyond, and at times against, the grosser and more usual bonds. One set
+is as much due to exercise and effect as the other. The other reason is
+that in meeting novel problems the mental set or attitude is likely to
+be one which rejects one after another response as their unfitness to
+satisfy a certain desideratum appears. What remains as the apparent
+course of thought includes only a few of the many bonds which did
+operate, but which, for the most part, were unsatisfying to the ruling
+attitude or adjustment.
+
+Successful responses to novel data, associations by similarity and
+purposive behavior are in only apparent opposition to the fundamental
+laws of associative learning. Really they are beautiful examples of it.
+Man's successful responses to novel data--as when he argues that the
+diagonal on a right triangle of 796.278 mm. base and 137.294 mm.
+altitude will be 808.022 mm., or that Mary Jones, born this morning,
+will sometime die--are due to habits, notably the habits of response to
+certain elements or features, under the laws of piecemeal activity and
+assimilation.
+
+Nothing is less like the mysterious operations of a faculty of reasoning
+transcending the laws of connection-forming, than the behavior of men in
+response to novel situations. Let children who have hitherto confronted
+only such arithmetical tasks, in addition and subtraction with one- and
+two-place numbers and multiplication with one-place numbers, as those
+exemplified in the first line below, be told to do the examples shown in
+the second line.
+
+  ADD    ADD     ADD     SUBT.   SUBT.  MULTIPLY   MULTIPLY   MULTIPLY
+   8      37      35       8      37       8          9           6
+   5      24      68       5      24       5          7           3
+  --      --      23      --      --      --         --          --
+                  19
+                  --
+
+          MULTIPLY                MULTIPLY                MULTIPLY
+             32                      43                      34
+             23                      22                      26
+             --                      --                      --
+
+They will add the numbers, or subtract the lower from the upper number,
+or multiply 3 x 2 and 2 x 3, etc., getting 66, 86, and 624, or respond
+to the element of 'Multiply' attached to the two-place numbers by "I
+can't" or "I don't know what to do," or the like; or, if one is a child
+of great ability, he may consider the 'Multiply' element and the bigness
+of the numbers, be reminded by these two aspects of the situation of the
+fact that
+
+  '9
+   9 multiply'
+  --
+
+gave only 81, and that
+
+  '10
+   10 multiply'
+   --
+
+gave only 100, or the like; and so may report an intelligent and
+justified "I can't," or reject the plan of 3 x 2 and 2 x 3, with 66, 86,
+and 624 for answers, as unsatisfactory. What the children will do will,
+in every case, be a product of the elements in the situation that are
+potent with them, the responses which these evoke, and the further
+associates which these responses in turn evoke. If the child were one of
+sufficient genius, he might infer the procedure to be followed as a
+result of his knowledge of the principles of decimal notation and the
+meaning of 'Multiply,' responding correctly to the 'place-value' element
+of each digit and adding his 6 tens and 9 tens, 20 twos and 3 thirties;
+but if he did thus invent the shorthand addition of a collection of
+twenty-three collections, each of 32 units, he would still do it by the
+operation of bonds, subtle but real.
+
+Association by similarity is, as James showed long ago, simply the
+tendency of an element to provoke the responses which have been bound to
+it. _abcde_ leads to _vwxyz_ because _a_ has been bound to _vwxyz_ by
+original nature, exercise, or effect.
+
+Purposive behavior is the most important case of the influence of the
+attitude or set or adjustment of an organism in determining (1) which
+bonds shall act, and (2) which results shall satisfy. James early
+described the former fact, showing that the mechanism of habit can give
+the directedness or purposefulness in thought's products, provided that
+mechanism includes something paralleling the problem, the aim, or need,
+in question.
+
+The second fact, that the set or attitude of the man helps to determine
+which bonds shall satisfy, and which shall annoy, has commonly been
+somewhat obscured by vague assertions that the selection and retention
+is of what is "in point," or is "the right one," or is "appropriate," or
+the like. It is thus asserted, or at least hinted, that "the will," "the
+voluntary attention," "the consciousness of the problem," and other such
+entities are endowed with magic power to decide what is the "right" or
+"useful" bond and to kill off the others. The facts are that in
+purposive thinking and action, as everywhere else, bonds are selected
+and retained by the satisfyingness, and are killed off by the
+discomfort, which they produce; and that the potency of the man's set or
+attitude to make this satisfy and that annoy--to put certain
+conduction-units in readiness to act and others in unreadiness--is in
+every way as important as its potency to set certain conduction-units in
+actual operation.
+
+Reasoning is not a radically different sort of force operating against
+habit but the organization and cooeperation of many habits, thinking
+facts together. Reasoning is not the negation of ordinary bonds, but
+the action of many of them, especially of bonds with subtle elements of
+the situation. Some outside power does not enter to select and
+criticize; the pupil's own total repertory of bonds relevant to the
+problem is what selects and rejects. An unsuitable idea is not killed
+off by some _actus purus_ of intellect, but by the ideas which it itself
+calls up, in connection with the total set of mind of the pupil, and
+which show it to be inadequate.
+
+Almost nothing in arithmetic need be taught as a matter of mere
+unreasoning habit or memory, nor need anything, first taught as a
+principle, ever become a matter of mere habit or memory. 5 x 4 = 20
+should not be learned as an isolated fact, nor remembered as we remember
+that Jones' telephone number is 648 J 2. Almost everything in arithmetic
+should be taught as a habit that has connections with habits already
+acquired and will work in an organization with other habits to come. The
+use of this organized hierarchy of habits to solve novel problems is
+reasoning.
+
+
+
+
+CHAPTER XI
+
+ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL
+
+
+THE UTILIZATION OF INSTINCTIVE INTERESTS
+
+The activities essential to acquiring ability in arithmetic can rely on
+little in man's instinctive equipment beyond the purely intellectual
+tendencies of curiosity and the satisfyingness of thought for thought's
+sake, and the general enjoyment of success rather than failure in an
+enterprise to which one sets oneself. It is only by a certain amount of
+artifice that we can enlist other vehement inborn interests of childhood
+in the service of arithmetical knowledge and skill. When this can be
+done at no cost the gain is great. For example, marching in files of
+two, in files of three, in files of four, etc., raising the arms once,
+two times, three times, showing a foot, a yard, an inch with the hands,
+and the like are admirable because learning the meanings of numbers thus
+acquires some of the zest of the passion for physical action. Even in
+late grades chances to make pictures showing the relations of fractional
+parts, to cut strips, to fold paper, and the like will be useful.
+
+Various social instincts can be utilized in matches after the pattern of
+the spelling match, contests between rows, certain number games, and the
+like. The scoring of both the play and the work of the classroom is a
+useful field for control by the teacher of arithmetic.
+
+Hunt ['12] has noted the more important games which have some
+considerable amount of arithmetical training as a by-product and which
+are more or less suitable for class use. Flynn ['12] has described
+games, most of them for home use, which give very definite arithmetical
+drill, though in many cases the drills are rather behind the needs of
+children old enough to understand and like the game itself.
+
+It is possible to utilize the interests in mystery, tricks, and puzzles
+so as to arouse a certain form of respect for arithmetic and also to get
+computational work done. I quote one simple case from Miss Selkin's
+admirable collection ['12, p. 69 f.]:--
+
+  I. ADDITION
+
+  "We must admit that there is nothing particularly interesting in
+  a long column of numbers to be added. Let the teacher, however,
+  suggest that he can write the answer at sight, and the task will
+  assume a totally different aspect.
+
+  "A very simple number trick of this kind can be performed by
+  making use of the principle of complementary addition. The
+  arithmetical complement of a number with respect to a larger
+  number is the difference between these two numbers. Most
+  interesting results can be obtained by using complements with
+  respect to 9.
+
+  "The children may be called upon to suggest several numbers of
+  two, three, or more digits. Below these write an equal number of
+  addends and immediately announce the answer. The children,
+  impressed by this apparently rapid addition, will set to work
+  enthusiastically to test the results of this lightning
+  calculation.
+
+  "Example:--     357 }                          999
+                  682 }  A                       x 3
+                  793 }                         ----
+                                                2997
+
+                  642 }
+                  317 }  B
+                  206 }
+
+  "Explanation:--The addends in group A are written down at
+  random or suggested by the class. Those in group B are their
+  complements. To write the first number in group B we look at the
+  first number in group A and, starting at the left write 6, the
+  complement of 3 with respect to 9; 4, the complement of 5; 2, the
+  complement of 7. The second and third addends in group B are
+  derived in the same way. Since we have three addends in each
+  group, the problem reduces itself to multiplying 999 by 3, or to
+  taking 3000 - 3. Any number of addends may be used and each addend
+  may consist of any number of digits."
+
+Respect for arithmetic as a source of tricks and magic is very much less
+important than respect for its everyday services; and computation to
+test such tricks is likely to be undertaken zealously only by the abler
+pupils. Consequently this source of interest should probably be used
+only sparingly, and perhaps the teacher should give such exhibitions
+only as a reward for efficiency in the regular work. For example, if the
+work for a week is well done in four days the fifth day might be given
+up to some semi-arithmetical entertainment, such as the demonstration of
+an adding machine, the story of primitive methods of counting, team
+races in computation, an exhibition of lightning calculation and
+intellectual sleight-of-hand by the teacher, or the voluntary study of
+arithmetical puzzles.
+
+The interest in achievement, in success, mentioned above is stronger in
+children than is often realized and makes advisable the systematic use
+of the practice experiment as a method of teaching much of arithmetic.
+Children who thus compete with their own past records, keeping an exact
+score from week to week, make notable progress and enjoy hard work in
+making it.
+
+
+THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES
+
+Negatively the difficulty of the work that pupils should be expected to
+do is conditioned by the gradual maturing of their capacities. Other
+things being equal, the common custom of reserving hard things for late
+in the elementary school course is, of course, sound. It seems probable
+that little is gained by using any of the child's time for arithmetic
+before grade 2, though there are many arithmetical facts that he can
+learn in grade 1. Postponement of systematic work in arithmetic to grade
+3 or even grade 4 is allowable if better things are offered. With proper
+textbooks and oral and written exercises, however, a child in grades 2
+and 3 can spend time profitably on arithmetical work. When all children
+can be held in school through the eighth grade it does not much matter
+whether arithmetic is begun early or late. If, however, many children
+are to leave in grades 5 and 6 as now, we may think it wise to provide
+somehow that certain minima of arithmetical ability be given them.
+
+There are, so far as is known, no special times and seasons at which the
+human animal by inner growth is specially ripe for one or another
+section or aspect of arithmetic, except in so far as the general inner
+growth of intellectual powers makes the more abstruse and complex tasks
+suitable to later and later years.
+
+Indeed, very few of even the most enthusiastic devotees of the
+recapitulation theory or culture-epoch theory have attempted to apply
+either to the learning of arithmetic, and Branford is the only
+mathematician, so far as I know, who has advocated such application,
+even tempered by elaborate shiftings and reversals of the racial order.
+He says:--
+
+  "Thus, for each age of the individual life--infancy, childhood,
+  school, college--may be selected from the racial history
+  the most appropriate form in which mathematical experience
+  can be assimilated. Thus the capacity of the infant and early
+  childhood is comparable with the capacity of animal consciousness
+  and primitive man. The mathematics suitable to later childhood
+  and boyhood (and, of course, girlhood) is comparable with Archaean
+  mathematics passing on through Greek and Hindu to mediaeval
+  European mathematics; while the student is become sufficiently
+  mature to begin the assimilation of modern and highly abstract
+  European thought. The filling in of details must necessarily
+  be left to the individual teacher, and also, within some such
+  broadly marked limits, the precise order of the marshalling of the
+  material for each age. For, though, on the whole, mathematical
+  development has gone forward, yet there have been lapses from
+  advances already made. Witness the practical world-loss of much
+  valuable Hindu thought, and, for long centuries, the neglect of
+  Greek thought: witness the world-loss of the invention by the
+  Babylonians of the Zero, until re-invented by the Hindus, passed
+  on by them to the Arabs, and by these to Europe.
+
+  "Moreover, many blunders and false starts and false principles
+  have marked the whole course of development. In a phrase, rivers
+  have their backwaters. But it is precisely the teacher's function
+  to avoid such racial mistakes, to take short cuts ultimately
+  discovered, and to guide the young along the road ultimately found
+  most accessible with such halts and retracings--returns up
+  side-cuts--as the mental peculiarities of the pupils demand.
+
+  "All this, the practical realization of the spirit of the principle,
+  is to be wisely left to the mathematical teacher, familiar with the
+  history of mathematical science and with the particular limitations
+  of his pupils and himself." ['08, p. 245.]
+
+The latitude of modification suggested by Branford reduces the guidance
+to be derived from racial history to almost _nil_. Also it is apparent
+that the racial history in the case of arithmetical achievement is
+entirely a matter of acquisition and social transmission. Man's original
+nature is destitute of all arithmetical ideas. The human germs do not
+know even that one and one make two!
+
+
+INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL
+
+A scientific plan for teaching arithmetic would begin with an exact
+inventory of the knowledge and skill which the pupils already possessed.
+Our ordinary notions of what a child knows at entrance to grade 1, or
+grade 2, or grade 3, and of what a first-grade child or second-grade
+child can do, are not adequate. If they were, we should not find
+reputable textbooks arranging to teach elaborately facts already
+sufficiently well known to over three quarters of the pupils when they
+enter school. Nor should we find other textbooks presupposing in their
+first fifty pages a knowledge of words which not half of the children
+can read even at the end of the 2 B grade.
+
+We do find just such evidence that ordinary ideas about the abilities of
+children at the beginning of systematic school training in arithmetic
+may be in gross error. For example, a reputable and in many ways
+admirable recent book has fourteen pages of exercises to teach the
+meaning of two and the fact that one and one make two! As an example of
+the reverse error, consider putting all these words in the first
+twenty-five pages of a beginner's book:--_absentees, attendance, blanks,
+continue, copy, during, examples, grouped, memorize, perfect, similar,
+splints, therefore, total_!
+
+Little, almost nothing, has been done toward providing an exact
+inventory compared with what needs to be done. We may note here (1) the
+facts relevant to arithmetic found by Stanley Hall, Hartmann, and others
+in their general investigations of the knowledge possessed by children
+at entrance to school, (2) the facts concerning the power of children to
+perceive differences in length, area, size of collection, and
+organization within a collection such as is shown in Fig. 24, and
+certain facts and theories about early awareness of number.
+
+In the Berlin inquiry of 1869, knowledge of the meaning of two, three,
+and four appeared in 74, 74, and 73 percent of the children upon
+entrance to school. Some of those recorded as ignorant probably really
+knew, but failed to understand that they were expected to reply or were
+shy. Only 85 percent were recorded as knowing their fathers' names.
+Seven eighths as many children knew the meanings of two, three, and four
+as knew their fathers' names. In a similar but more careful experiment
+with Boston children in September, 1880, Stanley Hall found that 92
+percent knew three, 83 percent knew four, and 71-1/2 percent knew five.
+Three was known about as well as the color red; four was known about as
+well as the color blue or yellow or green. Hartmann ['90] found that two
+thirds of the children entering school in Annaberg could count from one
+to ten. This is about as many as knew money, or the familiar objects of
+the town, or could repeat words spoken to them.
+
+    [Illustration: FIG. 24.--Objective presentation.]
+
+In the Stanford form of the Binet tests counting four pennies is given
+as an ability of the typical four-year-old. Counting 13 pennies
+correctly in at least one out of two trials, and knowing three of the
+four coins,--penny, nickel, dime, and quarter,--are given as abilities
+of the typical six-year-old.
+
+
+THE PERCEPTION OF NUMBER AND QUANTITY
+
+We know that educated adults can tell how many lines or dots, etc., they
+see in a single glance (with an exposure too short for the eye to move)
+up to four or more, according to the clearness of the objects and their
+grouping. For example, Nanu ['04] reports that when a number of bright
+circles on a dark background are shown to educated adults for only .033
+second, ten can be counted when arranged to form a parallelogram, but
+only five when arranged in a row. With certain groupings, of course,
+their 'perception' involves much inference, even conscious addition and
+multiplication. Similarly they can tell, up to twenty and beyond, the
+number of taps, notes, or other sounds in a series too rapid for single
+counting if the sounds are grouped in a convenient rhythm.
+
+These abilities are, however, the product of a long and elaborate
+learning, including the learning of arithmetic itself. Elementary
+psychology and common experience teach us that the mere observation of
+groups or quantities, no matter how clear their number quality appears
+to the person who already knows the meanings of numbers, does not of
+itself create the knowledge of the meanings of numbers in one who does
+not. The experiments of Messenger ['03] and Burnett ['06] showed that
+there is no direct intuitive apprehension even of two as distinct from
+one. We have to _learn_ to feel the two touches or see the two dots or
+lines as two.
+
+We do not know by exact measurements the growth in children of this
+ability to count or infer the number of elements in a collection seen or
+series heard. Still less do we know what the growth would be without
+the influence of school training in counting, grouping, adding, and
+multiplying. Many textbooks and teachers seem to overestimate it
+greatly. Not all educated adults can, apart from measurement, decide
+with surety which of these lines is the longer, or which of these areas
+is the larger, or whether this is a ninth or a tenth or an eleventh of a
+circle.
+
+    [Illustration]
+
+Children upon entering school have not been tested carefully in respect
+to judgments of length and area, but we know from such studies as
+Gilbert's ['94] that the difference required in their case is probably
+over twice that required for children of 13 or 14. In judging weights,
+for example, a difference of 6 is perceived as easily by children 13 to
+15 years of age as a difference of 15 by six-year-olds.
+
+A teacher who has adult powers of estimating length or area or weight
+and who also knows already which of the two is longer or larger or
+heavier, may use two lines to illustrate a difference which they really
+hide from the child. It is unlikely, for example, that the first of
+these lines ______________ ________________  would be recognized as
+shorter than the second by every child in a fourth-grade class, and it
+is extremely unlikely that it would be recognized as being 7/8 of the
+length of the latter, rather than 3/4 of it or 5/6 of it or 9/10 of it
+or 11/12 of it. If the two were shown to a second grade, with the
+question, "The first line is 7. How long is the other line?" there would
+be very many answers of 7 or 9; and these might be entirely correct
+arithmetically, the pupils' errors being all due to their inability to
+compare the lengths accurately.
+
+  _A_   ______________     ________________
+
+        ______________     ________________
+  _B_  |______________|   |________________|
+
+
+  _C_  |-|-|-|-|-|-|-|
+
+       |-|-|-|-|-|-|-|-|
+
+        __ __ __ __ __ __ __
+  _D_  |__|__|__|__|__|__|__|
+        __ __ __ __ __ __ __ __
+       |__|__|__|__|__|__|__|__|
+
+
+  _E_   .'\##|##/`.      .'\##|##/`.
+       /###\#|#/   \    /###\#|#/###\
+       |-----------|    |-----------|
+       \###/#|#\###/    \###/#|#\###/
+        `./##|##\.'      `./##|##\.'
+
+The quantities used should be such that their mere discrimination offers
+no difficulty even to a child of blunted sense powers. If 7/8 and 1 are
+to be compared, _A_ and _B_ are not allowable. _C_, _D_, and _E_ are
+much better.
+
+Teachers probably often underestimate or neglect the sensory
+difficulties of the tasks they assign and of the material they use to
+illustrate absolute and relative magnitudes. The result may be more
+pernicious when the pupils answer correctly than when they fail. For
+their correct answering may be due to their divination of what the
+teacher wants; and they may call a thing an inch larger to suit her
+which does not really seem larger to them at all. This, of course, is
+utterly destructive of their respect for arithmetic as an exact and
+matter-of-fact instrument. For example, if a teacher drew a series of
+lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this
+form--____ ______ and asked, "This is 20 inches long, how long is
+this?" she might, after some errors and correction thereof, finally
+secure successful response to all the lines by all the children. But
+their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be
+actually damaged by the exercise.
+
+
+THE EARLY AWARENESS OF NUMBER
+
+There has been some disagreement concerning the origin of awareness of
+number in the individual, in particular concerning the relative
+importance of the perception of how-many-ness and that of how-much-ness,
+of the perception of a defined aggregate and the perception of a defined
+ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and
+Decroly and Degand ['12].)
+
+The chief facts of significance for practice seem to be these: (1)
+Children with rare exceptions hear the names _one_, _two_, _three_,
+_four_, _half_, _twice_, _two times_, _more_, _less_, _as many as_,
+_again_, _first_, _second_, and _third_, long before they have analyzed
+out the qualities and relations to which these words refer so as to feel
+them at all clearly. (2) Their knowledge of the qualities and relations
+is developed in the main in close association with the use of these
+words to the child and by the child. (3) The ordinary experiences of the
+first five years so develop in the child awareness of the 'how many
+somethings' in various groups, of the relative magnitudes of two groups
+or quantities of any sort, and of groups and magnitudes as related to
+others in a series. For instance, if fairly gifted, a child comes, by
+the age of five, to see that a row of four cakes is an aggregate of
+four, seeing each cake as a part of the four and the four as the sum of
+its parts, to know that two of them are as many as the other two, that
+half of them would be two, and to think, when it is useful for him to do
+so, of four as a step beyond three on the way to five, or to think of
+hot as a step from warm on the way to very hot. The degree of
+development of these abilities depends upon the activity of the law of
+analysis in the individual and the character of his experiences.
+
+(4) He gets certain bad habits of response from the ambiguity of common
+usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears
+his parents or older children or others count pennies or rolls or eggs
+by saying one, two, three, four, and so on. He himself is perhaps misled
+into so counting. Thus the names properly belonging to a series of
+aggregations varying in amount come to be to him the names of the
+positions of the parts in a counted whole. This happens especially with
+numbers above 3 or 4, where the correct experience of the number as a
+name for the group has rarely been present. This attaching to the
+cardinal numbers above three or four the meanings of the ordinal numbers
+seems to affect many children on entrance to school. The numbering of
+pages in books, houses, streets, etc., and bad teaching of counting
+often prolong this error.
+
+(5) He also gets the habit, not necessarily bad, but often indirectly
+so, of using many names such as eight, nine, ten, eleven, fifteen, a
+hundred, a million, without any meaning.
+
+(6) The experiences of half, twice, three times as many, three times as
+long, etc., are rarer; even if they were not, they would still be less
+easily productive of the analysis of the proper abstract element than
+are the experiences of two, three, four, etc., in connection with
+aggregates of things each of which is usually called one, such as boys,
+girls, balls, apples. Experiences of the names, two, three, and four, in
+connection with two twos, two threes, two fours, are very rare.
+
+Hence, the names, two, three, etc., mean to these children in the main,
+"one something and one something," "one something usually called one,
+and one something usually called one, and another something usually
+called one," and more rarely and imperfectly "two times anything,"
+"three times anything," etc.
+
+With respect to Mr. Phillips' emphasis of the importance of the
+series-idea in children's minds, the matters of importance are: first,
+that the knowledge of a series of number names in order is of very
+little consequence to the teaching of arithmetic and of still less to
+the origin of awareness of number. Second, the habit of applying this
+series of words in counting in such a way that 8 is associated with the
+eighth thing, 9 with the ninth thing, etc., is of consequence because it
+does so much mischief. Third, the really valuable idea of the number
+series, the idea of a series of groups or of magnitudes varying by
+steps, is acquired later, as a result, not a cause, of awareness of
+numbers.
+
+With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers
+should be emphasized in schools, not because it is the main origin of
+the child's awareness of number, but because it is _not_, and because
+the ordinary practical issues of child life do _not_ adequately
+stimulate its action. It also seems both more economical and more
+scientific to introduce it through multiplication, division, and
+fractions rather than to insist that 4 and 5 shall from the start mean 4
+or 5 times anything that is called 1, for instance, that 8 inches shall
+be called 4 two-inches, or 10 cents, 5 two-cents. If I interpret
+Professor Dewey's writings correctly, he would agree that the use of
+inch, foot, yard, pint, quart, ounce, pound, glassful, cupful, handful,
+spoonful, cent, nickel, dime, and dollar gives a sufficient range of
+units for the first two school years. Teaching the meanings of 1/2 of 4,
+1/2 of 6, 1/2 of 8, 1/2 of 10, 1/2 of 20, 1/3 of 6, 1/3 of 9, 1/3 of 30,
+1/4 of 8, two 2s, five 2s, and the like, in early grades, each in
+connection with many different units of measure, provides a sufficient
+assurance that numbers will connect with relationships as well as with
+collections.
+
+
+
+
+CHAPTER XII
+
+INTEREST IN ARITHMETIC
+
+
+CENSUSES OF PUPILS' INTERESTS
+
+Arithmetic, although it makes little or no appeal to collecting,
+muscular manipulation, sensory curiosity, or the potent original
+interests in things and their mechanisms and people and their passions,
+is fairly well liked by children. The censuses of pupils' likes and
+dislikes that have been made are not models of scientific investigation,
+and the resulting percentages should not be used uncritically. They are,
+however, probably not on the average over-favorable to arithmetic in any
+unfair way. Some of their results are summarized below. In general they
+show arithmetic to be surpassed in interest clearly by only the manual
+arts (shopwork and manual training for boys, cooking and sewing for
+girls), drawing, certain forms of gymnastics, and history. It is about
+on a level with reading and science. It clearly surpasses grammar,
+language, spelling, geography, and religion.
+
+Lobsien ['03], who asked one hundred children in each of the first five
+grades (_Stufen_) of the elementary schools of Kiel, "Which part of the
+school work (literally, 'which instruction period') do you like best?"
+found arithmetic led only by drawing and gymnastics in the case of the
+boys, and only by handwork in the case of the girls.
+
+This is an exaggerated picture of the facts, since no count is made of
+those who especially dislike arithmetic. Arithmetic is as unpopular with
+some as it is popular with others. When full allowance is made for this,
+arithmetic still has popularity above the average. Stern ['05] asked,
+"Which subject do you like most?" and "Which subject do you like least?"
+The balance was greatly in favor of gymnastics for boys (28--1),
+handwork for girls (32--1-1/2), and drawing for both (16-1/2--6).
+Writing (6-1/2--4), arithmetic (14-1/2--13), history (9--6-1/2),
+reading (8-1/2--8), and singing (6--7-1/2) come next. Religion,
+nature study, physiology, geography, geometry, chemistry, language,
+and grammar are low.
+
+McKnight ['07] found with boys and girls in grades 7 and 8 of certain
+American cities that arithmetic was liked better than any of the school
+subjects except gymnastics and manual training. The vote as compared
+with history was:--
+
+  Arithmetic    327 liked greatly,     96 disliked greatly.
+  History       164 liked greatly,    113 disliked greatly.
+
+In a later study Lobsien ['09] had 6248 pupils from 9 to 15 years old
+representing all grades of the elementary school report, so far as they
+could, the subject most disliked, the subject most liked, the subject
+next most liked, and the subject next in order. No child was forced to
+report all of these four judgments, or even any of them. Lobsien counts
+the likes and the dislikes for each subject. Gymnastics, handwork, and
+cooking are by far the most popular. History and drawing are next,
+followed by arithmetic and reading. Below these are geography, writing,
+singing, nature study, biblical history, catechism, and three minor
+subjects.
+
+Lewis ['13] secured records from English children in elementary schools
+of the order of preference of all the studies listed below. He reports
+the results in the following table of percents:
+
+  ===================================================================
+                  | TOP THIRD OF | MIDDLE THIRD OF | LOWEST THIRD OF
+                  | STUDIES FOR  | STUDIES FOR     | STUDIES FOR
+                  | INTEREST     | INTEREST        | INTEREST
+  ----------------+--------------+-----------------+-----------------
+  Drawing         |      78      |       20        |        2
+  Manual Subjects |      66      |       26        |        8
+  History         |      64      |       24        |       12
+  Reading         |      53      |       38        |        9
+  Singing         |      32      |       48        |       20
+                  |              |                 |
+  Drill           |      20      |       55        |       25
+  Arithmetic      |      16      |       53        |       31
+  Science         |      23      |       37        |       40
+  Nature Study    |      16      |       36        |       48
+  Dictation       |       4      |       57        |       39
+                  |              |                 |
+  Composition     |      18      |       28        |       54
+  Scripture       |       4      |       38        |       58
+  Recitation      |       9      |       23        |       68
+  Geography       |       4      |       24        |       72
+  Grammar         |      --      |        6        |       94
+  ===================================================================
+
+Brandell ['13] obtained data from 2137 Swedish children in Stockholm
+(327), Norrkoeping (870), and Gothenburg (940).
+
+In general he found, as others have, that handwork, shopwork for boys
+and household work for girls, and drawing were reported as much better
+liked than arithmetic. So also was history, and (in this he differs from
+most students of this matter) so were reading and nature study.
+Gymnastics he finds less liked than arithmetic. Religion, geography,
+language, spelling, and writing are, as in other studies, much less
+popular than arithmetic.
+
+Other studies are by Lilius ['11] in Finland, Walsemann ['07],
+Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern ['13 and '14],
+in Germany. They confirm the general results stated.
+
+The reasons for the good showing that arithmetic makes are probably the
+strength of its appeal to the interest in definite achievement, success,
+doing what one attempts to do; and of its appeal, in grades 5 to 8, to
+the practical interest of getting on in the world, acquiring abilities
+that the world pays for. Of these, the former is in my opinion much the
+more potent interest. Arithmetic satisfies it especially well, because,
+more than any other of the 'intellectual' studies of the elementary
+school, it permits the pupil to see his own progress and determine his
+own success or failure.
+
+The most important applications of the psychology of satisfiers and
+annoyers to arithmetic will therefore be in the direction of utilizing
+still more effectively this interest in achievement. Next in importance
+come the plans to attach to arithmetical learning the satisfyingness of
+bodily action, play, sociability, cheerfulness, and the like, and of
+significance as a means of securing other desired ends than arithmetical
+abilities themselves. Next come plans to relieve arithmetical learning
+from certain discomforts such as the eyestrain of some computations and
+excessive copying of figures. These will be discussed here in the
+inverse order.
+
+
+RELIEVING EYESTRAIN
+
+At present arithmetical work is, hour for hour, probably more of a tax
+upon the eyes than reading. The task of copying numbers from a book to a
+sheet of paper is one of the very hardest tasks that the eyes of a pupil
+in the elementary schools have to perform. A certain amount of such
+work is desirable to teach a child to write numbers, to copy exactly,
+and to organize material in shape for computation. But beyond that,
+there is no more reason for a pupil to copy every number with which he
+is to compute than for him to copy every word he is to read. The
+meaningless drudgery of copying figures should be mitigated by arranging
+much work in the form of exercises like those shown on pages 216, 217,
+and 218, and by having many of the textbook examples in addition,
+subtraction, and multiplication done with a slip of paper laid below the
+numbers, the answers being written on it. There is not only a resulting
+gain in interest, but also a very great saving of time for the pupil
+(very often copying an example more than quadruples the time required to
+get its answer), and a much greater efficiency in supervision.
+Arithmetical errors are not confused with errors of copying,[16] and the
+teacher's task of following a pupil's work on the page is reduced to a
+minimum, each pupil having put the same part of the day's work in just
+the same place. The use of well-printed and well-spaced pages of
+exercises relieves the eyestrain of working with badly made gray
+figures, unevenly and too closely or too widely spaced. I reproduce in
+Fig. 25 specimens taken at random from one hundred random samples of
+arithmetical work by pupils in grade 8. Contrast the task of the eyes in
+working with these and their task in working with pages 216 to 218. The
+customary method of always copying the numbers to be used in computation
+from blackboard or book to a sheet of paper is an utterly unjustifiable
+cruelty and waste.
+
+  [16] Courtis finds in the case of addition that "of all the
+       individuals making mistakes at any given time in a class,
+       at least one third, and usually two thirds, will be making
+       mistakes in carrying or copying."
+
+    [Illustration: FIG. 25_a_.--Specimens taken at random from the
+    computation work of eighth-grade pupils. This computation
+    occurred in a genuine test. In the original gray of the pencil
+    marks the work is still harder to make out.]
+
+    [Illustration: FIG. 25_b_.--Specimens taken at random from the
+    computation work of eighth-grade pupils. This computation
+    occurred in a genuine test. In the original gray of the pencil
+    marks the work is still harder to make out.]
+
+Write the products:--
+
+ A. 3   4s=           B. 5   7s=           C. 9   2s=
+    5   2s=              8   3s=              4   4s=
+    7   2s=              4   2s=              2   7s=
+    1   6 =              4   5s=              6   4s=
+    1   3 =              4   7s=              5   5s=
+    3   7s=              5   9s=              3   6s=
+    4   1s=              7   5s=              3   2s=
+    6   8s=              7   1s=              3   9s=
+    9   8s=              6   3s=              5   1s=
+    4   3s=              4   9s=              8   6s=
+    2   4s=              3   5s=              8   4s=
+    2   2s=              9   6s=              8   5s=
+    8   7s=              2   5s=              7   9s=
+    5   8s=              5   4s=              6   2s=
+    7   6s=              8   2s=              7   4s=
+    7   3s=              8   9s=              9   3s=
+
+ D. 4   20s =     E.    9   60s =       F.    40 x 2 = 80
+    4  200s =           9  600s =             20 x 2 =
+    6   30s =           5   30s =             30 x 2 =
+    6  300s =           5  300s =             40 x 2 =
+    7 x  50 =           8 x  20 =             20 x 3 =
+    7 x 500 =           8 x 200 =             30 x 3 =
+    3 x  40 =           2 x  70 =            300 x 3 = 900
+    3 x 400 =           2 x 700 =            300 x 2 =
+
+Write the missing numbers: (_r_ stands for remainder.)
+
+  25 = .... 3s and .... _r_.
+  25 = .... 4s  "  .... _r_.
+  25 = .... 5s  "  .... _r_.
+  25 = .... 6s  "  .... _r_.
+  25 = .... 7s  "  .... _r_.
+  25 = .... 8s  "  .... _r_.
+  25 = .... 9s  "  .... _r_.
+
+  26 = .... 3s and .... _r_.
+  26 = .... 4s  "  .... _r_.
+  26 = .... 5s  "  .... _r_.
+  26 = .... 6s  "  .... _r_.
+  26 = .... 7s  "  .... _r_.
+  26 = .... 8s  "  .... _r_.
+  26 = .... 9s  "  .... _r_.
+
+  30 = .... 4s and .... _r_.
+  30 = .... 5s  "  .... _r_.
+  30 = .... 6s  "  .... _r_.
+  30 = .... 7s  "  .... _r_.
+  30 = .... 8s  "  .... _r_.
+  30 = .... 9s  "  .... _r_.
+
+  31 = .... 4s and .... _r_.
+  31 = .... 5s  "  .... _r_.
+  31 = .... 6s  "  .... _r_.
+  31 = .... 7s  "  .... _r_.
+  31 = .... 8s  "  .... _r_.
+  31 = .... 9s  "  .... _r_.
+
+Write the whole numbers or mixed numbers which these fractions equal:--
+
+      5      4      9      4      7
+      -      -      -      -      -
+      4      3      5      2      3
+
+      7      5      11     3      8
+      -      -      --     -      -
+      4      3      8      2      8
+
+      8      6      9      9      16
+      -      -      -      -      --
+      4      3      8      4      8
+
+      11     7      13     8      6
+      --     -      --     -      -
+      4      5      8      5      6
+
+Write the missing figures:--
+
+  6         2         8          1          2
+  - = -     - = -     -- = -     - = --     - = -
+  8   4     4   2     10   5     5   10     3   6
+
+Write the missing numerators:--
+
+  1
+  - = --  -  --  -  --  -  --
+  2   12  8  10  4  16  6  14
+
+  1
+  - = --  -  --  -  --  --  --
+  3   12  9  18  6  15  24  21
+
+  1
+  - = --  --  -  --  --  --  --
+  4   12  16  8  24  20  28  32
+
+  1
+  - = --  --  --  --  --  --  --
+  5   10  20  15  25  40  35  30
+
+  2
+  - = --  --  --  -  --  --  -
+  3   12  18  21  6  15  24  9
+
+  3
+  - = -  --  --  --  --  --  --
+  4   8  16  12  20  24  32  28
+
+Find the products. Cancel when you can:--
+
+  5            11           2
+  -- x 4 =     -- x 3 =     - x 5 =
+  16           12           3
+
+  7            8            1
+  -- x 8 =     - x 15 =     - x 8 =
+  12           5            6
+
+
+SIGNIFICANCE FOR RELATED ACTIVITIES
+
+The use of bodily action, social games, and the like was discussed in
+the section on original tendencies. "Significance as a means of securing
+other desired ends than arithmetical learning itself" is therefore our
+next topic. Such significance can be given to arithmetical work by using
+that work as a means to present and future success in problems of
+sports, housekeeping, shopwork, dressmaking, self-management, other
+school studies than arithmetic, and general school life and affairs.
+Significance as a means to future ends alone can also be more clearly
+and extensively attached to it than it now is.
+
+Whatever is done to supply greater strength of motive in studying
+arithmetic must be carefully devised so as not to get a strong but wrong
+motive, so as not to get abundant interest but in something other than
+arithmetic, and so as not to kill the goose that after all lays the
+golden eggs--the interest in intellectual activity and achievement
+itself. It is easy to secure an interest in laying out a baseball
+diamond, measuring ingredients for a cake, making a balloon of a certain
+capacity, or deciding the added cost of an extra trimming of ribbon for
+one's dress. The problem is to _attach_ that interest to arithmetical
+learning. Nor should a teacher be satisfied with attaching the interest
+as a mere tail that steers the kite, so long as it stays on, or as a
+sugar-coating that deceives the pupil into swallowing the pill, or as an
+anodyne whose dose must be increased and increased if it is to retain
+its power. Until the interest permeates the arithmetical activity itself
+our task is only partly done, and perhaps is made harder for the next
+time.
+
+One important means of really interfusing the arithmetical learning
+itself with these derived interests is to lead the pupil to seek the
+help of arithmetic himself--to lead him, in Dewey's phrase, to 'feel the
+need'--to take the 'problem' attitude--and thus appreciate the
+technique which he actively hunts for to satisfy the need. In so far as
+arithmetical learning is organized to satisfy the practical demands of
+the pupil's life at the time, he should, so to speak, come part way to
+get its help.
+
+Even if we do not make the most skillful use possible of these interests
+derived from the quantitative problems of sports, housekeeping,
+shopwork, dressmaking, self-management, other school studies, and school
+life and affairs, the gain will still be considerable. To have them in
+mind will certainly preserve us from giving to children of grades 3 and
+4 problems so devoid of relation to their interests as those shown
+below, all found (in 1910) in thirty successive pages of a book of
+excellent repute:--
+
+  A chair has 4 legs. How many legs have 8 chairs? 5 chairs?
+
+  A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?
+
+              (Eight more of the same sort.)
+
+  In 1890 New York had 1,513,501 inhabitants, Milwaukee had
+  206,308, Boston had 447,720, San Francisco 297,990. How many
+  had these cities together?
+
+              (Five more of the same sort.)
+
+  Milton was born in 1608 and died in 1674. How many years
+  did he live?
+
+             (Several others of the same sort.)
+
+  The population of a certain city was 35,629 in 1880 and 106,670
+  in 1890. Find the increase.
+
+              (Several others of this sort.)
+
+  A number of others about the words in various inaugural addresses
+  and the Psalms in the Bible.
+
+It also seems probable that with enough care other systematic plans of
+textbooks can be much improved in this respect. From every point of
+view, for example, the early work in arithmetic should be adapted to
+some extent to the healthy childish interests in home affairs, the
+behavior of other children, and the activities of material things,
+animals, and plants.
+
+TABLE 9
+
+FREQUENCY OF APPEARANCE OF CERTAIN WORDS ABOUT FAMILY LIFE, PLAY, AND
+ACTION IN EIGHT ELEMENTARY TEXTBOOKS IN ARITHMETIC, pp. 1-50.
+
+  ================================================================
+                  |  A  |  B  |  C  |  D  |  E  |  F  |  G  |  H
+  ----------------+-----+-----+-----+-----+-----+-----+-----+-----
+  baby            |     |     |     |  2  |     |  4  |     |
+  brother         |  2  |     |  6  |  1  |  1  |     |  1  |
+  family          |     |     |  2  |     |  2  |     |  4  |
+  father          |  1  |     |  3  |  5  |     |  2  |  1  |
+  help            |     |     |     |     |     |     |     |
+  home            |  2  |     |  4  |  4  |  2  |  2  |  7  |  1
+  mother          |  4  |  2  |  9  |  5  |     |  5  |  1  |  7
+  sister          |     |     |  1  |  2  |  2  |  9  |  1  |  1
+                  |     |     |     |     |     |     |     |
+  fork            |     |     |     |     |     |     |     |
+  knife           |     |     |     |     |     |     |     |
+  plate           |  4  |  2  |     |  2  |     |  1  |     |
+  spoon           |     |     |     |     |     |     |     |
+                  |     |     |     |     |     |     |     |
+  doll            | 10  |  1  | 10  |  6  |     | 10  |     |  9
+  game            |  1  |     |     |  3  |     |     |  5  |  5
+  jump            |     |     |     |     |     |     |     |  4
+  marbles         | 10  |  4  | 10  |     | 10  |     |  1  |
+  play            |     |     |  1  |     |     |  3  |     |
+  run             |     |     |     |     |     |  1  |     |  3
+  sing            |     |     |     |     |     |     |     |
+  tag             |     |     |     |     |     |     |     |
+  toy             |     |     |     |     |     |     |     |  1
+                  |     |     |     |     |     |     |     |
+  car             |     |     |  2  |  4  |     |  2  |  3  |  1
+  cut             |     |     | 10  |     |  6  |  2  |     |  8
+  dig             |     |     |     |     |     |     |  2  |
+  flower          |  1  |     |     |  4  |  1  |  1  |  2  |
+  grow            |     |     |     |  1  |     |     |     |
+  plant           |     |     |  2  |     |     |     |     |
+  seed            |     |     |     |  3  |     |     |  1  |
+  string          |     |     |     |     |  1  | 10  |  1  |  1
+  wheel           |  5  |     |     |     |     | 10  |     |
+  ================================================================
+
+The words used by textbooks give some indication of how far this aim is
+being realized, or rather of how far short we are of realizing it.
+Consider, for example, the words home, mother, father, brother, sister,
+help, plate, knife, fork, spoon, play, game, toy, tag, marbles, doll,
+run, jump, sing, plant, seed, grow, flower, car, wheel, string, cut,
+dig. The frequency of appearance in the first fifty pages of eight
+beginners' arithmetics was as shown in Table 9. The eight columns refer
+to the eight books (the first fifty pages of each). The numbers refer to
+the number of times the word in question appeared, the number 10 meaning
+10 _or more_ times in the fifty pages. Plurals, past tenses, and the
+like were counted. _Help_, _fork_, _knife_, _spoon_, _jump_, _sing_, and
+_tag_ did not appear at all! _Toy_ and _grow_ appeared each once in the
+400 pages! _Play_, _run_, _dig_, _plant_, and _seed_ appeared once in a
+hundred or more pages. _Baby_ did not appear as often as _buggy_.
+_Family_ appeared no oftener than _fence_ or _Friday_. _Father_ appears
+about a third as often as _farmer_.
+
+Book A shows only 10 of these thirty words in the fifty pages; book B
+only 4; book C only 12; and books D, E, F, G, and H only 13, 8, 14, 13,
+10, respectively. The total number of appearances (counting the 10s as
+only 10 in each case) is 40 for A, 9 for B, 60 for C, 42 for D, 25 for
+E, 62 for F, 30 for G, and 37 for H. The five words--apple, egg, Mary,
+milk, and orange--are used oftener than all these thirty together.
+
+If it appeared that this apparent neglect of childish affairs and
+interests was deliberate to provide for a more systematic treatment of
+pure arithmetic, a better gradation of problems, and a better
+preparation for later genuine use than could be attained if the author
+of the textbook were tied to the child's apron strings, the neglect
+could be defended. It is not at all certain that children in grade 2 get
+much more enjoyment or ability from adding the costs of purchases for
+Christmas or Fourth of July, or multiplying the number of cakes each
+child is to have at a party by the number of children who are to be
+there, than from adding gravestones or multiplying the number of hairs
+of bald-headed men. When, however, there is nothing gained by
+substituting remote facts for those of familiar concern to children, the
+safe policy is surely to favor the latter. In general, the neglect of
+childish data does not seem to be due to provision for some other end,
+but to the same inertia of tradition which has carried over the problems
+of laying walls and digging wells into city schools whose children never
+saw a stone wall or dug well.
+
+       *       *       *       *       *
+
+I shall not go into details concerning the arrangement of courses of
+study, textbooks, and lesson-plans to make desirable connections between
+arithmetical learning and sports, housework, shopwork, and the rest. It
+may be worth while, however, to explain the term _self-management_,
+since this source of genuine problems of real concern to the pupils has
+been overlooked by most writers.
+
+By self-management is meant the pupil's use of his time, his abilities,
+his knowledge, and the like. By the time he reaches grade 5, and to some
+extent before then, a boy should keep some account of himself, of how
+long it takes him to do specified tasks, of how much he gets done in a
+specified time at a certain sort of work and with how many errors, of
+how much improvement he makes month by month, of which things he can do
+best, and the like. Such objective, matter-of-fact, quantitative study
+of one's behavior is not a stimulus to morbid introspection or egotism;
+it is one of the best preventives of these. To treat oneself
+impersonally is one of the essential elements of mental balance and
+health. It need not, and should not, encourage priggishness. On the
+contrary, this matter-of-fact study of what one is and does may well
+replace a certain amount of the exhortations and admonitions concerning
+what one ought to do and be. All this is still truer for a girl.
+
+The demands which such an accounting of one's own activities make of
+arithmetic have the special value of connecting directly with the
+advanced work in computation. They involve the use of large numbers,
+decimals, averaging, percentages, approximations, and other facts and
+processes which the pupil has to learn for later life, but to which his
+childish activities as wage-earner, buyer and seller, or shopworker from
+10 to 14 do not lead. Children have little money, but they have time in
+thousands of units! They do not get discounts or bonuses from commercial
+houses, but they can discount their quantity of examples done for the
+errors made, and credit themselves with bonuses of all sorts for extra
+achievements.
+
+
+INTRINSIC INTEREST IN ARITHMETICAL LEARNING
+
+There remains the most important increase of interest in arithmetical
+learning--an increase in the interest directly bound to achievement and
+success in arithmetic itself. "Arithmetic," says David Eugene Smith, "is
+a game and all boys and girls are players." It should not be a _mere_
+game for them and they should not _merely_ play, but their unpractical
+interest in doing it because they can do it and can see how well they do
+do it is one of the school's most precious assets. Any healthy means to
+give this interest more and better stimulus should therefore be eagerly
+sought and cherished.
+
+Two such means have been suggested in other connections. The first is
+the extension of training in checking and verifying work so that the
+pupil may work to a standard of approximately 100% success, and may
+know how nearly he is attaining it. The second is the use of
+standardized practice material and tests, whereby the pupil may measure
+himself against his own past, and have a clear, vivid, and trustworthy
+idea of just how much better or faster he can do the same tasks than he
+could do a month or a year ago, and of just how much harder things he
+can do now than then.
+
+Another means of stimulating the essential interest in quantitative
+thinking itself is the arrangement of the work so that real arithmetical
+thinking is encouraged more than mere imitation and assiduity. This
+means the avoidance of long series of applied problems all of one type
+to be solved in the same way, the avoidance of miscellaneous series and
+review series which are almost verbatim repetitions of past problems,
+and in general the avoidance of excessive repetition of any one
+problem-situation. Stimulation to real arithmetical thinking is weak
+when a whole day's problem work requires no choice of methods, or when a
+review simply repeats without any step of organization or progress, or
+when a pupil meets a situation (say the 'buy _x_ things at _y_ per
+thing, how much pay' situation) for the five-hundredth time.
+
+Another matter worthy of attention in this connection is the unwise
+tendency to omit or present in diluted form some of the topics that
+appeal most to real intellectual interests, just because they are hard.
+The best illustration, perhaps, is the problem of ratio or "How many
+times as large (long, heavy, expensive, etc.) as _x_ is _y_?" Mastery of
+the 'times as' relation is hard to acquire, but it is well worth
+acquiring, not only because of its strong intellectual appeal, but also
+because of its prime importance in the applications of arithmetic to
+science. In the older arithmetics it was confused by pedantries and
+verbal difficulties and penalized by unreal problems about fractions of
+men doing parts of a job in strange and devious times. Freed from these,
+it should be reinstated, beginning as early as grade 5 with such simple
+exercises as those shown below and progressing to the problems of food
+values, nutritive ratios, gears, speeds, and the like in grade 8.
+
+  John is 4 years old.
+  Fred is 6 years old.
+  Mary is 8 years old.
+  Nell is 10 years old.
+  Alice is 12 years old.
+  Bert is 15 years old.
+
+  Who is twice as old as John?
+  Who is half as old as Alice?
+  Who is three times as old as John?
+  Who is one and one half times as old as Nell?
+  Who is two thirds as old as Fred?
+      etc., etc., etc.
+
+  Alice is .... times as old as John.
+  John is .... as old as Mary.
+  Fred is .... times as old as John.
+  Alice is .... times as old as Fred.
+  Fred is .... as old as Mary.
+      etc., etc., etc.
+
+Finally it should be remembered that all improvements in making
+arithmetic worth learning and helping the pupil to learn it will in the
+long run add to its interest. Pupils like to learn, to achieve, to gain
+mastery. Success is interesting. If the measures recommended in the
+previous chapters are carried out, there will be little need to entice
+pupils to take arithmetic or to sugar-coat it with illegitimate
+attractions.
+
+
+
+
+CHAPTER XIII
+
+THE CONDITIONS OF LEARNING
+
+
+We shall consider in this chapter the influence of time of day, size of
+class, and amount of time devoted to arithmetic in the school program,
+the hygiene of the eyes in arithmetical work, the use of concrete
+objects, and the use of sounds, sights, and thoughts as situations and
+of speech and writing and thought as responses.[17]
+
+  [17] Facts concerning the conditions of learning in general will
+       be found in the author's _Educational Psychology_, Vol. 2,
+       Chapter 8, or in the _Educational Psychology, Briefer Course_,
+       Chapter 15.
+
+
+EXTERNAL CONDITIONS
+
+Computation of one or another sort has been used by several
+investigators as a test of efficiency at different times in the day.
+When freed from the effects of practice on the one hand and lack of
+interest due to repetition on the other, the results uniformly show an
+increase in speed late in the school session with a falling off in
+accuracy that about balances it.[18] There is no wisdom in putting
+arithmetic early in the session because of its _difficulty_. Lively and
+sociable exercises in mental arithmetic with oral answers in fact seem
+to be admirably fitted for use late in the session. Except for the
+general principles (1) of starting the day with work that will set a
+good standard of cheerful, efficient production and (2) of getting the
+least interesting features of the day's work done fairly early in the
+day, psychology permits practical exigencies to rule the program, so far
+as present knowledge extends. Adequate measurements of the effect of
+time of day on _improvement_ have not been made, but there is no reason
+to believe that any one time between 9 A.M. and 4 P.M. is appreciably
+more favorable to arithmetical learning than to learning geography,
+history, spelling, and the like.
+
+  [18] See Thorndike ['00], King ['07], and Heck ['13].
+
+The influence of size of class upon progress in school studies is very
+difficult to measure because (1) within the same city system the average
+of the six (or more) sizes of class that a pupil has experienced will
+tend to approximate closely to the corresponding average for any other
+child; because further (2) there may be a tendency of supervisory
+officers to assign more pupils to the better teachers; and because
+(3) separate systems which differ in respect to size of class probably
+differ in other respects also so that their differences in achievement
+may be referable to totally different differences.
+
+Elliott ['14] has made a beginning by noting size of class during the
+year of test in connection with his own measures of the achievements of
+seventeen hundred pupils, supplemented by records from over four hundred
+other classes. As might be expected from the facts just stated, he finds
+no appreciable difference between classes of different sizes within the
+same school system, the effect of the few months in a small class being
+swamped by the antecedents or concomitants thereof.
+
+The effect of the amount of time devoted to arithmetic in the school
+program has been studied extensively by Rice ['02 and '03] and Stone
+['08].
+
+Dr. Rice ['02] measured the arithmetical ability of some 6000 children
+in 18 different schools in 7 different cities. The results of these
+measurements are summarized in Table 10. This table "gives two averages
+for each grade as well as for each school as a whole. Thus, the school
+at the top shows averages of 80.0 and 83.1, and the one at the bottom,
+25.3 and 31.5. The first represents the percentage of answers which were
+absolutely correct; the second shows what per cent of the problems were
+correct in principle, _i.e._ the average that would have been received
+if no mechanical errors had been made."
+
+The facts of Dr. Rice's table show that there is a positive relation
+between the general standing of a school system in the tests and the
+amount of time devoted to arithmetic by its program. The relation is
+not close, however, being that expressed by a correlation coefficient
+of .36-1/2. Within any one school system there is no relation between
+the standing of a particular school and the amount of time devoted to
+arithmetic in that school's program. It must be kept in mind that the
+amount of time given in the school program may be counterbalanced by
+emphasizing work at home and during study periods, or, on the other
+hand, may be a symptom of correspondingly small or great emphasis on
+arithmetic in work set for the study periods at home.
+
+A still more elaborate investigation of this same topic was made by
+Stone ['08]. I quote somewhat fully from it, since it is an instructive
+sample of the sort of studies that will doubtless soon be made in the
+case of every elementary school subject. He found that school systems
+differed notably in the achievements made by their sixth-grade pupils in
+his tests of computation (the so-called 'fundamentals') and of the
+solution of verbally described problems (the so-called 'reasoning'). The
+facts were as shown in Table 11.
+
+TABLE  10
+
+AVERAGES FOR INDIVIDUAL SCHOOLS IN ARITHMETIC
+
+  KEY    A: CITY
+         B: SCHOOL
+         C: Result
+         D: Principle
+         E: Percent of Mechanical Errors
+         F: Minutes Daily
+
+  ===========================================================
+     |   |6TH YEAR |7TH YEAR |8TH YEAR |SCHOOL AVERAGE |
+     |   |----+----+----+----+----+----+----+----+-----+-----
+   A | B | C  | D  | C  | D  | C  | D  | C  | D  |  E  |  F
+  ---+---+----+----+----+----+----+----+----+----+-----+-----
+  III| 1 |79.3|80.3|81.1|82.3|91.7|93.9|80.0|83.1| 3.7 |  53
+    I| 1 |80.4|81.5|64.2|67.2|80.9|82.8|76.6|80.3| 4.6 |  60
+    I| 2 |80.9|83.4|43.5|50.9|72.7|79.1|69.3|75.1| 7.7 |  25
+    I| 3 |72.2|74.0|63.5|66.2|74.5|76.6|67.8|72.2| 6.1 |  45
+    I| 4 |69.9|72.2|54.6|57.8|66.5|69.1|64.3|70.3| 8.5 |  45
+   II| 1 |71.2|75.3|33.6|35.7|36.8|40.0|60.2|64.8| 7.1 |  60
+  III| 2 |43.7|45.0|53.9|56.7|51.1|53.1|54.5|58.9| 7.4 |  60
+   IV| 1 |58.9|60.4|31.2|34.1|41.6|43.5|55.1|58.4| 5.6 |  60
+   IV| 2 |59.8|63.1| -- | -- |22.5|22.5|53.9|58.8| 8.3 |  --
+   IV| 3 |54.9|58.1|35.2|38.6|43.5|45.0|51.5|57.6|10.5 |  60
+   IV| 4 |42.3|45.1|16.1|19.2|48.7|48.7|42.8|48.2|11.2 |  --
+    V| 1 |44.1|48.7|29.2|32.5|51.1|58.3|45.9|51.3|10.5 |  40
+   VI| 1 |68.3|71.3|33.5|36.6|26.9|30.7|39.0|42.9| 9.0 |  33
+   VI| 2 |46.1|49.5|19.5|24.2|30.2|40.6|36.5|43.6|16.2 |  30
+   VI| 3 |34.5|36.4|30.5|35.1|23.3|24.1|36.0|42.5|15.2 |  48
+  VII| 1 |35.2|37.7|29.1|32.5|25.1|27.2|40.5|45.9|11.7 |  42
+  VII| 2 |35.2|38.7|15.0|16.4|19.6|21.2|36.5|40.6|10.1 |  75
+  VII| 3 |27.6|33.7| 8.9|10.1|11.3|11.3|25.3|31.5|19.6 |  45
+  ===========================================================
+
+High achievement by a system in computation went with high achievement
+in solving the problems, the correlation being about .50; and the
+system that scored high in addition or subtraction or multiplication or
+division usually showed closely similar excellence in the other three,
+the correlations being about .90.
+
+TABLE 11
+
+SCORES MADE BY THE SIXTH-GRADE PUPILS OF EACH OF TWENTY-SIX SCHOOL
+SYSTEMS
+
+  =================================================
+  SYSTEM | SCORE IN TESTS WITH | SCORE IN TESTS IN
+         |      PROBLEMS       |     COMPUTING
+  -------+---------------------+-------------------
+    23   |         356         |       1841
+    24   |         429         |       3513
+    17   |         444         |       3042
+     4   |         464         |       3563
+    25   |         464         |       2167
+    22   |         468         |       2311
+    16   |         469         |       3707
+    20   |         491         |       2168
+    18   |         509         |       3758
+    15   |         532         |       2779
+     3   |         533         |       2845
+     8   |         538         |       2747
+     6   |         550         |       3173
+     1   |         552         |       2935
+    10   |         601         |       2749
+     2   |         615         |       2958
+    21   |         627         |       2951
+    13   |         636         |       3049
+    14   |         661         |       3561
+     9   |         691         |       3404
+     7   |         734         |       3782
+    12   |         736         |       3410
+    11   |         759         |       3261
+    26   |         791         |       3682
+    19   |         848         |       4099
+     5   |         914         |       3569
+  =================================================
+
+Of the conditions under which arithmetical learning took place, the one
+most elaborately studied was the amount of time devoted to arithmetic.
+On the basis of replies by principals of schools to certain questions,
+he gave each of the twenty-six school systems a measure for the
+probable time spent on arithmetic up through grade 6. Leaving home study
+out of account, there seems to be little or no correlation between the
+amount of time a system devotes to arithmetic and its score in
+problem-solving, and not much more between time expenditure and score in
+computation. With home study included there is little relation to the
+achievement of the system in solving problems, but there is a clear
+effect on achievement in computation. The facts as given by Stone are:--
+
+TABLE 12
+
+CORRELATION OF TIME EXPENDITURES WITH ABILITIES
+
+  Without Home Study      { Reasoning and Time Expenditure        -.01
+                          { Fundamentals and Time Expenditure      .09
+
+  Including Home Study    { Reasoning and Time Expenditure         .13
+                          { Fundamentals and Time Expenditure      .49
+
+These correlations, it should be borne in mind, are for school systems,
+not for individual pupils. It might be that, though the system which
+devoted the most time to arithmetic did not show corresponding
+superiority in the product over the system devoting only half as much
+time, the pupils within the system did achieve in exact proportion to
+the time they gave to study. Neither correlation would permit inference
+concerning the effect of different amounts of time spent by the same
+pupil.
+
+Stone considered also the printed announcements of the courses of study
+in arithmetic in these twenty-six systems. Nineteen judges rated these
+announced courses of study for excellence according to the instructions
+quoted below:--
+
+CONCERNING THE RATING OF COURSES OF STUDY
+
+Judges please read before scoring
+
+I. Some Factors Determining Relative Excellence.
+
+(N. B. The following enumeration is meant to be suggestive rather than
+complete or exclusive. And each scorer is urged to rely primarily on his
+own judgment.)
+
+  1. Helpfulness to the teacher in teaching the subject matter outlined.
+
+  2. Social value or concreteness of sources of problems.
+
+  3. The arrangement of subject matter.
+
+  4. The provision made for adequate drill.
+
+  5. A reasonable minimum requirement with suggestions for valuable
+     additional work.
+
+  6. The relative values of any predominating so-called methods--such as
+     Speer, Grube, etc.
+
+  7. The place of oral or so-called mental arithmetic.
+
+  8. The merit of textbook references.
+
+II. Cautions and Directions.
+
+(Judges please follow as implicitly as possible.)
+
+  1. Include references to textbooks as parts of the Course of Study.
+
+     This necessitates judging the parts of the texts referred to.
+
+  2. As far as possible become equally familiar with all courses before
+     scoring any.
+
+  3. When you are ready to begin to score, (1) arrange in serial
+     order according to excellence, (2) starting with the middle one
+     score it 50, then score above and below 50 according as courses
+     are better or poorer, indicating relative differences in
+     excellence by relative differences in scores, _i.e._ in so far
+     as you find that the courses differ by about equal steps, score
+     those better than the middle one 51, 52, etc., and those poorer
+     49, 48, etc., but if you find that the courses differ by
+     unequal steps show these inequalities by omitting numbers.
+
+  4. Write ratings on the slip of paper attached to each course.
+
+The systems whose courses of study were thus rated highest did not
+manifest any greater achievement in Stone's tests than the rest. The
+thirteen with the most approved announcements of courses of study were
+in fact a little inferior in achievement to the other thirteen, and the
+correlation coefficients were slightly negative.
+
+Stone also compared eighteen systems where there was supervision of the
+work by superintendents or supervisors as well as by principals with
+four systems where the principals and teachers had no such help. The
+scores in his tests were very much lower in the four latter cities.
+
+
+THE HYGIENE OF THE EYES IN ARITHMETIC
+
+We have already noted that the task of reading and copying numbers is
+one of the hardest that the eyes have to perform in the elementary
+school, and that it should be alleviated by arranging much of the work
+so that only answers need be written by the pupil. The figures to be
+read and copied should obviously be in type of suitable size and style,
+so arranged and spaced on the page or blackboard as to cause a minimum
+of effort and strain.
+
+    [Illustration: FIG. 26.--Type too large.]
+
+    [Illustration: FIG. 27.--12-point, 11-point, and 10-point type.]
+
+_Size._--Type may be too large as well as too small, though the latter
+is the commoner error. If it is too large, as in Fig. 26, which is a
+duplicate of type actually used in a form of practice pad, the eye has
+to make too many fixations to take in a given content. All things
+considered, 12-point type in grades 3 and 4, 11-point in grades 5 and 6,
+and 10-point in grades 7 and 8 seem the most desirable sizes. These are
+shown in Fig. 27. Too small type occurs oftenest in fractions and in the
+dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30
+are samples from actual school practice. Samples of the desirable size
+are shown in Figs. 31 and 32. The technique of modern typesetting makes
+it very difficult and expensive to make fractions of the horizontal type
+
+  (1  3  5
+   -  -  -
+   4, 8, 6)
+
+large enough without making the whole-number figures with which they
+are mingled too large or giving an uncouth appearance to the total.
+Consequently fractions somewhat smaller than are desirable may have
+to be used occasionally in textbooks.[19] There is no valid excuse,
+however, for the excessively small fractions which often are made in
+blackboard work.
+
+  [19] A special type could be constructed that would use a large
+       type body, say 14 point, with integers in 10 or 12 point and
+       fractions much larger than now.
+
+    [Illustration: FIG. 28.--Type of measurements too small.
+
+        This is a picture of Mary's garden. How many feet is it
+        around the garden?]
+
+    [Illustration: FIG. 29.--Type too small.]
+
+    [Illustration: FIG. 30.--Numbers too small and badly designed.]
+
+    [Illustration: FIG. 31.--Figure 28 with suitable numbers.]
+
+    [Illustration: FIG. 32.--Figure 30 with suitable numbers.]
+
+_Style._--The ordinary type forms often have 3 and 8 so made as to
+require strain to distinguish them. 5 is sometimes easily confused with
+3 and even with 8. 1, 4, and 7 may be less easily distinguishable than
+is desirable. Figure 33 shows a specially good type in which each figure
+is represented by its essential[20] features without any distracting
+shading or knobs or turns. Figure 34 shows some of the types in common
+use. There are no demonstrably great differences amongst these. In
+fractions there is a notable gain from using the slant form (2/3, 3/4)
+for exercises in addition and subtraction, and for almost all mixed
+numbers. This appears clearly to the eye in the comparison of Fig. 35
+below, where the same fractions all in 10-point type are displayed in
+horizontal and in slant form. The figures in the slant form are in
+general larger and the space between them and the fraction-line is
+wider. Also the slant form makes it easier for the eye to examine the
+denominators to see whether reductions are necessary. Except for a few
+cases to show that the operations can be done just as truly with the
+horizontal forms, the book and the blackboard should display mixed
+numbers and fractions to be added or subtracted in the slant form. The
+slant line should be at an angle of approximately 45 degrees. Pupils
+should be taught to use this form in their own work of this sort.
+
+  [20] It will be still better if the 4 is replaced by an open-top 4.
+
+When script figures are presented they should be of simple design,
+showing clearly the essential features of the figure, the line being
+everywhere of equal or nearly equal width (that is, without shading, and
+without ornamentation or eccentricity of any sort). The opening of the 3
+should be wide to prevent confusion with 8; the top of the 3 should be
+curved to aid its differentiation from 5; the down stroke of the 9
+should be almost or quite straight; the 1, 4, 7, and 9 should be clearly
+distinguishable. There are many ways of distinguishing them clearly, the
+best probably being to use the straight line for 1, the open 4 with
+clear angularity, a wide top to the 7, and a clearly closed curve for
+the top of the 9.
+
+    [Illustration: FIG. 33.--Block type; a very desirable type except
+    that it is somewhat too heavy.]
+
+    [Illustration: FIG. 34.--Common styles of printed numbers.]
+
+    [Illustration: FIG. 35.--Diagonal and horizontal fractions
+    compared.]
+
+    [Illustration: FIG. 36.--Good vertical spacing.]
+
+    [Illustration: FIG. 37.--Bad vertical spacing.]
+
+    [Illustration: FIGS. 38 (above) and 39 (below).--Good and bad
+    left-right spacing.]
+
+The pupil's writing of figures should be clear. He will thereby be saved
+eyestrain and errors in his school work as well as given a valuable
+ability for life. Handwriting of figures is used enormously in spite of
+the development of typewriters; illegible figures are commonly more
+harmful than illegible letters or words, since the context far less
+often tells what the figure is intended to be; the habit of making clear
+figures is not so hard to acquire, since they are written unjoined and
+require only the automatic action of ten minor acts of skill. The
+schools have missed a great opportunity in this respect. Whereas the
+hand writing of words is often better than it needs to be for life's
+purposes, the writing of figures is usually much worse. The figures
+presented in books on penmanship are also commonly bad, showing neglect
+or misunderstanding of the matter on the part of leaders in penmanship.
+
+_Spacing._--Spacing up and down the column is rarely too wide, but very
+often too narrow. The specimens shown in Figs. 36 and 37 show good
+practice contrasted with the common fault.
+
+Spacing from right to left is generally fairly satisfactory in books,
+though there is a bad tendency to adopt some one routine throughout and
+so to miss chances to use reductions and increases of spacing so as to
+help the eye and the mind in special cases. Specimens of good and bad
+spacing are shown in Figs. 38 and 39. In the work of the pupils, the
+spacing from right to left is often too narrow. This crowding of
+letters, together with unevenness of spacing, adds notably to the task
+of eye and mind.
+
+_The composition or make-up of the page._--Other things being equal,
+that arrangement of the page is best which helps a child most to keep
+his place on a page and to find it after having looked away to work on
+the paper on which he computes, or for other good reasons. A good page
+and a bad page in this respect are shown in Figs. 40 and 41.
+
+    [Illustration: FIG. 40.--A page well made up to suit the action
+    of the eye.]
+
+    [Illustration: FIG. 41.--The same matter as in Fig. 40, much
+    less well made up.]
+
+_Objective presentations._--Pictures, diagrams, maps, and other
+presentations should not tax the eye unduly,
+
+  (_a_) by requiring too fine distinctions, or
+
+  (_b_) by inconvenient arrangement of the data, preventing easy
+        counting, measuring, comparison, or whatever the task is, or
+
+  (_c_) by putting too many facts in one picture so that the eye
+        and mind, when trying to make out any one, are confused by the
+        others.
+
+Illustrations of bad practices in these respects are shown in Figs. 42
+to 52. A few specimens of work well arranged for the eye are shown in
+Figs. 53 to 56.
+
+Good rules to remember are:--
+
+Other things being equal, make distinctions by the clearest method, fit
+material to the tendency of the eye to see an 'eyeful' at a time
+(roughly 1-1/2 inch by 1/2 inch in a book; 1-1/2 ft. by 1/2 ft. on the
+blackboard), and let one picture teach only one fact or relation, or
+such facts and relations as do not interfere in perception.
+
+The general conditions of seating, illumination, paper, and the like are
+even more important when the eyes are used with numbers than when they
+are used with words.
+
+    [Illustration: FIG. 42.--Try to count the rungs on the ladder,
+    or the shocks in the wagon.]
+
+    [Illustration: FIG. 43.--How many oars do you see? How many
+    birds? How many fish?]
+
+    [Illustration: FIG. 44.--Count the birds in each of the three
+    flocks of birds.]
+
+    [Illustration: FIG. 45.--Note the lack of clear division of the
+    hundreds. Consider the difficulty of counting one of these
+    columns of dots.]
+
+    [Illustration: FIG. 46.--What do you suppose these pictures are
+    intended to show?]
+
+    [Illustration: FIG. 47.--Would a beginner know that after
+    THIRTEEN he was to switch around and begin at the other end?
+    Could you read the SIX of TWENTY-SIX if you did not already know
+    what it ought to be? What meaning would all the brackets have
+    for a little child in grade 2? Does this picture illustrate or
+    obfuscate?]
+
+    [Illustration: FIG. 48.--How long did it take you to find out
+    what these pictures mean?]
+
+    [Illustration: FIG. 49.--Count the figures in the first row,
+    using your eyes alone; have some one make lines of 10, 11, 12,
+    13, and more repetitions of this figure spaced closely as here.
+    Count 20 or 30 such lines, using the eye unaided by fingers,
+    pencil, etc. ]
+
+    [Illustration: FIG. 50.--Can you answer the question without
+    measuring? Could a child of seven or eight?]
+
+    [Illustration: FIG. 51.--What are these drawings intended to
+    show? Why do they show the facts only obscurely and dubiously?]
+
+    [Illustration: FIG. 52.--What are these drawings intended to
+    show? What simple change would make them show the facts much
+    more clearly?]
+
+    [Illustration: FIG. 53.--Arranged in convenient "eye-fulls."]
+
+    [Illustration: FIG. 54.--Clear, simple, and easy of comparison.]
+
+    [Illustration: FIG. 55.--Clear, simple, and well spaced.]
+
+    [Illustration: FIG. 56.--Well arranged, though a little wider
+    spacing between the squares would make it even better.]
+
+
+THE USE OF CONCRETE OBJECTS IN ARITHMETIC
+
+We mean by concrete objects actual things, events, and relations
+presented to sense, in contrast to words and numbers and symbols which
+mean or stand for these objects or for more abstract qualities and
+relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart
+measures, bank books, and checks are such concrete things. A foot rule
+put successively along the three thirds of a yard rule, a bell rung five
+times, and a pound weight balancing sixteen ounce weights are such
+concrete events. A pint beside a quart, an inch beside a foot, an apple
+shown cut in halves display such concrete relations to a pupil who is
+attentive to the issue.
+
+Concrete presentations are obviously useful in arithmetic to teach
+meanings under the general law that a word or number or sign or symbol
+acquires meaning by being connected with actual things, events,
+qualities, and relations. We have also noted their usefulness as means
+to verifying the results of thinking and computing, as when a pupil,
+having solved, "How many badges each 5 inches long can be made from
+3-1/3 yd. of ribbon?" by using 10 x 12/5, draws a line 3-1/3 yd. long
+and divides it into 5-inch lengths.
+
+Concrete experiences are useful whenever the meaning of a number, like 9
+or 7/8 or .004, or of an operation, like multiplying or dividing or
+cubing, or of some term, like rectangle or hypothenuse or discount, or
+some procedure, like voting or insuring property against fire or
+borrowing money from a bank, is absent or incomplete or faulty. Concrete
+work thus is by no means confined to the primary grades but may be
+appropriate at all stages when new facts, relations, and procedures are
+to be taught.
+
+How much concrete material shall be presented will depend upon the fact
+or relation or procedure which is to be made intelligible, and the
+ability and knowledge of the pupil. Thus 'one half' will in general
+require less concrete illustration than 'five sixths'; and five sixths
+will require less in the case of a bright child who already knows 2/3,
+3/4, 3/8, 5/8, 7/8, 2/5, 3/5, and 4/5 than in the case of a dull child
+or one who only knows 2/3 and 3/4. As a general rule the same topic will
+require less concrete material the later it appears in the school
+course. If the meanings of the numbers are taught in grade 2 instead of
+grade 1, there will be less need of blocks, counters, splints, beans,
+and the like. If 1-1/2 + 1/2 = 2 is taught early in grade 3, there will
+be more gain from the use of 1-1/2 inches and 1/2 inch on the foot rule
+than if the same relations were taught in connection with the general
+addition of like fractions late in grade 4. Sometimes the understanding
+can be had either by connecting the idea with the reality directly, or
+by connecting the two indirectly _via_ some other idea. The amount of
+concrete material to be used will depend on its relative advantage per
+unit of time spent. Thus it might be more economical to connect 5/12,
+7/12, and 11/12 with real meanings indirectly by calling up the
+resemblance to the 2/3, 3/4, 3/8, 5/8, 7/8, 2/5, 3/5, 4/5, and 5/6
+already studied, than by showing 5/12 of an apple, 7/12 of a yard, 11/12
+of a foot, and the like.
+
+In general the economical course is to test the understanding of the
+matter from time to time, using more concrete material if it is needed,
+but being careful to encourage pupils to proceed to the abstract ideas
+and general principles as fast as they can. It is wearisome and
+debauching to pupils' intellects for them to be put through elaborate
+concrete experiences to get a meaning which they could have got
+themselves by pure thought. We should also remember that the new idea,
+say of the meaning of decimal fractions, will be improved and clarified
+by using it (see page 183 f.), so that the attainment of a _perfect_
+conception of decimal fractions before doing anything with them is
+unnecessary and probably very wasteful.
+
+A few illustrations may make these principles more instructive.
+
+(_a_) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000,
+need more concrete aids than are commonly given. Guessing contests about
+the value in dollars of the school building and other buildings, the
+area of the schoolroom floor and other surfaces in square inches, the
+number of minutes in a week, and year, and the like, together with
+proper computations and measurements, are very useful to reenforce the
+concrete presentations and supply genuine problems in multiplication and
+subtraction with large numbers.
+
+(_b_) Numbers very much smaller than one, such as 1/32, 1/64, .04,
+and .002, also need some concrete aids. A diagram like that of
+Fig. 57 is useful.
+
+(_c_) _Majority_ and _plurality_ should be understood by every citizen.
+They can be understood without concrete aid, but an actual vote is well
+worth while for the gain in vividness and surety.
+
+    [Illustration: FIG. 57.--Concrete aid to understanding fractions
+    with large denominators. A = 1/1000 sq. ft.; B = 1/100 sq. ft.;
+    C = 1/50 sq. ft.; D = 1/10 sq. ft.]
+
+(_d_) Insurance against loss by fire can be taught by explanation and
+analogy alone, but it will be economical to have some actual insuring
+and payment of premiums and a genuine loss which is reimbursed.
+
+(_e_) Four play banks in the corners of the room, receiving deposits,
+cashing checks, and later discounting notes will give good educational
+value for the time spent.
+
+(_f_) Trade discount, on the contrary, hardly requires more concrete
+illustration than is found in the very problems to which it is applied.
+
+(_g_) The process of finding the number of square units in a rectangle
+by multiplying with the appropriate numbers representing length and
+width is probably rather hindered than helped by the ordinary objective
+presentation as an introduction. The usual form of objective
+introduction is as follows:--
+
+    [Illustration: FIG. 58.]
+
+    How long is this rectangle? How large is each square? How many
+    square inches are there in the top row? How many rows are
+    there? How many square inches are there in the whole rectangle?
+    Since there are three rows each containing 4 square inches, we
+    have 3 x 4 square inches = 12 square inches.
+
+    Draw a rectangle 7 inches long and 2 inches wide. If you divide
+    it into inch squares how many rows will there be? How many inch
+    squares will there be in each row? How many square inches are
+    there in the rectangle?
+
+    [Illustration: FIG. 59.]
+
+It is better actually to hide the individual square units as in Fig. 59.
+There are four reasons: (1) The concrete rows and columns rather
+distract attention from the essential thing to be learned. This is not
+that "_x_ rows one square wide, _y_ squares in a row will make _xy_
+squares in all," but that "by using proper units and the proper
+operation the area of any rectangle can be found from its length and
+width." (2) Children have little difficulty in learning to multiply
+rather than add, subtract, or divide when computing area. (3) The habit
+so formed holds good for areas like 1-2/3 by 4-1/2, with fractional
+dimensions, in which any effort to count up the areas of rows is very
+troublesome and confusing. (4) The notion that a square inch is an area
+1' by 1' rather than 1/2' by 2' or 1/3 in. by 3 in. or 1-1/2 in. by 2/3
+in. is likely to be formed too emphatically if much time is spent upon
+the sort of concrete presentation shown above. It is then better to use
+concrete counting of rows of small areas as a means of _verification
+after_ the procedure is learned, than as a means of deriving it.
+
+There has been, especially in Germany, much argument concerning what
+sort of number-pictures (that is, arrangement of dots, lines, or the
+like, as shown in Fig. 60) is best for use in connection with the number
+names in the early years of the teaching of arithmetic.
+
+Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and
+others have measured the accuracy of children in estimating the number
+of dots in arrangements of one or more of these different types.[21]
+Many writers interpret a difference in favor of estimating, say, the
+square arrangements of Born or Lay as meaning that such is the best
+arrangement to use in teaching. The inference is, however, unjustified.
+That certain number-pictures are easier to estimate numerically does not
+necessarily mean that they are more instructive in learning. One set may
+be easier to estimate just because they are more familiar, having been
+oftener experienced. Even if the favored set was so after equal
+experience with all sets, accuracy of estimation would be a sign of
+superiority for use in instruction only if all other things were equal
+(or in favor of the arrangement in question). Obviously the way to
+decide which of these is best to use in teaching is by using them in
+teaching and measuring all relevant results, not by merely recording
+which of them are most accurately estimated in certain time exposures.
+
+  [21] For an account in English of their main findings see
+       Howell ['14], pp. 149-251.
+
+It may be noted that the Born, Lay, and Freeman pictures have claims for
+special consideration on grounds of probable instructiveness. Since they
+are also superior in the tests in respect to accuracy of estimate,
+choice should probably be made from these three by any teacher who
+wishes to connect one set of number-pictures systematically with the
+number names, as by drills with the blackboard or with cards.
+
+    [Illustration: FIG. 60.--Various proposed arrangements of dots
+    for use in teaching the meanings of the numbers 1 to 10.]
+
+Such drills are probably useful if undertaken with zeal, and if kept as
+supplementary to more realistic objective work with play money, children
+marching, material to be distributed, garden-plot lengths to be
+measured, and the like, and if so administered that the pupils soon get
+the generalized abstract meaning of the numbers freed from dependence on
+an inner picture of any sort. This freedom is so important that it may
+make the use of many types of number-pictures advisable rather than the
+use of the one which in and of itself is best.
+
+As Meumann says: "Perceptual reckoning can be overdone. It had its chief
+significance for the surety and clearness of the first foundation of
+arithmetical instruction. If, however, it is continued after the first
+operations become familiar to the child, and extended to operations
+which develop from these elementary ones, it necessarily works as a
+retarding force and holds back the natural development of arithmetic.
+This moves on to work with abstract number and with mechanical
+association and reproduction." ['07, Vol. 2, p. 357.]
+
+Such drills are commonly overdone by those who make use of them, being
+given too often, and continued after their instructiveness has waned,
+and used instead of more significant, interesting, and varied work in
+counting and estimating and measuring real things. Consequently, there
+is now rather a prejudice against them in our better schools. They
+should probably be reinstated but to a moderate and judicious use.
+
+
+ORAL, MENTAL, AND WRITTEN ARITHMETIC
+
+There has been much dispute over the relative merits of oral and written
+work in arithmetic--a question which is much confused by the different
+meanings of 'oral' and 'written.' _Oral_ has meant (1) work where the
+situations are presented orally and the pupil's final responses are
+given orally, or (2) work where the situations are presented orally and
+the pupils' final responses are written or partly written and partly
+oral, or (3) work where the situations are presented in writing or print
+and the final responses are oral. _Written_ has meant (1) work where the
+situations are presented in writing or print and the final responses are
+made in writing, or (2) work where also many of the intermediate
+responses are written, or (3) work where the situations are presented
+orally but the final responses and a large percentage of the
+intermediate computational responses are written. There are other
+meanings than these.
+
+It is better to drop these very ambiguous terms and ask clearly what are
+the merits and demerits, in the case of any specified arithmetical work,
+of auditory and of visual presentation of the situations, and of saying
+and of writing each specified step in the response.
+
+The disputes over mental _versus_ written arithmetic are also confused
+by ambiguities in the use of 'mental.' Mental has been used to mean
+"done without pencil and paper" and also "done with few overt
+responses, either written or spoken, between the setting of the task and
+the announcement of the answer." In neither case is the word _mental_
+specially appropriate as a description of the total fact. As before, we
+should ask clearly, "What are the merits and demerits of making certain
+specified intermediate responses in inner speech or imaged sounds or
+visual images or imageless thought--that is, _without_ actual writing or
+overt speech?"
+
+It may be said at the outset that oral, written, and inner presentations
+of initial situations, oral, written, and inner announcements of final
+responses, and oral, written, and inner management of intermediate
+processes have varying degrees of merit according to the particular
+arithmetical exercise, pupil, and context. Devotion to oralness or
+mentalness as such is simply fanatical. Various combinations, such as
+the written presentation of the situation with inner management of the
+intermediate responses and oral announcement of the final response have
+their special merits for particular cases.
+
+These merits the reader can evaluate for himself for any given sort of
+work for a given class by considering: (1) The amount of practice
+received by the class per hour spent; (2) the ease of correction of the
+work; (3) the ease of understanding the tasks; (4) the prevention of
+cheating; (5) the cheerfulness and sociability of the work; (6) the
+freedom from eyestrain, and other less important desiderata.
+
+It should be noted that the stock schemes A, B, C, and D below are only
+a few of the many that are possible and that schemes E, F, G, and H have
+special merits.
+
+    PRESENTATION OF         MANAGEMENT OF           ANNOUNCEMENT OF
+    INITIAL SITUATION       INTERMEDIATE PROCESSES  FINAL RESPONSE
+
+  A. Printed or written      Written                 Written
+
+  B.    "          "         Inner                   Oral by one pupil,
+                                                     inner by the rest
+
+  C. Oral (by teacher)       Written                 Written
+
+  D.   "         "           Inner                   Oral by one pupil,
+                                                     inner by the rest
+
+  E. As in A or C            A mixture, the pupil    As in A or B or H
+                            writing what he needs
+
+  F. The real situation      As in E                 As in A or B or H
+     itself, in part at
+     least
+
+  G. Both read by the pupil  As in E                 As in A or B or H
+     and put orally by the
+     teacher
+
+  H. As in A or C or G       As in E                 Written by all
+                                                     pupils, announced
+                                                     orally by one pupil
+
+The common practice of either having no use made of pencil and paper or
+having all computations and even much verbal analysis written out
+elaborately for examination is unfavorable for learning. The demands
+which life itself will make of arithmetical knowledge and skill will
+range from tasks done with every percentage of written work from zero up
+to the case where every main result obtained by thought is recorded for
+later use by further thought. In school the best way is that which, for
+the pupils in question, has the best total effect upon quality of
+product, speed, and ease of production, reenforcement of training
+already given, and preparation for training to be given. There is
+nothing intellectually criminal about using a pencil as well as inner
+thought; on the other hand there is no magical value in writing out for
+the teacher's inspection figures that the pupil does not need in order
+to attain, preserve, verify, or correct his result.
+
+The common practice of having the final responses of all _easy_ tasks
+given orally has no sure justification. On the contrary, the great
+advantage of having all pupils really do the work should be secured in
+the easy work more than anywhere else. If the time cost of copying the
+figures is eliminated by the simple plan of having them printed, and if
+the supervision cost of examining the papers is eliminated by having the
+pupils correct each other's work in these easy tasks, written answers
+are often superior to oral except for the elements of sociability and
+'go' and freedom from eyestrain of the oral exercise. Such written work
+provides the gifted pupils with from two to ten times as much practice
+as they would get in an oral drill on the same material, supposing them
+to give inner answers to every exercise done by the class as a whole; it
+makes sure that the dull pupils who would rarely get an inner answer at
+the rate demanded by the oral exercise, do as much as they are able to
+do.
+
+Two arguments often made for the oral statement of problems by the
+teacher are that problems so put are better understood, especially in
+the grades up through the fifth, and that the problems are more likely
+to be genuine and related to the life the pupils know. When these
+statements are true, the first is a still better argument for having the
+pupils read the problems _aided by the teacher's oral statement of
+them_. For the difficulty is largely that the pupils cannot read well
+enough; and it is better to help them to surmount the difficulty rather
+than simply evade it. The second is not an argument for oralness
+_versus_ writtenness, but for good problems _versus_ bad; the teacher
+who makes up such good problems should, in fact, take special care to
+write them down for later use, which may be by voice or by the
+blackboard or by printed sheet, as is best.
+
+
+
+
+CHAPTER XIV
+
+THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE
+
+
+Dewey, and others following him, have emphasized the desirability of
+having pupils do their work as active seekers, conscious of problems
+whose solution satisfies some real need of their own natures. Other
+things being equal, it is unwise, they argue, for pupils to be led along
+blindfold as it were by the teacher and textbook, not knowing where they
+are going or why they are going there. They ought rather to have some
+living purpose, and be zealous for its attainment.
+
+This doctrine is in general sound, as we shall see, but it is often
+misused as a defense of practices which neglect the formation of
+fundamental habits, or as a recommendation to practices which are quite
+unworkable under ordinary classroom conditions. So it seems probable
+that its nature and limitations are not thoroughly known, even to its
+followers, and that a rather detailed treatment of it should be given
+here.
+
+
+ILLUSTRATIVE CASES
+
+Consider first some cases where time spent in making pupils understand
+the end to be attained before attacking the task by which it is
+attained, or care about attaining the end (well or ill understood) is
+well spent.
+
+It is well for a pupil who has learned (1) the meanings of the numbers
+one to ten, (2) how to count a collection of ten or less, and (3) how to
+measure in inches a magnitude of ten, nine, eight inches, etc., to be
+confronted with the problem of true adding without counting or
+measuring, as in 'hidden' addition and measurement by inference. For
+example, the teacher has three pencils counted and put under a book; has
+two more counted and put under the book; and asks, "How many pencils are
+there under the book?" Answers, when obtained, are verified or refuted
+by actual counting and measuring.
+
+The time here is well spent because the children can do the necessary
+thinking if the tasks are well chosen; because they are thereby
+prevented from beginning their study of addition by the bad habit of
+pseudo-adding by looking at the two groups of objects and counting their
+number instead of real adding, that is, thinking of the two numbers and
+inferring their sum; and further, because facing the problem of adding
+as a real problem is in the end more economical for learning arithmetic
+and for intellectual training in general than being enticed into adding
+by objective or other processes which conceal the difficulty while
+helping the pupil to master it.
+
+The manipulation of short multiplication may be introduced by
+confronting the pupils with such problems as, "How to tell how many
+Uneeda biscuit there are in four boxes, by opening only one box."
+Correct solutions by addition should be accepted. Correct solutions by
+multiplication, if any gifted children think of this way, should be
+accepted, even if the children cannot justify their procedure.
+(Inferring the manipulation from the place-values of numbers is beyond
+all save the most gifted and probably beyond them.) Correct solution by
+multiplication by some child who happens to have learned it elsewhere
+should be accepted. Let the main proof of the trustworthiness of the
+manipulation be by measurement and by addition. Proof by the stock
+arguments from the place-values of numbers may also be used. If no child
+hits on the manipulation in question, the problem of finding the length
+_without_ adding may be set. If they still fail, the problem may be made
+easier by being put as "4 times 22 gives the answer. Write down what you
+think 4 times 22 will be." Other reductions of the difficulty of the
+problem may be made, or the teacher may give the answer without very
+great harm being done. The important requirement is that the pupils
+should be aware of the problem and treat the manipulation as a solution
+of it, not as a form of educational ceremonial which they learn to
+satisfy the whims of parents and teachers. In the case of any particular
+class a situation that is more appealing to the pupils' practical
+interests than the situation used here can probably be devised.
+
+The time spent in this way is well spent (1) because all but the very
+dull pupils can solve the problem in some way, (2) because the
+significance of the manipulation as an economy over addition is worth
+bringing out, and (3) because there is no way of beginning training in
+short multiplication that is much better.
+
+In the same fashion multiplication by two-place numbers may be
+introduced by confronting pupils with the problem of the number of
+sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square
+inches in 35 square feet, or the number of days in 32 years, or whatever
+similar problem can be brought up so as to be felt as a problem.
+
+Suppose that it is the 35 square feet. Solutions by (5 x 144) +
+(30 x 144), however arranged, or by (10 x 144) + (10 x 144) +
+(10 x 144) + (5 x 144), or by 3500 + (35 x 40) + (35 x 4), or by
+7 x (5 x 144), however arranged, should all be listed for verification
+or rejection. The pupils need not be required to justify their
+procedures by a verbal statement. Answers like 432,720, or 720,432,
+or 1152, or 4220, or 3220 should be listed for verification or
+rejection. Verification may be by a mixture of short multiplication
+and objective work, or by a mixture of short multiplication and
+addition, or by addition abbreviated by taking ten 144s as 1440, or
+even (for very stupid pupils) by the authority of the teacher. Or the
+manipulation in cases like 53 x 9 or 84 x 7 may be verified by the
+reverse short multiplication. The deductive proof of the correctness
+of the manipulation may be given in whole or in part in connection
+with exercises like
+
+  10 x   2 =          30 x  14 =
+  10 x   3 =           3 x  44 =
+  10 x   4 =          30 x  44 =
+  10 x  14 =           3 x 144 =
+  10 x  44 =          20 x 144 =
+  10 x 144 =          40 x 144 =
+  20 x   2 =          30 x 144 =
+  20 x   3 =           5 x 144 =
+  30 x   3 =          35 = 30 + ....
+  30 x   4 =          30 x 144 added to 5 x 144 =
+   3 x  14 =
+
+Certain wrong answers may be shown to be wrong in many ways; _e.g._,
+432,720 is too big, for 35 times a thousand square inches is only
+35,000; 1152 is too small, for 35 times a hundred square inches would be
+3500, or more than 1152.
+
+The time spent in realizing the problem here is fairly well spent
+because (1) any successful original manipulation in this case
+represents an excellent exercise of thought, because (2) failures show
+that it is useless to juggle the figures at random, and because (3) the
+previous experience with short multiplication makes it possible for the
+pupils to realize the problem in a very few minutes. It may, however, be
+still better to give the pupils the right method just as soon as the
+problem is realized, without having them spend more time in trying to
+solve it. Thus:--
+
+1 square foot has 144 square inches. How many square inches are there in
+35 square feet (marked out in chalk on the floor as a piece 10 ft. x 3
+ft. plus a piece 5 ft. x 1 ft.)?
+
+1 yard = 36 inches. How many inches long is this wall (found by measure
+to be 13 yards)?
+
+Here is a quick way to find the answers:--
+
+   144
+    35
+  ----
+   720
+  432
+  ----
+  5040 sq. inches in 35 sq. ft.
+
+   36
+   13
+  ---
+  108
+  36
+  ---
+  468 inches in 13 yd.
+
+Consider now the following introduction to dividing by a decimal:--
+
+  Dividing by a Decimal
+
+  1. How many minutes will it take a motorcycle, to go 12.675 miles
+     at the rate of .75 mi. per minute?
+
+                                      16.9
+                                      ______
+                                  .75|12.675
+                                       7 5
+                                       ---
+                                       5 17
+                                       4 50
+                                       ----
+                                         675
+                                         675
+                                         ---
+
+  2. Check by multiplying 16.9 by .75.
+
+  3. How do you know that the quotient cannot be as little as 1.69?
+
+  4. How do you know that the quotient cannot be as large as 169?
+
+  5. Find the quotient for 3.75 / 1.5.
+
+  6. Check your result by multiplying the quotient by the divisor.
+
+  7. How do you know that the quotient cannot be .25 or 25?
+                                  ____
+  8. Look at this problem.     .25|7.5
+
+     How do you know that 3.0 is wrong for the quotient?
+
+     How do you know that 300 is wrong for the quotient?
+
+  State which quotient is right for each of these:--
+
+      .021 or .21 or 2.1 or 21 or 210
+         ______
+  9.  1.8|3.78
+
+
+      .021 or .21 or 21 or 210
+         ______
+  10. 1.8|37.8
+
+
+      .03 or .3 or 3 or 30 or 300
+          ______
+  11. 1.25|37.5
+
+
+      .03 or .3 or 3 or 30 or 300
+          ______
+  12. 12.5|37.5
+
+
+      .05 or .5 or 5 or 50 or 500
+          ______
+  13. 1.25|6.25
+
+
+      .05 or .5 or 5 or 50 or 500
+          ______
+  14. 12.5|6.25
+
+
+  15. Is this rule true? If it is true, learn it.
+
+        #In a correct result, the number of decimal places in
+        the divisor and quotient together equals the number
+        of decimal places in the dividend.#
+
+These and similar exercises excite the problem attitude in children _who
+have a general interest in getting right answers_. Such a series
+carefully arranged is a desirable introduction to a statement of the
+rule for placing the decimal point in division with decimals. For it
+attracts attention to the general principle (divisor x quotient should
+equal dividend), which is more important than the rule for convenient
+location of the decimal point, and it gives training in placing the
+point by inspection of the divisor, quotient, and dividend, which
+suffices for nineteen out of twenty cases that the pupil will ever
+encounter outside of school. He is likely to remember this method by
+inspection long after he has forgotten the fixed rule.
+
+It is well for the pupil to be introduced to many arithmetical facts by
+way of problems about their common uses. The clockface, the railroad
+distance table in hundredths of a mile, the cyclometer and speedometer,
+the recipe, and the like offer problems which enlist his interest and
+energy and also connect the resulting arithmetical learning with the
+activities where it is needed. There is no time cost, but a time-saving,
+for the learning as a means to the solution of the problems is quicker
+than the mere learning of the arithmetical facts by themselves alone. A
+few samples of such procedure are shown below:--
+
+  GRADE 3
+
+  To be Done at Home
+
+  Look at a watch. Has it any hands besides the hour hand and the
+  minute hand? Find out all that you can about how a watch tells
+  seconds, how long a second is, and how many seconds make a minute.
+
+
+  GRADE 5
+
+  Measuring Rainfall
+
+         =Rainfall per Week=
+    (=cu. in. per sq. in. of area=)
+    June   1-7      1.056
+           8-14     1.103
+          15-21     1.040
+          22-28      .960
+          29-July 5  .915
+    July   6-12      .782
+          13-19      .790
+          20-26      .670
+          27-Aug. 2  .503
+    Aug.   3-9       .512
+          10-16      .240
+          17-23      .215
+          24-30      .811
+
+  1. In which weeks was the rainfall 1 or more?
+
+  2. Which week of August had the largest rainfall for that month?
+
+  3. Which was the driest week of the summer? (Driest means with
+     the least rainfall.)
+
+  4. Which week was the next to the driest?
+
+  5. In which weeks was the rainfall between .800 and 1.000?
+
+  6. Look down the table and estimate whether the average rainfall for
+     one week was about .5, or about .6, or about .7, or about .8, or
+     about .9.
+
+  Dairy Records
+
+    =Record of Star Elsie=
+
+          Pounds of Milk     Butter-Fat per Pound of Milk
+      Jan.    1742                  .0461
+      Feb.    1690                  .0485
+      Mar.    1574                  .0504
+      Apr.    1226                  .0490
+      May     1202                  .0466
+      June    1251                  .0481
+
+  Read this record of the milk given by the cow Star Elsie. The first
+  column tells the number of pounds of milk given by Star Elsie each
+  month. The second column tells what fraction of a pound of butter-fat
+  each pound of milk contained.
+
+  1. Read the first line, saying, "In January this cow gave 1742 pounds
+      of milk. There were 461 ten thousandths of a pound of butter-fat
+      per pound of milk." Read the other lines in the same way.
+
+  2. How many pounds of butter-fat did the cow produce in Jan.?
+      3. In Feb.? 4. In Mar.? 5. In Apr.? 6. In May? 7. In June?
+
+
+  GRADE 5 OR LATER
+
+  Using Recipes to Make Larger or Smaller Quantities
+
+  I. State how much you would use of each material in the following
+  recipes: (_a_) To make double the quantity. (_b_) To make half the
+  quantity. (_c_) To make 1-1/2 times the quantity. You may use pencil
+  and paper when you cannot find the right amount mentally.
+
+    1. PEANUT PENUCHE
+
+      1 tablespoon butter
+      2 cups brown sugar
+      1/3 cup milk or cream
+      3/4 cup chopped peanuts
+      1/3 teaspoon salt
+
+    2. MOLASSES CANDY
+
+      1/2 cup butter
+      2 cups sugar
+      1 cup molasses
+      1-1/2 cups boiling water
+
+    3. RAISIN OPERA CARAMELS
+
+      2 cups light brown sugar
+      7/8 cup thin cream
+      1/2 cup raisins
+
+    4. WALNUT MOLASSES SQUARES
+
+      2 tablespoons butter
+      1 cup molasses
+      1/3 cup sugar
+      1/2 cup walnut meats
+
+    5. RECEPTION ROLLS
+
+      1 cup scalded milk
+      1-1/2 tablespoons sugar
+      1 teaspoon salt
+      1/4 cup lard
+      1 yeast cake
+      1/4 cup lukewarm water
+      White of 1 egg
+      3-1/2 cups flour
+
+    6. GRAHAM RAISED LOAF
+
+      2 cups milk
+      6 tablespoons molasses
+      1-1/2 teaspoons salt
+      1/3 yeast cake
+      1/4 cup lukewarm water
+      2 cups sifted Graham flour
+      1/2 cup Graham bran
+      3/4 cup flour (to knead)
+
+  II. How much would you use of each material in the following recipes:
+  (_a_) To make 2/3 as large a quantity? (_b_) To make 1-1/3 times as
+  much? (_c_) To make 2-1/2 times as much?
+
+    1. ENGLISH DUMPLINGS
+
+      1/2 pound beef suet
+      1-1/4 cups flour
+      3 teaspoons baking powder
+      1 teaspoon salt
+      1/2 teaspoon pepper
+      1 teaspoon minced parsley
+      1/2 cup cold water
+
+    2. WHITE MOUNTAIN ANGEL CAKE
+
+      1-1/2 cups egg whites
+      1-1/2 cups sugar
+      1 teaspoon cream of tartar
+      1 cup bread flour
+      1/4 teaspoon salt
+      1 teaspoon vanilla
+
+In many cases arithmetical facts and principles can be well taught in
+connection with some problem or project which is not arithmetical, but
+which has special potency to arouse an intellectual activity in the
+pupil which by some ingenuity can be directed to arithmetical learning.
+Playing store is the most fundamental case. Planning for a party, seeing
+who wins a game of bean bag, understanding the calendar for a month,
+selecting Christmas presents, planning a picnic, arranging a garden, the
+clock, the watch with second hand, and drawing very simple maps are
+situations suggesting problems which may bring a living purpose into
+arithmetical learning in grade 2. These are all available under ordinary
+conditions of class instruction. A sample of such problems for a higher
+grade (6) is shown below.
+
+  Estimating Areas
+
+  The children in the geography class had a contest in estimating
+  the areas of different surfaces. Each child wrote his estimates
+  for each of these maps, A, B, C, D, and E. (Only C and D are
+  shown here.) In the arithmetic class they learned how to find
+  the exact areas. Then they compared their estimates with the
+  exact areas to find who came nearest.
+
+    [Illustration]
+
+  Write your estimates for A, B, C, D, and E. Then study the
+  next 6 pages and learn how to find the exact areas.
+
+  (The next 6 pages comprise training in the mensuration of
+  parallelograms and triangles.)
+
+In some cases the affairs of individual pupils include problems which
+may be used to guide the individual in question to a zealous study of
+arithmetic as a means of achieving his purpose--of making a canoe,
+surveying an island, keeping the accounts of a Girls' Canning Club, or
+the like. It requires much time and very great skill to direct the work
+of thirty or more pupils each busy with a special type of his own, so as
+to make the work instructive for each, but in some cases the expense of
+time and skill is justified.
+
+
+GENERAL PRINCIPLES
+
+In general what should be meant when one says that it is desirable to
+have pupils in the problem-attitude when they are studying arithmetic is
+substantially as follows:--
+
+_First._--Information that comes as an answer to questions is better
+attended to, understood, and remembered than information that just
+comes.
+
+_Second._--Similarly, movements that come as a step toward achieving an
+end that the pupil has in view are better connected with their
+appropriate situations, and such connections are longer retained, than
+is the case with movements that just happen.
+
+_Third._--The more the pupil is set toward getting the question answered
+or getting the end achieved, the greater is the satisfyingness attached
+to the bonds of knowledge or skill which mean progress thereto.
+
+_Fourth._--It is bad policy to rely exclusively on the purely
+intellectualistic problems of "How can I do this?" "How can I get the
+right answer?" "What is the reason for this?" "Is there a better way to
+do that?" and the like. It is bad policy to supplement these
+intellectualistic problems by only the remote problems of "How can I be
+fitted to earn a higher wage?" "How can I make sure of graduating?" "How
+can I please my parents?" and the like. The purely intellectualistic
+problems have too weak an appeal for many pupils; the remote problems
+are weak so long as they are remote and, what is worse, may be deprived
+of the strength that they would have in due time if we attempt to use
+them too soon. It is the extreme of bad policy to neglect those personal
+and practical problems furnished by life outside the class in arithmetic
+the solution of which can really be furthered by arithmetic then and
+there. It is good policy to spend time in establishing certain mental
+sets--stimulating, or even creating, certain needs--setting up problems
+themselves--when the time so spent brings a sufficient improvement in
+the quality and quantity of the pupils' interest in arithmetical
+learning.
+
+_Fifth._--It would be still worse policy to rely exclusively on
+problems arising outside arithmetic. To learn arithmetic is itself a
+series of problems of intrinsic interest and worth to healthy-minded
+children. The need for ability to multiply with United States money or
+to add fractions or to compute percents may be as truly vital and
+engaging as the need for skill to make a party dress or for money to buy
+it or for time to play baseball. The intellectualistic needs and
+problems should be considered along with all others, and given whatever
+weight their educational value deserves.
+
+
+DIFFICULTY AND SUCCESS AS STIMULI
+
+There are certain misconceptions of the doctrine of the
+problem-attitude. The most noteworthy is that difficulty--temporary
+failure--an inadequacy of already existing bonds--is the essential and
+necessary stimulus to thinking and learning. Dewey himself does not, as
+I understand him, mean this, but he has been interpreted as meaning it
+by some of his followers.[22]
+
+  [22] In his _How We Think_.
+
+Difficulty--temporary failure, inadequacy of existing bonds--on the
+contrary does nothing whatsoever in and of itself; and what is done by
+the annoying lack of success which sometimes accompanies difficulty
+sometimes hinders thinking and learning.
+
+Mere difficulty, mere failure, mere inadequacy of existing bonds, does
+nothing. It is hard for me to add three eight-place numbers at a glance;
+I have failed to find as effective illustrations for pages 276 to 277 as
+I wished; my existing sensori-motor connections are inadequate to
+playing a golf course in 65. But these events and conditions have done
+nothing to stimulate me in respect to the behavior in question. In the
+first of the three there is no annoying lack and no dynamic influence at
+all; in the second there was to some degree an annoying lack--a slight
+irritation at not getting just what I wanted,--and this might have
+impelled me to further thinking (though it did not, and getting one
+tiptop illustration would as a rule stimulate me to hunt for others more
+than failing to get such). In the third case the lack of the 65 does not
+annoy me or have any noteworthy dynamic effect. The lack of 90 instead
+of 95-100 is annoying and is at times a stimulus to further learning,
+though not nearly so strong a stimulus as the attainment of the 90 would
+be! At other times this annoying lack is distinctly inhibitory--a
+stimulus to ceasing to learn. In the intellectual life the inhibitory
+effect seems far the commoner of the two. Not getting answers seems as a
+rule to make us stop trying to get them. The annoying lack of success
+with a theoretical problem most often makes us desert it for problems to
+whose solution the existing bonds promise to be more adequate.
+
+The real issue in all this concerns the relative strength, in the
+pupil's intellectual life, of the "negative reaction" of behavior in
+general. An animal whose life processes are interfered with so that an
+annoying state of affairs is set up, changes his behavior, making one
+after another responses as his instincts and learned tendencies
+prescribe, until the annoying state of affairs is terminated, or the
+animal dies, or suffers the annoyance as less than the alternatives
+which his responses have produced. When the annoying state of affairs is
+characterized by the failure of things as they are to minister to a
+craving--as in cases of hunger, loneliness, sex-pursuit, and the
+like,--we have stimulus to action by an annoying lack or need, with
+relief from action by the satisfaction of the need.
+
+Such is in some measure true of man's intellectual life. In recalling a
+forgotten name, in solving certain puzzles, or in simplifying an
+algebraic complex, there is an annoying lack of the name, solution, or
+factor, a trial of one after another response, until the annoyance is
+relieved by success or made less potent by fatigue or distraction. Even
+here the _difficulty_ does not do anything--but only the annoying
+interference with our intellectual peace by the problem. Further,
+although for the particular problem, the annoying lack stimulates, and
+the successful attainment stops thinking, the later and more important
+general effect on thinking is the reverse. Successful attainment stops
+our thinking _on that problem_ but makes us more predisposed later to
+thinking _in general_.
+
+Overt negative reaction, however, plays a relatively small part in man's
+intellectual life. Filling intellectual voids or relieving intellectual
+strains in this way is much less frequent than being stimulated
+positively by things seen, words read, and past connections acting under
+modified circumstances. The notion of thinking as coming to a lack,
+filling it, meeting an obstacle, dodging it, being held up by a
+difficulty and overcoming it, is so one-sided as to verge on phantasy.
+The overt lacks, strains, and difficulties come perhaps once in five
+hours of smooth straightforward use and adaptation of existing
+connections, and they might as truly be called hindrances to
+thought--barriers which past successes help the thinker to surmount.
+Problems themselves come more often as cherished issues which new facts
+reveal, and whose contemplation the thinker enjoys, than as strains or
+lacks or 'problems which I need to solve.' It is just as true that the
+thinker gets many of his problems as results from, or bonuses along
+with, his information, as that he gets much of his information as
+results of his efforts to solve problems.
+
+As between difficulty and success, success is in the long run more
+productive of thinking. Necessity is not the mother of invention.
+Knowledge of previous inventions is the mother; original ability is the
+father. The solutions of previous problems are more potent in producing
+both new problems and their solutions than is the mere awareness of
+problems and desire to have them solved.
+
+In the case of arithmetic, learning to cancel instead of getting the
+product of the dividends and the product of the divisors and dividing
+the former by the latter, is a clear case of very valuable learning,
+with ease emphasized rather than difficulty, with the adequacy of
+existing bonds (when slightly redirected) as the prime feature of the
+process rather than their inadequacy, and with no sense of failure or
+lack or conflict. It would be absurd to spend time in arousing in the
+pupil, before beginning cancellation, a sense of a difficulty--viz.,
+that the full multiplying and dividing takes longer than one would like.
+A pupil in grade 4 or 5 might well contemplate that difficulty for years
+to no advantage. He should at once begin to cancel and prove by checking
+that errorless cancellation always gives the right answer. To emphasize
+before teaching cancellation the inadequacy of the old full multiplying
+and dividing would, moreover, not only be uneconomical as a means to
+teaching cancellation; it would amount to casting needless slurs on
+valuable past acquisitions, and it would, scientifically, be false.
+For, until a pupil has learned to cancel, the old full multiplying is
+not inadequate; it is admirable in every respect. The issue of its
+inadequacy does not truly appear until the new method is found. It is
+the best way until the better way is mastered.
+
+In the same way it is unwise to spend time in making pupils aware of the
+annoying lacks to be supplied by the multiplication tables, the division
+tables, the casting out of nines, or the use of the product of the
+length and breadth of a rectangle as its area, the unit being changed to
+the square erected on the linear unit as base. The annoying lack will
+be unproductive, while the learning takes place readily as a
+modification of existing habits, and is sufficiently appreciated as soon
+as it does take place. The multiplication tables come when instead of
+merely counting by 7s from 0 up saying "7, 14, 21," etc., the pupil
+counts by 7s from 0 up saying "Two sevens make 14, three sevens make 21,
+four sevens make 28," etc. The division tables come as easy selections
+from the known multiplications; the casting out of nines comes as an
+easy device. The computation of the area of a rectangle is best
+facilitated, not by awareness of the lack of a process for doing it, but
+by awareness of the success of the process as verified objectively.
+
+In all these cases, too, the pupil would be misled if we aroused first a
+sense of the inadequacy of counting, adding, and objective division, an
+awareness of the difficulties which the multiplication and division
+tables and nines device and area theorem relieve. The displaced
+processes are admirable and no unnecessary fault should be found with
+them, and they are _not_ inadequate until the shorter ways have been
+learned.
+
+
+FALSE INFERENCES
+
+One false inference about the problem-attitude is that the pupil should
+always understand the aim or issue before beginning to form the bonds
+which give the method or process that provides the solution. On the
+contrary, he will often get the process more easily and value it more
+highly if he is taught what it is _for_ gradually while he is learning
+it. The system of decimal notation, for example, may better be taken
+first as a mere fact, just as we teach a child to talk without trying
+first to have him understand the value of verbal intercourse, or to keep
+clean without trying first to have him understand the bacteriological
+consequences of filth.
+
+A second inference--that the pupil should always be taught to care about
+an issue and crave a process for managing it before beginning to learn
+the process--is equally false. On the contrary, the best way to become
+interested in certain issues and the ways of handling them is to learn
+the process--even to learn it by sheer habituation--and then note what
+it does for us. Such is the case with ".1666-2/3 x = divide by 6,"
+".333-1/3 x = divide by 3," "multiply by .875 = divide the number by 8
+and subtract the quotient from the number."
+
+A third unwise tendency is to degrade the mere giving of information--to
+belittle the value of facts acquired in any other way than in the course
+of deliberate effort by the pupil to relieve a problem or conflict or
+difficulty. As a protest against merely verbal knowledge, and merely
+memoriter knowledge, and neglect of the active, questioning search for
+knowledge, this tendency to belittle mere facts has been healthy, but as
+a general doctrine it is itself equally one-sided. Mere facts not got by
+the pupil's thinking are often of enormous value. They may stimulate to
+active thinking just as truly as that may stimulate to the reception of
+facts. In arithmetic, for example, the names of the numbers, the use of
+the fractional form to signify that the upper number is divided by the
+lower number, the early use of the decimal point in U. S. money to
+distinguish dollars from cents, and the meanings of "each," "whole,"
+"part," "together," "in all," "sum," "difference," "product,"
+"quotient," and the like are self-justifying facts.
+
+A fourth false inference is that whatever teaching makes the pupil face
+a question and think out its answer is thereby justified. This is not
+necessarily so unless the question is a worthy one and the answer that
+is thought out an intrinsically valuable one and the process of thinking
+used one that is appropriate for that pupil for that question. Merely
+to think may be of little value. To rely much on formal discipline is
+just as pernicious here as elsewhere. The tendency to emphasize the
+methods of learning arithmetic at the expense of what is learned is
+likely to lead to abuses different in nature but as bad in effect as
+that to which the emphasis on disciplinary rather than content value has
+led in the study of languages and grammar, or in the old puzzle problems
+of arithmetic.
+
+The last false inference that I shall discuss here is the inference that
+most of the problems by which arithmetical learning is stimulated had
+better be external to arithmetic itself--problems about Noah's Ark or
+Easter Flowers or the Merry Go Round or A Trip down the Rhine.
+
+Outside interests should be kept in mind, as has been abundantly
+illustrated in this volume, but it is folly to neglect the power, even
+for very young or for very stupid children, of the problem "How can I
+get the right answer?" Children do have intellectual interests. They do
+like dominoes, checkers, anagrams, and riddles as truly as playing tag,
+picking flowers, and baking cake. With carefully graded work that is
+within their powers they like to learn to add, subtract, multiply, and
+divide with integers, fractions, and decimals, and to work out
+quantitative relations.
+
+In some measure, learning arithmetic is like learning to typewrite. The
+learner of the latter has little desire to present attractive-looking
+excuses for being late, or to save expense for paper. He has no desire
+to hoard copies of such and such literary gems. He may gain zeal from
+the fact that a school party is to be given and invitations are to be
+sent out, but the problem "To typewrite better" is after all his main
+problem. Learning arithmetic is in some measure a game whose moves are
+motivated by the general set of the mind toward victory--winning right
+answers. As a ball-player learns to throw the ball accurately to
+first-base, not primarily because of any particular problem concerning
+getting rid of the ball, or having the man at first-base possess it, or
+putting out an opponent against whom he has a grudge, but because that
+skill is required by the game as a whole, so the pupil, in some measure,
+learns the technique of arithmetic, not because of particular concrete
+problems whose solutions it furnishes, but because that technique is
+required by the game of arithmetic--a game that has intrinsic worth and
+many general recommendations.
+
+
+
+
+CHAPTER XV
+
+INDIVIDUAL DIFFERENCES
+
+
+The general facts concerning individual variations in abilities--that
+the variations are large, that they are continuous, and that for
+children of the same age they usually cluster around one typical or
+modal ability, becoming less and less frequent as we pass to very high
+or very low degrees of the ability--are all well illustrated by
+arithmetical abilities.
+
+
+NATURE AND AMOUNT
+
+The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In
+these diagrams each space along the baseline represents a certain score
+or degree of ability, and the height of the surface above it represents
+the number of individuals obtaining that score. Thus in Fig. 61, 63 out
+of 1000 soldiers had no correct answer, 36 out of 1000 had one correct
+answer, 49 had two, 55 had three, 67 had four, and so on, in a test with
+problems (stated in words).
+
+Figure 61 shows that these adults varied from no problems solved
+correctly to eighteen, around eight as a central tendency. Figure 62
+shows that children of the same year-age (they were also from the same
+neighborhood and in the same school) varied from under 40 to over 200
+figures correct. Figure 63 shows that even among children who have all
+reached the same school grade and so had rather similar educational
+opportunities in arithmetic, the variation is still very great. It
+requires a range from 15 to over 30 examples right to include even nine
+tenths of them.
+
+    [Illustration: FIG. 61.--The scores of 1000 soldiers in the
+    National Army born in English-speaking countries, in Test 2 of the
+    Army Alpha. The score is the number of correct answers obtained in
+    five minutes. Probably 10 to 15 percent of these men were unable to
+    read or able to read only very easy sentences at a very slow rate.
+    Data furnished by the Division of Psychology in the office of the
+    Surgeon General.]
+
+It should, however, be noted that if each individual had been scored by
+the average of his work on eight or ten different days instead of by his
+work in just one test, the variability would have been somewhat less
+than appears in Figs. 61, 62, and 63.
+
+    [Illustration: FIG. 62.--The scores of 100 11-year-old pupils
+    in a test of computation. Estimated from the data given by Burt
+    ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the
+    number of correct figures.]
+
+It is also the case that if each individual had been scored, not in
+problem-solving alone or division alone, but in an elaborate examination
+on the whole field of arithmetic, the variability would have been
+somewhat less than appears in Figs. 61, 62, and 63. On the other hand,
+if the officers and the soldiers rejected for feeblemindedness had been
+included in Fig. 61, if the 11-year-olds in special classes for the very
+dull had been included in Fig. 62, and if all children who had been to
+school six years had been included in Fig. 63, no matter what grade they
+had reached, the effect would have been to _increase_ the variability.
+
+    [Illustration: FIG. 63.--The scores of pupils in grade 6 in city
+    schools in the Woody Division Test A. The score is the number of
+    correct answers obtained in 20 minutes. From Woody ['16, p. 61].]
+
+In spite of the effort by school officers to collect in any one school
+grade those somewhat equal in ability or in achievement or in a mixture
+of the two, the population of the same grades in the same school system
+shows a very wide range in any arithmetical ability. This is partly
+because promotion is on a more general basis than arithmetical ability
+so that some very able arithmeticians are deliberately held back on
+account of other deficiencies, and some very incompetent arithmeticians
+are advanced on account of other excellencies. It is partly because of
+general inaccuracy in classifying and promoting pupils.
+
+In a composite score made up of the sum of the scores in Woody
+tests,--Add. A, Subt. A, Mult. A, and Div. A, and two tests in
+problem-solving (ten and six graded problems, with maximum attainable
+credits of 30 and 18), Kruse ['18] found facts from which I compute
+those of Table 13, and Figs. 64 to 66, for pupils all having the
+training of the same city system, one which sought to grade its pupils
+very carefully.
+
+    [Illustration: FIGS. 64, 65, and 66.--The scores of pupils in
+    grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a
+    composite of tests in computation and problem-solving. The time
+    was about 120 minutes. The maximum score attainable was 196.]
+
+The overlapping of grade upon grade should be noted. Of the pupils in
+grade 6 about 18 percent do better than the average pupil in grade 7,
+and about 7 percent do better than the average pupil in grade 8. Of the
+pupils in grade 8 about 33 percent do worse than the average pupil in
+grade 7 and about 12 percent do worse than the average pupil in grade 6.
+
+TABLE 13
+
+RELATIVE FREQUENCIES OF SCORES IN AN EXTENSIVE TEAM OF ARITHMETICAL
+TESTS.[23] IN PERCENTS
+
+  ==============================================
+     SCORE    |  GRADE 6  |  GRADE 7  |  GRADE 8
+  ------------+-----------+-----------+---------
+    70 to  79 |    1.3    |     .9    |     .4
+    80  "  89 |    5.5    |    2.3    |     .4
+    90  "  99 |   10.6    |    4.3    |    2.9
+   100  " 109 |   19.4    |    5.2    |    4.4
+   110  " 119 |   19.8    |   18.5    |    5.8
+   120  " 129 |   23.5    |   16.2    |   16.8
+   130  " 139 |   12.6    |   17.5    |   16.8
+   140  " 149 |    4.6    |   13.9    |   22.9
+   150  " 159 |    1.7    |   13.6    |   17.1
+   160  " 169 |    1.2    |    4.8    |    9.4
+   170  " 179 |           |    2.5    |    3.3
+  ==============================================
+
+  [23] Compiled from data on p. 89 of Kruse ['18].
+
+
+DIFFERENCES WITHIN ONE CLASS
+
+The variation within a single class for which a single teacher has to
+provide is great. Even when teaching is departmental and promotion is by
+subjects, and when also the school is a large one and classification
+within a grade is by ability--there may be a wide range for any given
+special component ability. Under ordinary circumstances the range is so
+great as to be one of the chief limiting conditions for the teaching of
+arithmetic. Many methods appropriate to the top quarter of the class
+will be almost useless for the bottom quarter, and _vice versa_.
+
+    [Illustration: FIGS. 67 and 68.--The scores of ten 6 B classes in
+    a 12-minute test in computation with integers (the Courtis Test 7).
+    The score is the number of units done. Certain long tasks are
+    counted as two units.]
+
+Figures 67 and 68 show the scores of ten classes taken at random from
+ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of
+computation done in 12 minutes. Observe the very wide variation present
+in the case of every class. The variation within a class would be
+somewhat reduced if each pupil were measured by his average in eight or
+ten such tests given on different days. If a rather generous allowance
+is made for this we still have a variation in speed as great as that
+shown in Fig. 69, as the fact to be expected for a class of thirty-two 6
+B pupils.
+
+    [Illustration: FIG. 69.--A conservative estimate of the amount of
+    variation to be expected within a single class of 32 pupils in
+    grade 6, in the number of units done in Courtis Test 7 when all
+    chance variations are eliminated.]
+
+The variations within a class in respect to what processes are
+understood so as to be done with only occasional errors may be
+illustrated further as follows:--A teacher in grade 4 at or near the
+middle of the year in a city doing the customary work in arithmetic will
+probably find some pupil in her class who cannot do column addition even
+without carrying, or the easiest written subtraction
+
+  (8   9      78)
+  (5   3  or  37)
+  (-   -      --),
+
+who does not know his multiplication tables or how to derive them, or
+understand the meanings of + - x and /, or have any useful ideas
+whatever about division.
+
+There will probably be some child in the class who can do such work as
+that shown below, and with very few errors.
+
+  Add       3/8 + 5/8 + 7/8 + 1/8       2-1/2       1/6 + 3/8
+                                        6-3/8
+                                        3-3/4
+                                        -----
+
+  Subtract     10.00                4 yd.    1 ft.   6 in.
+                3.49                2 yd.    2 ft.   3 in.
+               -----                ----------------------
+
+  Multiply    1-1/4 x 8           16              145
+                                   2-5/8          206
+                                  ------          ---
+                   _______               _____
+  Divide          2)13.50              25)9750
+
+The invention of means of teaching thirty so different children at once
+with the maximum help and minimum hindrance from their different
+capacities and acquisitions is one of the great opportunities for
+applied science.
+
+Courtis, emphasizing the social demand for a certain moderate
+arithmetical attainment in the case of nearly all elementary school
+children of, say, grade 6, has urged that definite special means be
+taken to bring the deficient children up to certain standards, without
+causing undesirable 'overlearning' by the more gifted children. Certain
+experimental work to this end has been carried out by him and others,
+but probably much more must be done before an authoritative program for
+securing certain minimum standards for all or nearly all pupils can be
+arranged.
+
+
+THE CAUSES OF INDIVIDUAL DIFFERENCES
+
+The differences found among children of the same grade in the same city
+are due in large measure to inborn differences in their original
+natures. If, by a miracle, the children studied by Courtis, or by Woody,
+or by Kruse had all received exactly the same nurture from birth to
+date, they would still have varied greatly in arithmetical ability,
+perhaps almost as much as they now do vary.
+
+The evidence for this is the general evidence that variation in original
+nature is responsible for much of the eventual variation found in
+intellectual and moral traits, plus certain special evidence in the case
+of arithmetical abilities themselves.
+
+Thorndike found ['05] that in tests with addition and multiplication
+twins were very much more alike than siblings[24] two or three years
+apart in age, though the resemblance in home and school training in
+arithmetic should be nearly as great for the latter as for the former.
+Also the young twins (9-11) showed as close a resemblance in addition
+and multiplication as the older twins (12-15), although the similarities
+of training in arithmetic have had twice as long to operate in the
+latter case.
+
+  [24] Siblings is used for children of the same parents.
+
+If the differences found, say among children in grade 6 in addition,
+were due to differences in the quantity and quality of training in
+addition which they have had, then by giving each of them 200 minutes of
+additional identical training the differences should be reduced. For the
+200 minutes of identical training is a step toward equalizing training.
+It has been found in many investigations of the matter that when we make
+training in arithmetic more nearly equal for any group the variation
+within the group is not reduced.
+
+On the contrary, equalizing training seems rather to increase
+differences. The superior individual seems to have attained his
+superiority by his own superiority of nature rather than by superior
+past training, for, during a period of equal training for all, he
+increases his lead. For example, compare the gains of different
+individuals due to about 300 minutes of practice in mental
+multiplication of a three-place number by a three-place number shown
+in Table 14 below, from data obtained by the author ['08].[25]
+
+  [25] Similar results have been obtained in the case of arithmetical
+       and other abilities by Thorndike ['08, '10, '15, '16], Whitley
+       ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and
+       Thorndike ['13], Hahn and Thorndike ['14], and on a very
+       large scale by Race in a study as yet unpublished.
+
+TABLE 14
+
+THE EFFECT OF EQUAL AMOUNTS OF PRACTICE UPON INDIVIDUAL DIFFERENCE IN
+THE MULTIPLICATION OF THREE-PLACE NUMBERS
+
+  ====================================================================
+                                     |     AMOUNT     | PERCENTAGE OF
+                                     |                |CORRECT FIGURES
+                                     |----------------+---------------
+                                     | Initial |      | Initial |
+                                     |  Score  | Gain |  Score  | Gain
+  -----------------------------------+---------+------+---------+-----
+  Initially highest five individuals |    85   |  61  |    70   |  18
+            next    five     "       |    56   |  51  |    68   |  10
+            next    six      "       |    46   |  22  |    74   |   8
+            next    six      "       |    38   |   8  |    58   |  12
+            next    six      "       |    29   |  24  |    56   |  14
+  ====================================================================
+
+
+THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES
+
+Achievement in arithmetic depends upon a number of different abilities.
+For example, accuracy in copying numbers depends upon eyesight, ability
+to perceive visual details, and short-term memory for these. Long
+column addition depends chiefly upon great strength of the addition
+combinations especially in higher decades, 'carrying,' and keeping one's
+place in the column. The solution of problems framed in words requires
+understanding of language, the analysis of the situation described into
+its elements, the selection of the right elements for use at each step
+and their use in the right relations.
+
+Since the abilities which together constitute arithmetical ability are
+thus specialized, the individual who is the best of a thousand of his
+age or grade in respect to, say, adding integers, may occupy different
+stations, perhaps from 1st to 600th, in multiplying with integers,
+placing the decimal point in division with decimals, solving novel
+problems, copying figures, etc., etc. Such specialization is in part due
+to his having had, relatively to the others in the thousand, more or
+better training in certain of these abilities than in others, and to
+various circumstances of life which have caused him to have, relatively
+to the others in the thousand, greater interest in certain of these
+achievements than in others. The specialization is not wholly due
+thereto, however. Certain inborn characteristics of an individual
+predispose him to different degrees of superiority or inferiority to
+other men in different features of arithmetic.
+
+We measure the extent to which ability of one sort goes with or fails to
+go with ability of some other sort by the coefficient of correlation
+between the two. If every individual keeps the same rank in the second
+ability--if the individual who is the best of the thousand in one is the
+best of the group in the other, and so on down the list--the correlation
+is 1.00. In proportion as the ranks of individuals vary in the two
+abilities the coefficient drops from 1.00, a coefficient of 0 meaning
+that the best individual in ability A is no more likely to be in first
+place in ability B than to be in any other rank.
+
+The meanings of coefficients of correlation of .90, .70, .50, and 0 are
+shown by Tables 15, 16, 17 and 18.[26]
+
+  [26] Unless he has a thorough understanding of the underlying
+       theory, the student should be very cautious in making
+       inferences from coefficients of correlation.
+
+TABLE 15
+
+ DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .90
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |     |     |     |     |  .1 |  .4 | 1.8 | 6.6 |22.4 |68.7
+  2d tenth  |     |     |  .1 |  .4 | 1.4 | 4.7 |11.5 |23.5 |36.0 |22.4
+  3d tenth  |     |  .1 |  .5 | 2.1 | 5.8 |12.8 |21.1 |27.4 |23.5 | 6.6
+  4th tenth |     |  .4 | 2.1 | 6.4 |12.8 |20.1 |23.8 |21.2 |11.5 | 1.8
+  5th tenth |  .1 | 1.4 | 5.8 |12.8 |19.3 |22.6 |20.1 |12.8 | 4.7 |  .4
+  6th tenth |  .4 | 4.7 |12.8 |20.1 |22.6 |19.3 |12.8 | 5.8 | 1.4 |  .1
+  7th tenth | 1.8 |11.5 |21.2 |23.8 |20.1 |12.8 | 6.4 | 2.1 |  .4 |
+  8th tenth | 6.6 |23.5 |27.4 |21.1 |12.8 | 5.8 | 2.1 |  .5 |  .1 |
+  9th tenth |22.4 |36.0 |23.5 |11.5 | 4.7 | 1.4 |  .4 |  .1 |     |
+  10th tenth|68.7 |22.4 | 6.6 | 1.8 |  .4 |  .1 |     |     |     |
+  ======================================================================
+
+TABLE 16
+
+DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .70
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |     |  .2 |  .7 | 1.5 | 2.8 | 4.8 | 8.0 |13.0 |22.3 |46.7
+  2d tenth  |  .2 | 1.2 | 2.6 | 4.5 | 7.0 | 9.8 |13.4 |17.3 |21.7 |22.3
+  3d tenth  |  .7 | 2.6 | 5.0 | 7.3 |10.0 |12.5 |14.9 |16.7 |17.3 |13.0
+  4th tenth | 1.5 | 4.5 | 7.3 | 9.8 |12.0 |13.7 |14.8 |14.9 |13.4 | 8.0
+  5th tenth | 2.8 | 7.0 |10.0 |12.0 |13.4 |14.0 |13.7 |12.5 | 9.8 | 4.8
+  6th tenth | 4.8 | 9.8 |12.5 |13.7 |14.0 |13.4 |12.0 |10.0 | 7.0 | 2.8
+  7th tenth | 8.0 |13.4 |14.9 |14.8 |13.7 |12.0 | 9.8 | 7.3 | 4.5 | 1.5
+  8th tenth |13.0 |17.3 |16.7 |14.9 |12.5 |10.0 | 7.3 | 5.0 | 2.6 |  .7
+  9th tenth |22.3 |21.7 |17.3 |13.4 | 9.8 | 7.0 | 4.5 | 2.6 | 1.2 |  .2
+  10th tenth|46.7 |22.3 |13.0 | 8.0 | 4.8 | 2.8 | 1.5 |  .7 |  .2 |
+  ======================================================================
+
+TABLE 17
+
+DISTRIBUTION OF ARRAYS OF SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .50
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |  .8 | 2.0 | 3.2 | 4.6 | 6.2 | 8.1 |10.5 |13.9 |18.0 |31.8
+  2d tenth  | 2.0 | 4.1 | 5.7 | 7.3 | 8.8 |10.5 |12.2 |14.1 |16.4 |18.9
+  3d tenth  | 3.2 | 5.7 | 7.4 | 8.9 |10.0 |11.2 |12.3 |13.3 |14.1 |13.9
+  4th tenth | 4.6 | 7.3 | 8.8 | 9.9 |10.8 |11.6 |12.0 |12.3 |12.2 |10.5
+  5th tenth | 6.2 | 8.8 |10.0 |10.8 |11.3 |11.5 |11.6 |11.2 |10.5 | 8.1
+  6th tenth | 8.1 |10.5 |11.2 |11.6 |11.5 |11.3 |10.8 |10.0 | 8.8 | 6.2
+  7th tenth |10.5 |12.2 |12.3 |12.0 |11.6 |10.8 | 9.9 | 8.8 | 7.5 | 4.6
+  8th tenth |13.9 |14.1 |13.3 |12.3 |11.2 |10.0 | 8.8 | 7.4 | 5.7 | 3.2
+  9th tenth |18.9 |16.4 |14.1 |12.2 |10.5 | 8.8 | 7.3 | 5.7 | 4.1 | 2.0
+  10th tenth|31.8 |18.9 |13.9 |10.5 | 8.1 | 6.2 | 4.6 | 3.2 | 2.0 |  .8
+  ======================================================================
+
+TABLE 18
+
+DISTRIBUTION OF ARRAYS, IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .0
+
+  ======================================================================
+            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
+  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
+  1st tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  2d tenth  |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  3d tenth  |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  4th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  5th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  6th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  7th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  8th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  9th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  10th tenth|10   |10   |10   |10   |10   |10   |10   |10   |10   |10
+  ======================================================================
+
+The significance of any coefficient of correlation depends upon the
+group of individuals for which it is determined. A correlation of .40
+between computation and problem-solving in eighth-grade pupils of 14
+years would mean a much closer real relation than a correlation of .40
+in all 14-year-olds, and a very, very much closer relation than a
+correlation of .40 for all children 8 to 15.
+
+Unless the individuals concerned are very elaborately tested on several
+days, the correlations obtained are "attenuated" toward 0 by the
+"accidental" errors in the original measurements. This effect was not
+known until 1904; consequently the correlations in the earlier studies
+of arithmetic are all too low.
+
+In general, the correlation between ability in any one important feature
+of computation and ability in any other important feature of computation
+is high. If we make enough tests to measure each individual exactly
+in:--
+
+  (_A_) Subtraction with integers and decimals,
+  (_B_) Multiplication with integers and decimals,
+  (_C_) Division with integers and decimals,
+  (_D_) Multiplication and division with common fractions, and
+  (_E_) Computing with percents,
+
+we shall probably find the intercorrelations for a thousand 14-year-olds
+to be near .90. Addition of integers (_F_) will, however, correlate less
+closely with any of the above, being apparently dependent on simpler and
+more isolated abilities.
+
+The correlation between problem-solving (_G_) and computation will be
+very much less, probably not over .60.
+
+It should be noted that even when the correlation is as high as .90,
+there will be some individuals very high in one ability and very low in
+the other. Such disparities are to some extent, as Courtis ['13, pp.
+67-75] and Cobb ['17] have argued, due to inborn characteristics of the
+individual in question which predispose him to very special sorts of
+strength and weakness. They are often due, however, to defects in his
+learning whereby he has acquired more ability than he needs in one line
+of work or has failed to acquire some needed ability which was well
+within his capacity.
+
+In general, all correlations between an individual's divergence from the
+common type or average of his age for one arithmetical function, and his
+divergences from the average for any other arithmetical function, are
+positive. The correlation due to original capacity more than
+counterbalances the effects that robbing Peter to pay Paul may have.
+
+Speed and accuracy are thus positively correlated. The individuals who
+do the most work in ten minutes will be above the average in a test of
+accuracy. The common notion that speed is opposed to accuracy is correct
+when it means that the same person will tend to make more errors if he
+works at too rapid a rate; but it is entirely wrong when it means that
+the kind of person who works more rapidly than the average person is
+likely to be less accurate than the average person.
+
+Interest in arithmetic and ability at arithmetic are probably correlated
+positively in the sense that the pupil who has more interest than other
+pupils of his age tends in the long run to have more ability than they.
+They are certainly correlated in the sense that the pupil who 'likes'
+arithmetic better than geography or history tends to have relatively
+more ability in arithmetic, or, in other words, that the pupil who is
+more gifted at arithmetic than at drawing or English tends also to like
+it better than he likes these. These correlations are high.
+
+It is correct then to think of mathematical ability as, in a sense, a
+unitary ability of which any one individual may have much or little,
+most individuals possessing a moderate amount of it. This is
+consistent, however, with the occasional appearance of individuals
+possessed of very great talents for this or that particular feature of
+mathematical ability and equally notable deficiencies in other features.
+
+Finally it may be noted that ability in arithmetic, though occasionally
+found in men otherwise very stupid, is usually associated with superior
+intelligence in dealing with ideas and symbols of all sorts, and is one
+of the best early indications thereof.
+
+
+
+
+BIBLIOGRAPHY OF REFERENCES MADE IN THE TEXT
+
+
+  Ames, A. F., and McLellan, J. F.; '00
+    Public School Arithmetic.
+
+  Ballou, F. W.; '16
+    Determining the Achievements of Pupils in the Addition of Fractions.
+    School Document No. 3, 1916, Boston Public Schools.
+
+  Brandell, G.; '13
+    Skolbarns intressen.  Translated ['15] by W. Stern as, Das Interesse
+    der Schulkinder an den Unterrichtsfaechern.
+
+  Brandford, B.; '08
+    A Study of Mathematical Education.
+
+  Brown, J. C.; '11, '12
+    An Investigation on the Value of Drill Work in the Fundamental
+    Operations in Arithmetic. Journal of Educational Psychology, vol. 2,
+    pp. 81-88, vol. 3, pp. 485-492 and 561-570.
+
+  Brown, J. C. and Coffman, L. D.; '14
+    How to Teach Arithmetic.
+
+  Burgerstein, L.; '91
+    Die Arbeitscurve einer Schulstunde. Zeitschrift fuer
+    Schulgesundheitspflege, vol. 4, pp. 543-562 and 607-627.
+
+  Burnett, C. J.; '06
+    The Estimation of Number. Harvard Psychological Studies, vol. 2,
+    pp. 349-404.
+
+  Burt, C.; '17
+    The Distribution and Relations of Educational Abilities. Report of
+    The London County Council, No. 1868.
+
+
+  Chapman, J. C.; '14
+    Individual Differences in Ability and Improvement and Their
+    Correlations. Teachers College Contributions to Education, No. 63.
+
+  Chapman, J. C.; '17
+    The Scientific Measurement of Classroom Products. (With G. P. Rush.)
+
+  Cobb, M. V.; '17
+    A Preliminary Study of the Inheritance of Arithmetical Abilities.
+    Jour. of Educational Psychology, vol. 8, pp. 1-20. Jan., 1917.
+
+  Coffman, L. D., and Brown, J. C.; '14
+    How to Teach Arithmetic.
+
+  Coffman, L. D., and Jessup, W. A.; '16
+    The Supervision of Arithmetic.
+
+  Courtis, S. A.; '09, '10, '11
+    Measurement of Growth and Efficiency in Arithmetic. Elementary
+    School Teacher, vol. 10, pp. 58-74 and 177-199, vol. 11,
+    pp. 171-185, 360-370, and 528-539.
+
+  Courtis, S. A.; '11, '12
+    Report on Educational Aspects of the Public School System of the
+    City of New York. Part II, Subdivision 1, Section D. Report on the
+    Courtis Tests in Arithmetic.
+
+  Courtis, S. A.; '13
+    Courtis Standard Tests. Second Annual Accounting.
+
+  Courtis, S. A.; '14
+    Manual of Instructions for Giving and Scoring the Courtis Standard
+    Tests in the Three R's. Department of Comparative Research, 82
+    Elliot St., Detroit, Mich., 1914.
+
+  Decroly, M., and Degand, J.; '12
+    L'evolution des notions de quantites continues et discontinues chez
+    l'enfant. Archives de psychologie, vol. 12, pp. 81-121.
+
+  Degand, J.  _See_ Decroly.
+
+  De Voss, J. C. _See_ Monroe, De Voss, and Kelly.
+
+  Dewey, J.; '10
+    How We Think.
+
+  Dewey, J., and McLellan, J. A.; '95
+    Psychology of Number and Its Applications to Methods of Teaching
+    Arithmetic.
+
+  Donovan, M. E., and Thorndike, E. L.; '13
+    Improvement in a Practice Experiment under School Conditions.
+    American Journal of Psychology, vol. 24, pp. 426-428.
+
+  Elliott, C. H.; '14
+    Variation in the Achievements of Pupils. Teachers College, Columbia
+    University, Contributions to Education, No. 72.
+
+  Flynn, F. J.; '12
+    Mathematical Games--Adaptations from Games Old and New. Teachers
+    College Record, vol. 13, pp. 399-412.
+
+  Freeman, F. N.; '10
+    Untersuchungen ueber den Aufmerksamkeitsumfang und die
+    Zahlauffassung. Paedagogische-Psychologische Arbeiten, I, 88-168.
+
+  Friedrich, J.; '97
+    Untersuchungen ueber die Einfluesse der Arbeitsdauer und die
+    Arbeitspausen auf die geistige Leistungsfaehigkeit der Schulkinder.
+    Zeitschrift fuer Psychologie, vol. 13, pp. 1-53.
+
+  Gilbert, J. A.; '94
+    Researches on the Mental and Physical Development of School
+    Children. Studies from the Yale Psychological Laboratory, vol. 2,
+    pp. 40-100.
+
+  Greenleaf, B.; '73
+    Practical Arithmetic.
+
+  Hahn, H. H., and Thorndike, E. L.; '14
+    Some Results of Practice in Addition under School Conditions.
+    Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.
+
+  Hall, G. S.; '83
+    The Contents of Children's Minds on Entering School. Princeton
+    Review, vol. II, pp. 249-272, May, 1883. Reprinted in Aspects of
+    Child Life and Education, 1907.
+
+  Hartmann, B.; '90
+    Die Analyze des Kindlichen Gedanken-Kreises als die Naturgemaessedes
+    Ersten Schulunterrichts, 1890.
+
+  Heck, W. H.; '13
+    A Study of Mental Fatigue.
+
+  Heck, W. H.; '13
+    A Second Study in Mental Fatigue in the Daily School Program.
+    Psychological Clinic, vol. 7, pp. 29-34.
+
+  Hoffmann, P.; '11
+    Das Interesse der Schueler an den Unterrichtsfaechern. Zeitschrift
+    fuer paedagogische Psychologie, XII, 458-470.
+
+  Hoke, K. J., and Wilson, G. M.; '20
+    How to Measure.
+
+  Holmes, M. E.; '95
+    The Fatigue of a School Hour. Pedagogical Seminary, vol. 3,
+    pp. 213-234.
+
+  Howell, H. B.; '14
+    A Foundation Study in the Pedagogy of Arithmetic.
+
+  Hunt, C. W.; '12
+    Play and Recreation in Arithmetic. Teachers College Record, vol. 13,
+    pp. 388-398.
+
+  Jessup, W. A., and Coffman, L. D.; '16
+    The Supervision of Arithmetic.
+
+  Kelly, F. J. _See_ Monroe, De Voss and Kelly.
+
+  King, A. C.; '07
+    The Daily Program in Elementary Schools. MSS.
+
+  Kirby, T. J.; '13
+    Practice in the Case of School Children. Teachers College
+    Contributions to Education, No. 58.
+
+  Klapper, P.; '16
+    The Teaching of Arithmetic.
+
+  Kruse, P. J.; '18
+    The Overlapping of Attainments in Certain Sixth, Seventh, and Eighth
+    Grades. Teachers College, Columbia University, Contributions to
+    Education, No. 92.
+
+  Laser, H.; '94
+    Ueber geistige Ermuedung beim Schulunterricht. Zeitschrift fuer
+    Schulgesundheitspflege, vol. 7, pp. 2-22.
+
+  Lay, W. A.; '98
+    Fuehrer durch den ersten Rechenunterricht.
+
+  Lay, W. A.; '07
+    Fuehrer durch den Rechenunterricht der Unterstufe.
+
+  Lewis, E. O.; '13
+    Popular and Unpopular School-Subjects. The Journal of Experimental
+    Pedagogy, vol. 2, pp. 89-98.
+
+  Lobsien, M.; '03
+    Kinderideale.  Zeitschrift fuer paedagogische Psychologie, V, 323-344
+    and 457-494.
+
+  Lobsien, M.; '09
+    Beliebtheit und Unbeliebtheit der Unterrichtsfaecher. Paedagogisches
+    Magazin, Heft 361.
+
+  McCall, W. A.; '21
+    How to Measure in Education.
+
+  McDougle, E. C.; '14
+    A Contribution to the Pedagogy of Arithmetic. Pedagogical Seminary,
+    vol. 21, pp. 161-218.
+
+  McKnight, J.A.; '07
+    Differentiation of the Curriculum in the Upper Grammar Grades.
+    MSS. in the library of Teachers College, Columbia University.
+
+  McLellan, J.A., and Dewey, J.; '95
+    Psychology of Number and Its Applications to Methods of Teaching.
+
+  McLellan, J.A., and Ames, A.F.; '00
+    Public School Arithmetic.
+
+  Messenger, J.F.; '03
+    The Perception of Number. Psychological Review, Monograph Supplement
+    No. 22.
+
+  Meumann, E.; '07
+    Vorlesungen zur Einfuehrung in die experimentelle Paedagogik.
+
+  Mitchell, H.E.; '20
+    Unpublished studies of the uses of arithmetic in factories, shops,
+    farms, and the like.
+
+  Monroe, W.S., De Voss, J.C., and Kelly, F.J.; '17
+    Educational Tests and Measurements.
+
+  Nanu, H.A.; '04
+    Zur Psychologie der Zahl Auffassung.
+
+  National Intelligence Tests; '20
+    Scale A, Form 1, Edition 1.
+
+  Phillips, D.E.; '97
+    Number and Its Application Psychologically Considered. Pedagogical
+    Seminary, vol. 5, pp. 221-281.
+
+  Pommer, O.; '14
+    Die Erforschung der Beliebtheit der Unterrichtsfaecher.
+    Ihre psychologischen Grundlagen und ihre paedagog. Bedeutung. VII.
+    Jahresber. des k.k. Ssaatsgymn. im XVIII Bez. v. Wien.
+
+  Rice, J.M.; '02
+    Test in Arithmetic. Forum, vol. 34, pp. 281-297.
+
+  Rice, J.M.; '03
+    Causes of Success and Failure in Arithmetic.
+    Forum, vol. 34, pp. 437-452.
+
+  Rush, G.P.; '17
+    The Scientific Measurement of Classroom Products. (With J. C.
+    Chapman.)
+
+  Seekel, E.; '14
+    Ueber die Beziehung zwischen der Beliebtheit und der Schwierigkeit
+    der Schulfaecher. Ergebnisse einer Erhebung. Zeitschrift fuer
+    Angewandte Psychologie 9. S. 268-277.
+
+  Selkin, F. B.; '12
+    Number Games Bordering on Arithmetic and Algebra.
+    Teachers College Record, vol. 13, pp. 452-493.
+
+  Smith, D. E.; '01
+    The Teaching of Elementary Mathematics.
+
+  Smith, D. E.; '11
+    The Teaching of Arithmetic.
+
+  Speer, W. W.; '97
+    Arithmetic: Elementary for Pupils.
+
+  Starch, D.; '11
+    Transfer of Training in Arithmetical Operations.
+    Journal of Educational Psychology, vol. 2, pp. 306-310.
+
+  Starch, D.; '16
+    Educational Measurements.
+
+  Stern, W.; '05
+    Ueber Beliebtheit und Unbeliebtheit der Schulfaecher.
+    Zeitschrift fuer paedagogische Psychologie, VII, 267-296.
+
+  Stern, C., and Stern, W.; '13
+    Beliebtheit und Schwierigkeit der Schulfaecher.
+    (Freie Schulgemeinde Wickersdorf.)
+    Auf Grund der von Herrn Luserke beschafften Materialien bearbeitet.
+    In: "Die Ausstellung zur vergleichenden Jungendkunde
+    der Geschlechter in Breslau."
+    Arbeit 7 des Bundes fuer Schulreform. S. 24-26.
+
+  Stern, W.; '14
+    Zur vergleichenden Jugendkunde der Geschlechter.
+    Vortrag. III. Deutsch. Kongr. f. Jugendkunde
+    usw. Arbeiten 8 des Bundes fuer Schulreform. S. 17-38.
+
+  Stone, C.W.; '08
+    Arithmetical Abilities and Some Factors Determining Them.
+    Teachers College Contributions to Education, No. 19.
+
+  Suzzallo, H.; '11
+    The Teaching of Primary Arithmetic.
+
+  Thorndike, E.L.; '00
+    Mental Fatigue. Psychological Review, vol. 7, pp. 466-482 and
+    547-579.
+
+  Thorndike, E.L.; '08
+    The Effect of Practice in the Case of a Purely Intellectual
+    Function. American Journal of Psychology, vol. 10, pp. 374-384.
+
+  Thorndike, E.L.; '10
+    Practice in the case of Addition.
+    American Journal of Psychology, vol. 21, pp. 483-486.
+
+  Thorndike, E.L., and Donovan, M.E.; '13
+    Improvement in a Practice Experiment under School Conditions.
+    American Journal of Psychology, vol. 24, pp. 426-428.
+
+  Thorndike, E.L., and Donovan, M.E., and Hahn, H.H.; '14
+    Some Results of Practice in Addition under School Conditions.
+    Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.
+
+  Thorndike, E.L.; '15
+    The Relation between Initial Ability and Improvement in a
+    Substitution Test. School and Society, vol. 12, p. 429.
+
+  Thorndike, E.L.; '16
+    Notes on Practice, Improvability and the Curve of Work.
+    American Journal of Psychology, vol 27, pp. 550-565.
+
+  Walsh, J.H.; '06
+    Grammar School Arithmetic.
+
+  Wells, F.L.; '12
+    The Relation of Practice to Individual Differences.
+    American Journal of Psychology, vol. 23, pp. 75-88.
+
+  White, E. E.; '83
+    A New Elementary Arithmetic.
+
+  Whitley, M. T.; '11
+    An Empirical Study of Certain Tests for Individual Differences.
+    Archives of Psychology, No. 19.
+
+  Wiederkehr, G.; '07
+    Statistiche Untersuchungen ueber die
+    Art und den Grad des Interesses bei Kindern der Volksschule.
+    Neue Bahnen, vol. 19, pp. 241-251, 289-299.
+
+  Wilson, G. M.; '19
+    A Survey of the Social and Business Usage of Arithmetic. Teachers
+    College Contributions to Education, No. 100.
+
+  Wilson, G. M., and Hoke, K. J.; '20
+    How to Measure.
+
+  Woody, C.; '10
+    Measurements of Some Achievements in Arithmetic. Teachers College
+    Contributions to Education, No. 80.
+
+
+
+
+INDEX
+
+
+  Abilities, arithmetical, nature of, 1 ff.;
+    measurement of, 27 ff.;
+    constitution of, 51 ff.;
+    organization of, 137 ff.
+
+  Abstract numbers, 85 ff.
+
+  Abstraction, 169 ff.
+
+  Accuracy, in relation to speed, 31;
+    in fundamental operations, 102 ff.
+
+  Addition, measurement of, 27 ff., 34;
+    constitution of, 52 f.;
+    habit in relation to, 71 f.;
+    in the higher decades, 75 f.;
+    accuracy in, 108 f.;
+    amount of practice in, 122 ff.;
+    interest in 196 f.
+
+  Aims of the teaching of arithmetic, 23 f.
+
+  AMES, A. F., 89
+
+  Analysis, learning by, 169 ff.;
+    systematic and opportunistic stimuli to, 178 f.;
+    gradual progress in, 180 ff.
+
+  Area, 257 f., 275
+
+  Arithmetic, sociology of, 24 ff.
+
+  Arithmetical abilities. _See_ Abilities.
+
+  Arithmetical language, 8 f., 19, 89 ff., 94 ff.
+
+  Arithmetical learning, before school, 199 ff.;
+    conditions of, 227 ff.;
+    in relation to time of day, 227 ff.;
+    in relation to time devoted to arithmetic, 228 ff.
+
+  Arithmetical reasoning. _See_ Reasoning.
+
+  Arithmetical terms, 8, 19
+
+  Averages, 40 f.; 135 f.
+
+
+  BALLOU, F. W., 34, 38
+
+  Banking, 256 f.
+
+  BINET, A., 201
+
+  Bonds, selection of, 70 ff.;
+    strength of, 102 ff.;
+    for temporary service, 111 ff.;
+    order of formation of, 141 ff.
+    _See also_ Habits.
+
+  BRANDELL, G., 211
+
+  BRANDFORD, B., 198 f.
+
+  BROWN, J. C., xvi, 103
+
+  BURGERSTEIN, L., 103
+
+  BURNETT, C. J., 202
+
+  BURT, C., 286
+
+
+  Cardinal and ordinal numbers confused, 206
+
+  Catch problems, 21 ff.
+
+  CHAPMAN, J. C., 49
+
+  Class, size of, in relation to arithmetical learning, 228;
+    variation within a, 289 ff.
+
+  COBB, M. V., 299
+
+  COFFMAN, L. D., xvi
+
+  Collection meaning of numbers, 3 ff.
+
+  Computation, measurements of, 33 ff.;
+    explanations of the processes in, 60 ff.;
+    accuracy in, 102 ff.
+    _See also_ Addition, Subtraction, Multiplication, Division,
+          Fractions, Decimal numbers, Percents.
+
+  Concomitants, law of varying, 172 ff.;
+    law of contrasting, 173 ff.
+
+  Concrete numbers, 85 ff.
+
+  Concrete objects, use of, 253 ff.
+
+  Conditions of arithmetical learning, 227 ff.
+
+  Constitution of arithmetical abilities, 51 ff.
+
+  Copying of numbers, eyestrain due to, 212 f.
+
+  Correlations of arithmetical abilities, 295 ff.
+
+  Courses of study, 232 f.
+
+  COURTIS, S. A., 28 ff., 43 ff., 49, 103, 291, 293, 299
+
+  Crutches, 112 f.
+
+  Culture-epoch theory, 198 f.
+
+
+  Dairy records, 273
+
+  Decimal numbers, uses of, 24 f.;
+    measurement of ability with, 36 ff.;
+    learning, 181 ff.;
+    division by, 270 f.
+
+  DE CROLY, M., 205
+
+  Deductive reasoning, 60 ff., 185 ff.
+
+  DEGAND, J., 205
+
+  Denominate numbers, 141 f., 147 f.
+
+  Described problems, 10 ff.
+
+  Development of knowledge of number, 205 ff.
+
+  DE VOSS, J. C., 49
+
+  DEWEY, J., 3, 83, 150, 205, 207, 208, 219, 266, 277
+
+  Differences in arithmetical ability, 285 ff.;
+    within a class, 289 ff.
+
+  Difficulty as a stimulus, 277 ff.
+
+  Drill, 102 ff.
+
+  Discipline, mental, 20
+
+  Distribution of practice, 156 ff.
+
+  Division, measurement of, 35 f., 37;
+    constitution of, 57 ff.;
+    deductive explanations of, 63, 64 f.;
+    inductive explanations of, 63 f., 65 f.;
+    habit in relation to, 72;
+    with remainders, 76;
+    with fractions, 78 ff.;
+    amount of practice in, 122 ff.;
+    distribution of practice in, 167;
+    use of the problem attitude in teaching, 270 f.
+
+  DONOVAN, M. E., 295
+
+
+  Elements, responses to, 169 ff.
+
+  Eleven, multiples of, 85
+
+  ELLIOTT, C. H., 228
+
+  Equation form, importance of, 77 f.
+
+  Explanations of the processes of computation, 60 ff.;
+    memory of, 115 f.;
+    time for giving, 154 ff.
+
+  Eyestrain in arithmetical work, 212 ff.
+
+
+  Facilitation, 143 ff.
+
+  Figures, printing of, 235 ff.;
+    writing of, 214 f., 241
+
+  FLYNN, F. J., 196
+
+  Fractions, uses of, 24 f.;
+    measurement of ability with, 36 ff.;
+    knowledge of the meaning of, 54 ff.
+
+  FREEMAN, F. N., 259, 261
+
+  FRIEDRICH, J., 103
+
+
+  Generalization, 169 ff.
+
+  GILBERT, J. A., 203
+
+  Graded tests, 28 ff., 36 ff.
+
+  Greatest common divisor, 88 f.
+
+
+  Habits, importance of, in arithmetical learning, 70 ff.;
+    now neglected, 75 ff.;
+    harmful or wasteful, 83 ff.; 91 ff.;
+    propaedeutic, 117 ff.;
+    organization of, 137 ff.;
+    arrangement of, 141 ff.
+
+  HAHN, H. H., 295
+
+  HALL, G. S., 200 f.
+
+  HARTMANN, B., 200 f.
+
+  HECK, W. H., 227
+
+  Heredity in arithmetical abilities, 293 ff.
+
+  Highest common factor, 88 f.
+
+  HOKE, K. J., 49
+
+  HOLMES, M. E., 103
+
+  HOWELL, H. B., 259
+
+  HUNT, C. W., 196
+
+  Hygiene of arithmetic, 212 ff., 234 ff.
+
+
+  Individual differences, 285 ff.
+
+  Inductive reasoning, 60 ff., 169 ff.
+
+  Insurance, 256
+
+  Interest as a principle determining the order of topics, 150 ff.
+
+  Interests, instinctive 195 ff.;
+    censuses of, 209 ff.;
+    neglect of childish, 226 ff.;
+    in self-management, 223 f.;
+    intrinsic, 224 ff.
+
+  Interference, 143 ff.
+
+  Inventories of arithmetical knowledge and skill, 199 ff.
+
+
+  JESSUP, W. A., xvi
+
+
+  KELLY, F. J., 49
+
+  KING, A. C., 103, 227
+
+  KIRBY, T. J., 76 f., 104, 295
+
+  KLAPPER, P., xvi
+
+  KRUSE, P. J., 289, 293
+
+
+  Ladder tests, 28 ff., 36 ff.
+
+  Language in arithmetic, 8 f., 19, 89 ff., 94 ff.
+
+  LASER, H., 103
+
+  LAY, W. A., 259, 261
+
+  Learning, nature of arithmetical, 1 ff.
+
+  Least common multiple, 88 f.
+
+  LEWIS, E. O., 210 f.
+
+  LOBSIEN, M., 209 f.
+
+
+  MCCALL, W. A., 49
+
+  MCDOUGLE, E. C., 85 ff.
+
+  MCKNIGHT, J. A., 210
+
+  MCLELLAN, J. A., 3, 83, 89, 205, 207
+
+  Manipulation of numbers, 60 ff.
+
+  Meaning, of numbers, 2 ff., 171;
+    of a fraction, 54 ff.;
+    of decimals, 181 f.
+
+  Measurement of arithmetical abilities, 27 ff.
+
+  Mental arithmetic, 262 ff.
+
+  MESSENGER, J. F., 202
+
+  Metric system, 147
+
+  MEUMANN, E., 261
+
+  MITCHELL, H. E., 24
+
+  MONROE, W. S., 49
+
+  Multiplication, measurement of, 35, 36;
+    constitution of, 51;
+    deductive explanations of, 61;
+    inductive explanations of, 61 f.;
+    with fractions, 78 ff.;
+    by eleven, 85;
+    amount of practice in, 122 ff.;
+    order of learning the elementary facts of, 144 f.;
+    distribution of practice in, 158 ff.;
+    use of the problem attitude in teaching, 267 ff.
+
+
+  NANU, H. A., 202
+
+  National Intelligence Tests, 49 f.
+
+  Negative reaction in intellectual life, 278 f.
+
+  Number pictures, 259 ff.
+
+  Numbers, meaning of, 2;
+    as measures of continuous quantities, 75;
+    abstract and concrete, 85 ff.;
+    denominate, 141 f., 147 f.;
+    use of large, 145 f.;
+    perception of, 205 ff.;
+    early awareness of, 205 ff.;
+    confusion of cardinal and ordinal, 206.
+    _See also_ Decimal numbers _and_ Fractions.
+
+
+  Objective aids, used for verification, 154;
+    in general, 243 ff.
+
+  Oral arithmetic, 262 ff.
+
+  Order of topics, 141 ff.
+
+  Ordinal numbers, confused with cardinal, 206
+
+  Original tendencies and arithmetic, 195 ff.
+
+  Overlearning, 134 ff.
+
+
+  Percents, 80 f.
+
+  Perception of number, 202 ff.
+
+  PHILLIPS, D. E., 3, 4, 205, 207
+
+  Pictures, hygiene of, 246 ff.;
+    number, 259 ff.
+
+  POMMER, O., 212
+
+  Practice, amount of, 122 ff.;
+    distribution of, 156 ff.
+
+  Precision in fundamental operations, 102 ff.
+
+  Problem attitude, 266 ff.
+
+  Problems, 9 ff.;
+    "catch," 21 ff.;
+    measurement of ability with, 42 ff.;
+    whose answer must be known in order to frame them, 93 f.;
+    verbal form of, 111 f.;
+    interest in, 220 ff.;
+    as introductions to arithmetical learning, 266 ff.
+
+  Propaedeutic bonds, 117 ff.
+
+  Purposive thinking, 193 ff.
+
+
+  Quantity, number and, 85 ff.;
+    perception of, 202 ff.
+
+
+  RACE, H., 295
+
+  Rainfall, 272
+
+  Ratio, 225 f.;
+    meaning of numbers, 3 ff.
+
+  Reaction, negative, 278 f.
+
+  Reality, in problems, 9 ff.
+
+  Reasoning, arithmetical, nature of, 19 ff.;
+    measurement of ability in, 42 ff.;
+    derivation of tables by, 58 f.;
+    about the rationale of computations, 60 ff.;
+    habit in relation to, 73 f., 190 ff.;
+    problems which provoke false, 100 f.;
+    the essentials of arithmetical, 185 ff.;
+    selection in, 187 ff.;
+    as the cooeperation of organized habits, 190 ff.
+
+  Recapitulation theory, 198 f.
+
+  Recipes, 273 f.
+
+  Rectangle, area of, 257 f.
+
+  RICE, J. M., 228 ff.
+
+  RUSH, G. P., 49
+
+
+  SEEKEL, E., 212
+
+  SELKIN, F. B., 196 f.
+
+  Sequence of topics, 141 ff.
+
+  Series meaning of numbers, 2 ff.
+
+  Size of class in relation to arithmetical learning, 228
+
+  SMITH, D. E., xvi, 224
+
+  Social instincts, use of, 195 f.
+
+  Sociology of arithmetic, 24 ff.
+
+  Speed in relation to accuracy, 31, 108
+
+  SPEER, W. W., 3, 5, 83
+
+  Spiral order, 141, 145
+
+  STARCH, D., 49, 295
+
+  STERN, W., 210, 212
+
+  STONE, C. W., 27 ff., 42 ff., 228 ff.
+
+  Subtraction, measurement of, 34 f.;
+    constitution of, 57 f.;
+    amount of practice in, 122 ff.
+
+  Supervision, 233 f.
+
+  SUZZALLO, H., xvi
+
+
+  Temporary bonds, 111 ff.
+
+  Terms, 113 f.
+
+  Tests of arithmetical abilities, 27 ff.
+
+  THORNDIKE, E. L., 34, 38 ff., 227, 294
+
+  Time, devoted to arithmetic, 228 ff.;
+    of day, in relation to arithmetical learning, 227 f.
+
+  Type, hygiene of, 235 ff.
+
+
+  Underlearning, 134 ff.
+
+  United States money, 148 ff.
+
+  Units of measure, arbitrary, 5, 83 f.
+
+
+  Variation, among individuals, 285 ff.
+
+  Variety, in teaching, 153
+
+  Verification, 81 f.;
+    aided by greater strength of the fundamental bonds, 107 ff.
+
+
+  WALSH, J. H., 11
+
+  WELLS, F. L., 295
+
+  WHITE, E. E., 5
+
+  WHITLEY, M. T., 295
+
+  WIEDERKEHR, G., 212
+
+  WILSON, G. M., 24, 49
+
+  WOODY, C., 29 ff., 52, 287, 293
+
+  Words. _See_ Language _and_ Terms.
+
+  Written arithmetic, 262 ff.
+
+
+  Zero in multiplication, 179 f.
+
+
+
+
+TRANSCRIBER'S NOTES:
+
+
+1. Passages in italics are surrounded by _underscores_.
+
+2. Passages in bold are indicated by #bold#.
+
+3. Mixed fractions are represented using forward slash and hyphen in
+this text version. For example, 5-1/2 represents five and a half.
+
+4. Images and footnotes have been moved from the middle of a paragraph
+to the closest paragraph break.
+
+5. Obvious errors in spelling and punctuation have been silently closed.
+
+
+
+
+
+
+End of the Project Gutenberg EBook of The Psychology of Arithmetic, by
+Edward L. Thorndike
+
+*** END OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC ***
+
+***** This file should be named 39300.txt or 39300.zip *****
+This and all associated files of various formats will be found in:
+        http://www.gutenberg.org/3/9/3/0/39300/
+
+Produced by Jonathan Ingram and the Online Distributed
+Proofreading Team at http://www.pgdp.net
+
+
+Updated editions will replace the previous one--the old editions
+will be renamed.
+
+Creating the works from public domain print editions means that no
+one owns a United States copyright in these works, so the Foundation
+(and you!) can copy and distribute it in the United States without
+permission and without paying copyright royalties.  Special rules,
+set forth in the General Terms of Use part of this license, apply to
+copying and distributing Project Gutenberg-tm electronic works to
+protect the PROJECT GUTENBERG-tm concept and trademark.  Project
+Gutenberg is a registered trademark, and may not be used if you
+charge for the eBooks, unless you receive specific permission.  If you
+do not charge anything for copies of this eBook, complying with the
+rules is very easy.  You may use this eBook for nearly any purpose
+such as creation of derivative works, reports, performances and
+research.  They may be modified and printed and given away--you may do
+practically ANYTHING with public domain eBooks.  Redistribution is
+subject to the trademark license, especially commercial
+redistribution.
+
+
+
+*** START: FULL LICENSE ***
+
+THE FULL PROJECT GUTENBERG LICENSE
+PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
+
+To protect the Project Gutenberg-tm mission of promoting the free
+distribution of electronic works, by using or distributing this work
+(or any other work associated in any way with the phrase "Project
+Gutenberg"), you agree to comply with all the terms of the Full Project
+Gutenberg-tm License (available with this file or online at
+http://gutenberg.org/license).
+
+
+Section 1.  General Terms of Use and Redistributing Project Gutenberg-tm
+electronic works
+
+1.A.  By reading or using any part of this Project Gutenberg-tm
+electronic work, you indicate that you have read, understand, agree to
+and accept all the terms of this license and intellectual property
+(trademark/copyright) agreement.  If you do not agree to abide by all
+the terms of this agreement, you must cease using and return or destroy
+all copies of Project Gutenberg-tm electronic works in your possession.
+If you paid a fee for obtaining a copy of or access to a Project
+Gutenberg-tm electronic work and you do not agree to be bound by the
+terms of this agreement, you may obtain a refund from the person or
+entity to whom you paid the fee as set forth in paragraph 1.E.8.
+
+1.B.  "Project Gutenberg" is a registered trademark.  It may only be
+used on or associated in any way with an electronic work by people who
+agree to be bound by the terms of this agreement.  There are a few
+things that you can do with most Project Gutenberg-tm electronic works
+even without complying with the full terms of this agreement.  See
+paragraph 1.C below.  There are a lot of things you can do with Project
+Gutenberg-tm electronic works if you follow the terms of this agreement
+and help preserve free future access to Project Gutenberg-tm electronic
+works.  See paragraph 1.E below.
+
+1.C.  The Project Gutenberg Literary Archive Foundation ("the Foundation"
+or PGLAF), owns a compilation copyright in the collection of Project
+Gutenberg-tm electronic works.  Nearly all the individual works in the
+collection are in the public domain in the United States.  If an
+individual work is in the public domain in the United States and you are
+located in the United States, we do not claim a right to prevent you from
+copying, distributing, performing, displaying or creating derivative
+works based on the work as long as all references to Project Gutenberg
+are removed.  Of course, we hope that you will support the Project
+Gutenberg-tm mission of promoting free access to electronic works by
+freely sharing Project Gutenberg-tm works in compliance with the terms of
+this agreement for keeping the Project Gutenberg-tm name associated with
+the work.  You can easily comply with the terms of this agreement by
+keeping this work in the same format with its attached full Project
+Gutenberg-tm License when you share it without charge with others.
+
+1.D.  The copyright laws of the place where you are located also govern
+what you can do with this work.  Copyright laws in most countries are in
+a constant state of change.  If you are outside the United States, check
+the laws of your country in addition to the terms of this agreement
+before downloading, copying, displaying, performing, distributing or
+creating derivative works based on this work or any other Project
+Gutenberg-tm work.  The Foundation makes no representations concerning
+the copyright status of any work in any country outside the United
+States.
+
+1.E.  Unless you have removed all references to Project Gutenberg:
+
+1.E.1.  The following sentence, with active links to, or other immediate
+access to, the full Project Gutenberg-tm License must appear prominently
+whenever any copy of a Project Gutenberg-tm work (any work on which the
+phrase "Project Gutenberg" appears, or with which the phrase "Project
+Gutenberg" is associated) is accessed, displayed, performed, viewed,
+copied or distributed:
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+1.E.2.  If an individual Project Gutenberg-tm electronic work is derived
+from the public domain (does not contain a notice indicating that it is
+posted with permission of the copyright holder), the work can be copied
+and distributed to anyone in the United States without paying any fees
+or charges.  If you are redistributing or providing access to a work
+with the phrase "Project Gutenberg" associated with or appearing on the
+work, you must comply either with the requirements of paragraphs 1.E.1
+through 1.E.7 or obtain permission for the use of the work and the
+Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or
+1.E.9.
+
+1.E.3.  If an individual Project Gutenberg-tm electronic work is posted
+with the permission of the copyright holder, your use and distribution
+must comply with both paragraphs 1.E.1 through 1.E.7 and any additional
+terms imposed by the copyright holder.  Additional terms will be linked
+to the Project Gutenberg-tm License for all works posted with the
+permission of the copyright holder found at the beginning of this work.
+
+1.E.4.  Do not unlink or detach or remove the full Project Gutenberg-tm
+License terms from this work, or any files containing a part of this
+work or any other work associated with Project Gutenberg-tm.
+
+1.E.5.  Do not copy, display, perform, distribute or redistribute this
+electronic work, or any part of this electronic work, without
+prominently displaying the sentence set forth in paragraph 1.E.1 with
+active links or immediate access to the full terms of the Project
+Gutenberg-tm License.
+
+1.E.6.  You may convert to and distribute this work in any binary,
+compressed, marked up, nonproprietary or proprietary form, including any
+word processing or hypertext form.  However, if you provide access to or
+distribute copies of a Project Gutenberg-tm work in a format other than
+"Plain Vanilla ASCII" or other format used in the official version
+posted on the official Project Gutenberg-tm web site (www.gutenberg.org),
+you must, at no additional cost, fee or expense to the user, provide a
+copy, a means of exporting a copy, or a means of obtaining a copy upon
+request, of the work in its original "Plain Vanilla ASCII" or other
+form.  Any alternate format must include the full Project Gutenberg-tm
+License as specified in paragraph 1.E.1.
+
+1.E.7.  Do not charge a fee for access to, viewing, displaying,
+performing, copying or distributing any Project Gutenberg-tm works
+unless you comply with paragraph 1.E.8 or 1.E.9.
+
+1.E.8.  You may charge a reasonable fee for copies of or providing
+access to or distributing Project Gutenberg-tm electronic works provided
+that
+
+- You pay a royalty fee of 20% of the gross profits you derive from
+     the use of Project Gutenberg-tm works calculated using the method
+     you already use to calculate your applicable taxes.  The fee is
+     owed to the owner of the Project Gutenberg-tm trademark, but he
+     has agreed to donate royalties under this paragraph to the
+     Project Gutenberg Literary Archive Foundation.  Royalty payments
+     must be paid within 60 days following each date on which you
+     prepare (or are legally required to prepare) your periodic tax
+     returns.  Royalty payments should be clearly marked as such and
+     sent to the Project Gutenberg Literary Archive Foundation at the
+     address specified in Section 4, "Information about donations to
+     the Project Gutenberg Literary Archive Foundation."
+
+- You provide a full refund of any money paid by a user who notifies
+     you in writing (or by e-mail) within 30 days of receipt that s/he
+     does not agree to the terms of the full Project Gutenberg-tm
+     License.  You must require such a user to return or
+     destroy all copies of the works possessed in a physical medium
+     and discontinue all use of and all access to other copies of
+     Project Gutenberg-tm works.
+
+- You provide, in accordance with paragraph 1.F.3, a full refund of any
+     money paid for a work or a replacement copy, if a defect in the
+     electronic work is discovered and reported to you within 90 days
+     of receipt of the work.
+
+- You comply with all other terms of this agreement for free
+     distribution of Project Gutenberg-tm works.
+
+1.E.9.  If you wish to charge a fee or distribute a Project Gutenberg-tm
+electronic work or group of works on different terms than are set
+forth in this agreement, you must obtain permission in writing from
+both the Project Gutenberg Literary Archive Foundation and Michael
+Hart, the owner of the Project Gutenberg-tm trademark.  Contact the
+Foundation as set forth in Section 3 below.
+
+1.F.
+
+1.F.1.  Project Gutenberg volunteers and employees expend considerable
+effort to identify, do copyright research on, transcribe and proofread
+public domain works in creating the Project Gutenberg-tm
+collection.  Despite these efforts, Project Gutenberg-tm electronic
+works, and the medium on which they may be stored, may contain
+"Defects," such as, but not limited to, incomplete, inaccurate or
+corrupt data, transcription errors, a copyright or other intellectual
+property infringement, a defective or damaged disk or other medium, a
+computer virus, or computer codes that damage or cannot be read by
+your equipment.
+
+1.F.2.  LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
+of Replacement or Refund" described in paragraph 1.F.3, the Project
+Gutenberg Literary Archive Foundation, the owner of the Project
+Gutenberg-tm trademark, and any other party distributing a Project
+Gutenberg-tm electronic work under this agreement, disclaim all
+liability to you for damages, costs and expenses, including legal
+fees.  YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
+LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
+PROVIDED IN PARAGRAPH 1.F.3.  YOU AGREE THAT THE FOUNDATION, THE
+TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
+LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
+INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
+DAMAGE.
+
+1.F.3.  LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
+defect in this electronic work within 90 days of receiving it, you can
+receive a refund of the money (if any) you paid for it by sending a
+written explanation to the person you received the work from.  If you
+received the work on a physical medium, you must return the medium with
+your written explanation.  The person or entity that provided you with
+the defective work may elect to provide a replacement copy in lieu of a
+refund.  If you received the work electronically, the person or entity
+providing it to you may choose to give you a second opportunity to
+receive the work electronically in lieu of a refund.  If the second copy
+is also defective, you may demand a refund in writing without further
+opportunities to fix the problem.
+
+1.F.4.  Except for the limited right of replacement or refund set forth
+in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER
+WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
+WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE.
+
+1.F.5.  Some states do not allow disclaimers of certain implied
+warranties or the exclusion or limitation of certain types of damages.
+If any disclaimer or limitation set forth in this agreement violates the
+law of the state applicable to this agreement, the agreement shall be
+interpreted to make the maximum disclaimer or limitation permitted by
+the applicable state law.  The invalidity or unenforceability of any
+provision of this agreement shall not void the remaining provisions.
+
+1.F.6.  INDEMNITY - You agree to indemnify and hold the Foundation, the
+trademark owner, any agent or employee of the Foundation, anyone
+providing copies of Project Gutenberg-tm electronic works in accordance
+with this agreement, and any volunteers associated with the production,
+promotion and distribution of Project Gutenberg-tm electronic works,
+harmless from all liability, costs and expenses, including legal fees,
+that arise directly or indirectly from any of the following which you do
+or cause to occur: (a) distribution of this or any Project Gutenberg-tm
+work, (b) alteration, modification, or additions or deletions to any
+Project Gutenberg-tm work, and (c) any Defect you cause.
+
+
+Section  2.  Information about the Mission of Project Gutenberg-tm
+
+Project Gutenberg-tm is synonymous with the free distribution of
+electronic works in formats readable by the widest variety of computers
+including obsolete, old, middle-aged and new computers.  It exists
+because of the efforts of hundreds of volunteers and donations from
+people in all walks of life.
+
+Volunteers and financial support to provide volunteers with the
+assistance they need, are critical to reaching Project Gutenberg-tm's
+goals and ensuring that the Project Gutenberg-tm collection will
+remain freely available for generations to come.  In 2001, the Project
+Gutenberg Literary Archive Foundation was created to provide a secure
+and permanent future for Project Gutenberg-tm and future generations.
+To learn more about the Project Gutenberg Literary Archive Foundation
+and how your efforts and donations can help, see Sections 3 and 4
+and the Foundation web page at http://www.pglaf.org.
+
+
+Section 3.  Information about the Project Gutenberg Literary Archive
+Foundation
+
+The Project Gutenberg Literary Archive Foundation is a non profit
+501(c)(3) educational corporation organized under the laws of the
+state of Mississippi and granted tax exempt status by the Internal
+Revenue Service.  The Foundation's EIN or federal tax identification
+number is 64-6221541.  Its 501(c)(3) letter is posted at
+http://pglaf.org/fundraising.  Contributions to the Project Gutenberg
+Literary Archive Foundation are tax deductible to the full extent
+permitted by U.S. federal laws and your state's laws.
+
+The Foundation's principal office is located at 4557 Melan Dr. S.
+Fairbanks, AK, 99712., but its volunteers and employees are scattered
+throughout numerous locations.  Its business office is located at
+809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email
+business@pglaf.org.  Email contact links and up to date contact
+information can be found at the Foundation's web site and official
+page at http://pglaf.org
+
+For additional contact information:
+     Dr. Gregory B. Newby
+     Chief Executive and Director
+     gbnewby@pglaf.org
+
+
+Section 4.  Information about Donations to the Project Gutenberg
+Literary Archive Foundation
+
+Project Gutenberg-tm depends upon and cannot survive without wide
+spread public support and donations to carry out its mission of
+increasing the number of public domain and licensed works that can be
+freely distributed in machine readable form accessible by the widest
+array of equipment including outdated equipment.  Many small donations
+($1 to $5,000) are particularly important to maintaining tax exempt
+status with the IRS.
+
+The Foundation is committed to complying with the laws regulating
+charities and charitable donations in all 50 states of the United
+States.  Compliance requirements are not uniform and it takes a
+considerable effort, much paperwork and many fees to meet and keep up
+with these requirements.  We do not solicit donations in locations
+where we have not received written confirmation of compliance.  To
+SEND DONATIONS or determine the status of compliance for any
+particular state visit http://pglaf.org
+
+While we cannot and do not solicit contributions from states where we
+have not met the solicitation requirements, we know of no prohibition
+against accepting unsolicited donations from donors in such states who
+approach us with offers to donate.
+
+International donations are gratefully accepted, but we cannot make
+any statements concerning tax treatment of donations received from
+outside the United States.  U.S. laws alone swamp our small staff.
+
+Please check the Project Gutenberg Web pages for current donation
+methods and addresses.  Donations are accepted in a number of other
+ways including checks, online payments and credit card donations.
+To donate, please visit: http://pglaf.org/donate
+
+
+Section 5.  General Information About Project Gutenberg-tm electronic
+works.
+
+Professor Michael S. Hart is the originator of the Project Gutenberg-tm
+concept of a library of electronic works that could be freely shared
+with anyone.  For thirty years, he produced and distributed Project
+Gutenberg-tm eBooks with only a loose network of volunteer support.
+
+
+Project Gutenberg-tm eBooks are often created from several printed
+editions, all of which are confirmed as Public Domain in the U.S.
+unless a copyright notice is included.  Thus, we do not necessarily
+keep eBooks in compliance with any particular paper edition.
+
+
+Most people start at our Web site which has the main PG search facility:
+
+     http://www.gutenberg.org
+
+This Web site includes information about Project Gutenberg-tm,
+including how to make donations to the Project Gutenberg Literary
+Archive Foundation, how to help produce our new eBooks, and how to
+subscribe to our email newsletter to hear about new eBooks.
diff --git a/39300.zip b/39300.zip
new file mode 100644
index 0000000..ffe8e0a
--- /dev/null
+++ b/39300.zip
diff --git a/LICENSE.txt b/LICENSE.txt
new file mode 100644
index 0000000..6312041
--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,11 @@
+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
diff --git a/README.md b/README.md
new file mode 100644
index 0000000..3e66096
--- /dev/null
+++ b/README.md
@@ -0,0 +1,2 @@
+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #39300 (https://www.gutenberg.org/ebooks/39300)