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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. 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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. 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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. 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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. 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