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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:12:24 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:12:24 -0700 |
| commit | afc4d58b75cd1f1c35ab18d6162b54d373af1e6f (patch) | |
| tree | c3c211900f26ab4a32594b36688e8a65f18c97a0 /39300-h | |
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diff --git a/39300-h/39300-h.htm b/39300-h/39300-h.htm new file mode 100644 index 0000000..8fe1cbf --- /dev/null +++ b/39300-h/39300-h.htm @@ -0,0 +1,13736 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> + +<html xmlns="http://www.w3.org/1999/xhtml"> + <head> + <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1" /> + <title> + The Project Gutenberg eBook of The Psychology of Arithmetic, by Edward L. Thorndike. + </title> + <style type="text/css"> + + p { margin-top: .75em; + text-indent: 1em; + text-align: justify; + margin-bottom: .75em; + } + h1,h2,h3,h4,h5,h6 { + text-align: center; /* all headings centered */ + margin-top: 2em; + margin-bottom: 1em; + width: 80%; + margin-left: auto; + margin-right: auto; + clear: both; + } + hr { width: 33%; + margin-top: 2em; + margin-bottom: 2em; + margin-left: auto; + margin-right: auto; + clear: both; + } + + table {margin-left: auto; margin-right: auto; border-collapse: collapse;} + + body{margin-left: 10%; + margin-right: 10%; + } + + /* sup { vertical-align: 0.8ex; font-size:95%; } */ +/* sub { vertical-align: -0.6ex; font-size:95%; } */ + span.inline_eqn { white-space: nowrap; } + .pagenum { /* uncomment the next line for invisible page numbers */ + /* visibility: hidden; */ + position: absolute; + left: 92%; + font-size: smaller; + text-align: right; + } /* page numbers */ + + .dblockquot{font-size: 1.1em; font-weight: bold;} + .iblockquot{font-style: italic;} + .blockquot{margin-left: 10%; margin-right: 10%;} + .pblockquot{margin-top: 2em; margin-bottom: 2em; font-size: 0.9em;} + .sblockquot{font-size: 0.9em; margin-left: 2em;} + .nblockquot{text-indent: -3em; margin-left: 3em;} + .nnblockquot{text-indent: -1.5em; margin-left: 2.5em;} + + .spacer {padding: 1em; text-align: left;} + td {padding-right: 1em; padding-left: 1em; } + th {padding-right: 2em; padding-left: 2em; } + .bb {border-bottom: solid 1px;} + .bl {border-left: solid 1px;} + .bt {border-top: solid 1px;} + .br {border-right: solid 1px;} + .bbt {border-bottom: solid 1px; border-top: solid 1px;} + .bbox {border: solid 1px;} + .bbox1 {border: solid 1px; font-size: 85%; padding-right: 1em; padding-left: 1em;} + + .center {text-align: center; text-indent: 0em;} + .smcap {font-variant: small-caps;} + .u {text-decoration: underline;} + .overline {text-decoration: overline;} + .sz30 {font-size: 3em;} + .ft05 {font-size: 60%;} + .noidt {text-indent: 0em;} + .alignleft {float: left;} + .alignright {float: right;} + + .caption {font-weight: bold; text-align: center;} + .tabcap {font-weight: bold; text-align: center; margin-top: 1.5em;} + + .figcenter {margin: auto; text-align: center;} + + .figleft {float: left; clear: left; margin-left: 0; margin-bottom: 1em; margin-top: + 0em; margin-right: 1em; padding: 0; text-align: center;} + + .figright {float: right; clear: right; margin-left: 1em; margin-bottom: 1em; + margin-top: 1em; margin-right: 0; padding: 0; text-align: center;} + + .footnotes {border: dashed 1px;} + .footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em; margin-top: 1.5em;} + .footnote .label {position: absolute; right: 82%; text-align: right;} + .fnanchor {vertical-align: super; font-size: .8em; text-decoration: none;} + + .poem {margin-left:10%; margin-right:10%; text-align: left;} + .poem br {display: none;} + .poem .stanza {margin: 1em 0em 1em 0em;} + .poem span.i0 {display: block; margin-left: 0em; padding-left: 3em; text-indent: -3em;} + .poem span.i2 {display: block; margin-left: 2em; padding-left: 3em; text-indent: -3em;} + .poem span.i4 {display: block; margin-left: 4em; padding-left: 3em; text-indent: -3em;} + + .notebox {border: solid 2px; padding: 1em; margin-left: 15%; margin-right: 15%; background: #CCCCB2;} + </style> + </head> +<body> + + +<pre> + +Project Gutenberg's The Psychology of Arithmetic, by Edward L. Thorndike + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org/license + + +Title: The Psychology of Arithmetic + +Author: Edward L. Thorndike + +Release Date: March 29, 2012 [EBook #39300] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE PSYCHOLOGY OF ARITHMETIC *** + + + + +Produced by Jonathan Ingram and the Online Distributed +Proofreading Team at http://www.pgdp.net + + + + + + +</pre> + + + + +<div class="figcenter" style="width: 216px;"> +<img src="images/publogo.png" width="216" height="90" alt="Logo" title="Logo" /> +</div> + +<p class="center"> +THE MACMILLAN COMPANY<br /> + +<small>NEW YORK · BOSTON · CHICAGO · DALLAS<br /> +ATLANTA · SAN FRANCISCO</small><br /><br /> + +MACMILLAN & CO., LIMITED<br /> + +<small>LONDON · BOMBAY · CALCUTTA<br /> +MELBOURNE</small><br /><br /> + +THE MACMILLAN COMPANY<br /> +OF CANADA, <span class="smcap">Limited</span><br /> +<small>TORONTO</small></p> + + +<hr style="width: 65%;" /> +<h1>THE PSYCHOLOGY OF<br /> +ARITHMETIC</h1> + + + +<h3><small>BY<br /><br /></small> +<big>EDWARD L. THORNDIKE</big></h3> + +<p class="center">TEACHERS COLLEGE, COLUMBIA<br /> +UNIVERSITY</p> + +<p> </p> + +<p class="center"><big>New York<br /> +THE MACMILLAN COMPANY<br /> +1929</big><br /> + +<i>All rights reserved</i></p> + + +<hr style="width: 65%;" /> +<p class="center"><span class="smcap">Copyright</span>, 1922,<br /> +<span class="smcap">By</span> THE MACMILLAN COMPANY.</p> +<hr style="width: 10%;" /> +<p class="center">Set up and electrotyped. Published January, 1922.<br /> +Reprinted October, 1924; May, 1926; August, 1927; October, 1929.</p> + + +<p> </p> +<p class="center"><b>· PRINTED IN THE UNITED STATES OF AMERICA ·</b></p> + + +<hr style="width: 100%;" /> +<p><span class='pagenum'>[Pg v]</span></p> +<h2>PREFACE</h2> + + +<p>Within recent years there have been three lines of advance +in psychology which are of notable significance for teaching. +The first is the new point of view concerning the +general process of learning. We now understand that +learning is essentially the formation of connections or +bonds between situations and responses, that the satisfyingness +of the result is the chief force that forms them, and +that habit rules in the realm of thought as truly and as +fully as in the realm of action.</p> + +<p>The second is the great increase in knowledge of the +amount, rate, and conditions of improvement in those +organized groups or hierarchies of habits which we call +abilities, such as ability to add or ability to read. Practice +and improvement are no longer vague generalities, but +concern changes which are definable and measurable by +standard tests and scales.</p> + +<p>The third is the better understanding of the so-called +"higher processes" of analysis, abstraction, the formation +of general notions, and reasoning. The older view of a +mental chemistry whereby sensations were compounded +into percepts, percepts were duplicated by images, percepts +and images were amalgamated into abstractions and concepts, +and these were manipulated by reasoning, has given +way to the understanding of the laws of response to elements +or aspects of situations and to many situations or elements +thereof in combination. James' view of reasoning as +"selection of essentials" and "thinking things together" +<span class='pagenum'>[Pg vi]</span> +in a revised and clarified form has important applications +in the teaching of all the school subjects.</p> + +<p>This book presents the applications of this newer dynamic +psychology to the teaching of arithmetic. Its contents are +substantially what have been included in a course of lectures +on the psychology of the elementary school subjects given +by the author for some years to students of elementary +education at Teachers College. Many of these former +students, now in supervisory charge of elementary schools, +have urged that these lectures be made available to teachers +in general. So they are now published in spite of the +author's desire to clarify and reinforce certain matters by +further researches.</p> + +<p>A word of explanation is necessary concerning the exercises +and problems cited to illustrate various matters, especially +erroneous pedagogy. These are all genuine, having +their source in actual textbooks, courses of study, state +examinations, and the like. To avoid any possibility of +invidious comparisons they are not quotations, but equivalent +problems such as represent accurately the spirit and +intent of the originals.</p> + +<p>I take pleasure in acknowledging the courtesy of Mr. S. A. +Courtis, Ginn and Company, D. C. Heath and Company, +The Macmillan Company, The Oxford University Press, +Rand, McNally and Company, Dr. C. W. Stone, The +Teachers College Bureau of Publications, and The World +Book Company, in permitting various quotations.</p> + +<p style='text-align:right'> +<span class="smcap">Edward L. Thorndike.</span><br /> +</p> + +<p> + <span class="smcap">Teachers College</span><br /> +<span class="smcap">Columbia University</span><br /> + April 1, 1920</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'>[Pg vii]</span></p> +<h2>CONTENTS</h2> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="Contents"> +<tr><td align='right'><small>CHAPTER</small></td><td> </td><td align='right'><small>PAGE</small></td></tr> +<tr><td align='right'> </td><td align='left'><a href="#GENERAL_INTRODUCTION"><span class="smcap">Introduction: The Psychology of the Elementary School Subjects</span></a></td><td align='right'><a href="#Page_xi">xi</a></td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_I">I.</a></td><td align='left'><span class="smcap">The Nature of Arithmetical Abilities</span></td><td align='right'><a href="#Page_1">1</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Knowledge of the Meanings of Numbers</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Arithmetical Language</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Problem Solving</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Arithmetical Reasoning</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Summary</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Sociology of Arithmetic</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_II">II.</a></td><td align='left'><span class="smcap">The Measurement of Arithmetical Abilities</span></td><td align='right'><a href="#Page_27">27</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> A Sample Measurement of an Arithmetical Ability</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Ability to Add Integers</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Measurements of Ability in Computation</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Measurements of Ability in Applied Arithmetic: the Solution of Problems</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_III">III.</a></td><td align='left'><span class="smcap">The Constitution of Arithmetical Abilities</span></td><td align='right'><a href="#Page_51">51</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Elementary Functions of Arithmetical Learning</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Knowledge of the Meaning of a Fraction</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Learning the Processes of Computation</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_IV">IV.</a></td><td align='left'><span class="smcap">The Constitution of Arithmetical Abilities</span> (<i>continued</i>)</td><td align='right'><a href="#Page_70">70</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Selection of the Bonds to Be Formed</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Importance of Habit Formation</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Desirable Bonds Now Often Neglected</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Wasteful and Harmful Bonds</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Guiding Principles</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_V">V.</a></td><td align='left'><span class="smcap">The Psychology of Drill in Arithmetic: The Strength of Bonds</span><span class='pagenum'>[Pg viii]</span></td><td align='right'><a href="#Page_102">102</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Need of Stronger Elementary Bonds</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Early Mastery</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Strength of Bonds for Temporary Service</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Strength of Bonds with Technical Facts and Terms</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Strength of Bonds Concerning the Reasons for Arithmetical Processes</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Propædeutic Bonds</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_VI">VI.</a></td><td align='left'><span class="smcap">The Psychology of Drill in Arithmetic: The Amount of Practice and the Organization of Abilities</span></td><td align='right'><a href="#Page_122">122</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Amount of Practice</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Under-learning and Over-learning</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Organization of Abilities</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_VII">VII.</a></td><td align='left'><span class="smcap">The Sequence of Topics: The Order of Formation of Bonds</span></td><td align='right'><a href="#Page_141">141</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Conventional <i>versus</i> Effective Orders</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Decreasing Interference and Increasing Facilitation</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Interest</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> General Principles</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_VIII">VIII.</a></td><td align='left'><span class="smcap">The Distribution of Practice</span></td><td align='right'><a href="#Page_156">156</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Problem</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Sample Distributions</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Possible Improvements</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_IX">IX.</a></td><td align='left'><span class="smcap">The Psychology of Thinking: Abstract Ideas and General Notions in Arithmetic</span></td><td align='right'><a href="#Page_169">169</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Responses to Elements and Classes</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Facilitating the Analysis of Elements</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Systematic and Opportunistic Stimuli to Analysis</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Adaptations to Elementary-school Pupils</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_X">X.</a></td><td align='left'><span class="smcap">The Psychology of Thinking: Reasoning in Arithmetic</span></td><td align='right'><a href="#Page_185">185</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Essentials of Arithmetical Reasoning</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Reasoning as the Coöperation of Organized Habits</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_XI">XI.</a></td><td align='left'><span class="smcap">Original Tendencies and Acquisitions before School</span><span class='pagenum'>[Pg ix]</span></td><td align='right'><a href="#Page_195">195</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Utilization of Instinctive Interests</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Order of Development of Original Tendencies</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Inventories of Arithmetical Knowledge and Skill</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Perception of Number and Quantity</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Early Awareness of Number</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_XII">XII.</a></td><td align='left'><span class="smcap">Interest in Arithmetic</span></td><td align='right'><a href="#Page_209">209</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Censuses of Pupils' Interests</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Relieving Eye Strain</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Significance for Related Activities</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Intrinsic Interest in Arithmetical Learning</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_XIII">XIII.</a></td><td align='left'><span class="smcap">The Conditions of Learning</span></td><td align='right'><a href="#Page_227">227</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> External Conditions</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Hygiene of the Eyes in Arithmetic</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Use of Concrete Objects in Arithmetic</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Oral, Mental, and Written Arithmetic</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_XIV">XIV.</a></td><td align='left'><span class="smcap">The Conditions of Learning: the Problem Attitude</span></td><td align='right'><a href="#Page_266">266</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Illustrative Cases</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> General Principles</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Difficulty and Success as Stimuli</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> False Inferences</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'><a href="#CHAPTER_XV">XV.</a></td><td align='left'><span class="smcap">Individual Differences</span></td><td align='right'><a href="#Page_285">285</a></td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Nature and Amount</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> Differences within One Class</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Causes of Individual Differences</td></tr> +<tr><td align='left'> </td><td align='left' colspan='2'> The Interrelations of Individual Differences</td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'> </td><td align='left'><a href="#BIBLIOGRAPHY"><span class="smcap">Bibliography of References</span></a></td><td align='right'><a href="#Page_302">302</a></td></tr> +<tr><td align='left' colspan='3'> </td></tr> +<tr><td align='right'> </td><td align='left'><a href="#INDEX"><span class="smcap">Index</span></a></td><td align='right'><a href="#Page_311">311</a></td></tr> +</table></div> + + + +<p><span class='pagenum'><a name="Page_x" id="Page_x">[Pg x]</a></span></p> + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_xi" id="Page_xi">[Pg xi]</a></span></p> +<h2><a name="GENERAL_INTRODUCTION" id="GENERAL_INTRODUCTION"></a>GENERAL INTRODUCTION</h2> + +<h2>THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL +SUBJECTS</h2> + + +<p>The psychology of the elementary school subjects is +concerned with the connections whereby a child is able to +respond to the sight of printed words by thoughts of their +meanings, to the thought of "six and eight" by thinking +"fourteen," to certain sorts of stories, poems, songs, and +pictures by appreciation thereof, to certain situations by +acts of skill, to certain others by acts of courtesy and justice, +and so on and on through the series of situations and responses +which are provided by the systematic training of +the school subjects and the less systematic training of +school life during their study. The aims of elementary +education, when fully defined, will be found to be the production +of changes in human nature represented by an almost +countless list of connections or bonds whereby the pupil +thinks or feels or acts in certain ways in response to the +situations the school has organized and is influenced to think +and feel and act similarly to similar situations when life +outside of school confronts him with them.</p> + +<p>We are not at present able to define the work of the elementary +school in detail as the formation of such and such +bonds between certain detached situations and certain +specified responses. As elsewhere in human learning, we +are at present forced to think somewhat vaguely in terms +of mental functions, like "ability to read the vernacular," +"ability to spell common words," "ability to add, subtract, +<span class='pagenum'><a name="Page_xii" id="Page_xii">[Pg xii]</a></span> +multiply, and divide with integers," "knowledge of +the history of the United States," "honesty in examinations," +and "appreciation of good music," defined by some +general results obtained rather than by the elementary +bonds which constitute them.</p> + +<p>The psychology of the school subjects begins where our +common sense knowledge of these functions leaves off and +tries to define the knowledge, interest, power, skill, or ideal +in question more adequately, to measure improvement in +it, to analyze it into its constituent bonds, to decide what +bonds need to be formed and in what order as means to the +most economical attainment of the desired improvement, +to survey the original tendencies and the tendencies already +acquired before entrance to school which help or hinder +progress in the elementary school subjects, to examine +the motives that are or may be used to make the desired +connections satisfying, to examine any other special conditions +of improvement, and to note any facts concerning +individual differences that are of special importance to the +conduct of elementary school work.</p> + +<p>Put in terms of problems, the task of the psychology of +the elementary school subjects is, in each case:—</p> + +<p>(1) <i>What is the function?</i> For example, just what is +"ability to read"? Just what does "the understanding of +decimal notation" mean? Just what are "the moral effects +to be sought from the teaching of literature"?</p> + +<p>(2) <i>How are degrees of ability or attainment, and degrees of +progress or improvement in the function or a part of the function +measured?</i> For example, how can we determine how well +a pupil should write, or how hard words we expect him +to spell, or what good taste we expect him to show? +How can we define to ourselves what knowledge of the +meaning of a fraction we shall try to secure in grade +4?<span class='pagenum'><a name="Page_xiii" id="Page_xiii">[Pg xiii]</a></span></p> + +<p>(3) <i>What can be done toward reducing the function to terms +of particular situation-response connections, whose formation +can be more surely and easily controlled?</i> For example, how +far does ability to spell involve the formation one by one +of bonds between the thought of almost every word in the +language and the thought of that word's letters in their +correct order; and how far does, say, the bond leading from +the situation of the sound of <i>ceive</i> in <i>receive</i> and <i>deceive</i> to +their correct spelling insure the correct spelling of that part +of <i>perceive</i>? Does "ability to add" involve special bonds +leading from "27 and 4" to "31," from "27 and 5" to "32," +and "27 and 6" to "33"; or will the bonds leading from "7 +and 4" to "11," "7 and 5" to "12" and "7 and 6" to +"13" (each plus a simple inference) serve as well? What +are the situations and responses that represent in actual +behavior the quality that we call school patriotism?</p> + +<p>(4) <i>In almost every case a certain desired change of knowledge +or skill or power can be attained by any one of several +sets of bonds. Which of them is the best? What are the +advantages of each?</i> For example, learning to add may include +the bonds "0 and 0 are 0," "0 and 1 are 1," "0 and +2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these +may be all left unformed, the pupil being taught the habits +of entering 0 as the sum of a column that is composed of +zeros and otherwise neglecting 0 in addition. Are the rules +of usage worth teaching as a means toward correct speech, +or is the time better spent in detailed practice in correct +speech itself?</p> + +<p>(5) <i>A bond to be formed may be formed in any one of many +degrees of strength. Which of these is, at any given stage of +learning the subject, the most desirable, all things considered?</i> +For example, shall the dates of all the early settlements +of North America be learned so that the exact year will be +<span class='pagenum'><a name="Page_xiv" id="Page_xiv">[Pg xiv]</a></span> +remembered for ten years, or so that the exact date will be +remembered for ten minutes and the date with an error plus +or minus of ten years will be remembered for a year or two? +Shall the tables of inches, feet, and yards, and pints, quarts, +and gallons be learned at their first appearance so as to be +remembered for a year, or shall they be learned only well +enough to be usable in the work of that week, which in +turn fixes them to last for a month or so? Should a pupil +in the first year of study of French have such perfect +connections between the sounds of French words and their +meanings that he can understand simple sentences containing +them spoken at an ordinary rate of speaking? Or is +slow speech permissible, and even imperative, on the part of +the teacher, with gradual increase of rate?</p> + +<p>(6) <i>In almost every case, any set of bonds may produce the +desired change when presented in any one of several orders. +Which is the best order? What are the advantages of each?</i> +Certain systems for teaching handwriting perfect the elementary +movements one at a time and then teach their +combination in words and sentences. Others begin and +continue with the complex movement-series that actual +words require. What do the latter lose and gain? The +bonds constituting knowledge of the metric system are +now formed late in the pupil's course. Would it be better +if they were formed early as a means of facilitating knowledge +of decimal fractions?</p> + +<p>(7) <i>What are the original tendencies and pre-school acquisitions +upon which the connection-forming of the elementary +school may be based or which it has to counteract?</i> For +example, if a pupil knows the meaning of a heard word, +he may read it understandingly from getting its sound, +as by phonic reconstruction. What words does the average +beginner so know? What are the individual differences in +<span class='pagenum'><a name="Page_xv" id="Page_xv">[Pg xv]</a></span> +this respect? What do the instincts of gregariousness, +attention-getting, approval, and helpfulness recommend +concerning group-work <i>versus</i> individual-work, and concerning +the size of a group that is most desirable? The original +tendency of the eyes is certainly not to move along a line +from left to right of a page, then back in one sweep and along +the next line. What is their original tendency when confronted +with the printed page, and what must we do with it +in teaching reading?</p> + +<p>(8) <i>What armament of satisfiers and annoyers, of positive +and negative interests and motives, stands ready for use in the +formation of the intrinsically uninteresting connections between +black marks and meanings, numerical exercises and their +answers, words and their spelling, and the like?</i> School +practice has tried, more or less at random, incentives and +deterrents from quasi-physical pain to the most sentimental +fondling, from sheer cajolery to philosophical argument, +from appeals to assumed savage and primitive traits to +appeals to the interest in automobiles, flying-machines, and +wireless telegraphy. Can not psychology give some rules +for guidance, or at least limit experimentation to its more +hopeful fields?</p> + +<p>(9) <i>The general conditions of efficient learning are described +in manuals of educational psychology. How do these +apply in the case of each task of the elementary school?</i> For +example, the arrangement of school drills in addition and +in short division in the form of practice experiments has +been found very effective in producing interest in the work +and in improvement at it. In what other arithmetical +functions may we expect the same?</p> + +<p>(10) <i>Beside the general principles concerning the nature and +causation of individual differences, there must obviously be, +in existence or obtainable as a possible result of proper investigation,</i> +<span class='pagenum'><a name="Page_xvi" id="Page_xvi">[Pg xvi]</a></span> +<i>a great fund of knowledge of special differences relevant +to the learning of reading, spelling, geography, arithmetic, and +the like. What are the facts as far as known? What are the +means of learning more of them?</i> Courtis finds that a child +may be specially strong in addition and yet be specially +weak in subtraction in comparison with others of his age +and grade. It even seems that such subtle and intricate +tendencies are inherited. How far is such specialization +the rule? Is it, for example, the case that a child may have +a special gift for spelling certain sorts of words, for drawing +faces rather than flowers, for learning ancient history rather +than modern?</p> + +<p> </p> + +<p>Such are our problems: this volume discusses them in the +case of arithmetic. The student who wishes to relate the +discussion to the general pedagogy of arithmetic may +profitably read, in connection with this volume: The +Teaching of Elementary Mathematics, by D. E. Smith +['01], The Teaching of Primary Arithmetic, by H. Suzzallo +['11], How to Teach Arithmetic, by J. C. Brown and L. D. +Coffman ['14], The Teaching of Arithmetic, by Paul Klapper +['16], and The New Methods in Arithmetic, by the author +['21].</p> + + +<hr style="width: 100%;" /> +<h2>THE PSYCHOLOGY OF ARITHMETIC</h2> + + +<hr style="width: 100%;" /> +<h1>THE PSYCHOLOGY OF<br /> +ARITHMETIC</h1> + +<hr style="width: 15%;" /> + + +<p><span class='pagenum'><a name="Page_1" id="Page_1">[Pg 1]</a></span></p> +<h2><a name="CHAPTER_I" id="CHAPTER_I"></a>CHAPTER I</h2> + +<h2>THE NATURE OF ARITHMETICAL ABILITIES</h2> + + +<p>According to common sense, the task of the elementary +school is to teach:—(1) the meanings of numbers, (2) the +nature of our system of decimal notation, (3) the meanings +of addition, subtraction, multiplication, and division, and +(4) the nature and relations of certain common measures; +to secure (5) the ability to add, subtract, multiply, and +divide with integers, common and decimal fractions, and +denominate numbers, (6) the ability to apply the knowledge +and power represented by (1) to (5) in solving problems, +and (7) certain specific abilities to solve problems concerning +percentage, interest, and other common occurrences in +business life.</p> + +<p>This statement of the functions to be developed and improved +is sound and useful so far as it goes, but it does not +go far enough to make the task entirely clear. If teachers +had nothing but the statement above as a guide to what +changes they were to make in their pupils, they would often +leave out important features of arithmetical training, and<span class='pagenum'><a name="Page_2" id="Page_2">[Pg 2]</a></span> +put in forms of training that a wise educational plan would +not tolerate. It is also the case that different leaders in +arithmetical teaching, though they might all subscribe to +the general statement of the previous paragraph, certainly +do not in practice have identical notions of what arithmetic +should be for the elementary school pupil.</p> + +<p>The ordinary view of the nature of arithmetical learning +is obscure or inadequate in four respects. It does not define +what 'knowledge of the meanings of numbers' is; it does +not take account of the very large amount of teaching of +<i>language</i> which is done and should be done as a part of the +teaching of arithmetic; it does not distinguish between the +ability to meet certain quantitative problems as life offers +them and the ability to meet the problems provided by +textbooks and courses of study; it leaves 'the ability to +apply arithmetical knowledge and power' as a rather +mystical general faculty to be improved by some educational +magic. The four necessary amendments may be discussed +briefly.</p> + + +<h4>KNOWLEDGE OF THE MEANINGS OF NUMBERS</h4> + +<p>Knowledge of the meanings of the numbers from one to +ten may mean knowledge that 'one' means a single thing +of the sort named, that two means one more than one, that +three means one more than two, and so on. This we may +call the <i>series</i> meaning. To know the meaning of 'six' in +this sense is to know that it is one more than five and one +less than seven—that it is between five and seven in the +number series. Or we may mean by knowledge of the +meanings of numbers, knowledge that two fits a collection of +two units, that three fits a collection of three units, and so +on, each number being a name for a certain sized collection +of discrete things, such as apples, pennies, boys, balls, fingers,<span class='pagenum'><a name="Page_3" id="Page_3">[Pg 3]</a></span> +and the other customary objects of enumeration in the +primary school. This we may call the <i>collection</i> meaning. +To know the meaning of six in this sense is to be able to name +correctly any collection of six separate, easily distinguishable +individual objects. In the third place, knowledge of the +numbers from one to ten may mean knowledge that two is +twice whatever is called one, that three is three times whatever +is one, and so on. This is, of course, the <i>ratio</i> meaning. +To know the meaning of six in this sense is to know that +if _________ is one, a line half a foot long is six, that if +[ __ ] is one, [ ____________ ] is about six, while +if [ _ ] is one, [ ______ ] is about six, and the like. In +the fourth place, the meaning of a number may be a smaller +or larger fraction of its <i>implications</i>—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 '<i>nucleus of facts</i>' or <i>relational</i> +meaning of a number.</p> + +<p>Ordinary school practice has commonly accepted the +second meaning as that which it is the task of the school to +teach beginners, but each of the other meanings has been +alleged to be the essential one—the series idea by Phillips +['97], the ratio idea by McLellan and Dewey ['95] and Speer +['97], and the relational idea by Grube and his followers.</p> + +<p>This diversity of views concerning what the function is +that is to be improved in the case of learning the meanings +of the numbers one to ten is not a trifling matter of definition, +but produces very great differences in school practice. +Consider, for example, the predominant value assigned to +counting by Phillips in the passage quoted below, and the +<span class='pagenum'><a name="Page_4" id="Page_4">[Pg 4]</a></span> +samples of the sort of work at which children were kept +employed for months by too ardent followers of Speer and +Grube.</p> + + +<h4>THE SERIES IDEA OVEREMPHASIZED</h4> + +<div class="pblockquot"> +<p>"This is essentially the counting period, and any words that +can be arranged into a series furnish all that is necessary. Counting +is fundamental, and counting that is spontaneous, free from +sensible observation, and from the strain of reason. A study of +these original methods shows that multiplication was developed +out of counting, and not from addition as nearly all textbooks +treat it. Multiplication is counting. When children count by +4's, etc., they accent the same as counting gymnastics or music. +When a child now counts on its fingers it simply reproduces a +stage in the growth of the civilization of all nations.</p> + +<p>I would emphasize again that during the counting period there +is a somewhat spontaneous development of the number series-idea +which Preyer has discussed in his Arithmogenesis; that an +immense momentum is given by a systematic series of names; +and that these names are generally first learned and applied to +objects later. A lady teacher told me that the Superintendent +did not wish the teachers to allow the children to count on their +fingers, but she failed to see why counting with horse-chestnuts +was any better. Her children could hardly avoid using their +fingers in counting other objects yet they followed the series to +100 without hesitation or reference to their fingers. This spontaneous +counting period, or naming and following the series, should +precede its application to objects." [D.E. Phillips, '97, p. 238.]</p> +</div> + +<h4>THE RATIO IDEA OVEREMPHASIZED</h4> + +<div class="figcenter" style="width: 448px;"> +<img src="images/fig1.jpg" width="448" height="195" alt="Fig. 1." title="Fig. 1." /> +<span class="caption"><span class="smcap">Fig. 1.</span></span> +</div> + +<p><span class='pagenum'><a name="Page_5" id="Page_5">[Pg 5]</a></span></p> + +<div class="pblockquot"><p>"Ratios.—<b>1.</b> Select solids having the relation, or ratio, of +<i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>o</i>, <i>e</i>.</p> + +<p><b>2.</b> Name the solids, <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>o</i>, <i>e</i>.</p> + +<p>The means of expressing must be as freely supplied as the means +of discovery. The pupil is not expected to invent terms.</p> + +<p><b>3.</b> Tell all you can about the relation of these units.</p> + +<p><b>4.</b> Unite units and tell what the sum equals.</p> + +<p><b>5.</b> Make statements like this: <i>o</i> less <i>e</i> equals <i>b</i>.</p> + +<p><b>6.</b> <i>c</i> can be separated into how many <i>d</i>'s? into how +many <i>b</i>'s?</p> + +<p><b>7.</b> <i>c</i> can be separated into how many <i>b</i>'s? What is the name +of the largest unit that can be found in both <i>c</i> and <i>d</i> an exact +number of times?</p> + +<p><b>8.</b> Each of the other units equals what part of <i>c</i>?</p> + +<p><b>9.</b> If <i>b</i> is 1, what is each of the other units?</p> + +<p><b>10.</b> If <i>a</i> is 1, what is each of the other units?</p> + +<p><b>11.</b> If <i>b</i> is 1, how many 1's are there in each of the other units?</p> + +<p><b>12.</b> If <i>d</i> is 1, how many 1's and parts of 1 in each of the other units?</p> + +<p><b>13.</b> 2 is the relation of what units?</p> + +<p><b>14.</b> 3 is the relation of what units?</p> + +<p><b>15.</b> <sup>1</sup>⁄<sub>2</sub> is the relation of what units?</p> + +<p><b>16.</b> <sup>2</sup>⁄<sub>3</sub> is the relation of what units?</p> + +<p><b>17.</b> Which units have the relation <sup>3</sup>⁄<sub>2</sub>?</p> + +<p><b>18.</b> Which unit is 3 times as large as <sup>1</sup>⁄<sub>2</sub> of <i>b</i>?</p> + +<p><b>19.</b> <i>c</i> equals 6 times <sup>1</sup>⁄<sub>3</sub> of what unit?</p> + +<p><b>20.</b> <sup>1</sup>⁄<sub>3</sub> of what unit equals <sup>1</sup>⁄<sub>6</sub> of <i>c</i>?</p> + +<p><b>21.</b> What equals <sup>1</sup>⁄<sub>2</sub> of <i>c</i>? <i>d</i> equals how many sixths of <i>c</i>?</p> + +<p><b>22.</b> <i>o</i> equals 5 times <sup>1</sup>⁄<sub>3</sub> of what unit?</p> + +<p><b>23.</b> <sup>1</sup>⁄<sub>3</sub> of what unit equals <sup>1</sup>⁄<sub>5</sub> of <i>o</i>?</p> + +<p><b>24.</b> <sup>2</sup>⁄<sub>3</sub> of <i>d</i> equals what unit? <i>b</i> equals how many thirds of <i>d</i>?</p> + +<p><b>25.</b> 2 is the ratio of <i>d</i> to <sup>1</sup>⁄<sub>3</sub> of what unit? 3 is the ratio of <i>d</i> to <sup>1</sup>⁄<sub>2</sub> +of what unit?</p> + +<p><b>26.</b> <i>d</i> equals <sup>3</sup>⁄<sub>4</sub> of what unit? <sup>3</sup>⁄<sub>4</sub> is the ratio of what units?" +[Speer, '97, p. 9f.]</p></div> + + +<h4>THE RELATIONAL IDEA OVEREMPHASIZED</h4> + +<div class="pblockquot"> +<p>An inspection of books of the eighties which followed the +"Grube method" (for example, the <i>New Elementary Arithmetic</i> +by E.E. White ['83]) will show undue emphasis on the relational +ideas. There will be over a hundred and fifty successive tasks all, +or nearly all, on + 7 and − 7. There will be much written work +of the sort shown below:</p></div> +<p><span class='pagenum'><a name="Page_6" id="Page_6">[Pg 6]</a></span></p> + +<div class="pblockquot"><p><i>Add:</i></p></div> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="60%"> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>4</td><td align='center'>4</td></tr> +<tr><td align='center'>4</td><td align='center'>1</td><td align='center'>2</td></tr> +<tr><td align='center'>——</td><td align='center'>——</td><td align='center'>——</td></tr> +</table></div> + +<div class="pblockquot"> +<p class="noidt">which must have sorely tried the eyes of all concerned. Pupils +are taught to "give the analysis and synthesis of each of the nine +digits." Yet the author states that he does not carry the principle +of the Grube method "to the extreme of useless repetition +and mechanism."</p></div> + +<p>It should be obvious that all four meanings have claims +upon the attention of the elementary school. Four is the +thing between three and five in the number series; it is the +name for a certain sized collection of discrete objects; it is +also the name for a continuous magnitude equal to four +units—for four quarts of milk in a gallon pail as truly as +for four separate quart-pails of milk; it is also, if we know it +well, the thing got by adding one to three or subtracting +six from ten or taking two two's or half of eight. To know +the meaning of a number means to know somewhat about +it in all of these respects. The difficulty has been the +narrow vision of the extremists. A child must not be left +interminably counting; in fact the one-more-ness of the +number series can almost be had as a by-product. A child +must not be restricted to exercises with collections objectified +<span class='pagenum'><a name="Page_7" id="Page_7">[Pg 7]</a></span> +as in Fig. 2 or stated in words as so many apples, oranges, +hats, pens, etc., when work with measurement of continuous +quantities with varying units—inches, feet, yards, glassfuls, +<span class='pagenum'><a name="Page_8" id="Page_8">[Pg 8]</a></span> +pints, quarts, seconds, minutes, hours, and the like—is so +easy and so significant. On the other hand, the elaboration +of artificial problems with fictitious units of measure just to +have relative magnitudes as in the exercises on page 5 is +a wasteful sacrifice. Similarly, special drills emphasizing +the fact that eighteen is eleven and seven, twelve and six, +three less than twenty-one, and the like, are simply idolatrous; +these facts about eighteen, so far as they are needed, +are better learned in the course of actual column-addition +and -subtraction.</p> + + +<div class="figcenter" style="width: 548px;"> +<img src="images/fig2.jpg" width="548" height="740" alt="Fig. 2." title="Fig. 2." /> +<span class="caption"><span class="smcap">Fig. 2.</span></span> +</div> + + +<h4>ARITHMETICAL LANGUAGE</h4> + +<p>The second improvement to be made in the ordinary +notion of what the functions to be improved are in the case +of arithmetic is to include among these functions the knowledge +of certain words. The understanding of such words +as <i>both</i>, <i>all</i>, <i>in all</i>, <i>together</i>, <i>less</i>, <i>difference</i>, <i>sum</i>, <i>whole</i>, <i>part</i>, +<i>equal</i>, <i>buy</i>, <i>sell</i>, <i>have left</i>, <i>measure</i>, <i>is contained in</i>, and the +like, is necessary in arithmetic as truly as is the understanding +of numbers themselves. It must be provided +for by the school; for pre-school and extra-school training +does not furnish it, or furnishes it too late. It can be +provided for much better in connection with the teaching +of arithmetic than in connection with the teaching of +English.</p> + +<p>It has not been provided for. An examination of the first +fifty pages of eight recent textbooks for beginners in arithmetic +reveals very slight attention to this matter at the +best and no attention at all in some cases. Three of the +books do not even use the word <i>sum</i>, and one uses it only +once in the fifty pages. In all the four hundred pages +the word <i>difference</i> occurs only twenty times. When the +words are used, no great ingenuity or care appears in +<span class='pagenum'><a name="Page_9" id="Page_9">[Pg 9]</a></span> +the means of making sure that their meanings are understood.</p> + +<p>The chief reason why it has not been provided for is precisely +that the common notion of what the functions are that +arithmetic is to develop has left out of account this function +of intelligent response to quantitative terms, other than the +names of the numbers and processes.</p> + +<p>Knowledge of language over a much wider range is a +necessary element in arithmetical ability in so far as the +latter includes ability to solve verbally stated problems. +As arithmetic is now taught, it does include that ability, and +a large part of the time of wise teaching is given to improving +the function 'knowing what a problem states and what it asks +for.' Since, however, this understanding of verbally stated +problems may not be an absolutely necessary element of +arithmetic, it is best to defer its consideration until we have +seen what the general function of problem-solving is.</p> + + +<h4>PROBLEM-SOLVING</h4> + +<p>The third respect in which the function, 'ability in arithmetic,' +needs clearer definition, is this 'problem-solving.' +The aim of the elementary school is to provide for correct +and economical response to genuine problems, such as +knowing the total due for certain real quantities at certain +real prices, knowing the correct change to give or get, +keeping household accounts, calculating wages due, computing +areas, percentages, and discounts, estimating quantities +needed of certain materials to make certain household +or shop products, and the like. Life brings these problems +usually either with a real situation (as when one buys and +counts the cost and his change), or with a situation that one +imagines or describes to himself (as when one figures out +how much money he must save per week to be able to buy +<span class='pagenum'><a name="Page_10" id="Page_10">[Pg 10]</a></span> +a forty-dollar bicycle before a certain date). Sometimes, +however, the problem is described in words to the person +who must solve it by another person (as when a life insurance +agent says, 'You pay only 25 cents a week from now till—and +you get $250 then'; or when an employer says, 'Your +wages would be 9 dollars a week, with luncheon furnished +and bonuses of such and such amounts'). Sometimes also +the problem is described in printed or written words to the +person who must solve it (as in an advertisement or in the +letter of a customer asking for an estimate on this or that). +The problem may be in part real, in part imagined or described +to oneself, and in part described to one orally or in +printed or written words (as when the proposed articles +for purchase lie before one, the amount of money one +has in the bank is imagined, the shopkeeper offers 10 +percent discount, and the printed price list is there to be +read).</p> + +<p>To fit pupils to solve these real, personally imagined, or +self-described problems, and 'described-by-another' problems, +schools have relied almost exclusively on training with +problems of the last sort only. The following page taken +almost at random from one of the best recent textbooks +could be paralleled by thousands of others; and the oral +problems put by teachers have, as a rule, no real situation +supporting them.</p> + +<div class="pblockquot"><p><b>1.</b> At 70 cents per 100 pounds, what will be the amount of +duty on an invoice of 3622 steel rails, each rail being 27 feet long +and weighing 60 pounds to the yard?</p> + +<p><b>2.</b> A man had property valued at $6500. What will be his +taxes at the rate of $10.80 per $1000?</p> + +<p><b>3.</b> Multiply seventy thousand fourteen hundred-thousandths +by one hundred nine millionths, and divide the product by five +hundred forty-five.</p> + +<p><b>4.</b> What number multiplied by 43¾ will produce 265<sup>5</sup>⁄<sub>8</sub>?</p> + +<p><span class='pagenum'><a name="Page_11" id="Page_11">[Pg 11]</a></span><b>5.</b> +What decimal of a bushel is 3 quarts?</p> + +<p><b>6.</b> A man sells <sup>5</sup>⁄<sub>8</sub> of an acre of land for $93.75. What would +be the value of his farm of 150¾ acres at the same rate?</p> + +<p><b>7.</b> A coal dealer buys 375 tons coal at $4.25 per ton of 2240 +pounds. He sells it at $4.50 per ton of 2000 pounds. What is his +profit?</p> + +<p><b>8.</b> Bought 60 yards of cloth at the rate of 2 yards for $5, and +80 yards more at the rate of 4 yards for $9. I immediately sold +the whole of it at the rate of 5 yards for $12. How much did I +gain?</p> + +<p><b>9.</b> A man purchased 40 bushels of apples at $1.50 per bushel. +Twenty-five hundredths of them were damaged, and he sold them +at 20 cents per peck. He sold the remainder at 50 cents per peck. +How much did he gain or lose?</p> + +<p><b>10.</b> If oranges are 37½ cents per dozen, how many boxes, each +containing 480, can be bought for $60?</p> + +<p><b>11.</b> A man can do a piece of work in 18¾ days. What part of +it can he do in 6<sup>2</sup>⁄<sub>3</sub> days?</p> + +<p><b>12.</b> How old to-day is a boy that was born Oct. 29, 1896? +[Walsh, '06, Part I, p. 165.]</p></div> + +<p>As a result, teachers and textbook writers have come to +think of the functions of solving arithmetical problems as +identical with the function of solving the described problems +which they give in school in books, examination papers, and +the like. If they do not think explicitly that this is so, they +still act in training and in testing pupils as if it were so.</p> + +<p>It is not. Problems should be solved in school to the +end that pupils may solve the problems which life offers. +To know what change one should receive after a given real +purchase, to keep one's accounts accurately, to adapt a +recipe for six so as to make enough of the article for four +persons, to estimate the amount of seed required for a plot +of a given size from the statement of the amount required +per acre, to make with surety the applications that the +household, small stores, and ordinary trades require—such +is the ability that the elementary school should develop. +<span class='pagenum'><a name="Page_12" id="Page_12">[Pg 12]</a></span> +Other things being equal, the school should set problems in +arithmetic which life then and later will set, should favor +the situations which life itself offers and the responses which +life itself demands.</p> + +<p>Other things are not always equal. The same amount of +time and effort will often be more productive toward the +final end if directed during school to 'made-up' problems. +The keeping of personal financial accounts as a school +exercise is usually impracticable, partly because some of the +children have no earnings or allowance—no accounts to +keep, and partly because the task of supervising work when +each child has a different problem is too great for the teacher. +The use of real household and shop problems will be easy +only when the school program includes the household arts +and industrial education, and when these subjects themselves +are taught so as to improve the functions used by +real life. Very often the most efficient course is to make +sure that arithmetical procedures are applied to the real +and personally initiated problems which they fit, by having +a certain number of such problems arise and be solved; +then to make sure that the similarity between these real +problems and certain described problems of the textbook +or teacher's giving is appreciated; and then to give the +needed drill work with described problems. In many cases +the school practice is fairly well justified in assuming that +solving described problems will prepare the pupil to solve +the corresponding real problems actually much better than +the same amount of time spent on the real problems themselves.</p> + +<p>All this is true, yet the general principle remains that, +other things being equal, the school should favor real situations, +should present issues as life will present them.</p> + +<p>Where other things make the use of verbally described +<span class='pagenum'><a name="Page_13" id="Page_13">[Pg 13]</a></span> +problems of the ordinary type desirable, these should be +chosen so as to give a maximum of preparation for the real +applications of arithmetic in life. We should not, for +example, carelessly use any problem that comes to mind +in applying a certain principle, but should stop to consider +just what the situations of life really require +and show clearly the application of that principle. +For example, contrast these two problems applying cancellation:—</p> + +<div class="pblockquot"><p>A. A man sold 24 lambs at $18 apiece on each of six days, +and bought 8 pounds of metal with the proceeds. How much did +he pay per ounce for the metal?</p> + +<p>B. How tall must a rectangular tank 16" long by 8" wide be +to hold as much as a rectangular tank 24" by 18" by 6"?</p></div> + +<p class="noidt">The first problem not only presents a situation that +would rarely or never occur, but also takes a way to find +the answer that would not, in that situation, be taken +since the price set by another would determine the +amount.</p> + +<p>Much thought and ingenuity should in the future be +expended in eliminating problems whose solution does not +improve the real function to be improved by applied arithmetic, +or improves it at too great cost, and in devising +problems which prepare directly for life's demands and still +can fit into a curriculum that can be administered by one +teacher in charge of thirty or forty pupils, under the limitations +of school life.</p> + +<p>The following illustrations will to some extent show concretely +what the ability to apply the knowledge and power +represented by abstract or pure arithmetic—the so-called +fundamentals—in solving problems should mean and what +it should not mean.</p> + +<p><span class='pagenum'><a name="Page_14" id="Page_14">[Pg 14]</a></span></p> + +<h4><i>Samples of Desirable Applications of Arithmetic in Problems +where the Situation is Actually Present to +Sense in Whole or in Part</i></h4> + +<p>Keeping the scores and deciding which side beat and by +how much in appropriate classroom games, spelling matches, +and the like.</p> + +<p>Computing costs, making and inspecting change, taking +inventories, and the like with a real or play store.</p> + +<p>Mapping the school garden, dividing it into allotments, +planning for the purchase of seeds, and the like.</p> + +<p>Measuring one's own achievement and progress in tests +of word-knowledge, spelling, addition, subtraction, speed of +writing, and the like. Measuring the rate of improvement +per hour of practice or per week of school life, and the like.</p> + +<p>Estimating costs of food cooked in the school kitchen, +articles made in the school shops, and the like.</p> + +<p>Computing the cost of telegrams, postage, expressage, +for a real message or package, from the published tariffs.</p> + +<p>Computing costs from mail order catalogues and the like.</p> + + +<h4><i>Samples of Desirable Applications of Arithmetic where the +Situation is Not Present to Sense</i></h4> + +<p>The samples given here all concern the subtraction of +fractions. Samples concerning any other arithmetical principle +may be found in the appropriate pages of any text +which contains problem-material selected with consideration +of life's needs.</p> + +<h4>A</h4> + +<div class="pblockquot"> +<p><b>1.</b> Dora is making jelly. The recipe calls for 24 cups of sugar +and she has only 21½. She has no time to go to the store +so she has to borrow the sugar from a neighbor. How +much must she get?</p> + +<p><i>Subtract</i></p> +</div> +<p><span class='pagenum'><a name="Page_15" id="Page_15">[Pg 15]</a></span></p> + + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'>24</td><td align='left'><i>Think "½ and ½ = 1." Write ½.</i></td></tr> +<tr><td align='left'>21½</td><td align='left'><i>Think "2 and 2 = 4." Write the 2.</i></td></tr> +<tr><td align='left'>———</td><td> </td></tr> +<tr><td align='left'> 2½</td><td> </td></tr> +</table></div> + +<div class="pblockquot"> +<p><b>2.</b> A box full of soap weighs 29½ lb. The empty box weighs +3½ lb. How much does the soap alone weigh?</p> + +<p><b>3.</b> On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt. +On July 15 there were just 11½ lb. left. How much had he +used in the two weeks?</p> + +<p><b>4.</b> Grace promised to pick 30 qt. blueberries for her mother. +So far she has picked 18½ qt. How many more quarts must +she pick?</p> +</div> + +<h4>B</h4> + +<div class="pblockquot"><p>This table of numbers tells +what Nell's baby sister Mary +weighed every two months from +the time she was born till she +was a year old.</p></div> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="30%"> +<tr><th colspan='2'>Weight of Mary Adams</th></tr> +<tr><td align='left'><b> When born</b></td><td align='right'>7<sup>3</sup>⁄<sub>8</sub> lb.</td></tr> +<tr><td align='left'><b> 2 months old</b></td><td align='right'>11<sup>1</sup>⁄<sub>4</sub> lb.</td></tr> +<tr><td align='left'><b> 4 months old</b></td><td align='right'>14<sup>1</sup>⁄<sub>8</sub> lb.</td></tr> +<tr><td align='left'><b> 6 months old</b></td><td align='right'>15<sup>3</sup>⁄<sub>4</sub> lb.</td></tr> +<tr><td align='left'><b> 8 months old</b></td><td align='right'>17<sup>5</sup>⁄<sub>8</sub> lb.</td></tr> +<tr><td align='left'><b>10 months old</b></td><td align='right'>19<sup>1</sup>⁄<sub>2</sub> lb.</td></tr> +<tr><td align='left'><b>12 months old</b></td><td align='right'>21<sup>3</sup>⁄<sub>8</sub> lb.</td></tr> +</table></div> + +<div class="pblockquot"> +<p><b>1.</b> How much did the Adams baby gain in the first two months?</p> + +<p><b>2.</b> How much did the Adams baby gain in the second two months?</p> + +<p><b>3.</b> In the third two months?</p> +<p><b>4.</b> In the fourth two months?</p> + +<p><b>5.</b> From the time it was 8 months old till it was 10 months old?</p> + +<p><b>6.</b> In the last two months?</p> + +<p><b>7.</b> From the time it was born till it was 6 months old?</p> +</div> + + +<h4>C</h4> + +<div class="pblockquot"><p><b>1.</b> Helen's exact average for December was 87<sup>1</sup>⁄<sub>3</sub>. Kate's was +84<sup>1</sup>⁄<sub>2</sub>. How much higher was Helen's than Kate's?</p></div> + +<p class="center"><br /> +87<sup>1</sup>⁄<sub>3</sub><br /> +84<sup>1</sup>⁄<sub>2</sub><br /> +——— +</p> + +<div class="pblockquot"><p>How do you think of <sup>1</sup>⁄<sub>2</sub> and <sup>1</sup>⁄<sub>3</sub>?</p> +<p>How do you think of 1<sup>2</sup>⁄<sub>6</sub>?</p> +<p>How do you change the 4?</p> + +<p><b>2.</b> Find the exact average for each girl in the following list. +Write the answers clearly so that you can see them easily. You +will use them in solving problems 3, 4, 5, 6, 7, and 8.</p></div> + + +<p><span class='pagenum'><a name="Page_16" id="Page_16">[Pg 16]</a></span></p> +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><th> </th><th>Alice</th><th>Dora</th><th>Emma</th><th>Grace</th><th>Louise</th><th>Mary</th><th>Nell</th><th>Rebecca</th></tr> +<tr><td align='left'><b>Reading</b></td><td align='center'>91</td><td align='center'>87</td><td align='center'>83</td><td align='center'>81</td><td align='center'>79</td><td align='center'>77</td><td align='center'>76</td><td align='center'>73</td></tr> +<tr><td align='left'><b>Language</b></td><td align='center'>88</td><td align='center'>78</td><td align='center'>82</td><td align='center'>79</td><td align='center'>73</td><td align='center'>78</td><td align='center'>73</td><td align='center'>75</td></tr> +<tr><td align='left'><b>Arithmetic</b></td><td align='center'>89</td><td align='center'>85</td><td align='center'>79</td><td align='center'>75</td><td align='center'>84</td><td align='center'>87</td><td align='center'>89</td><td align='center'>80</td></tr> +<tr><td align='left'><b>Spelling</b></td><td align='center'>90</td><td align='center'>79</td><td align='center'>75</td><td align='center'>80</td><td align='center'>82</td><td align='center'>91</td><td align='center'>68</td><td align='center'>81</td></tr> +<tr><td align='left'><b>Geography</b></td><td align='center'>91</td><td align='center'>87</td><td align='center'>83</td><td align='center'>75</td><td align='center'>78</td><td align='center'>85</td><td align='center'>73</td><td align='center'>79</td></tr> +<tr><td align='left'><b>Writing</b></td><td align='center'>90</td><td align='center'>88</td><td align='center'>75</td><td align='center'>72</td><td align='center'>93</td><td align='center'>92</td><td align='center'>95</td><td align='center'>78</td></tr> +</table></div> + +<div class="pblockquot"> +<p><b>3.</b> Which girl had the highest average?</p> + +<p><b>4.</b> How much higher was her average than the next highest?</p> + +<p><b>5.</b> How much difference was there between the highest and the +lowest girl?</p> + +<p><b>6.</b> Was Emma's average higher or lower than Louise's? How +much?</p> + +<p><b>7.</b> How much difference was there between Alice's average and +Dora's?</p> + +<p><b>8.</b> How much difference was there between Mary's average and +Nell's?</p> + +<p><b>9.</b> Write five other problems about these averages, and solve +each of them.</p> +</div> + +<h4><i>Samples of Undesirable Applications of Arithmetic</i><a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a></h4> + +<div class="pblockquot"> +<p>Will has XXI marbles, XII jackstones, XXXVI pieces of +string. How many things had he?</p> + +<p>George's kite rose CDXXXV feet and Tom's went LXIII feet +higher. How high did Tom's kite rise?</p> + +<p>If from DCIV we take CCIV the result will be a number IV +times as large as the number of dollars Mr. Dane paid for his horse. +How much did he pay for his horse?</p> + +<p>Hannah has <sup>5</sup>⁄<sub>8</sub> of a dollar, Susie <sup>7</sup>⁄<sub>25</sub>, Nellie <sup>3</sup>⁄<sub>4</sub>, Norah <sup>13</sup>⁄<sub>16</sub>. How +much money have they all together?</p> + +<p>A man saves 3<sup>17</sup>⁄<sub>80</sub> dollars a week. How much does he save in +a year?</p> + +<p>A tree fell and was broken into 4 pieces, 13<sup>1</sup>⁄<sub>6</sub> feet, 10<sup>3</sup>⁄<sub>7</sub> feet, 8<sup>1</sup>⁄<sub>2</sub> +feet, and 7<sup>16</sup>⁄<sub>21</sub> feet long. How tall was the tree?<span class='pagenum'><a name="Page_17" id="Page_17">[Pg 17]</a></span></p> + +<p>Annie's father gave her 20 apples to divide among her friends. +She gave each one 2<sup>2</sup>⁄<sub>9</sub> apples apiece. How many playmates had she?</p> + +<p>John had 17<sup>2</sup>⁄<sub>5</sub> apples. He divided his whole apples into fifths. +How many pieces had he in all?</p> + +<p>A landlady has 3<sup>3</sup>⁄<sub>7</sub> pies to be divided among her 8 boarders. +How much will each boarder receive?</p> + +<p>There are twenty columns of spelling words in Mary's lesson +and 16 words in each column. How many words are in her +lesson?</p> + +<p>There are 9 nuts in a pint. How many pints in a pile of +5,888,673 nuts?</p> + +<p>The Adams school contains eight rooms; each room contains +48 pupils; if each pupil has eight cents, how much have they +together?</p> + +<p>A pile of wood in the form of a cube contains 15½ cords. What +are the dimensions to the nearest inch?</p> + +<p>A man 6 ft. high weighs 175 lb. How tall is his wife who is of +similar build, and weighs 125 lb.?</p> + +<p>A stick of timber is in the shape of the frustum of a square +pyramid, the lower base being 22 in. square and the upper 14 in. +square. How many cubic feet in the log, if it is 22 ft. long?</p> + +<p>Mr. Ames, being asked his age, replied: "If you cube one half +of my age and add 41,472 to the result, the sum will be one half the +cube of my age. How old am I?"</p></div> + +<p>These samples, just given, of the kind of problem-solving +that should not be emphasized in school training refer in +some cases to books of forty years back, but the following +represent the results of a collection made in 1910 from books +then in excellent repute. It required only about an hour to +collect them; and I am confident that a thousand such +problems describing situations that the pupil will never +encounter in real life, or putting questions that he will never +be asked in real life, could easily be found in any ten textbooks +of the decade from 1900 to 1910.</p> + +<div class="pblockquot"> +<p>If there are 250 kernels of corn on one ear, how many are there +on 24 ears of corn the same size?</p> + +<p>Maud is four times as old as her sister, who is 4 years old. What +is the sum of their ages?</p> +<p><span class='pagenum'><a name="Page_18" id="Page_18">[Pg 18]</a></span></p> +<p>If the first century began with the year 1, with what year does +it end?</p> + +<p>Every spider has 8 compound eyes. How many eyes have 21 +spiders?</p> + +<p>A nail 4 inches long is driven through a board so that it projects +1.695 inches on one side and 1.428 on the other. How thick is the +board?</p> + +<p>Find the perimeter of an envelope 5 in. by 3¼ in.</p> + +<p>How many minutes in <sup>5</sup>⁄<sub>9</sub> of <sup>9</sup>⁄<sub>4</sub> of an hour?</p> + +<p>Mrs. Knox is <sup>3</sup>⁄<sub>4</sub> as old as Mr. Knox, who is 48 years old. Their +son Edward is <sup>4</sup>⁄<sub>9</sub> as old as his mother. How old is Edward?</p> + +<p>Suppose a pie to be exactly round and 10½ miles in diameter. +If it were cut into 6 equal pieces, how long would the curved edge +of each piece be?</p> + +<p>8<sup>1</sup>⁄<sub>3</sub>% of a class of 36 boys were absent on a rainy day. 33<sup>1</sup>⁄<sub>3</sub>% +of those present went out of the room to the school yard. How +many were left in the room?</p> + +<p>Just after a ton of hay was weighed in market, a horse ate one +pound of it. What was the ratio of what he ate to what was left?</p> + +<p>If a fan having 15 rays opens out so that the outer rays form a +straight line, how many degrees are there between any two adjacent +rays?</p> + +<p>One half of the distance between St. Louis and New Orleans +is 280 miles more than <sup>1</sup>⁄<sub>10</sub> of the distance; what is the distance +between these places?</p> + +<p>If the pressure of the atmosphere is 14.7 lb. per square inch what +is the pressure on the top of a table 1¼ yd. long and <sup>2</sup>⁄<sub>3</sub> yd. wide?</p> + +<p><sup>13</sup>⁄<sub>28</sub> of the total acreage of barley in 1900 was 100,000 acres; +what was the total acreage?</p> + +<p>What is the least number of bananas that a mother can exactly +divide between her 2 sons, or among her 4 daughters, or among all +her children?</p> + +<p>If Alice were two years older than four times her actual age she +would be as old as her aunt, who is 38 years old. How old is +Alice?</p> + +<p>Three men walk around a circular island, the circumference of +which is 360 miles. A walks 15 miles a day, B 18 miles a day, and +C 24 miles a day. If they start together and walk in the same direction, +how many days will elapse before they will be together again?</p></div> + +<p>With only thirty or forty dollars a year to spend on a +pupil's education, of which perhaps eight dollars are spent<span class='pagenum'><a name="Page_19" id="Page_19">[Pg 19]</a></span> +on improving his arithmetical abilities, the immediate +guidance of his responses to real situations and personally +initiated problems has to be supplemented largely by +guidance of his responses to problems described in words, +diagrams, pictures, and the like. Of these latter, words will +be used most often. As a consequence the understanding +of the words used in these descriptions becomes a part +of the ability required in arithmetic. Such word knowledge +is also required in so far as the problems to be solved in real +life are at times described, as in advertisements, business +letters, and the like.</p> + +<p>This is recognized by everybody in the case of words like +<i>remainder</i>, <i>profit</i>, <i>loss</i>, <i>gain</i>, <i>interest</i>, <i>cubic capacity</i>, <i>gross</i>, +<i>net</i>, and <i>discount</i>, but holds equally of <i>let</i>, <i>suppose</i>, <i>balance</i>, +<i>average</i>, <i>total</i>, <i>borrowed</i>, <i>retained</i>, and many such semi-technical +words, and may hold also of hundreds of other +words unless the textbook and teacher are careful to use +only words and sentence structures which daily life and the +class work in English have made well known to the pupils. +To apply arithmetic to a problem a pupil must understand +what the problem is; problem-solving depends on problem-reading. +In actual school practice training in problem-reading +will be less and less necessary as we get rid of problems +to be solved simply for the sake of solving them, +unnecessarily unreal problems, and clumsy descriptions, +but it will remain to some extent as an important joint task +for the 'arithmetic' and 'reading' of the elementary school.</p> + + +<h4>ARITHMETICAL REASONING</h4> + +<p>The last respect in which the nature of arithmetical +abilities requires definition concerns arithmetical reasoning. +An adequate treatment of the reasoning that may be +expected of pupils in the elementary school and of the most +<span class='pagenum'><a name="Page_20" id="Page_20">[Pg 20]</a></span> +efficient ways to encourage and improve it cannot be given +until we have studied the formation of habits. For reasoning +is essentially the organization and control of habits of +thought. Certain matters may, however, be decided here. +The first concerns the use of computation and problems +merely for discipline,—that is, the emphasis on training +in reasoning regardless of whether the problem is otherwise +worth reasoning about. It used to be thought that the mind +was a set of faculties or abilities or powers which grew strong +and competent by being exercised in a certain way, no +matter on what they were exercised. Problems that could +not occur in life, and were entirely devoid of any worthy +interest, save the intellectual interest in solving them, were +supposed to be nearly or quite as useful in training the mind +to reason as the genuine problems of the home, shop, or +trade. Anything that gave the mind a chance to reason +would do; and pupils labored to find when the minute hand +and hour hand would be together, or how many sheep a +shepherd had if half of what he had plus ten was one third +of twice what he had!</p> + +<p>We now know that the training depends largely on the +particular data used, so that efficient discipline in reasoning +requires that the pupil reason about matters of real importance. +There is no magic essence or faculty of reasoning that +works in general and irrespective of the particular facts and relations +reasoned about. So we should try to find problems which +not only stimulate the pupil to reason, but also direct his +reasoning in useful channels and reward it by results that are +of real significance. We should replace the purely disciplinary +problems by problems that are also valuable as special +training for important particular situations of life. Reasoning +sought for reasoning's sake alone is too wasteful an expenditure +of time and is also likely to be inferior as reasoning.</p> + +<p><span class='pagenum'><a name="Page_21" id="Page_21">[Pg 21]</a></span></p> +<p>The second matter concerns the relative merits of 'catch' +problems, where the pupil has to go against some customary +habit of thinking, and what we may call 'routine' problems, +where the regular ways of thinking that have served him in +the past will, except for some blunder, guide him rightly.</p> + +<p>Consider, for example, these four problems:</p> + +<div class="pblockquot"><p><b>1.</b> "A man bought ten dozen eggs for $2.50 and sold them for +30 cents a dozen. How many cents did he lose?"</p> + +<p><b>2.</b> "I went into Smith's store at 9 <span class="smcap">A.M.</span> and remained until +10 <span class="smcap">A.M.</span> I bought six yards of gingham at 40 cents a yard and +three yards of muslin at 20 cents a yard and gave a $5.00 bill. +How long was I in the store?"</p> + +<p><b>3.</b> "What must you divide 48 by to get half of twice 6?"</p> + +<p><b>4.</b> "What must you add to 19 to get 30?"</p></div> + +<p>The 'catch' problem is now in disrepute, the wise teacher +feeling by a sort of intuition that to willfully require a pupil +to reason to a result sharply contrary to that to which previous +habits lead him is risky. The four illustrations just +given show, however, that mere 'catchiness' or 'contra-previous-habit-ness' +in a problem is not enough to condemn +it. The fourth problem is a catch problem, but so useful a +one that it has been adopted in many modern books as a +routine drill! The first problem, on the contrary, all, save +those who demand no higher criterion for a problem than +that it make the pupil 'think,' would reject. It demands the +reversal of fixed habits <i>to no valid purpose</i>; for in life the +question in such case would never (or almost never) be +'How many cents did he lose?' but 'What was the result?' +or simply 'What of it?' This problem weakens without excuse +the child's confidence in the training he has had. Problems +like (2) are given by teachers of excellent reputation, +but probably do more harm than good. If a pupil should +interrupt his teacher during the recitation in arithmetic by +<span class='pagenum'><a name="Page_22" id="Page_22">[Pg 22]</a></span> +saying, "I got up at 7 o'clock to multiply 9 by 2¾ and got 24¾ +for my answer; was that the right time to get up?" the +teacher would not thank fortune for the stimulus to thought +but would think the child a fool. Such catch questions may +be fairly useful as an object lesson on the value of search +for the essential element in a situation if a great variety of +them are given one after another with routine problems +intermixed and with warning of the general nature of the +exercise at the beginning. Even so, it should be remembered +that reasoning should be chiefly a force organizing habits, +not opposing them; and also that there are enough bad +habits to be opposed to give all necessary training. Fabricated +puzzle situations wherein a peculiar hidden element +of the situation makes the good habits called up by the +situation misleading are useful therefore rather as a relief +and amusing variation in arithmetical work than as stimuli +to thought.</p> + +<p>Problems like the third quoted above we might call puzzling +rather than 'catch' problems. They have value as +drills in analysis of a situation into its elements that will +amuse the gifted children, and as tests of certain abilities. +They also require that of many confusing habits, the right one +be chosen, rather than that ordinary habits be set aside by +some hidden element in the situation. Not enough is known +about their effect to enable us to decide whether or not the +elementary school should include special facility with them +as one of the arithmetical functions that it specially trains.</p> + +<p>The fourth 'catch' quoted above, which all would admit +is a good problem, is good because it opposes a good habit +for the sake of another good habit, forces the analysis of an +element whose analysis life very much requires, and does it +with no obvious waste. It is not safe to leave a child with +the one habit of responding to 'add, 19, 30' by 49, for in<span class='pagenum'><a name="Page_23" id="Page_23">[Pg 23]</a></span> +life the 'have 19, must get .... to have 30' situation is very +frequent and important.</p> + +<p>On the whole, the ordinary problems which ordinary life +proffers seem to be the sort that should be reasoned out, +though the elementary school may include the less noxious +forms of pure mental gymnastics for those pupils who like +them.</p> + + +<h4>SUMMARY</h4> + +<p>These discussions of the meanings of numbers, the linguistic +demands of arithmetic, the distinction between +scholastic and real applications of arithmetic, and the possible +restrictions of training in reasoning,—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:—</p> + +<p>(1) Working knowledge of the meanings of numbers as +names for certain sized collections, for certain relative magnitudes, +the magnitude of unity being known, and for certain +centers or nuclei of relations to other numbers.</p> + +<p>(2) Working knowledge of the system of decimal notation.</p> + +<p>(3) Working knowledge of the meanings of addition, +subtraction, multiplication, and division.</p> + +<p>(4) Working knowledge of the nature and relations of +certain common measures.</p> + +<p>(5) Working ability to add, subtract, multiply, and divide +with integers, common and decimal fractions, and denominate +numbers, all being real positive numbers.</p> + +<p>(6) Working knowledge of words, symbols, diagrams, and<span class='pagenum'><a name="Page_24" id="Page_24">[Pg 24]</a></span> +the like as required by life's simpler arithmetical demands +or by economical preparation therefor.</p> + +<p>(7) The ability to apply all the above as required by life's +simpler arithmetical demands or by economical preparation +therefor, including (7 <i>a</i>) certain specific abilities to solve +problems concerning areas of rectangles, volumes of rectangular +solids, percents, interest, and certain other common +occurrences in household, factory, and business life.</p> + + +<h4>THE SOCIOLOGY OF ARITHMETIC</h4> + +<div class="pblockquot"> +<p>The phrase 'life's simpler arithmetical demands' is necessarily +left vague. Just what use is being made of arithmetic in this +country in 1920 by each person therein, we know only very roughly. +What may be called a 'sociology' of arithmetic is very much needed +to investigate this matter. For rare or difficult demands the elementary +school should not prepare; there are too many other +desirable abilities that it should improve.</p> + +<p>A most interesting beginning at such an inventory of the actual +uses of arithmetic has been made by Wilson ['19] and Mitchell.<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> +Although their studies need to be much extended and checked by +other methods of inquiry, two main facts seem fairly certain.</p> + +<p>First, the great majority of people in the great majority of their +doings use only very elementary arithmetical processes. In +1737 cases of addition reported by Wilson, seven eighths were of +five numbers or less. Over half of the multipliers reported were +one-figure numbers. Over 95 per cent of the fractions operated +with were included in this list: <sup>1</sup>⁄<sub>2</sub> <sup>1</sup>⁄<sub>4</sub> <sup>3</sup>⁄<sub>4</sub> +<sup>1</sup>⁄<sub>3</sub> <sup>2</sup>⁄<sub>3</sub> <sup>1</sup>⁄<sub>8</sub> <sup>3</sup>⁄<sub>8</sub> +<sup>1</sup>⁄<sub>5</sub> <sup>2</sup>⁄<sub>5</sub> <sup>4</sup>⁄<sub>5</sub>. Three fourths +of all the cases reported were simple one-step computations with +integers or United States money.</p> + +<p>Second, they often use these very elementary processes, not +because such are the quickest and most convenient, but because +they have lost, or maybe never had, mastery of the more advanced +processes which would do the work better. The 5 and 10 cent +stores, the counter with "Anything on this counter for 25¢," and +the arrangements for payments on the installment plan are familiar +instances of human avoidance of arithmetic. Wilson found very +slight use of decimals; and Mitchell found men computing with +<span class='pagenum'><a name="Page_25" id="Page_25">[Pg 25]</a></span> +49ths as common fractions when the use of decimals would have +been more efficient. If given 120 seconds to do a test like that +shown below, leading lawyers, physicians, manufacturers, and +business men and their wives will, according to my experience, get +only about half the work right. Many women, finding on their +meat bill "7<sup>3</sup>⁄<sub>8</sub> lb. roast beef $2.36," will spend time and money to +telephone the butcher asking how much roast beef was per pound, +because they have no sure power in dividing by a mixed number.</p> +</div> + +<div class="pblockquot"> + +<p class="tabcap">Test</p> + +<p>Perform the operations indicated. Express all fractions in +answers in lowest terms.</p> + +<div class="blockquot"><p> +<i>Add:</i></p> + +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'><sup>3</sup>⁄<sub>4</sub> + <sup>1</sup>⁄<sub>6</sub> + .25 </td> +<td align='left' class='bb'> + 4 yr. 6 mo.<br /> + 1 yr. 2 mo.<br /> + 6 yr. 9 mo.<br /> + 3 yr. 6 mo.<br /> + 4 yr. 5 mo. +</td></tr> +</table> + + +<p><i>Subtract:</i></p> + +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'>8.6 − 6.05007 +<sup>7</sup>⁄<sub>8</sub> − <sup>2</sup>⁄<sub>3</sub> = +5<sup>7</sup>⁄<sub>16</sub> − 2<sup>3</sup>⁄<sub>16</sub> =</td></tr> +</table> + + +<p><i>Multiply:</i></p> + +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'>7 × 8 × 4½ = </td> +<td align='left' valign='top' class='bb'>29 ft.</td><td align='left' class='bb'>6 in.<br />8</td></tr> +</table> + +<p><i>Divide:</i></p> +<p class="center">4½ ÷ 7 =</p> +</div> + +<p>It seems probable that the school training in arithmetic of the +past has not given enough attention to perfecting the more elementary +abilities. And we shall later find further evidence of this. +On the other hand, the fact that people in general do not at present +use a process may not mean that they ought not to use it.</p> + +<p>Life's simpler arithmetical demands certainly do not include +matters like the rules for finding cube root or true discount, which +no sensible person uses. They should not include matters like +computing the lateral surface or volume of pyramids and cones, +or knowing the customs of plasterers and paper hangers, which +are used only by highly specialized trades. They should not include +matters like interest on call loans, usury, exact interest, and<span class='pagenum'><a name="Page_26" id="Page_26">[Pg 26]</a></span> +the rediscounting of notes, which concern only brokers, bank clerks, +and rich men. They should not include the technique of customs +which are vanishing from efficient practice, such as simple interest +on amount for times longer than a year, days of grace, or extremes +and means in proportions. They should not include any elaborate +practice with very large numbers, or decimals beyond thousandths, +or the addition and subtraction of fractions which not one person +in a hundred has to add or subtract oftener than once a year.</p> + +<p>When we have an adequate sociology of arithmetic, stating +accurately just who should use each arithmetical ability and how +often, we shall be able to define the task of the elementary school +in this respect. For the present, we may proceed by common +sense, guided by two limiting rules. The first is,—"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."</p> +</div> + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_27" id="Page_27">[Pg 27]</a></span></p> +<h2><a name="CHAPTER_II" id="CHAPTER_II"></a>CHAPTER II</h2> + +<h2>THE MEASUREMENT OF ARITHMETICAL ABILITIES</h2> + + +<p>One of the best ways to clear up notions of what the +functions are which schools should develop and improve +is to get measures of them. If any given knowledge or +skill or power or ideal exists, it exists in some amount. A +series of amounts of it, varying from less to more, defines +the ability itself in a way that no general verbal description +can do. Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., +etc., helps to tell us what we mean by weight. By finding a +series of words like <i>only</i>, <i>smoke</i>, <i>another</i>, <i>pretty</i>, <i>answer</i>, <i>tailor</i>, +<i>circus</i>, <i>telephone</i>, <i>saucy</i>, and <i>beginning</i>, which are spelled +correctly by known and decreasing percentages of children +of the same age, or of the same school grade, we know better +what we mean by 'spelling-difficulty.' Indeed, until we +can measure the efficiency and improvement of a function, +we are likely to be vague and loose in our ideas of what the +function is.</p> + + +<h4>A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: +THE ABILITY TO ADD INTEGERS</h4> + +<p>Consider first, as a sample, the measurement of ability +to add integers.</p> + +<p>The following were the examples used in the measurements +made by Stone ['08]:<span class='pagenum'><a name="Page_28" id="Page_28">[Pg 28]</a></span></p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="40%"> +<tr><td align='right'> </td><td align='right'>596</td><td align='right'>4695</td></tr> +<tr><td align='right'> </td><td align='right'>428</td><td align='right'>872</td></tr> +<tr><td align='right'>2375</td><td align='right'>94</td><td align='right'>7948</td></tr> +<tr><td align='right'>4052</td><td align='right'>75</td><td align='right'>6786</td></tr> +<tr><td align='right'>6354</td><td align='right'>304</td><td align='right'>567</td></tr> +<tr><td align='right'>260</td><td align='right'>645</td><td align='right'>858</td></tr> +<tr><td align='right'>5041</td><td align='right'>984</td><td align='right'>9447</td></tr> +<tr><td align='right'>1543</td><td align='right'>897</td><td align='right'>7499</td></tr> +<tr><td align='right'>———</td><td align='right'>———</td><td align='right'>———</td></tr> +</table></div> + +<p>The scoring was as follows: Credit of 1 for each column +added correctly. Stone combined measures of other abilities +with this in a total score for amount done correctly in 12 +minutes. Stone also scored the correctness of the additions +in certain work in multiplication.</p> + +<p>Courtis uses a sheet of twenty-four tasks or 'examples,' +each consisting of the addition of nine three-place numbers +as shown below. Eight minutes is allowed. He scores the +amount done by the number of examples, and also scores +the number of examples done correctly, but does not suggest +any combination of these two into a general-efficiency +score.</p> + +<p class="center"><br /> +927<br /> +379<br /> +756<br /> +837<br /> +924<br /> +110<br /> +854<br /> +965<br /> +344<br /> +———<br /> +</p> + +<p>The author long ago proposed that pupils be measured +also with series like <i>a</i> to <i>g</i> shown below, in which the difficulty +increases step by step.</p> + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>a.</i></td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>1</td><td align='right'>2</td></tr> +<tr><td align='right'></td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>4</td><td align='right'>5</td><td align='right'>5</td><td align='right'>1</td></tr> +<tr><td align='right'></td><td align='right'>4</td><td align='right'>2</td><td align='right'>3</td><td align='right'>3</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>2</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> +<p><span class='pagenum'><a name="Page_29" id="Page_29">[Pg 29]</a></span></p> + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>b.</i></td><td align='right'>21</td><td align='right'>32</td><td align='right'>12</td><td align='right'>24</td><td align='right'>34</td><td align='right'>34</td><td align='right'>22</td><td align='right'>12</td></tr> +<tr><td align='right'></td><td align='right'>23</td><td align='right'>12</td><td align='right'>52</td><td align='right'>31</td><td align='right'>33</td><td align='right'>12</td><td align='right'>23</td><td align='right'>13</td></tr> +<tr><td align='right'></td><td align='right'>24</td><td align='right'>25</td><td align='right'>15</td><td align='right'>14</td><td align='right'>32</td><td align='right'>23</td><td align='right'>43</td><td align='right'>61</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> + + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>c.</i></td><td align='right'>22</td><td align='right'>3</td><td align='right'>4</td><td align='right'>35</td><td align='right'>32</td><td align='right'>83</td><td align='right'>22</td><td align='right'> 3</td></tr> +<tr><td align='right'></td><td align='right'> 3</td><td align='right'>31</td><td align='right'>3</td><td align='right'>2</td><td align='right'>33</td><td align='right'>11</td><td align='right'>3</td><td align='right'>21</td></tr> +<tr><td align='right'></td><td align='right'>38</td><td align='right'>45</td><td align='right'>52</td><td align='right'>52</td><td align='right'>2</td><td align='right'>4</td><td align='right'>33</td><td align='right'>64</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> + + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>d.</i></td><td align='right'>30</td><td align='right'>20</td><td align='right'>10</td><td align='right'>22</td><td align='right'>10</td><td align='right'>20</td><td align='right'>52</td><td align='right'>12</td></tr> +<tr><td align='right'></td><td align='right'>20</td><td align='right'>50</td><td align='right'>40</td><td align='right'>43</td><td align='right'>30</td><td align='right'>4</td><td align='right'>6</td><td align='right'>22</td></tr> +<tr><td align='right'></td><td align='right'>40</td><td align='right'>17</td><td align='right'>24</td><td align='right'>13</td><td align='right'>40</td><td align='right'>23</td><td align='right'>30</td><td align='right'>44</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> + + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>e.</i></td><td align='right'></td><td align='right'>4</td><td align='right'>5</td><td align='right'>20</td><td align='right'>12</td><td align='right'>12</td><td align='right'>20</td><td align='right'>10</td></tr> +<tr><td align='right'></td><td align='right'>20</td><td align='right'>30</td><td align='right'>3</td><td align='right'>40</td><td align='right'>4</td><td align='right'>11</td><td align='right'>20</td><td align='right'>20</td></tr> +<tr><td align='right'></td><td align='right'>10</td><td align='right'>30</td><td align='right'>20</td><td align='right'>4</td><td align='right'>1</td><td align='right'>23</td><td align='right'>7</td><td align='right'>2</td></tr> +<tr><td align='right'></td><td align='right'>20</td><td align='right'>2</td><td align='right'>40</td><td align='right'>23</td><td align='right'>40</td><td align='right'>11</td><td align='right'>10</td><td align='right'>30</td></tr> +<tr><td align='right'></td><td align='right'>20</td><td align='right'>20</td><td align='right'>10</td><td align='right'>11</td><td align='right'>20</td><td align='right'>22</td><td align='right'>30</td><td align='right'>25</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> + + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>f.</i></td><td align='right'></td><td align='right'></td><td align='right'>19</td><td align='right'>9</td><td align='right'></td><td align='right'></td><td align='right'>9</td></tr> +<tr><td align='right'></td><td align='right'>14</td><td align='right'>2</td><td align='right'>19</td><td align='right'>24</td><td align='right'>9</td><td align='right'>4</td><td align='right'>13</td></tr> +<tr><td align='right'></td><td align='right'>9</td><td align='right'>14</td><td align='right'>13</td><td align='right'>12</td><td align='right'>13</td><td align='right'>13</td><td align='right'>9</td><td align='right'>14</td></tr> +<tr><td align='right'></td><td align='right'>17</td><td align='right'>23</td><td align='right'>13</td><td align='right'>15</td><td align='right'>15</td><td align='right'>34</td><td align='right'>12</td><td align='right'>25</td></tr> +<tr><td align='right'></td><td align='right'>26</td><td align='right'>29</td><td align='right'>18</td><td align='right'>19</td><td align='right'>25</td><td align='right'>28</td><td align='right'>18</td><td align='right'>39</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> + + +<p> </p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><td align='right'><i>g.</i></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'>13</td></tr> +<tr><td align='right'></td><td align='right'></td><td align='right'>13</td><td align='right'></td><td align='right'>9</td><td align='right'>14</td><td align='right'>12</td><td align='right'>9</td></tr> +<tr><td align='right'></td><td align='right'></td><td align='right'>9</td><td align='right'></td><td align='right'>13</td><td align='right'>12</td><td align='right'>9</td><td align='right'>14</td><td align='right'>24</td></tr> +<tr><td align='right'></td><td align='right'>23</td><td align='right'>19</td><td align='right'>19</td><td align='right'>29</td><td align='right'>9</td><td align='right'>9</td><td align='right'>13</td><td align='right'>21</td></tr> +<tr><td align='right'></td><td align='right'>28</td><td align='right'>26</td><td align='right'>26</td><td align='right'>14</td><td align='right'>8</td><td align='right'>8</td><td align='right'>29</td><td align='right'>23</td></tr> +<tr><td align='right'></td><td align='right'>29</td><td align='right'>16</td><td align='right'>15</td><td align='right'>19</td><td align='right'>17</td><td align='right'>19</td><td align='right'>19</td><td align='right'>22</td></tr> +<tr><td align='right'></td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td><td align='right'>—</td></tr> +</table></div> + +<p>Woody ['16] has constructed his well-known tests on this +principle, though he uses only one example at each step of +difficulty instead of eight or ten as suggested above. His +test, so far as addition of integers goes, is:—</p> +<p><span class='pagenum'><a name="Page_30" id="Page_30">[Pg 30]</a></span></p> + +<p class="tabcap">SERIES A. ADDITION SCALE (in part)</p> +<p class="tabcap">By Clifford Woody</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><th align='right'>(1)</th><th align='right'>(2)</th><th align='right'>(3)</th><th align='right'>(4)</th><th align='right'>(5)</th><th align='right'>(6)</th><th align='center'>(7)</th><th align='center'>(8)</th><th align='right'>(9)</th></tr> +<tr> + <td align='right'>2<br />3<br />—</td> + <td align='right'>2<br />4<br />3<br />—</td> + <td align='right'>17<br />2<br />—</td> + <td align='right'>53<br />45<br />—</td> + <td align='right'>72<br />26<br />—</td> + <td align='right'>60<br />37<br />—</td> + <td align='center'>3 + 1 =</td> + <td align='center'>2 + 5 + 1 =</td> + <td align='right'>20<br />10<br />2<br />30<br />25<br />—</td> +</tr> +<tr><th align='right'>(10)</th><th align='right'>(11)</th><th align='right'>(12)</th><th align='right'>(13)</th><th align='center'>(14)</th><th align='right'>(15)</th><th align='right'>(16)</th><th align='right'>(17)</th><th align='right'>(18)</th></tr> +<tr> + <td align='right'>21<br />33<br />35<br />—</td> + <td align='right'>32<br />59<br />17<br />—</td> + <td align='right'>43<br />1<br />2<br />13<br />—</td> + <td align='right'>23<br />25<br />16<br />—</td> + <td align='center'>25 + 42 =</td> + <td align='right'>100<br />33<br />45<br />201<br />46<br />—</td> + <td align='right'>9<br />24<br />12<br />15<br />19<br />—</td> + <td align='right'>199<br />194<br />295<br />156<br />——</td> + <td align='right'>2563<br />1387<br />4954<br />2065<br />——</td> +</tr> +<tr><th align='right'>(19)</th><th align='right'>(20)</th><th align='right'>(21)</th><th align='right'>(22)</th></tr> +<tr> + <td align='right'>$ .75<br />1.25<br />.49<br />——</td> + <td align='right'>$12.50<br />16.75<br />15.75<br />——</td> + <td align='right'>$8.00<br />5.75<br />2.33<br />4.16<br />.94<br />6.32<br />——</td> + <td align='right'>547<br />197<br />685<br />678<br />456<br />393<br />525<br />240<br />152<br />——</td> +</tr> +</table></div> + +<p><span class='pagenum'><a name="Page_31" id="Page_31">[Pg 31]</a></span></p> + +<p>In his original report, Woody gives no scheme for scoring +an individual, wisely assuming that, with so few samples at +each degree of difficulty, a pupil's score would be too unreliable +for individual diagnosis. The test is reliable for a +class; and for a class Woody used the degree of difficulty +such that a stated fraction of the class can do the work +correctly, if twenty minutes is allowed for the thirty-eight +examples of the entire test.</p> + +<p>The measurement of even so simple a matter as the efficiency +of a pupil's responses to these tests in adding integers +is really rather complex. There is first of all the problem +of combining speed and accuracy into some single estimate. +Stone gives no credit for a column unless it is correctly +added. Courtis evades the difficulty by reporting both +number done and number correct. The author's scheme, +which gives specified weights to speed and accuracy at each +step of the series, involves a rather intricate computation.</p> + +<p>This difficulty of equating speed and accuracy in adding +means precisely that we have inadequate notions of what the +ability is that the elementary school should improve. Until, +for example, we have decided whether, for a given group +of pupils, fifteen Courtis attempts with ten right, is or is +not a better achievement than ten Courtis attempts with +nine right, we have not decided just what the business of +the teacher of addition is, in the case of that group of pupils.</p> + +<p>There is also the difficulty of comparing results when +short and long columns are used. Correctness with a short +column, say of five figures, testifies to knowledge of the process +and to the power to do four successive single additions +without error. Correctness with a long column, say of ten +digits, testifies to knowledge of the process and to the +power to do nine successive single additions without error. +Now if a pupil's precision was such that on the average he<span class='pagenum'><a name="Page_32" id="Page_32">[Pg 32]</a></span> +made one mistake in eight single additions, he would get +about half of his five-digit columns right and almost none of +his ten-digit columns right. (He would do this, that is, +if he added in the customary way. If he were taught to +check results by repeated addition, by adding in half-columns +and the like, his percentages of accurate answers might be +greatly increased in both cases and be made approximately +equal.) Length of column in a test of addition under +ordinary conditions thus automatically overweights precision +in the single additions as compared with knowledge of +the process, and ability at carrying.</p> + +<p>Further, in the case of a column of whatever size, the +result as ordinarily scored does not distinguish between one, +two, three, or more (up to the limit) errors in the single +additions. Yet, obviously, a pupil who, adding with ten-digit +columns, has half of his answer-figures wrong, probably +often makes two or more errors within a column, whereas a +pupil who has only one column-answer in ten wrong, probably +almost never makes more than one error within a +column. A short-column test is then advisable as a means +of interpreting the results of a long-column test.</p> + +<p>Finally, the choice of a short-column or of a long-column +test is indicative of the measurer's notion of the kind of +efficiency the world properly demands of the school. Twenty +years ago the author would have been readier to accept a +long-column test than he now is. In the world at large, +long-column addition is being more and more done by +machine, though it persists still in great frequency in the +bookkeeping of weekly and monthly accounts in local +groceries, butcher shops, and the like.</p> + +<p>The search for a measure of ability to add thus puts the +problem of speed <i>versus</i> precision, and of short-column +<i>versus</i> long-column additions clearly before us. The latter<span class='pagenum'><a name="Page_33" id="Page_33">[Pg 33]</a></span> +problem has hardly been realized at all by the ordinary +definitions of ability to add.</p> + +<p>It may be said further that the measurement of ability to +add gives the scientific student a shock by the lack of precision +found everywhere in schools. Of what value is it to a +graduate of the elementary school to be able to add with +examples like those of the Courtis test, getting only eight +out of ten right? Nobody would pay a computer for that +ability. The pupil could not keep his own accounts with it. +The supposed disciplinary value of habits of precision +runs the risk of turning negative in such a case. It appears, +at least to the author, imperative that checking should be +taught and required until a pupil can add single columns of +ten digits with not over one wrong answer in twenty columns. +Speed is useful, especially indirectly as an indication of +control of the separate higher-decade additions, but the +social demand for addition below a certain standard of +precision is <i>nil</i>, and its disciplinary value is <i>nil</i> or negative. +This will be made a matter of further study later.</p> + + +<h4>MEASUREMENTS OF ABILITIES IN COMPUTATION</h4> + +<p>Measurements of these abilities may be of two sorts—(1) +of the speed and accuracy shown in doing one same sort +of task, as illustrated by the Courtis test for addition shown +on page 28; and (2) of how hard a task can be done perfectly +(or with some specified precision) within a certain +assigned time or less, as illustrated by the author's rough +test for addition shown on pages 28 and 29, and by the +Woody tests, when extended to include alternative forms.</p> + +<p>The Courtis tests, originated as an improvement on the +Stone tests and since elaborated by the persistent devotion +of their author, are a standard instrument of the first +sort for measuring the so-called 'fundamental' arithmetical<span class='pagenum'><a name="Page_34" id="Page_34">[Pg 34]</a></span> +abilities with integers. They are shown on this and the +following page.</p> + +<p>Tests of the second sort are the Woody tests, which +include operations with integers, common and decimal +fractions, and denominate numbers, the Ballou test for +common fractions ['16], and the "Ladder" exercises of the +Thorndike arithmetics. Some of these are shown on pages +36 to 41.</p> + + +<p class="tabcap">Courtis Test</p> + +<p class="tabcap">Arithmetic. Test No. 1. Addition</p> + +<p class="center">Series B</p> + +<div class="pblockquot"> +<p>You will be given eight minutes to find the answers to as many +of these addition examples as possible. Write the answers on this +paper directly underneath the examples. You are not expected +to be able to do them all. You will be marked for both speed and +accuracy, but it is more important to have your answers right than +to try a great many examples.</p></div> + + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="80%"> +<tr><td align='right'>927</td><td align='right'>297</td><td align='right'>136</td><td align='right'>486</td><td align='right'>384</td><td align='right'>176</td><td align='right'>277</td><td align='right'>837</td></tr> +<tr><td align='right'>379</td><td align='right'>925</td><td align='right'>340</td><td align='right'>765</td><td align='right'>477</td><td align='right'>783</td><td align='right'>445</td><td align='right'>882</td></tr> +<tr><td align='right'>756</td><td align='right'>473</td><td align='right'>988</td><td align='right'>524</td><td align='right'>881</td><td align='right'>697</td><td align='right'>682</td><td align='right'>959</td></tr> +<tr><td align='right'>837</td><td align='right'>983</td><td align='right'>386</td><td align='right'>140</td><td align='right'>266</td><td align='right'>200</td><td align='right'>594</td><td align='right'>603</td></tr> +<tr><td align='right'>924</td><td align='right'>315</td><td align='right'>353</td><td align='right'>812</td><td align='right'>679</td><td align='right'>366</td><td align='right'>481</td><td align='right'>118</td></tr> +<tr><td align='right'>110</td><td align='right'>661</td><td align='right'>904</td><td align='right'>466</td><td align='right'>241</td><td align='right'>851</td><td align='right'>778</td><td align='right'>781</td></tr> +<tr><td align='right'>854</td><td align='right'>794</td><td align='right'>547</td><td align='right'>355</td><td align='right'>796</td><td align='right'>535</td><td align='right'>849</td><td align='right'>756</td></tr> +<tr><td align='right'>965</td><td align='right'>177</td><td align='right'>192</td><td align='right'>834</td><td align='right'>850</td><td align='right'>323</td><td align='right'>157</td><td align='right'>222</td></tr> +<tr><td align='right'>344</td><td align='right'>124</td><td align='right'>439</td><td align='right'>567</td><td align='right'>733</td><td align='right'>229</td><td align='right'>953</td><td align='right'>525</td></tr> +<tr><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td></tr> +</table></div> + +<div class="pblockquot"> +<p class="noidt">and sixteen more addition examples of nine three-place numbers.</p> +</div> + +<p class="tabcap">Courtis Test</p> + +<p class="tabcap">Arithmetic. Test No. 2. Subtraction</p> + +<p class="center">Series B</p> + +<div class="pblockquot"> +<p>You will be given four minutes to find the answers to as many +of these subtraction examples as possible. Write the answers +on this paper directly underneath the examples. You are not<span class='pagenum'><a name="Page_35" id="Page_35">[Pg 35]</a></span> +expected to be able to do them all. You will be marked for both +speed and accuracy, but it is more important to have your answers +right than to try a great many examples.</p></div> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="80%"> +<tr><td align='right'>107795491</td><td align='right'>75088824</td><td align='right'>91500053</td><td align='right'>87939983</td></tr> +<tr><td align='right'>77197029</td><td align='right'>57406394</td><td align='right'>19901563</td><td align='right'>72207316</td></tr> +<tr><td align='right'>—————</td><td align='right'>—————</td><td align='right'>—————</td><td align='right'>—————</td></tr> +</table></div> +<div class="pblockquot"><p class="noidt">and twenty more tasks of the same sort.</p></div> + + +<p class="tabcap">Courtis Test</p> + +<p class="tabcap">Arithmetic. Test No. 3. Multiplication</p> + +<p class="center">Series B</p> + +<div class="pblockquot"> +<p>You will be given six minutes to work as many of these multiplication +examples as possible. You are not expected to be able +to do them all. Do your work directly on this paper; use no other. +You will be marked for both speed and accuracy, but it is more +important to get correct answers than to try a large number of +examples.</p></div> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="80%"> +<tr><td align='right'>8246</td><td align='right'>7843</td><td align='right'>4837</td><td align='right'>3478</td><td align='right'>6482</td></tr> +<tr><td align='right'>29</td><td align='right'>702</td><td align='right'>83</td><td align='right'>15</td><td align='right'>46</td></tr> +<tr><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td></tr> +</table></div> + +<div class="pblockquot"> +<p class="noidt">and twenty more multiplication examples of the same sort.</p> +</div> + + +<p class="tabcap">Courtis Test</p> + +<p class="tabcap">Arithmetic. Test No. 4. Division</p> + +<p class="center">Series B</p> + +<div class="pblockquot"> +<p>You will be given eight minutes to work as many of these division +examples as possible. You are not expected to be able to +do them all. Do your work directly on this paper; use no other. +You will be marked for both speed and accuracy, but it is more +important to get correct answers than to try a large number of +examples.</p></div> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='right'>25 <span class='overline'>) 6775</span></td><td align='right'>94 <span class='overline'>) 85352</span></td><td align='right'>37 <span class='overline'>) 9990</span></td><td align='right'>86 <span class='overline'>) 80066</span></td></tr> +</table></div> + +<div class="pblockquot"><p class="noidt">and twenty more division examples of the same sort.</p></div> + +<p><span class='pagenum'><a name="Page_36" id="Page_36">[Pg 36]</a></span></p> + +<p class="tabcap">SERIES B. MULTIPLICATION SCALE</p> +<p class="tabcap">By Clifford Woody</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><th align='right'>(1)</th><th align='right'>(3)</th><th align='right'>(4)</th><th align='right'>(5)</th></tr> +<tr> + <td align='right'>3 × 7 =</td> + <td align='right'>2 × 3 =</td> + <td align='right'>4 × 8 =</td> + <td align='right'>23<br />3<br />—</td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='right'>(8)</th><th align='right'>(9)</th><th align='right'>(11)</th><th align='right'>(12)</th></tr> +<tr> + <td align='right'>50<br />3<br />—</td> + <td align='right'>254<br />6<br />—</td> + <td align='right'>1036<br />8<br />—</td> + <td align='right'>5096<br />6<br />—</td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='right'>(13)</th><th align='right'>(16)</th><th align='right'>(18)</th><th align='right'>(20)</th></tr> +<tr> + <td align='right'>8754<br />8<br />——</td> + <td align='right'>7898<br />9<br />——</td> + <td align='right'>24<br />234<br />——</td> + <td align='right'>287<br />.05<br />——</td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='right'>(24)</th><th align='right'>(26)</th><th align='right'>(27)</th><th align='right'>(29)</th></tr> +<tr> + <td align='right'>16 <br />2<sup>5</sup>⁄<sub>8</sub><br />——</td> + <td align='right'>9742<br />59<br />——</td> + <td align='right'>6.25<br />3.2<br />——</td> + <td align='right'><sup>1</sup>⁄<sub>8</sub> × 2 =</td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='right'>(33)</th><th align='right'>(35)</th><th align='right'>(37)</th><th align='right'>(38)</th></tr> +<tr> + <td align='right'>2½ × 3½ =</td> + <td align='right'>987¾<br />25 <br />———</td> + <td align='right'>2¼ × 4½ × 1½ =</td> + <td align='right'>.0963<sup>1</sup>⁄<sub>8</sub><br />.084 <br />——</td> +</tr> +</table></div> + + + + +<p><span class='pagenum'><a name="Page_37" id="Page_37">[Pg 37]</a></span></p> + +<p class="tabcap">SERIES B. DIVISION SCALE<br /> +By Clifford Woody</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><th align='center'>(1)</th><th align='center'>(2)</th><th align='center'>(7)</th><th align='center'>(8)</th></tr> +<tr> + <td align='center'>3 )<span class="overline"> 6 </span></td> + <td align='center'>9 )<span class="overline"> 27 </span></td> + <td align='center'>4 ÷ 2 =</td> + <td align='center'>9 )<span class="overline"> 0 </span></td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='center'>(11)</th><th align='center'>(14)</th><th align='center'>(15)</th><th align='center'>(17)</th></tr> +<tr> + <td align='center'>2 )<span class="overline"> 13 </span></td> + <td align='center'>8 )<span class="overline"> 5856 </span></td> + <td align='center'>¼ of 128 =</td> + <td align='center'>50 ÷ 7 =</td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='center'>(19)</th><th align='center'>(23)</th><th align='center'>(27)</th><th align='center'>(28)</th></tr> +<tr> + <td align='center'>248 ÷ 7 =</td> + <td align='center'>23 )<span class="overline"> 469 </span></td> + <td align='center'><sup>7</sup>⁄<sub>8</sub> of 624 =</td> + <td align='center'>.003 )<span class="overline"> .0936 </span></td> +</tr> +<tr><td colspan='4'> </td></tr> +<tr><th align='center'>(30)</th><th align='center'>(34)</th><th align='center'>(36)</th></tr> +<tr> + <td align='center'><sup>3</sup>⁄<sub>4</sub> ÷ 5 =</td> + <td align='center'>62.50 ÷ 1¼ =</td> + <td align='center'>9 )<span class="overline"> 69 lbs. 9 oz. </span></td> +</tr> +</table></div> + + +<p><span class='pagenum'><a name="Page_38" id="Page_38">[Pg 38]</a></span></p> + +<p class="tabcap">Ballou Test</p> +<p class="tabcap">Addition of Fractions</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="100%"> +<tr><th></th><th><i>Test 1</i></th><th></th><th><i>Test 2</i></th></tr> +<tr> + <td align='right'><b>(1)</b> ¼<br />¼<br />—</td> + <td align='right'><b>(2)</b> <sup>3</sup>⁄<sub>14</sub><br /><sup>1</sup>⁄<sub>14</sub><br />—</td> + <td align='right'><b>(1)</b> <sup>1</sup>⁄<sub>3</sub><br /><sup>1</sup>⁄<sub>6</sub><br />—</td> + <td align='right'><b>(2)</b> <sup>2</sup>⁄<sub>7</sub><br /><sup>3</sup>⁄<sub>14</sub><br />—</td> +</tr> + +<tr><th></th><th><i>Test 3</i></th><th></th><th><i>Test 4</i></th></tr> +<tr> + <td align='right'><b>(1)</b> <sup>3</sup>⁄<sub>5</sub><br /><sup>11</sup>⁄<sub>15</sub><br />—</td> + <td align='right'><b>(2)</b> <sup>5</sup>⁄<sub>6</sub><br /><sup>1</sup>⁄<sub>2</sub><br />—</td> + <td align='right'><b>(1)</b> <sup>1</sup>⁄<sub>7</sub><br /><sup>9</sup>⁄<sub>10</sub><br />—</td> + <td align='right'><b>(2)</b> <sup>7</sup>⁄<sub>9</sub><br /><sup>1</sup>⁄<sub>4</sub><br />—</td> +</tr> + +<tr><th></th><th><i>Test 5</i></th><th></th><th><i>Test 6</i></th></tr> +<tr> + <td align='right'><b>(1)</b> <sup>1</sup>⁄<sub>10</sub><br /><sup>1</sup>⁄<sub>6</sub><br />—</td> + <td align='right'><b>(2)</b> <sup>4</sup>⁄<sub>9</sub><br /><sup>5</sup>⁄<sub>12</sub><br />—</td> + <td align='right'><b>(1)</b> <sup>1</sup>⁄<sub>6</sub><br /><sup>9</sup>⁄<sub>10</sub><br />—</td> + <td align='right'><b>(2)</b> <sup>5</sup>⁄<sub>6</sub><br /><sup>3</sup>⁄<sub>8</sub><br />—</td> +</tr> + +</table></div> + + +<p class="tabcap">An Addition Ladder [Thorndike, '17, III, 5]</p> + +<p>Begin at the bottom of the ladder. See if you can climb to the +top without making a mistake. Be sure to copy the numbers +correctly.</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='left'><b>Step 6.</b></td><td align='left'><i>a.</i> Add 1<sup>1</sup>⁄<sub>3</sub> yd., <sup>7</sup>⁄<sub>8</sub> yd., 1¼ yd., <sup>3</sup>⁄<sub>4</sub> yd., <sup>7</sup>⁄<sub>8</sub> yd., and 1½ yd.</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> Add 62½¢, 66<sup>2</sup>⁄<sub>3</sub>¢, 56¼¢, 60¢, and 62½¢.</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> Add 1<sup>5</sup>⁄<sub>16</sub>, 1<sup>9</sup>⁄<sub>32</sub>, 1<sup>3</sup>⁄<sub>8</sub>, 1<sup>11</sup>⁄<sub>32</sub>, and 1<sup>7</sup>⁄<sub>16</sub>.</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> Add 1<sup>1</sup>⁄<sub>3</sub> yd., 1¼ yd., 1½ yd., 2 yd., <sup>3</sup>⁄<sub>4</sub> yd., and <sup>2</sup>⁄<sub>3</sub> yd.</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 5.</b></td><td align='left'><i>a.</i> Add 4 ft. 6½ in., 53¼ in., 5 ft. ½ in., 56¾ in., and 5 ft.</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> Add 7 lb., 6 lb. 11 oz., 7½ lb., 6 lb. 4½ oz., and 8½ lb.</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min., and 55 min.</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes, and 19 nickels.</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 4.</b></td><td align='left'><i>a.</i> Add .05½, .06, .04¾, .02¾, and .05¼.</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> Add .33<sup>1</sup>⁄<sub>3</sub>, .12½, .18, .16<sup>2</sup>⁄<sub>3</sub>, .08<sup>1</sup>⁄<sub>3</sub> and .15.</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> Add .08<sup>1</sup>⁄<sub>3</sub>, .06¼, .21, .03¾, and .16<sup>2</sup>⁄<sub>3</sub>.</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> Add .62, .64½, .66<sup>2</sup>⁄<sub>3</sub>, .10¼, and .68.</td></tr> +<tr><td align='left'> <span class='pagenum'><a name="Page_39" id="Page_39">[Pg 39]</a></span></td></tr> +<tr><td align='left'><b>Step 3.</b></td><td align='left'><i>a.</i> Add 7¼, 6½, 8<sup>3</sup>⁄<sub>8</sub>, 5¾, 9<sup>5</sup>⁄<sub>8</sub> and 3<sup>7</sup>⁄<sub>8</sub>.</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> Add 4<sup>5</sup>⁄<sub>8</sub>, 12, 7½, 8¾, 6 and 5¼.</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> Add 9¾, 5<sup>7</sup>⁄<sub>8</sub>, 4<sup>1</sup>⁄<sub>8</sub>, 6½, 7, 3<sup>5</sup>⁄<sub>8</sub>.</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> Add 12, 8½, 7<sup>1</sup>⁄<sub>3</sub>, 5, 6<sup>2</sup>⁄<sub>3</sub>, and 9½.</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 2.</b></td><td align='left'><i>a.</i> Add 12.04, .96, 4.7, 9.625, 3.25, and 20.</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> Add .58, 6.03, .079, 4.206, 2.75, and 10.4.</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> Add 52, 29.8, 41.07, 1.913, 2.6, and 110.</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05.</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 1.</b></td><td align='left'><i>a.</i> Add 10<sup>3</sup>⁄<sub>5</sub>, 11<sup>1</sup>⁄<sub>5</sub>, 10<sup>4</sup>⁄<sub>5</sub>, 11, 11<sup>2</sup>⁄<sub>5</sub>, 10<sup>3</sup>⁄<sub>5</sub>, and 11.</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> Add 7<sup>3</sup>⁄<sub>8</sub>, 6<sup>5</sup>⁄<sub>8</sub>, 8, 9<sup>1</sup>⁄<sub>8</sub>, 7<sup>7</sup>⁄<sub>8</sub>, 5<sup>3</sup>⁄<sub>8</sub>, and 8<sup>1</sup>⁄<sub>8</sub>.</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> Add 21½, 18¾, 31½, 19¼, 17¼, 22, and 16½.</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> Add 14<sup>5</sup>⁄<sub>12</sub>, 12<sup>7</sup>⁄<sub>12</sub>, 9<sup>11</sup>⁄<sub>12</sub>, 6<sup>1</sup>⁄<sub>12</sub>, and 5.</td></tr> +</table></div> + + + +<p class="tabcap">A Subtraction Ladder [Thorndike, '17, III, 11]</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><th><b>Step 9.</b></th></tr> +<tr><td></td> + <td align='left' colspan='2'><i>a.</i> 2.16 mi. − 1¾ mi.<br /> + <i>c.</i> 2 min. 10½ sec. − 93.4 sec.<br /> + <i>e.</i> 10 gal. 2½ qt. − 4.623 gal.</td> + <td align='left' colspan='2'><i>b.</i> 5.72 ft. − 5 ft. 3 in.<br /> + <i>d.</i> 30.28 A. − 10<sup>1</sup>⁄<sub>5</sub> A.<br /> </td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 8.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>25<sup>7</sup>⁄<sub>12</sub><br />12<sup>3</sup>⁄<sub>4</sub><br />———</td> + <td align='right'>10<sup>1</sup>⁄<sub>4</sub><br />7<sup>1</sup>⁄<sub>3</sub><br />———</td> + <td align='right'>9<sup>5</sup>⁄<sub>16</sub><br />6<sup>3</sup>⁄<sub>8</sub> <br />———</td> + <td align='right'>5<sup>7</sup>⁄<sub>16</sub><br />2<sup>3</sup>⁄<sub>4</sub> <br />———</td> + <td align='right'>4<sup>2</sup>⁄<sub>3</sub><br />1<sup>3</sup>⁄<sub>4</sub><br />———</td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 7.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>28<sup>3</sup>⁄<sub>4</sub><br />16<sup>1</sup>⁄<sub>8</sub><br />———</td> + <td align='right'>40<sup>1</sup>⁄<sub>2</sub><br />14<sup>3</sup>⁄<sub>8</sub><br />———</td> + <td align='right'>10<sup>1</sup>⁄<sub>4</sub><br />6<sup>1</sup>⁄<sub>2</sub><br />———</td> + <td align='right'>24<sup>1</sup>⁄<sub>3</sub><br />11<sup>1</sup>⁄<sub>2</sub><br />———</td> + <td align='right'>37<sup>1</sup>⁄<sub>2</sub><br />14<sup>3</sup>⁄<sub>4</sub><br />———</td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 6.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>10<sup>1</sup>⁄<sub>3</sub><br />4<sup>2</sup>⁄<sub>3</sub><br />———</td> + <td align='right'>7<sup>1</sup>⁄<sub>4</sub><br />2<sup>3</sup>⁄<sub>4</sub><br />———</td> + <td align='right'>15<sup>1</sup>⁄<sub>8</sub><br />6<sup>3</sup>⁄<sub>8</sub><br />———</td> + <td align='right'>12<sup>1</sup>⁄<sub>5</sub><br />11<sup>4</sup>⁄<sub>5</sub><br />———</td> + <td align='right'>4<sup>1</sup>⁄<sub>16</sub><br />2<sup>7</sup>⁄<sub>16</sub><br />———</td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 5.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>58<sup>4</sup>⁄<sub>5</sub><br />52<sup>1</sup>⁄<sub>5</sub><br />———</td> + <td align='right'>66<sup>2</sup>⁄<sub>3</sub><br />33<sup>1</sup>⁄<sub>3</sub><br />———</td> + <td align='right'>28<sup>7</sup>⁄<sub>8</sub><br />7<sup>5</sup>⁄<sub>8</sub><br />———</td> + <td align='right'>62½<br />37½<br />——</td> + <td align='right'>9<sup>7</sup>⁄<sub>12</sub><br />4<sup>5</sup>⁄<sub>12</sub><br />——</td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 4.</b></th></tr> +<tr><td></td> + <td align='left' colspan='2'><i>a.</i> 4 hr. − 2 hr. 17 min.<br /> + <i>c.</i> 1 lb. 5 oz. − 13 oz.<br /> + <i>e.</i> 1 bu. − 1 pk.</td> + <td align='left' colspan='2'><i>b.</i> 4 lb. 7 oz. − 2 lb. 11 oz.<br /> + <i>d.</i> 7 ft. − 2 ft. 8 in.<br /> </td> +</tr> +<tr><td align='left'> <span class='pagenum'><a name="Page_40" id="Page_40">[Pg 40]</a></span></td></tr> +<tr><th><b>Step 3.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>92 mi.<br />84.15 mi.<br />————</td> + <td align='right'>6735 mi.<br />6689 mi.<br />————</td> + <td align='right'>$3 − 89¢<br /><br />————</td> + <td align='right'>28.4 mi.<br />18.04 mi.<br />————</td> + <td align='right'>$508.40<br />208.62<br />————</td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 2.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>$25.00<br />9.36<br />———</td> + <td align='right'>$100.00<br />71.28<br />———</td> + <td align='right'>$750.00<br />736.50<br />———</td> + <td align='right'>6124 sq. mi.<br />2494 sq. mi.<br />—————</td> + <td align='right'>7846 sq. mi.<br />2789 sq. mi.<br />—————</td> +</tr> +<tr><td align='left'> </td></tr> +<tr><th><b>Step 1.</b></th><th><i>a</i></th><th><i>b</i></th><th><i>c</i></th><th><i>d</i></th><th><i>e</i></th></tr> +<tr><td></td> + <td align='right'>$18.64<br />7.40<br />———</td> + <td align='right'>$25.39<br />13.37<br />———</td> + <td align='right'>$56.70<br />45.60<br />———</td> + <td align='right'>819.4 mi.<br />209.2 mi.<br />————</td> + <td align='right'>67.55 mi.<br />36.14 mi.<br />————</td> +</tr> +</table></div> + +<p class="tabcap">An Average Ladder [Thorndike, '17, III, 132]</p> + +<p>Find the average of the quantities on each line. Begin with +<b>Step 1</b>. Climb to the top without making a mistake. Be sure +to copy the numbers correctly. Extend the division to two +decimal places if necessary.</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='left'><b>Step 6.</b></td><td align='left'><i>a</i>. 2<sup>2</sup>⁄<sub>3</sub>, 1<sup>7</sup>⁄<sub>8</sub>, 2¾, 4¼, 3<sup>5</sup>⁄<sub>8</sub>, 3½</td></tr> +<tr><td align='left'></td><td align='left'><i>b</i>. 62½¢, 66<sup>2</sup>⁄<sub>3</sub>¢, 40¢, 83<sup>1</sup>⁄<sub>3</sub>¢, $1.75, $2.25</td></tr> +<tr><td align='left'></td><td align='left'><i>c</i>. 3<sup>11</sup>⁄<sub>16</sub>, 3<sup>9</sup>⁄<sub>32</sub>, 3<sup>3</sup>⁄<sub>8</sub>, 3<sup>17</sup>⁄<sub>32</sub>, 3<sup>7</sup>⁄<sub>16</sub></td></tr> +<tr><td align='left'></td><td align='left'><i>d</i>. .17, 19, .16<sup>2</sup>⁄<sub>3</sub>, .15½, .23¼, .18</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 5.</b></td><td align='left'><i>a</i>. 5 ft. 3½ in., 61¼ in., 58¾ in., 4 ft. 11 in.</td></tr> +<tr><td align='left'></td><td align='left'><i>b</i>. 6 lb. 9 oz., 6 lb. 11 oz., 7¼ lb., 7<sup>3</sup>⁄<sub>8</sub> lb.</td></tr> +<tr><td align='left'></td><td align='left'><i>c</i>. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1¼ hr.</td></tr> +<tr><td align='left'></td><td align='left'><i>d</i>. 2.8 miles, 3½ miles, 2.72 miles</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 4.</b></td><td align='left'><i>a</i>. .03½, .06, .04¾, .05½, .05¼</td></tr> +<tr><td align='left'></td><td align='left'><i>b</i>. .043, .045, .049, .047, .046, .045</td></tr> +<tr><td align='left'></td><td align='left'><i>c</i>. 2.20, .87½, 1.18, .93¾, 1.2925, .80</td></tr> +<tr><td align='left'></td><td align='left'><i>d</i>. .14½, .12½, .33<sup>1</sup>⁄<sub>3</sub>, .16<sup>2</sup>⁄<sub>3</sub>, .15, .17</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 3.</b></td><td align='left'><i>a</i>. 5¼, 4½, 8<sup>3</sup>⁄<sub>8</sub>, 7¾, 6<sup>5</sup>⁄<sub>8</sub>, 9<sup>3</sup>⁄<sub>8</sub></td></tr> +<tr><td align='left'></td><td align='left'><i>b</i>. 9<sup>5</sup>⁄<sub>8</sub>, 12, 8½, 8¾, 6, 5¼, 9</td></tr> +<tr><td align='left'></td><td align='left'><i>c</i>. 9<sup>3</sup>⁄<sub>8</sub>, 5¾, 4<sup>1</sup>⁄<sub>8</sub>, 7½, 6</td></tr> +<tr><td align='left'></td><td align='left'><i>d</i>. 11, 9½, 10<sup>1</sup>⁄<sub>3</sub>, 13, 16<sup>2</sup>⁄<sub>3</sub>, 9½</td></tr> +<tr><td align='left'> <span class='pagenum'><a name="Page_41" id="Page_41">[Pg 41]</a></span></td></tr> +<tr><td align='left'><b>Step 2.</b></td><td align='left'><i>a.</i> 13.05, .97, 4.8, 10.625, 3.37</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> 68, 71.4, 59.8, 112, 96.1, 79.8</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> 2.079, 3.908, 4.165, 2.74</td></tr> +<tr><td align='left'> </td></tr> +<tr><td align='left'><b>Step 1.</b></td><td align='left'><i>a.</i> 4, 9½, 6, 5, 7½, 8, 10, 9</td></tr> +<tr><td align='left'></td><td align='left'><i>b.</i> 6, 5, 3.9, 7.1, 8</td></tr> +<tr><td align='left'></td><td align='left'><i>c.</i> 1086, 1141, 1059, 1302, 1284</td></tr> +<tr><td align='left'></td><td align='left'><i>d.</i> $100.82, $206.49, $317.25, $244.73</td></tr> +</table></div> + +<p>As such tests are widened to cover the whole task of the +elementary school in respect to arithmetic, and accepted by +competent authorities as adequate measures of achievement +in computing, they will give, as has been said, a working +definition of the task. The reader will observe, for example, +that work such as the following, though still found in many +textbooks and classrooms, does not, in general, appear in +the modern tests and scales.</p> + +<p>Reduce the following improper fractions to mixed numbers:—</p> + +<p class="center"> +<sup>19</sup>⁄<sub>13</sub> + +<sup>43</sup>⁄<sub>21</sub> + +<sup>176</sup>⁄<sub>25</sub> + +<sup>198</sup>⁄<sub>14</sub> +</p> + + +<p>Reduce to integral or mixed numbers:—</p> + +<p class="center"> +<sup>61381</sup>⁄<sub>37</sub> + +<sup>2134</sup>⁄<sub>67</sub> + +<sup>413</sup>⁄<sub>413</sub> + +<sup>697</sup>⁄<sub>225</sub> +</p> + + +<p>Simplify:—</p> + +<p class="center"> +<sup>3</sup>⁄<sub>4</sub> of +<sup>8</sup>⁄<sub>9</sub> of +<sup>3</sup>⁄<sub>5</sub> of +<sup>15</sup>⁄<sub>22</sub> +</p> + + +<p>Reduce to lowest terms:—</p> + +<p class="center"> +<sup>357</sup>⁄<sub>527</sub> + +<sup>264</sup>⁄<sub>312</sub> + +<sup>492</sup>⁄<sub>779</sub> + +<sup>418</sup>⁄<sub>874</sub> + +<sup>854</sup>⁄<sub>1769</sub> + +<sup>30</sup>⁄<sub>735</sub> + +<sup>44</sup>⁄<sub>242</sub> + +<sup>77</sup>⁄<sub>847</sub> + +<sup>18</sup>⁄<sub>243</sub> + +<sup>96</sup>⁄<sub>224</sub> +</p> + +<p>Find differences:—</p> + + +<div class='center'> +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr> +<td align='left'>6<sup>2</sup>⁄<sub>7</sub><br />3<sup>1</sup>⁄<sub>14</sub><br />——</td> +<td align='left'>8<sup>5</sup>⁄<sub>11</sub><br />5<sup>1</sup>⁄<sub>7</sub><br />——</td> +<td align='left'>8<sup>4</sup>⁄<sub>13</sub><br />3<sup>7</sup>⁄<sub>13</sub><br />——</td> +<td align='left'>5<sup>1</sup>⁄<sub>4</sub><br />2<sup>11</sup>⁄<sub>14</sub><br />——</td> +<td align='left'>7<sup>1</sup>⁄<sub>8</sub><br />2<sup>1</sup>⁄<sub>7</sub><br />——</td> +</tr> +</table></div> + + +<p>Square:—</p> + + +<p class="center"> +<sup>2</sup>⁄<sub>3</sub> + +<sup>4</sup>⁄<sub>5</sub> + +<sup>5</sup>⁄<sub>7</sub> + +<sup>6</sup>⁄<sub>9</sub> + +<sup>10</sup>⁄<sub>11</sub> + +<sup>12</sup>⁄<sub>13</sub> + +<sup>2</sup>⁄<sub>7</sub> + +<sup>15</sup>⁄<sub>16</sub> + +<sup>19</sup>⁄<sub>20</sub> + +<sup>17</sup>⁄<sub>18</sub> + +<sup>25</sup>⁄<sub>30</sub> + +<sup>41</sup>⁄<sub>53</sub> +</p> + +<p>Multiply:—</p> + +<p class="center"> +<sup>2</sup>⁄<sub>11</sub> × 33 + + 32 × <sup>3</sup>⁄<sub>14</sub> + + 39 × <sup>2</sup>⁄<sub>13</sub> + + 60 × <sup>11</sup>⁄<sub>28</sub> + + 77 × <sup>4</sup>⁄<sub>11</sub> + + 63 × <sup>2</sup>⁄<sub>27</sub> + +<br /> + +54 × <sup>8</sup>⁄<sub>45</sub> + +65 × <sup>3</sup>⁄<sub>13</sub> + +344<sup>16</sup>⁄<sub>21</sub> 432<sup>2</sup>⁄<sub>7</sub> +</p> + +<p><span class='pagenum'><a name="Page_42" id="Page_42">[Pg 42]</a></span></p> + +<h4>MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE +SOLUTION OF PROBLEMS</h4> + +<p>Stone ['08] measured achievement with the following +problems, fifteen minutes being the time allowed.</p> + +<p>"Solve as many of the following problems as you have +time for; work them in order as numbered:</p> + +<div class="pblockquot"><p>1. If you buy 2 tablets at 7 cents each and a book for 65 cents, +how much change should you receive from a two-dollar bill?</p> + +<p>2. John sold 4 Saturday Evening Posts at 5 cents each. He +kept <sup>1</sup>⁄<sub>2</sub> the money and with the other <sup>1</sup>⁄<sub>2</sub> he bought Sunday papers +at 2 cents each. How many did he buy?</p> + +<p>3. If James had 4 times as much money as George, he would +have $16. How much money has George?</p> + +<p>4. How many pencils can you buy for 50 cents at the rate of +2 for 5 cents? '</p> + +<p>5. The uniforms for a baseball nine cost $2.50 each. The +shoes cost $2 a pair. What was the total cost of uniforms and +shoes for the nine?</p> + +<p>6. In the schools of a certain city there are 2200 pupils; <sup>1</sup>⁄<sub>2</sub> +are in the primary grades, <sup>1</sup>⁄<sub>4</sub> in the grammar grades, <sup>1</sup>⁄<sub>8</sub> in the high +school, and the rest in the night school. How many pupils are +there in the night school?</p> + +<p>7. If 3½ tons of coal cost $21, what will 5½ tons cost?</p> + +<p>8. A news dealer bought some magazines for $1. He sold +them for $1.20, gaining 5 cents on each magazine. How many +magazines were there?</p> + +<p>9. A girl spent <sup>1</sup>⁄<sub>8</sub> of her money for car fare, and three times +as much for clothes. Half of what she had left was 80 cents. +How much money did she have at first?</p> + +<p>10. Two girls receive $2.10 for making buttonholes. One +makes 42, the other 28. How shall they divide the money?</p> + +<p>11. Mr. Brown paid one third of the cost of a building; Mr. +Johnson paid <sup>1</sup>⁄<sub>2</sub> the cost. Mr. Johnson received $500 more annual +rent than Mr. Brown. How much did each receive?</p> + +<p>12. A freight train left Albany for New York at 6 o'clock. An +express left on the same track at 8 o'clock. It went at the rate +of 40 miles an hour. At what time of day will it overtake the freight +train if the freight train stops after it has gone 56 miles?"</p></div> +<p><span class='pagenum'><a name="Page_43" id="Page_43">[Pg 43]</a></span></p> + +<p>The criteria he had in mind in selecting the problems +were as follows:—</p> + +<p>"The main purpose of the reasoning test is the determination +of the ability of VI A children to reason in arithmetic. +To this end, the problems, as selected and arranged, +are meant to embody the following conditions:—</p> + +<div class="pblockquot"> +<p>1. Situations equally concrete to all VI A children.</p> + +<p>2. Graduated difficulties.<br /> +<span style="margin-left: 2.5em;"><i>a.</i> As to arithmetical thinking.</span><br /> +<span style="margin-left: 2.5em;"><i>b.</i> As to familiarity with the situation presented.</span></p> + +<p>3. The omission of<br /> +<span style="margin-left: 2.5em;"><i>a.</i> Large numbers.</span><br /> +<span style="margin-left: 2.5em;"><i>b.</i> Particular memory requirements.</span><br /> +<span style="margin-left: 2.5em;"><i>c.</i> Catch problems.</span><br /> +<span style="margin-left: 2.5em;"><i>d.</i> All subject matter except whole numbers, fractions, and United States money.</span></p> +</div> + +<p>The test is purposely so long that only very rarely did any +pupil fully complete it in the fifteen minute limit."</p> + +<p>Credits were given of 1, for each of the first five problems, +1.4, 1.2, and 1.6 respectively for problems 6, 7, and 8, and of +2 for each of the others.</p> + +<p>Courtis sought to improve the Stone test of problem-solving, +replacing it by the two tests reproduced below.</p> + +<p class="tabcap">ARITHMETIC—Test No. 6. Speed Test—Reasoning</p> + +<div class="pblockquot"> + +<p><b>Do not work</b> the following examples. Read each example through, make +up your mind what operation you would use if you were going to work it, +then write the name of the operation selected in the blank space after the +example. Use the following abbreviations:—"Add." for addition, "Sub." +for subtraction, "Mul." for multiplication, and "Div." for division.</p> + +<div class='center'> +<table cellpadding="4" cellspacing="0" summary=""> +<tr><td align='justify'> </td><td class='bbox'><span class="smcap">Operation</span></td><td class='bbox'> </td></tr> + +<tr><td align='justify'>1. A girl brought a collection of 37 colored postal cards +to school one day, and gave away 19 cards to her friends. +How many cards did she have left to take home?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>2. Five boys played marbles. When the game was +over, each boy had the same number of marbles. If there +were 45 marbles altogether, how many did each boy have?</td> +<td class='bbox'> <span class='pagenum'><a name="Page_44" id="Page_44">[Pg 44]</a></span></td><td class='bbox'> </td></tr> + +<tr><td align='justify'>3. A girl, watching from a window, saw 27 automobiles +pass the school the first hour, and 33 the second. How +many autos passed by the school in the two hours?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>4. In a certain school there were eight rooms and each +room had seats for 50 children. When all the places were +taken, how many children were there in the school?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>5. A club of boys sent their treasurer to buy baseballs. +They gave him $3.15 to spend. How many balls did they +expect him to buy, if the balls cost 45¢. apiece?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>6. A teacher weighed all the girls in a certain grade. If +one girl weighed 79 pounds and another 110 pounds, how +many pounds heavier was one girl than the other?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>7. A girl wanted to buy a 5-pound box of candy to give +as a present to a friend. She decided to get the kind worth +35¢. a pound. What did she pay for the present?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>8. One day in vacation a boy went on a fishing trip and +caught 12 fish in the morning, and 7 in the afternoon. How +many fish did he catch altogether?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>9. A boy lived 15 blocks east of a school; his chum lived +on the same street, but 11 blocks west of the school. How +many blocks apart were the two boys' houses?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>10. A girl was 5 times as strong as her small sister. If +the little girl could lift a weight of 20 pounds, how large a +weight could the older girl lift?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>11. The children of a school gave a sleigh-ride party. +There were 270 children to go on the ride and each sleigh +held 30 children. How many sleighs were needed?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>12. In September there were 43 children in the eighth +grade of a certain school; by June there were 59. How +many children entered the grade during the year?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>13. A girl who lived 17 blocks away walked to school and +back twice a day. What was the total number of blocks the +girl walked each day in going to and from school?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>14. A boy who made 67¢. a day carrying papers, was +hired to run on a long errand for which he received 50¢. +What was the total amount the boy earned that day?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='right'>Total Right</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +</table></div> + + +<p>(Two more similar problems follow.)</p> + +<p>Test 6 and Test 8 are from the Courtis Standard Test. Used by permission +of S. A. Courtis.</p></div> +<p><span class='pagenum'><a name="Page_45" id="Page_45">[Pg 45]</a></span></p> + + +<p class="tabcap">ARITHMETIC—Test No. 8. Reasoning</p> + +<div class="pblockquot"> +<p>In the blank space below, work as many of the following examples as possible +in the time allowed. Work them in order as numbered, entering each answer +in the "answer" column before commencing a new example. Do not work +on any other paper.</p> + +<div class='center'> +<table cellpadding="4" cellspacing="0" summary=""> +<tr><td align='justify'> </td><td class='bbox'><span class="smcap">Answer</span></td><td class='bbox'> </td></tr> + +<tr><td align='justify'>1. The children in a certain school gave a Christmas +party. One of the presents was a box of candy. In filling +the boxes, one grade used 16 pounds of candy, another 17 +pounds, a third 12 pounds, and a fourth 13 pounds. What +did the candy cost at 26¢. a pound?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>2. A school in a certain city used 2516 pieces of chalk +in 37 school days. Three new rooms were opened, each +room holding 50 children, and the school was then found +to use 84 sticks of chalk per day. How many more sticks +of chalk were used per day than at first?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>3. Several boys went on a bicycle trip of 1500 miles. +The first week they rode 374 miles, the second week 264 +miles, the third 423 miles, the fourth 401 miles. They +finished the trip the next week. How many miles did they +ride the last week?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>4. Forty-five boys were hired to pick apples from 15 trees +in an apple orchard. In 50 minutes each boy had picked +48 choice apples. If all the apples picked were packed away +carefully in 8 boxes of equal size, how many apples were put +in each box?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>5. In a certain school 216 children gave a sleigh-ride +party. They rented 7 sleighs at a cost of $30.00 and paid +$24.00 for the refreshments. The party travelled 15 miles +in 2½ hours and had a very pleasant time. What was each +child's share of the expense?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>6. A girl found, by careful counting, that there were +2400 letters on one page of her history, and only 2295 letters +on a page of her reader. How many more letters had she +read in one book than in the other if she had read 47 pages +in each of the books?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>7. Each of 59 rooms in the schools of a certain city contributed +25 presents to a Christmas entertainment for poor +children. The stores of the city gave 1986 other articles for +presents. What was the total number of presents given +away at the entertainment?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='justify'>8. Forty-eight children from a certain school paid 10¢. +apiece to ride 7 miles on the cars to a woods. There in a +few hours they gathered 2765 nuts. 605 of these were bad, +but the rest were shared equally among the children. How +many good nuts did each one get?</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> + +<tr><td align='right'>Total</td> +<td class='bbox'> </td><td class='bbox'> </td></tr> +</table></div> + +</div> + +<p><span class='pagenum'><a name="Page_46" id="Page_46">[Pg 46]</a></span></p> + +<p>These proposed measures of ability to apply arithmetic +illustrate very nicely the differences of opinion concerning +what applied arithmetic and arithmetical reasoning should be. +The thinker who emphasizes the fact that in life out of school +the situation demanding quantitative treatment is usually +real rather than described, will condemn a test all of whose +constituents are <i>described</i> problems. Unless we are excessively +hopeful concerning the transfer of ideas of method +and procedure from one mental function to another we shall +protest against the artificiality of No. 3 of the Stone series, +and of the entire Courtis Test 8 except No. 4. The Courtis +speed-reasoning test (No. 6) is a striking example of the mixture +of ability to understand quantitative relations with +the ability to understand words. Consider these five, for +example, in comparison with the revised versions attached.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></p> + +<div class="pblockquot"><p>1. The children of a school gave a sleigh-ride party. There +were 9 sleighs, and each sleigh held 30 children. How many +children were there in the party?</p> + +<p><span class="smcap">Revision.</span> <i>If one sleigh holds 30 children, 9 sleighs hold .... +children.</i></p> + +<p>2. Two school-girls played a number-game. The score of the +girl that lost was 57 points and she was beaten by 16 points. +What was the score of the girl that won?</p> + +<p><span class="smcap">Revision.</span> <i>Mary and Nell played a game. Mary had a score +of 57. Nell beat Mary by 16. Nell had a score of</i> ....</p> + +<p>3. A girl counted the automobiles that passed a school. The +total was 60 in two hours. If the girl saw 27 pass the first hour +how many did she see the second?</p> + +<p><span class="smcap">Revision.</span> <i>In two hours a girl saw 60 automobiles. She saw 27 +the first hour. She saw .... the second hour.</i></p> + +<p>4. On a playground there were five equal groups of children +each playing a different game. If there were 75 children all together, +how many were there in each group?</p> + +<p><span class="smcap">Revision.</span> <i>75 pounds of salt just filled five boxes. The boxes +were exactly alike. There were .... pounds in a box.</i></p> + +<p><span class='pagenum'><a name="Page_47" id="Page_47">[Pg 47]</a></span></p> + +<p>5. A teacher weighed all the children in a certain grade. One +girl weighed 70 pounds. Her older sister was 49 pounds heavier. +How many pounds did the sister weigh?</p> + +<p><span class="smcap">Revision.</span> <i>Mary weighs 70 lb. Jane weighs 49 pounds more +than Mary. Jane weighs .... pounds.</i></p></div> + +<p>The distinction between a problem described as clearly +and simply as possible and the same problem put awkwardly +or in ill-known words or willfully obscured should be regarded; +and as a rule measurements of ability to apply arithmetic +should eschew all needless obscurity or purely linguistic +difficulty. For example,</p> + +<div class="blockquot"><p><i>A boy bought a two-cent stamp. He gave the man in the store 10 +cents. The right change was .... cents.</i></p></div> + +<p class="noidt">is better as a test than</p> + +<div class="blockquot"><p><i>If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in +payment, what change should he be expected to receive in return?</i></p></div> + +<p>The distinction between the description of a <i>bona fide</i> +problem that a human being might be called on to solve out +of school and the description of imaginary possibilities or +puzzles should also be considered. Nos. 3 and 9 of Stone +are bad because to frame the problems one must first know +the answers, so that in reality there could never be any +point in solving them. It is probably safe to say that +nobody in the world ever did or ever will or ever should +find the number of apples in a box by the task of No. 4 +of the Courtis Test 8.</p> + +<p>This attaches no blame to Dr. Stone or to Mr. Courtis. +Until very recently we were all so used to the artificial +problems of the traditional sort that we did not expect +anything better; and so blind to the language demands of +described problems that we did not see their very great +influence. Courtis himself has been active in reform and +has pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.<span class='pagenum'><a name="Page_48" id="Page_48">[Pg 48]</a></span></p> + +<p>"Tests Nos. 6 and 8, the so-called reasoning tests, have +proved the least satisfactory of the series. The judgments +of various teachers and superintendents as to the inequalities +of the units in any one test, and of the differences between +the different editions of the same test, have proved the need +of investigating these questions. Tests of adults in many +lines of commercial work have yielded in many cases lower +scores than those of the average eighth grade children. At +the same time the scores of certain individuals of marked +ability have been high, and there appears to be a general +relation between ability in these tests and accuracy in the +abstract work. The most significant facts, however, have +been the difficulties experienced by teachers in attempting +to remedy the defects in reasoning. It is certain that the +tests measure abilities of value but the abilities are probably +not what they seem to be. In an attempt to measure the +value of different units, for instance, as many problems as possible +were constructed based upon a single situation. Twenty-one +varieties were secured by varying the relative form of +the question and the relative position of the different phrases. +One of these proved nineteen times as hard as another as measured +by the number of mistakes made by the children; yet +the cause of the difference was merely the changes in the +phrasing. This and other facts of the same kind seem to show +that Tests 6 and 8 measure mainly the ability to read."</p> + +<p>The scientific measurement of the abilities and achievements +concerned with applied arithmetic or problem-solving +is thus a matter for the future. In the case of described +problems a beginning has been made in the series which +form a part of the National Intelligence Tests ['20], one of +which is shown on page 49 f. In the case of problems with +real situations, nothing in systematic form is yet available.</p> + +<p>Systematic tests and scales, besides defining the abil<span class='pagenum'><a name="Page_49" id="Page_49">[Pg 49]</a></span>ities +we are to establish and improve, are of very great +service in measuring the status and improvement of individuals +and of classes, and the effects of various methods +of instruction and of study. They are thus helpful to +pupils, teachers, supervisors, and scientific investigators; +and are being more and more widely used every year. +Information concerning the merits of the different tests, +the procedure to follow in giving and scoring them, the age +and grade standards to be used in interpreting results, and +the like, is available in the manuals of Educational Measurement, +such as Courtis, <i>Manual of Instructions for Giving and +Scoring the Courtis Standard Tests in the Three R's</i> ['14]; +Starch, <i>Educational Measurements</i> ['16]; Chapman and +Rush, <i>Scientific Measurement of Classroom Products</i> ['17]; +Monroe, DeVoss, and Kelly, <i>Educational Tests and Measurements</i> +['17]; Wilson and Hoke, <i>How to Measure</i> ['20]; +and McCall, <i>How to Measure in Education</i> ['21].</p> + +<p class="noidt"><small>National Intelligence Tests.<br /> +Scale A. Form 1, Edition 1</small></p> + +<h4>TEST 1</h4> + +<p class="center">Find the answers as quickly as you can.<br /> +Write the answers on the dotted lines.<br /> +Use the side of the page to figure on.</p> + +<p class="noidt"><b>Begin here</b></p> + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td align='right'><b>1</b></td><td align='left'>Five cents make 1 nickel. How many nickels make a +dime?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>2</b></td><td align='left'>John paid 5 dollars for a watch and 3 dollars for a chain. +How many dollars did he pay for the watch and chain?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>3</b></td><td align='left'>Nell is 13 years old. Mary is 9 years old. How much +younger is Mary than Nell?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>4</b></td><td align='left'>One quart of ice cream is enough for 5 persons. How +many quarts of ice cream are needed for 25 persons?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>5</b></td><td align='left'>John's grandmother is 86 years old. If she lives, in +how many years will she be 100 years old?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>6</b></td><td align='left'>If a man gets $2.50 a day, what will he be paid for six +days' work?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>7</b></td><td align='left'>How many inches are there in a foot and a half?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>8</b></td><td align='left'>What is the cost of 12 cakes at 6 for 5 cents?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>9</b></td><td align='left'>The uniforms for a baseball team of nine boys cost $2.50 +each. The shoes cost $2 a pair. What was the total +cost of uniforms and shoes for the nine?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>10</b></td><td align='left'>A train that usually arrives at half-past ten was 17 +minutes late. When did it arrive?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>11</b></td><td align='left'>At 10¢ a yard, what is the cost of a piece 10½ ft. long?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>12</b></td><td align='left'>A man earns $6 a day half the time, $4.50 a day one +fourth of the time, and nothing on the remaining days +for a total period of 40 days. What did he earn in all +in the 40 days?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>13</b></td><td align='left'>What per cent of $800 is 4% of $1000?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>14</b></td><td align='left'>If 60 men need 1500 lb. flour per month, what is the +requirement per man per day counting a month as 30 +days?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>15</b></td><td align='left'>A car goes at the rate of a mile a minute. A truck goes 20 miles an hour. How many times as far will the car go as the truck in 10 seconds?</td><td align='right'><i>Answer</i>......</td></tr> +<tr><td align='right'><b>16</b></td><td align='left'>The area of the base (inside measure) of a cylindrical tank is 90 square feet. How tall must it be to hold 100 cubic yards?</td><td align='right'><i>Answer</i>......</td></tr> +</table></div> + +<p class="center"><small>From National Intelligence Tests by National Research Council.<br /> +Copyright, 1920, by The World Book Company, Yonkers-on-Hudson, New York.<br /> +Used by permission of the publishers.</small></p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_51" id="Page_51">[Pg 51]</a></span></p> +<h2><a name="CHAPTER_III" id="CHAPTER_III"></a>CHAPTER III</h2> + +<h3>THE CONSTITUTION OF ARITHMETICAL ABILITIES</h3> + + +<h4>THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING</h4> + +<p>It would be a useful work for some one to try to analyze +arithmetical learning into the unitary abilities which compose +it, showing just what, in detail, the mind has to do in +order to be prepared to pass a thorough test on the whole of +arithmetic. These unitary abilities would make a very +long list. Examination of a well-planned textbook will show +that such an ability as multiplication is treated as a composite +of the following: knowledge of the multiplications up +to 9 × 9; ability to multiply two (or more)-place numbers +by 2, 3, and 4 when 'carrying' is not required and no zeros +occur in the multiplicand; ability to multiply by 2, 3, ... 9, +with carrying; the ability to handle zeros in the multiplicand; +the ability to multiply with two-place numbers not ending +in zero; the ability to handle zero in the multiplier as +last number; the ability to multiply with three (or more)-place +numbers not including a zero; the ability to multiply +with three- and four-place numbers with zero in second +or third, or second and third, as well as in last place; the +ability to save time by annexing zeros; and so on and on +through a long list of further abilities required to multiply +with United States money, decimal fractions, common +fractions, mixed numbers, and denominate numbers.<span class='pagenum'><a name="Page_52" id="Page_52">[Pg 52]</a></span></p> + +<p>The units or 'steps' thus recognized by careful teaching +would make a long list, but it is probable that a still more +careful study of arithmetical ability as a hierarchy of mental +habits or connections would greatly increase the list. Consider, +for example, ordinary column addition. The majority +of teachers probably treat this as a simple application of the +knowledge of the additions to 9 + 9, plus understanding of +'carrying.' On the contrary there are at least seven processes +or minor functions involved in two-place column addition, +each of which is psychologically distinct and requires +distinct educational treatment.</p> + +<p>These are:—</p> + +<p class="nblockquot">A. Learning to keep one's place in the column as one adds.</p> + +<p class="nblockquot">B. Learning to keep in mind the result of each addition until +the next number is added to it.</p> + +<p class="nblockquot">C. Learning to add a seen to a thought-of number.</p> + +<p class="nblockquot">D. Learning to neglect an empty space in the columns.</p> + +<p class="nblockquot">E. Learning to neglect 0s in the columns.</p> + +<p class="nblockquot">F. Learning the application of the combinations to higher +decades may for the less gifted pupils involve as much +time and labor as learning all the original addition +tables. And even for the most gifted child the +formation of the connection '8 and 7 = 15' probably +never quite insures the presence of the connections +'38 and 7 = 45' and '18 + 7 = 25.'</p> + +<p class="nblockquot">G. Learning to write the figure signifying units rather than +the total sum of a column. In particular, learning +to write 0 in the cases where the sum of the column +is 10, 20, etc. Learning to 'carry' also involves in +itself at least two distinct processes, by whatever way +it is taught.</p> + +<p>We find evidence of such specialization of functions in +the results with such tests as Woody's. For example, +<span class='pagenum'><a name="Page_53" id="Page_53">[Pg 53]</a></span> +2 + 5 + 1 = .... surely involves abilities in part different from</p> + +<p class="center">2<br />4<br />3<br />—</p> + +<p class="noidt">because only 77 percent of children in grade 3 +do the former correctly, whereas 95 percent of children in +that grade do the latter correctly. In grade 2 the difference +is even more marked. In the case of subtraction</p> + +<p class="center">4<br />4<br />—</p> + +<p class="noidt">involves abilities different from those involved in</p> + +<p class="center">9<br />3<br />—,</p> + +<p class="noidt">being much less often solved correctly in grades 2 and 4.</p> + +<p class="center">6<br />0<br />—</p> + +<p class="noidt">is much harder than either of the above.</p> + +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'>43<br /> 1<br /> 2<br />13</td><td align='left' valign='bottom'>is much harder than</td><td align='left' valign='bottom'>21<br />33<br />35.</td></tr> +<tr><td align='left'>—</td><td></td><td align='left'>—</td></tr> +</table> + + +<p class="noidt">It may be said that these differences in difficulty are due to +different amounts of practice. This is probably not true, +but if it were, it would not change the argument; if the two +abilities were identical, the practice of one would improve +the other equally.</p> + +<p>I shall not undertake here this task of listing and describing +the elementary functions which constitute arithmetical +learning, partly because what they are is not fully known, +partly because in many cases a final ability may be constituted +in several different ways whose descriptions become +necessarily tedious, and partly because an adequate statement +of what is known would far outrun the space limits +of this chapter. Instead, I shall illustrate the results by +some samples.</p> +<p><span class='pagenum'><a name="Page_54" id="Page_54">[Pg 54]</a></span></p> + +<h4>KNOWLEDGE OF THE MEANING OF A FRACTION</h4> + +<p>As a first sample, consider knowledge of the meaning of +a fraction. Is the ability in question simply to understand +that a fraction is a statement of the number of parts, each +of a certain size, the upper number or numerator telling how +many parts are taken and the lower number or denominator +telling what fraction of unity each part is? And is the +educational treatment required simply to describe and +illustrate such a statement and have the pupils apply it to +the recognition of fractions and the interpretation of each +of them? And is the learning process (1) the formation +of the notions of part, size of part, number of part, (2) relating +the last two to the numbers in a fraction, and, as a +necessary consequence, (3) applying these notions adequately +whenever one encounters a fraction in operation?</p> + +<p>Precisely this was the notion a few generations ago. The +nature of fractions was taught as one principle, in one step, +and the habits of dealing with fractions were supposed to be +deduced from the general law of a fraction's nature. As a +result the subject of fractions had to be long delayed, was +studied at great cost of time and effort, and, even so, remained +a mystery to all save gifted pupils. These gifted +pupils probably of their own accord built up the ability +piecemeal out of constituent insights and habits.</p> + +<p>At all events, scientific teaching now does build up the +total ability as a fusion or organization of lesser abilities. +What these are will be seen best by examining the means +taken to get them. (1) First comes the association of ½ +of a pie, ½ of a cake, ½ of an apple, and such like with their +concrete meanings so that a pupil can properly name a +clearly designated half of an obvious unit like an orange, +pear, or piece of chalk. The same degree of understanding<span class='pagenum'><a name="Page_55" id="Page_55">[Pg 55]</a></span> +of <sup>1</sup>⁄<sub>4</sub>, <sup>1</sup>⁄<sub>8</sub>, <sup>1</sup>⁄<sub>3</sub>, <sup>1</sup>⁄<sub>6</sub>, and <sup>1</sup>⁄<sub>5</sub> is secured. The pupil is taught that +1 pie = 2 <sup>1</sup>⁄<sub>2</sub>s, 3 <sup>1</sup>⁄<sub>3</sub>s, +4 <sup>1</sup>⁄<sub>4</sub>s, 5 <sup>1</sup>⁄<sub>5</sub>s, 6 <sup>1</sup>⁄<sub>6</sub>s, and 8 <sup>1</sup>⁄<sub>8</sub>s; similarly for 1 +cake, 1 apple, and the like.</p> + +<p>So far he understands <sup>1</sup>⁄<sub><i>x</i></sub> of <i>y</i> in the sense of certain simple +parts of obviously unitary <i>y</i>s.</p> + +<p>(2) Next comes the association with ½ of an inch, ½ of a +foot, ½ of a glassful and other cases where <i>y</i> is not so obviously +a unitary object whose pieces still show their derivation +from it. Similarly for <sup>1</sup>⁄<sub>4</sub>, <sup>1</sup>⁄<sub>3</sub>, etc.</p> + +<p>(3) Next comes the association with <sup>1</sup>⁄<sub>2</sub> of a collection of +eight pieces of candy, <sup>1</sup>⁄<sub>3</sub> of a dozen eggs, <sup>1</sup>⁄<sub>5</sub> of a squad of ten +soldiers, etc., until <sup>1</sup>⁄<sub>2</sub>, <sup>1</sup>⁄<sub>3</sub>, <sup>1</sup>⁄<sub>4</sub>, <sup>1</sup>⁄<sub>5</sub>, <sup>1</sup>⁄<sub>6</sub>, and <sup>1</sup>⁄<sub>8</sub> are understood as +names of certain parts of a collection of objects.</p> + +<p>(4) Next comes the similar association when the nature +of the collection is left undefined, the pupil responding to<br /> +<sup>1</sup>⁄<sub>2</sub> of 6 is ..., <sup>1</sup>⁄<sub>4</sub> of 8 is ..., 2 is <sup>1</sup>⁄<sub>5</sub> of ...,<br /> +<sup>1</sup>⁄<sub>3</sub> of 6 is ..., <sup>1</sup>⁄<sub>3</sub> of 9 is ..., 2 is <sup>1</sup>⁄<sub>3</sub> of ..., and the like.</p> + +<p>Each of these abilities is justified in teaching by its intrinsic +merits, irrespective of its later service in helping to +constitute the general understanding of the meaning of a +fraction. The habits thus formed in grades 3 or 4 are of +constant service then and thereafter in and out of school.</p> + +<p>(5) With these comes the use of <sup>1</sup>⁄<sub>5</sub> of 10, 15, 20, etc., <sup>1</sup>⁄<sub>6</sub> of +12, 18, 42, etc., as a useful variety of drill on the division +tables, valuable in itself, and a means of making the notion +of a unit fraction more general by adding <sup>1</sup>⁄<sub>7</sub> and <sup>1</sup>⁄<sub>9</sub> to the +scheme.</p> + +<p>(6) Next comes the connection of <sup>3</sup>⁄<sub>4</sub>, <sup>2</sup>⁄<sub>5</sub>, <sup>3</sup>⁄<sub>5</sub>, <sup>4</sup>⁄<sub>5</sub>, <sup>2</sup>⁄<sub>3</sub>, <sup>1</sup>⁄<sub>6</sub>, <sup>5</sup>⁄<sub>6</sub>, +<sup>3</sup>⁄<sub>8</sub>, <sup>5</sup>⁄<sub>8</sub>, +<sup>7</sup>⁄<sub>8</sub>, <sup>3</sup>⁄<sub>10</sub>, <sup>7</sup>⁄<sub>10</sub>, +and <sup>9</sup>⁄<sub>10</sub>, each with its meaning as a certain part of +some conveniently divisible unit, and, (7) and (8), connections +between these fractions and their meanings as parts +of certain magnitudes (7) and collections (8) of convenient<span class='pagenum'><a name="Page_56" id="Page_56">[Pg 56]</a></span> +size, and (9) connections between these fractions and their +meanings when the nature of the magnitude or collection is +unstated, as in <sup>4</sup>⁄<sub>5</sub> of 15 = ..., <sup>5</sup>⁄<sub>8</sub> of 32 = ....</p> + +<p>(10) That the relation is general is shown by using it with +numbers requiring written division and multiplication, such +as <sup>7</sup>⁄<sub>8</sub> of 1736 = ..., and with United States money.</p> + +<p>Elements (6) to (10) again are useful even if the pupil never +goes farther in arithmetic. One of the commonest uses of +fractions is in calculating the cost of fractions of yards of +cloth, and fractions of pounds of meat, cheese, etc.</p> + +<p>The next step (11) is to understand to some extent the +principle that the value of any of these fractions is unaltered +by multiplying or dividing the numerator and denominator +by the same number. The drills in expressing fractions in +lower and higher terms which accomplish this are paralleled +by (12) and (13) simple exercises in adding and subtracting +fractions to show that fractions are quantities that can be +operated on like any quantities, and by (14) simple work +with mixed numbers (addition and subtraction and reductions), +and (15) improper fractions. All that is done with +improper fractions is (<i>a</i>) to have the pupil use a few of them +as he would any fractions and (<i>b</i>) to note their equivalent +mixed numbers. In (12), (13), and (14) only fractions of +the same denominators are added or subtracted, and in (12) +(13), (14), and (15) only fractions with 2, 3, 4, 5, 6, 8, or 10 +in the denominator are used. As hitherto, the work of (11) +to (15) is useful in and of itself. (16) Definitions are given of +the following type:—</p> + +<p>Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called +whole numbers.</p> + +<p>Numbers like <sup>7</sup>⁄<sub>8</sub>, <sup>1</sup>⁄<sub>5</sub>, <sup>2</sup>⁄<sub>3</sub>, +<sup>3</sup>⁄<sub>4</sub>, <sup>11</sup>⁄<sub>8</sub>, +<sup>7</sup>⁄<sub>6</sub>, <sup>1</sup>⁄<sub>3</sub>, <sup>4</sup>⁄<sub>3</sub>, <sup>1</sup>⁄<sub>8</sub>, +<sup>1</sup>⁄<sub>6</sub> are called fractions.</p> + +<p>Numbers like 5¼, 7<small><sup>3</sup>⁄<sub>8</sub></small>, 9½, +16<small><sup>4</sup>⁄<sub>5</sub></small>, +315<small><sup>7</sup>⁄<sub>8</sub></small>, +1<small><sup>1</sup>⁄<sub>3</sub></small>, +1<small><sup>2</sup>⁄<sub>3</sub></small> are called mixed +numbers.<span class='pagenum'><a name="Page_57" id="Page_57">[Pg 57]</a></span></p> + +<p>(17) The terms numerator and denominator are connected +with the upper and lower numbers composing a fraction.</p> + +<p>Building this somewhat elaborate series of minor abilities +seems to be a very roundabout way of getting knowledge +of the meaning of a fraction, and is, if we take no account +of what is got along with this knowledge. Taking account +of the intrinsically useful habits that are built up, one might +retort that the pupil gets his knowledge of the meaning of a +fraction at zero cost.</p> + + +<h4>KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES</h4> + +<p>Consider next the knowledge of the subtraction and division +'Tables.' The usual treatment presupposes that learning +them consists of forming independently the bonds:—</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='center'>3 − 1 = 2</td><td align='center'>4 ÷ 2 = 2</td></tr> +<tr><td align='center'>3 − 2 = 1</td><td align='center'>6 ÷ 2 = 3</td></tr> +<tr><td align='center'>4 − 1 = 3</td><td align='center'>6 ÷ 3 = 2</td></tr> +<tr> +<td align='center'> + .<br /> + .<br /> + .</td> +<td align='center'> + .<br /> + .<br /> + .</td> +</tr> +<tr><td align='center'>18 − 9 = 9</td><td align='center'>81 ÷ 9 = 9</td></tr> +</table></div> + +<p>In fact, however, these 126 bonds are not formed independently. +Except perhaps in the case of the dullest +twentieth of pupils, they are somewhat facilitated by the +already learned additions and multiplications. And by +proper arrangement of the learning they may be enormously +facilitated thereby. Indeed, we may replace the independent +memorizing of these facts by a set of instructive +<span class='pagenum'><a name="Page_58" id="Page_58">[Pg 58]</a></span>exercises wherein the pupil derives the subtractions from +the corresponding additions by simple acts of reasoning or +selective thinking. As soon as the additions giving sums of +9 or less are learned, let the pupil attack an exercise like +the following:—</p> + +<p>Write the missing numbers:—</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='center'>A</td><td align='center'>B</td><td align='center'>C</td><td align='center'>D</td></tr> +<tr><td align='left'>3 and ... are 5.</td><td align='left'>5 and ... are 8.</td><td align='left'>4 and ... are 5.</td><td align='left'>4 and ... are 8.</td></tr> +<tr><td align='left'>3 and ... are 9.</td><td align='left'>3 and ... are 6.</td><td align='left'>5 and ... are 6.</td><td align='left'>1 and ... are 7.</td></tr> +<tr><td align='left'>4 and ... are 7.</td><td align='left'>4 and ... are 9.</td><td align='left'>6 and ... are 9.</td><td align='left'>6 and ... are 7.</td></tr> +<tr><td align='left'>5 and ... are 7.</td><td align='left'>2 and ... = 6.</td><td align='left'>1 and ... are 8.</td><td align='left'>8 and ... are 9.</td></tr> +<tr><td align='left'>6 and ... are 8.</td><td align='left'>5 and ... = 9.</td><td align='left'>3 and ... are 7.</td><td align='left'>3 + ... are 4.</td></tr> +<tr><td align='left'>4 and ... are 6.</td><td align='left'>2 and ... = 7.</td><td align='left'>1 + ... are 3.</td><td align='left'>7 + ... are 8.</td></tr> +<tr><td align='left'>2 and ... are 5.</td><td align='left'>3 and ... = 8.</td><td align='left'>1 + ... are 5.</td><td align='left'>4 + ... are 9.</td></tr> +<tr><td align='left'>2 and ... = 8.</td><td align='left'>1 and ... = 4.</td><td align='left'>4 + ... are 8.</td><td align='left'>2 + ... are 3.</td></tr> +<tr><td align='left'>3 and ... = 6.</td><td align='left'>2 and ... = 4.</td><td align='left'>7 + ... are 9.</td><td align='left'>1 + ... are 9.</td></tr> +<tr><td align='left'>6 and ... = 9.</td><td align='left'>3 and ... = 8.</td><td align='left'>2 + ... = 4.</td><td align='left'>3 + ... = 6.</td></tr> +<tr><td align='left'>4 and ... = 6.</td><td align='left'>6 and ... = 7.</td><td align='left'>3 + ... = 8.</td><td align='left'>5 + ... = 9.</td></tr> +<tr><td align='left'>4 and ... = 7.</td><td align='left'>2 and ... = 5.</td><td align='left'>4 + ... = 5.</td><td align='left'>1 + ... = 3.</td></tr> +</table></div> + +<p>The task for reasoning is only to try, one after another, +numbers that seem promising and to select the right one +when found. With a little stimulus and direction children +can thus derive the subtractions up to those with 9 as the +larger number. Let them then be taught to do the same +with the printed forms:—</p> + +<p class="center">Subtract</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='center'>9</td><td align='center'>7</td><td align='center'>8</td><td align='center'>5</td><td align='center'>8</td><td align='center'>6</td></tr> +<tr><td align='center'>3</td><td align='center'>5</td><td align='center'>6</td><td align='center'>2</td><td align='center'>2</td><td align='center'>4</td><td align='center'>etc.</td></tr> +<tr><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td></tr> +</table></div> + +<p class="noidt">and 9 − 7 = ..., 9 − 5 = ..., 7 − 5 = ..., etc.</p> + +<p>In the case of the divisions, suppose that the pupil has +learned his first table and gained surety in such exercises +as:<span class='pagenum'><a name="Page_59" id="Page_59">[Pg 59]</a></span>—</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'>4 5s = ....</td><td align='left'>6 × 5 = ....</td><td align='left'>9 nickels = .... cents.</td></tr> +<tr><td align='left'>8 5s = ....</td><td align='left'>4 × 5 = ....</td><td align='left'>6 " = .... "</td></tr> +<tr><td align='left'>3 5s = ....</td><td align='left'>2 × 5 = ....</td><td align='left'>5 " = .... "</td></tr> +<tr><td align='left'>7 5s = ....</td><td align='left'>9 × 5 = ....</td><td align='left'>7 " = .... "</td></tr> +</table></div> + +<p class="noidt">If one ball costs 5 cents,<br /> +<span style="margin-left: 6em;">two balls cost .... cents,</span><br /> +<span style="margin-left: 6em;">three balls cost .... cents, etc.</span><br /> +</p> + +<p class="noidt">He may then be set at once to work at the answers to exercises +like the following:—</p> + +<p>Write the answers and the missing numbers:—</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='center'>A</td><td align='center'>B</td><td align='center'>C</td><td align='center'>D</td></tr> +<tr><td align='left'>.... 5s = 15</td><td align='left'>40 = .... 5s</td><td align='left'>.... × 5 = 25</td><td align='left'>20 cents = .... nickels.</td></tr> +<tr><td align='left'>.... 5s = 20</td><td align='left'>20 = .... 5s</td><td align='left'>.... × 5 = 50</td><td align='left'>30 cents = .... nickels.</td></tr> +<tr><td align='left'>.... 5s = 40</td><td align='left'>15 = .... 5s</td><td align='left'>.... × 5 = 35</td><td align='left'>15 cents = .... nickels.</td></tr> +<tr><td align='left'>.... 5s = 25</td><td align='left'>45 = .... 5s</td><td align='left'>.... × 5 = 10</td><td align='left'>40 cents = .... nickels.</td></tr> +<tr><td align='left'>.... 5s = 30</td><td align='left'>50 = .... 5s</td><td align='left'>.... × 5 = 40</td></tr> +<tr><td align='left'>.... 5s = 35</td><td align='left'>25 = .... 5s</td><td align='left'>.... × 5 = 45</td></tr> +</table></div> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td class="center">E</td></tr> +<tr><td align='left'>For 5 cents you can buy 1 small loaf of bread.<br /> +For 10 cents you can buy 2 small loaves of bread.<br /> +For 25 cents you can buy .... small loaves of bread.<br /> +For 45 cents you can buy .... small loaves of bread.<br /> +For 35 cents you can buy .... small loaves of bread. +</td></tr> +</table></div> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td class="center">F</td></tr> +<tr><td align='left'> 5 cents pays 1 car fare.<br /> +15 cents pays .... car fares.<br /> +10 cents pays .... car fares.<br /> +20 cents pays .... car fares. +</td></tr> +</table></div> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td class="center">G</td></tr> +<tr><td align='left'>How many 5 cent balls can you buy with 30 cents? ....<br /> +How many 5 cent balls can you buy with 35 cents? ....<br /> +How many 5 cent balls can you buy with 25 cents? ....<br /> +How many 5 cent balls can you buy with 15 cents? ....</td></tr> +</table></div> + + +<p>In the case of the meaning of a fraction, the ability, and +so the learning, is much more elaborate than common +<span class='pagenum'><a name="Page_60" id="Page_60">[Pg 60]</a></span> +practice has assumed; in the case of the subtraction and +division tables the learning is much less so. In neither case +is the learning either mere memorizing of facts or the mere +understanding of a principle <i>in abstracto</i> followed by its +application to concrete cases. It is (and this we shall find +true of almost all efficient learning in arithmetic) the formation +of connections and their use in such an order that each +helps the others to the maximum degree, and so that each +will do the maximum amount for arithmetical abilities +other than the one specially concerned, and for the general +competence of the learner.</p> + + +<h4>LEARNING THE PROCESSES OF COMPUTATION</h4> + +<p>As another instructive topic in the constitution of arithmetical +abilities, we may take the case of the reasoning involved +in understanding the manipulations of figures in two +(or more)-place addition and subtraction, multiplication and +division involving a two (or more)-place number, and the +manipulations of decimals in all four operations. The +psychology of these is of special interest and importance. +For there are two opposite explanations possible here, +leading to two opposite theories of teaching.</p> + +<p>The common explanation is that these methods of manipulation, +if understood at all, are understood as deductions from +the properties of our system of decimal notation. The other +is that they are understood partly as inductions from the +experience that they always give the right answer. The +first explanation leads to the common preliminary deductive +explanations of the textbooks. The other leads to explanations +by verification; <i>e.g.</i>, of addition by counting, of subtraction +by addition, of multiplication by addition, of division +by multiplication. Samples of these two sorts of +explanation are given below.</p> +<p><span class='pagenum'><a name="Page_61" id="Page_61">[Pg 61]</a></span></p> + + +<div class="pblockquot"> +<p class="center"><b>SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION</b></p> + +<p><span class="smcap">Multiplication</span> is the process of taking one number as many +times as there are units in another number.</p> + +<p>The <span class="smcap">Product</span> is the result of the multiplication.</p> + +<p>The <span class="smcap">Multiplicand</span> is the number to be taken.</p> + +<p>The <span class="smcap">Multiplier</span> is the number denoting how many times the +multiplicand is to be taken.</p> + +<p>The multiplier and multiplicand are the <span class="smcap">Factors</span>.</p> + + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='center' colspan='2'>Multiply 623 by 3<br /><br />OPERATION<br /> </td></tr> +<tr><td align='left'><i>Multiplicand</i> </td><td align='right'>623</td></tr> +<tr><td align='left'><i>Multiplier</i></td><td align='right'>3</td></tr> +<tr><td align='left'><i>Product</i></td><td align='right'><span class="overline">1869</span></td></tr> +</table></div> + +<p><small><span class="smcap">Explanation.</span>—For convenience we write the multiplier under the multiplicand, +and begin with units to multiply. 3 times 3 units are 9 units. We write the nine +units in units' place in the product. 3 times 2 tens are 6 tens. We write the 6 tens +in tens' place in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and +8 hundreds. The 1 thousand we write in thousands' place and the 8 hundreds in +hundreds' place in the product. Therefore, the product is 1 thousand 8 hundreds, +6 tens and 9 units, or 1869.</small></p> + + +<p class="center"><b>SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE +EXPLANATION</b></p> + +<p><b>1.</b> The children of the third grade are to have a picnic. 32 are +going. How many sandwiches will they need if each of +the 32 children has four sandwiches?</p> + +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr><td align='right'> <br /><i>32</i><br /><span class="u"> <i>4</i></span></td> +<td align='left'><i>Here is a quick way to find out</i>:—<br /> +<i>Think "4 × 2," write 8 under the 2 in the ones column.</i><br /> +<i>Think "4 × 3," write 12 under the 3 in the tens column.</i> +</td></tr> +</table> + +<p><b>2.</b> How many bananas will they need if each of the 32 children +has two bananas? 32 × 2 or 2 × 32 will give the answer.</p> + +<p><b>3.</b> How many little cakes will they need if each child has three +cakes? 32 × 3 or 3 × 32 will give the answer.</p> + +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr><td align='right'><i>32</i><br /><span class="u"> <i>3</i></span></td> +<td align='left'><i>3 × 2 = .... Where do you write the 6?</i><br /> +<i>3 × 3 = .... Where do you write the 9?</i> +</td></tr> +</table> +<p><span class='pagenum'><a name="Page_62" id="Page_62">[Pg 62]</a></span></p> + +<p><b>4.</b> Prove that 128, 64, and 96 are right by adding four 32s, +two 32s, and three 32s.</p> + +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr><td align='right' valign='bottom'>32<br />32<br />32<br /><span class="u">32</span></td> +<td align='right' valign='bottom'>32<br />32<br /><span class="u">32</span></td> +<td align='right' valign='bottom'>32<br /><span class="u">32</span></td></tr> +</table> + + +<p class="center"><b>Multiplication</b></p> + +<p>You <b>multiply</b> when you find the answers to questions like</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'>How many are 9 × 3?<br /> +How many are 3 × 32?<br /> +How many are 8 × 5?<br /> +How many are 4 × 42?</td></tr> +</table></div> + +<p> +<b>1.</b> Read these lines. Say the right numbers where the dots are:<br /> +<span style="margin-left: 4.5em;">If you <b>add</b> 3 to 32, you have .... 35 is the <b>sum</b>.</span><br /> +<span style="margin-left: 4.5em;">If you <b>subtract</b> 3 from 32, the result is .... 29 is the <b>difference</b> or <b>remainder</b>.</span><br /> +<span style="margin-left: 4.5em;">If you <b>multiply</b> 3 by 32 or 32 by 3, you have .... 96 is the <b>product</b>.</span><br /> +</p> + +<p>Find the products. Check your answers to the first line by +adding.</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%"> +<tr><td align='right'><b>2. </b></td><td align='right'><b>3. </b></td><td align='right'><b>4. </b></td><td align='right'><b>5. </b></td><td align='right'><b>6. </b></td><td align='right'><b>7. </b></td><td align='right'><b>8. </b></td><td align='right'><b>9. </b></td></tr> +<tr><td align='right'>41</td><td align='right'>33</td><td align='right'>42</td><td align='right'>44</td><td align='right'>53</td><td align='right'>43</td><td align='right'>34</td><td align='right'>24</td></tr> +<tr><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 4</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 2</span></td></tr> +<tr><td colspan='8'> </td></tr> +<tr><td align='right'><b>10. </b></td><td align='right'><b>11. </b></td><td align='right'><b>12. </b></td><td align='right'><b>13. </b></td><td align='right'><b>14. </b></td><td align='right'><b>15. </b></td><td align='right'><b>16. </b></td></tr> +<tr><td align='right'>43</td><td align='right'>52</td><td align='right'>32</td><td align='right'>23</td><td align='right'>41</td><td align='right'>51</td><td align='right'>14</td></tr> +<tr><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 4</span></td><td align='right'><span class="u"> 2</span></td></tr> +</table></div> + +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'><br /><b>17.</b></td></tr> +<tr><td align='right'><i>213</i><br /><span class="u"> <i>3</i></span><br /> </td> +<td align='left'><i>Write the 9 in the ones column.</i><br /> +<i>Write the 6 in the hundreds column.</i> <br /> +<i>Write the 3 in the tens column.</i></td> +<td align='left'><i>Check your answer by adding.</i></td> +<td align='left'>Add<br /><i>213</i><br /><i>213</i><br /><span class="u"><i>213</i></span><br /> </td> +</tr> +</table> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%"> +<tr><td align='right'><b>18. </b></td><td align='right'><b>19. </b></td><td align='right'><b>20. </b></td><td align='right'><b>21. </b></td><td align='right'><b>22. </b></td><td align='right'><b>23. </b></td><td align='right'><b>24. </b></td></tr> +<tr><td align='right'>214</td><td align='right'>312</td><td align='right'>432</td><td align='right'>231</td><td align='right'>132</td><td align='right'>314</td><td align='right'>243</td></tr> +<tr><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 3</span></td><td align='right'><span class="u"> 2</span></td><td align='right'><span class="u"> 2</span></td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_63" id="Page_63">[Pg 63]</a></span></p> + +<p class="center"><b>SHORT DIVISION: DEDUCTIVE EXPLANATION</b></p> + +<p class="center">Divide 1825 by 4</p> + +<p>Divisor 4 |<span class="u"> 1825 </span> Dividend<br /> +<span style="margin-left: 6.5em;">456¼</span><br /> +<span style="margin-left: 6em;">Quotient</span></p> + +<p><small><span class="smcap">Explanation.</span>—For convenience +we write the divisor at the left of +the dividend, and the quotient below +it, and begin at the left to divide. +<b>4</b> is not contained in 1 thousand +any thousand times, therefore +the quotient contains no unit of +any order higher than hundreds. +Consequently we find how many times 4 is contained in the hundreds of the dividend. +1 thousand and 8 hundreds are 18 hundreds. 4 is contained in 18 hundreds +4 hundred times and 2 hundreds remaining. We write the 4 hundreds in the quotient. +The 2 hundreds we consider as united with the 2 tens, making 22 tens. +4 is contained in 22 tens 5 tens times, and 2 tens remaining. We write the 5 tens +in the quotient, and the remaining 2 tens we consider as united with the 5 units, +making 25 units. 4 is contained in 25 units 6 units times and 1 unit remaining. +We write the 6 units in the quotient and indicate the division of the remainder, 1 +unit, by the divisor 4.</small></p> + +<p><small>Therefore the quotient of 1825 divided by 4 is 456¼, or 456 and 1 remainder.</small></p> + + +<p class="center"><b>SHORT DIVISION: INDUCTIVE EXPLANATION</b></p> + + +<p class="center"><b>Dividing Large Numbers</b></p> + +<p><b>1.</b> Tom, Dick, Will, and Fred put in 2 cents each to buy an +eight-cent bag of marbles. There are 128 marbles in it. +How many should each boy have, if they divide the +marbles equally among the four boys?</p> + +<p>4 |<span class="overline"> 128</span></p> + +<p><i>Think "12 = three 4s." Write the 3 over the 2 in the tens column.</i></p> + +<p><i>Think "8 = two 4s." Write the 2 over the 8 in the ones column.</i></p> + +<p><i>32 is right, because 4 × 32 = 128.</i></p> + + +<p><b>2.</b> Mary, Nell, and Alice are going to buy a book as a present +for their Sunday-school teacher. The present costs 69 +cents. How much should each girl pay, if they divide +the cost equally among the three girls?</p> + +<p>3 |<span class="overline"> 69</span></p> + +<p><i>Think "6 = .... 3s." Write the 2 over the 6 in the tens column</i>.<span class='pagenum'><a name="Page_64" id="Page_64">[Pg 64]</a></span></p> + +<p><i>Think "9 = .... 3s." Write the 3 over the 9 in the ones column.</i></p> + +<p><i>23 is right, for 3 × 23 = 69.</i></p> + +<p><b>3.</b> Divide the cost of a 96-cent present equally among three +girls. How much should each girl pay? +girls. How much should each girl pay? 3 |<span class="overline"> 96</span></p> + +<p><b>4.</b> Divide the cost of an 84-cent present equally among 4 girls. +How much should each girl pay?</p> + +<p><b>5.</b> Learn this: (Read ÷ as "<i>divided by</i>.")</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'><b>12 + 4 = 16.</b> </td><td align='left'><b>16 is the sum.</b></td></tr> +<tr><td align='left'><b>12 − 4 = 8.</b></td><td align='left'><b> 8 is the difference or remainder.</b></td></tr> +<tr><td align='left'><b>12 × 4 = 48.</b></td><td align='left'><b>48 is the product.</b></td></tr> +<tr><td align='left'><b>12 ÷ 4 = 3.</b></td><td align='left'><b> 3 is the quotient.</b></td></tr> +</table></div> +<p><b>6.</b> Find the quotients. Check your answers by multiplying.</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%"> +<tr><td align='left'>3 |<span class="overline"> 99</span></td><td align='left'>2 |<span class="overline"> 86</span></td><td align='left'>5 |<span class="overline"> 155</span></td> +<td align='left'>6 |<span class="overline"> 246</span></td><td align='left'>4 |<span class="overline"> 168</span></td><td align='left'>3 |<span class="overline"> 219</span></td></tr> +</table></div> +<p>[Uneven division is taught by the same general plan, extended.]</p> + + +<p class="center"><b>LONG DIVISION: DEDUCTIVE EXPLANATION</b></p> + + +<p class="center"><b>To Divide by Long Division</b></p> + +<p><b>1</b>. Let it be required to divide 34531 by 15.</p> + +<p class="center"><i>Operation</i></p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr> +<td align='right' valign='top'> <br />Divisor </td> +<td align='right' valign='top'>Divided<br /> +15 ) 34531 (<br /> +30 <br /> +<span class="overline"> 45</span> <br /> +<span class="u"> 45</span> <br /> +31 <br /> +<span class="u">30</span> <br /> +1 <br /> +</td> +<td align='left' valign='top'> <br />2302<sup>1</sup>⁄<sub>15</sub> Quotient</td> +</tr> +<tr><td align='left'></td><td align='right'>Remainder</td><td align='left'></td></tr> +</table></div> + + +<p>For convenience we write +the divisor at the left and the +quotient at the right of the +dividend, and begin to divide +as in Short Division.</p> + +<p>15 is contained in 3 ten-thousands +0 ten-thousands +times; therefore, there will +be 0 ten-thousands in the +quotient. Take 34 thousands; +15 is contained in +34 thousands 2 thousands +times; we write the 2 thousands in the quotient. 15 × 2 thousands += 30 thousands, which, subtracted from 34 thousands, +leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we +have 45 hundreds.</p> + +<p>15 in 45 hundreds 3 hundreds times; we write the 3 hundreds +in the quotient. 15 × 3 hundreds = 45 hundreds, which subtracted +from 45 hundreds, leaves nothing. Adding the 3 tens, we have +3 tens.<span class='pagenum'><a name="Page_65" id="Page_65">[Pg 65]</a></span></p> + +<p>15 in 3 tens 0 tens times; we write 0 tens in the quotient. +Adding to the three tens, which equal 30 units, the 1 unit, we +have 31 units.</p> + +<p>15 in 31 units 2 units times; we write the 2 units in the quotient. +15 × 2 units = 30 units, which, subtracted from 31 units, leaves 1 +unit as a remainder. Indicating the division of the 1 unit, we +annex the fractional expression, <sup>1</sup>⁄<sub>15</sub> unit, to the integral part of the +quotient.</p> + +<p>Therefore, 34531 divided by 15 is equal to 2302<sup>1</sup>⁄<sub>15</sub>.</p> + +<p class="noidt">[B. Greenleaf, <i>Practical Arithmetic</i>, '73, p. 49.]</p> + + +<p class="center"><b>LONG DIVISION: INDUCTIVE EXPLANATION</b></p> + +<p class="center"><b>Dividing by Large Numbers</b></p> + +<p><b>1</b>. Just before Christmas Frank's father sent 360 oranges to +be divided among the children in Frank's class. There +are 29 children. How many oranges should each child +receive? How many oranges will be left over?</p> + +<p><i>Here is the best way to find out:</i></p> + +<div class='center'> +<table border="0" cellpadding="5" cellspacing="0" summary=""> +<tr> +<td align='right' valign='top'><i> +12<br /> +29|<span class="overline">360</span><br /> +29 <br /> +<span class="overline"> 70</span><br /> +58<br /> +<span class="overline">12</span> +</i></td> +<td align='right' valign='top'><i><small>and 12 remainder</small></i></td> +<td align='left'> +<i>Think how many 29s there are in 36. 1 is right.<br /> +Write 1 over the 6 of 36. Multiply 29 by 1.<br /> +Write the 29 under the 36. Subtract 29 from 36.<br /> +Write the 0 of 360 after the 7.<br /> +Think how many 29s there are in 70. 2 is right.<br /> +Write 2 over the 0 of 360. Multiply 29 by 2.<br /> +Write the 58 under 70. Subtract 58 from 70.<br /> +There is 12 remainder.</i><br /> +<br /> +<i>Each child gets 12 oranges, and there are 12 left +over. This is right, for 12 multiplied by 29 = 348, +and 348 + 12 = 360</i>. +</td></tr> +</table></div> + + +<hr style='width: 45%;' /> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%"> +<tr> +<td align='left'><b> 8.</b><br />31 |<span class="overline"> 99,587</span></td> +<td align='left'><i>In No. 8, keep on dividing by 31 until you +have used the 5, the 8, and the 7, and have +four figures in the quotient.</i></td> +</tr> +</table></div> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%"> +<tr> +<td align='left'><b> 9.</b><br />22 |<span class="overline"> 253</span></td> +<td align='left'><b> 10.</b><br />22 |<span class="overline"> 2895</span></td> +<td align='left'><b> 11.</b><br />21 |<span class="overline"> 8891</span></td> +<td align='left'><b> 12.</b><br />22 |<span class="overline"> 290</span></td> +<td align='left'><b> 13.</b><br />32 |<span class="overline"> 16,368</span></td> +</tr> +</table></div> + +<p>Check your results for 9, 10, 11, 12, and 13.</p> + +<p> </p> +<p><span class='pagenum'><a name="Page_66" id="Page_66">[Pg 66]</a></span></p> + +<p><b>1</b>. The boys and girls of the Welfare Club plan to earn money +to buy a victrola. There are 23 boys and girls. They +can get a good second-hand victrola for $5.75. How +much must each earn if they divide the cost equally?</p> + +<p><i>Here is the best way to find out</i>:</p> + +<div class='center'> +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr> +<td align='right'><i> +$.25<br /> +23|<span class="overline">$5.75</span><br /> +46 <br /> +<span class="overline"> 115</span><br /> +<span class="u">115</span> +</i></td> +<td align='left'> +<i>Think how many 23s there are in 57. 2 is right.<br /> +Write 2 over the 7 of 57. Multiply 23 by 2.<br /> +Write 46 under 57 and subtract. Write the 5 of 575 after the 11.<br /> +Think how many 23s there are in 115. 5 is right.<br /> +Write 5 over the 5 of 575. Multiply 23 by 5.<br /> +Write the 115 under the 115 that is there and subtract.<br /> +There is no remainder.<br /> +Put $ and the decimal point where they belong.<br /> +Each child must earn 25 cents. This is right, for $.25 multiplied by 23 = $5.75</i>. +</td></tr> +</table></div> + +<p><b>2</b>. Divide $71.76 equally among 23 persons. How much is +each person's share?</p> + +<p><b>3</b>. Check your result for No. 2 by multiplying the quotient by +the divisor.</p> + +<p>Find the quotients. Check each quotient by multiplying it by +the divisor.</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="90%"> +<tr> +<td align='left'><b> 4.</b><br />23 |<span class="overline"> $99.13</span></td> +<td align='left'><b> 5.</b><br />25 |<span class="overline"> $18.50</span></td> +<td align='left'><b> 6.</b><br />21 |<span class="overline"> $129.15</span></td> +<td align='left'><b> 7.</b><br />13 |<span class="overline"> $29.25</span></td> +<td align='left'><b> 8.</b><br />32 |<span class="overline"> $73.92</span></td> +</tr> +</table></div> + +<p class="center"><b>1 bushel = 32 qt</b>.</p> + +<p><b>9</b>. How many bushels are there in 288 qt.? <b>10</b>. In 192 qt.? <b>11</b>. In 416 qt.?</p> +</div> + +<p>Crucial experiments are lacking, but there are several +lines of well-attested evidence. First of all, there can be +no doubt that the great majority of pupils learn these manipulations +at the start from the placing of units under units, +tens under tens, etc., in adding, to the placing of the decimal +point in division with decimals, by imitation and blind<span class='pagenum'><a name="Page_67" id="Page_67">[Pg 67]</a></span> +following of specific instructions, and that a very large proportion +of the pupils do not to the end, that is to the fifth +school-year, understand them as necessary deductions from +decimal notation. It also seems probable that this proportion +would not be much reduced no matter how ingeniously +and carefully the deductions were explained by textbooks +and teachers. Evidence of this fact will appear abundantly +to any one who will observe schoolroom life. It also appears +in the fact that after the properties of the decimal notation +have been thus used again and again; <i>e.g.</i>, for deducing +'carrying' in addition, 'borrowing' in subtraction, 'carrying' +in multiplication, the value of the digits in the partial product, +the value of each remainder in short division, the +value of the quotient figures in division, the addition, subtraction, +multiplication, and division of United States money, +and the placing of the decimal point in multiplication, no +competent teacher dares to rely upon the pupil, even though +he now has four or more years' experience with decimal notation, +to deduce the placing of the decimal point in division +with decimals. It may be an illusion, but one seems to +sense in the better textbooks a recognition of the futility +of the attempt to secure deductive derivations of those +manipulations. I refer to the brevity of the explanations +and their insertion in such a form that they will influence +the pupils' thinking as little as possible. At any rate the +fact is sure that most pupils do not learn the manipulations +by deductive reasoning, or understand them as necessary +consequences of abstract principles.</p> + +<p>It is a common opinion that the only alternative is knowing +them by rote. This, of course, is one common alternative, +but the other explanation suggests that understanding the +manipulations by inductive reasoning from their results +is another and an important alternative. The manipula<span class='pagenum'><a name="Page_68" id="Page_68">[Pg 68]</a></span>tions +of 'long' multiplication, for instance, learned by imitation +or mechanical drill, are found to give for 25 × <i>A</i> a +result about twice as large as for 13 × <i>A</i>, for 38 or 39 × <i>A</i> +a result about three times as large; for 115 × <i>A</i> a result +about ten times as large as for 11 × <i>A</i>. With even the very +dull pupils the procedure is verified at least to the extent +that it gives a result which the scientific expert in the case—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 <i>does</i>, but not that it <i>must</i> give the right +answer, the answer being 'right' because the teacher, the +answer-list, and collateral evidence assure him of it.</p> + +<p>I have taken the manipulation of the partial products as +an illustration because it is one of the least favored cases +for the explanation I am presenting. If we take the first +case where a manipulation may be deduced from decimal +notation, known merely by rote, or verified inductively, +namely, the addition of two-place numbers, it seems sure +that the mental processes just described are almost the +universal rule.</p> + +<p>Surely in our schools at present children add the 3 of 23 +to the 3 of 53 and the 2 of 23 to the 5 of 53 at the start, in +nine cases out of ten because they see the teacher do so and +are told to do so. They are protected from adding +3 + 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<span class='pagenum'><a name="Page_69" id="Page_69">[Pg 69]</a></span> +shown or told what they are to do. They are protected from +adding 3 + 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 +(<i>a</i> + <i>b</i>) + (<i>c</i> + <i>d</i>) = (<i>a</i> + <i>c</i>) + (<i>b</i> + <i>d</i>)!</p> + +<p>Just as surely all but the very dullest twentieth or so of +children come in the end to something more than rote +knowledge,—to <i>understand</i>, to <i>know</i> that the procedure +in question is right.</p> + +<p>Whether they know <i>why</i> 76 is right depends upon what is +meant by <i>why</i>. If it means that 76 is the result which +competent people agree upon, they do. If it means that 76 +is the result which would come from accurate counting they +perhaps know why as well as they would have, had they been +given full explanations of the relation of the procedure in +two-place addition to decimal notation. If <i>why</i> means +because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and (<i>a</i> + <i>b</i>) + (<i>c</i> + <i>d</i>) = (<i>a</i> + <i>c</i>) + (<i>b</i> + <i>d</i>), +they do not. Nor, I am tempted +to add, would most of them by any sort of teaching whatever.</p> + +<p>I conclude, therefore, that school children may and do +reason about and understand the manipulations of numbers +in this inductive, verifying way without being able to, or at +least without, under present conditions, finding it profitable +to derive them deductively. I believe, in fact, that pure +arithmetic <i>as it is learned and known</i> is largely an <i>inductive +science</i>. At one extreme is a minority to whom it is a series +of deductions from principles; at the other extreme is a +minority to whom it is a series of blind habits; between +the two is the great majority, representing every gradation +but centering about the type of the inductive thinker.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_70" id="Page_70">[Pg 70]</a></span></p> +<h2><a name="CHAPTER_IV" id="CHAPTER_IV"></a>CHAPTER IV</h2> + +<h3>THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED): +THE SELECTION OF THE BONDS TO BE +FORMED</h3> + + +<p>When the analysis of the mental functions involved in +arithmetical learning is made thorough it turns into the +question, 'What are the elementary bonds or connections +that constitute these functions?' and when the problem of +teaching arithmetic is regarded, as it should be in the light +of present psychology, as a problem in the development of a +hierarchy of intellectual habits, it becomes in large measure +a problem of the choice of the bonds to be formed and of the +discovery of the best order in which to form them and the +best means of forming each in that order.</p> + + +<h4>THE IMPORTANCE OF HABIT-FORMATION</h4> + +<p>The importance of habit-formation or connection-making +has been grossly underestimated by the majority of teachers +and writers of textbooks. For, in the first place, mastery +by deductive reasoning of such matters as 'carrying' in +addition, 'borrowing' in subtraction, the value of the digits +in the partial products in multiplication, the manipulation +of the figures in division, the placing of the decimal point +after multiplication or division with decimals, or the manipulation +of the figures in the multiplication and division of +<span class='pagenum'><a name="Page_71" id="Page_71">[Pg 71]</a></span> +fractions, is impossible or extremely unlikely in the case of +children of the ages and experience in question. They do +not as a rule deduce the method of manipulation from their +knowledge of decimal notation. Rather they learn about +decimal notation by carrying, borrowing, writing the last +figure of each partial product under the multiplier which +gives that product, etc. They learn the method of manipulating +numbers by seeing them employed, and by more or +less blindly acquiring them as associative habits.</p> + +<p>In the second place, we, who have already formed and +long used the right habits and are thereby protected against +the casual misleadings of unfortunate mental connections, +can hardly realize the force of mere association. When a +child writes sixteen as 61, or finds 428 as the sum of</p> + +<p class="center"> +15<br /> +19<br /> +16<br /> +<span class="u">18</span> +</p> + +<p class="noidt">or gives 642 as an answer to 27 × 36, or says that 4 divided +by ¼ = 1, we are tempted to consider him mentally perverse, +forgetting or perhaps never having understood that he +goes wrong for exactly the same general reason that we go +right; namely, the general law of habit-formation. If we +study the cases of 61 for 16, we shall find them occurring +in the work of pupils who after having been drilled in writing +26, 36, 46, 62, 63, and so on, in which the order of the six +in writing is the same as it is in speech, return to writing the +'teen numbers. If our language said onety-one for eleven +and onety-six for sixteen, we should probably never find +such errors except as 'lapses' or as the results of misperception +or lack of memory. They would then be more +frequent <i>before</i> the 20s, 30s, etc., were learned.</p> + +<p>If pupils are given much drill on written single column +addition involving the higher decades (each time writing +<span class='pagenum'><a name="Page_72" id="Page_72">[Pg 72]</a></span> +the two-figure sum), they are forming a habit of writing 28 +after the sum of 8, 6, 9, and 5 is reached; and it should not +surprise us if the pupil still occasionally writes the two-figure +sum for the first column though a second column is +to be added also. On the contrary, unless some counter +force influences him, he is absolutely sure to make this +mistake.</p> + +<p>The last mistake quoted (4 ÷ ¼ = 1) is interesting because +here we have possibly one of the cases where deduction from +psychology alone can give constructive aid to teaching. +Multiplication and division by fractions have been notorious +for their difficulty. The former is now alleviated by using <i>of</i> +instead of × until the new habit is fixed. The latter is still +approached with elaborate caution and with various means +of showing why one must 'invert and multiply' or 'multiply +by the reciprocal.'</p> + +<p>But in the author's opinion it seems clear that the difficulty +in multiplying and dividing by a fraction was not that +children felt any logical objections to canceling or inverting. +I fancy that the majority of them would cheerfully invert +any fraction three times over or cancel numbers at random +in a column if they were shown how to do so. But if you +are a youngster inexperienced in numerical abstractions +and if you have had <i>divide</i> connected with 'make smaller' +three thousand times and never once connected with 'make +bigger,' you are sure to be somewhat impelled to make the +number smaller the three thousand and first time you are +asked to divide it. Some of my readers will probably confess +that even now they feel a slight irritation or doubt in saying +or writing that <sup>16</sup>⁄<sub>1</sub> ÷ <sup>1</sup>⁄<sub>8</sub> = 128.</p> + +<p>The habits that have been confirmed by every multiplication +and division by integers are, in this particular of '<i>the +ratio of result to number operated upon</i>,' directly opposed to<span class='pagenum'><a name="Page_73" id="Page_73">[Pg 73]</a></span> +the formation of the habits required with fractions. And +that is, I believe, the main cause of the difficulty. Its +treatment then becomes easy, as will be shown later.</p> + +<p>These illustrations could be added to almost indefinitely, +especially in the case of the responses made to the so-called +'catch' problems. The fact is that the learner rarely can, +and almost never does, survey and analyze an arithmetical +situation and justify what he is going to do by articulate +deductions from principles. He usually feels the situation +more or less vaguely and responds to it as he has responded +to it or some situation like it in the past. Arithmetic is +to him not a logical doctrine which he applies to various +special instances, but a set of rather specialized habits of +behavior toward certain sorts of quantities and relations. +And in so far as he does come to know the doctrine it is +chiefly by doing the will of the master. This is true even +with the clearest expositions, the wisest use of objective +aids, and full encouragement of originality on the pupil's +part.</p> + +<p>Lest the last few paragraphs be misunderstood, I hasten to +add that the psychologists of to-day do not wish to make +the learning of arithmetic a mere matter of acquiring thousands +of disconnected habits, nor to decrease by one jot the +pupil's genuine comprehension of its general truths. They +wish him to reason not less than he has in the past, but more. +They find, however, that you do not secure reasoning in a +pupil by demanding it, and that his learning of a general +truth without the proper development of organized +habits back of it is likely to be, not a rational learning +of that general truth, but only a mechanical memorizing +of a verbal statement of it. They have come to +know that reasoning is not a magic force working in +independence of ordinary habits of thought, but an or<span class='pagenum'><a name="Page_74" id="Page_74">[Pg 74]</a></span>ganization +and coöperation of those very habits on a higher +level.</p> + +<p>The older pedagogy of arithmetic stated a general law or +truth or principle, ordered the pupil to learn it, and gave +him tasks to do which he could not do profitably unless he +understood the principle. It left him to build up himself +the particular habits needed to give him understanding and +mastery of the principle. The newer pedagogy is careful +to help him build up these connections or bonds ahead of +and along with the general truth or principle, so that he +can understand it better. The older pedagogy commanded +the pupil to reason and let him suffer the penalty of small +profit from the work if he did not. The newer provides +instructive experiences with numbers which will stimulate +the pupil to reason so far as he has the capacity, but will +still be profitable to him in concrete knowledge and skill, +even if he lacks the ability to develop the experiences into +a general understanding of the principles of numbers. The +newer pedagogy secures more reasoning in reality by not +pretending to secure so much.</p> + +<p>The newer pedagogy of arithmetic, then, scrutinizes +every element of knowledge, every connection made in the +mind of the learner, so as to choose those which provide the +most instructive experiences, those which will grow together +into an orderly, rational system of thinking about numbers +and quantitative facts. It is not enough for a problem +to be a test of understanding of a principle; it must also +be helpful in and of itself. It is not enough for an example +to be a case of some rule; it must help review and +consolidate habits already acquired or lead up to and +facilitate habits to be acquired. Every detail of the pupil's +work must do the maximum service in arithmetical +learning.</p> +<p><span class='pagenum'><a name="Page_75" id="Page_75">[Pg 75]</a></span></p> + +<h4>DESIRABLE BONDS NOW OFTEN NEGLECTED</h4> + +<p>As hitherto, I shall not try to list completely the elementary +bonds that the course of study in arithmetic should +provide for. The best means of preparing the student of +this topic for sound criticism and helpful invention is to let +him examine representative cases of bonds now often neglected +which should be formed and representative cases of +useless, or even harmful, bonds now often formed at considerable +waste of time and effort.</p> + +<p>(1) <i>Numbers as measures of continuous quantities.</i>—The +numbers one, two, three, 1, 2, 3, etc., should be connected +soon after the beginning of arithmetic each with the appropriate +amount of some continuous quantity like length or +volume or weight, as well as with the appropriate sized +collection of apples, counters, blocks, and the like. Lines +should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two +inches, three inches, etc.; weights should be lifted and called +one pound, two pounds, etc.; things should be measured in +glassfuls, handfuls, pints, and quarts. Otherwise the pupil +is likely to limit the meaning of, say, <i>four</i> to four sensibly +discrete things and to have difficulty in multiplication and +division. Measuring, or counting by insensibly marked off +repetitions of a unit, binds each number name to its meaning +as —— <i>times whatever 1 is</i>, more surely than mere counting +of the units in a collection can, and should reënforce the +latter.</p> + +<p>(2) <i>Additions in the higher decades.</i>—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,<span class='pagenum'><a name="Page_76" id="Page_76">[Pg 76]</a></span> +counting by 3s beginning with 1, with 2, and with 3, counting +by 4s beginning with 1, with 2, with 3, and with 4, and so on, +make easy beginnings in the formation of the decade connections. +Practice with isolated bonds should soon be added to +get freer use of the bonds. The work of column addition +should be checked for accuracy so that a pupil will continually +get beneficial practice rather than 'practice in +error.'</p> + +<p>(3) <i>The uneven divisions.</i>—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:—</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'> +10 = .... 2s<br /> +10 = .... 3s and .... rem.<br /> +10 = .... 4s and .... rem.<br /> +10 = .... 5s<br /> +11 = .... 2s and .... rem.<br /> +11 = .... 3s and .... rem.<br /> + .<br /> + .<br /> + .<br /> +89 = .... 9s and .... rem. +</td></tr> +</table></div> + +<p>These bonds must be formed before short division can be +efficient, are useful as a partial help toward selection of the +proper quotient figures in long division, and are the chief +instruments for one of the important problem series in +applied arithmetic,—"How many <i>x</i>s can I buy for <i>y</i> cents +at <i>z</i> cents per <i>x</i> and how much will I have left?" That +these bonds are at present sadly neglected is shown by Kirby +['13], who found that pupils in the last half of grade 3 and the<span class='pagenum'><a name="Page_77" id="Page_77">[Pg 77]</a></span> +first half of grade 4 could do only about four such examples +per minute (in a ten-minute test), and even at that rate +made far from perfect records, though they had been taught +the regular division tables. Sixty minutes of practice +resulted in a gain of nearly 75 percent in number done per +minute, with an increase in accuracy as well.</p> + +<p>(4) <i>The equation form</i>.—The equation form with an +unknown quantity to be determined, or a missing number +to be found, should be connected with its meaning and with +the problem attitude long before a pupil begins algebra, +and in the minds of pupils who never will study algebra.</p> + +<p>Children who have just barely learned to add and subtract +learn easily to do such work as the following:—</p> + +<p>Write the missing numbers:—</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='left'> +4 + 8 = ....<br /> +5 + .... = 14<br /> +.... + 3 = 11<br /> +.... = 5 + 2<br /> +16 = 7 + ....<br /> +12 = .... + 5 +</td></tr> +</table></div> + + +<p>The equation form is the simplest uniform way yet devised +to state a quantitative issue. It is capable of indefinite +extension if certain easily understood conventions about +parentheses and fraction signs are learned. It should be +employed widely in accounting and the treatment of commercial +problems, and would be except for outworn conventions. +It is a leading contribution of algebra to business and +industrial life. Arithmetic can make it nearly as well. +It saves more time in the case of drills on reducing fractions +to higher and lower terms alone than is required to learn +its meaning and use. To rewrite a quantitative problem<span class='pagenum'><a name="Page_78" id="Page_78">[Pg 78]</a></span> +as an equation and then make the easy selection of the +necessary technique to solve the equation is one of the most +universally useful intellectual devices known to man. The +words 'equals,' 'equal,' 'is,' 'are,' 'makes,' 'make,' 'gives,' +'give,' and their rarer equivalents should therefore early +give way on many occasions to the '=' which so far surpasses +them in ultimate convenience and simplicity.</p> + +<p>(5) <i>Addition and subtraction facts in the case of fractions.</i>—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.</p> + +<p>(6) <i>Fractional equivalents.</i>—Efficiency requires that in +the end the much used reductions should be firmly connected +with the situations where they are needed. They may as +well, therefore, be so connected from the beginning, with +the gain of making the general process far easier for the dull +pupils to master. We shall see later that, for all save the +very gifted pupils, the economical way to get an understanding +of arithmetical principles is not, usually, to learn a rule +and then apply it, but to perform instructive operations and, +in the course of performing them, to get insight into the +principles.</p> + +<p>(7) <i>Protective habits in multiplying and dividing with fractions.</i>—In +multiplying and dividing with fractions special +bonds should be formed to counteract the now harmful<span class='pagenum'><a name="Page_79" id="Page_79">[Pg 79]</a></span> +influence of the 'multiply = get a larger number' and 'divide = get +a smaller number' bonds which all work with integers +has been reënforcing.</p> + +<p>For example, at the beginning of the systematic work +with multiplication by a fraction, let the following be printed +clearly at the top of every relevant page of the textbook and +displayed on the blackboard:—</p> + +<p><i>When you multiply a number by anything more than 1 the +result is larger than the number.</i></p> + +<p><i>When you multiply a number by 1 the result is the same as the +number.</i></p> + +<p><i>When you multiply a number by anything less than 1 the +result is smaller than the number.</i></p> + +<p>Let the pupils establish the new habit by many such +exercises as:—</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary="" width="60%"> +<tr> +<td align='left'> +18 × 4 = ....<br /> +4 × 4 = ....<br /> +2 × 4 = ....<br /> +1 × 4 = ....<br /> +<sup>1</sup>⁄<sub>2</sub> × 4 = ....<br /> +<sup>1</sup>⁄<sub>4</sub> × 4 = ....<br /> +<sup>1</sup>⁄<sub>8</sub> × 4 = .... +</td> +<td align='left'> +9 × 2 = ....<br /> +6 × 2 = ....<br /> +3 × 2 = ....<br /> +1 × 2 = ....<br /> +<sup>1</sup>⁄<sub>3</sub> × 2 = ....<br /> +<sup>1</sup>⁄<sub>6</sub> × 2 = ....<br /> +<sup>1</sup>⁄<sub>9</sub> × 2 = .... +</td> +</tr> +</table></div> + +<p>In the case of division by a fraction the old harmful habit +should be counteracted and refined by similar rules and +exercises as follows:—</p> + +<p><i>When you divide a number by anything more than 1 the result +is smaller than the number.</i></p> + +<p><i>When you divide a number by 1 the result is the same as the +number.</i></p> + +<p><i>When you divide a number by anything less than 1 the result +is larger than the number.</i><span class='pagenum'><a name="Page_80" id="Page_80">[Pg 80]</a></span></p> + +<p>State the missing numbers:—</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary="" width="60%"> +<tr><td align='left'>8 = .... 4s</td><td align='left'>12 = .... 6s</td><td align='left'>9 = .... 9s</td></tr> +<tr><td align='left'>8 = .... 2s</td><td align='left'>12 = .... 4s</td><td align='left'>9 = .... 3s</td></tr> +<tr><td align='left'>8 = .... 1s</td><td align='left'>12 = .... 3s</td><td align='left'>9 = .... 1s</td></tr> +<tr><td align='left'>8 = .... <sup>1</sup>⁄<sub>2</sub>s</td><td align='left'>12 = .... 2s</td><td align='left'>9 = .... <sup>1</sup>⁄<sub>3</sub>s</td></tr> +<tr><td align='left'>8 = .... <sup>1</sup>⁄<sub>4</sub>s</td><td align='left'>12 = .... 1s</td><td align='left'>9 = .... <sup>1</sup>⁄<sub>9</sub>s</td></tr> +<tr><td align='left'>8 = .... <sup>1</sup>⁄<sub>8</sub>s</td><td align='left'>12 = .... <sup>1</sup>⁄<sub>2</sub>s</td></tr> +<tr><td align='left'></td><td align='left'>12 = .... <sup>1</sup>⁄<sub>3</sub>s</td></tr> +<tr><td align='left'></td><td align='left'>12 = .... <sup>1</sup>⁄<sub>4</sub>s</td></tr> +</table></div> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary="" width="60%"> +<tr><td align='left'>16 ÷ 16 =</td><td align='left'>9 ÷ 9 =</td><td align='left'>10 ÷ 10 =</td><td align='left'>12 ÷ 6 =</td></tr> +<tr><td align='left'>16 ÷ 8 =</td><td align='left'>9 ÷ 3 =</td><td align='left'>10 ÷ 5 =</td><td align='left'>12 ÷ 4 =</td></tr> +<tr><td align='left'>16 ÷ 4 =</td><td align='left'>9 ÷ 1 =</td><td align='left'>10 ÷ 1 =</td><td align='left'>12 ÷ 3 =</td></tr> +<tr><td align='left'>16 ÷ 2 =</td><td align='left'>9 ÷ <sup>1</sup>⁄<sub>3</sub> =</td><td align='left'>10 ÷ <sup>1</sup>⁄<sub>5</sub> =</td><td align='left'>12 ÷ 2 =</td></tr> +<tr><td align='left'>16 ÷ 1 =</td><td align='left'>9 ÷ <sup>1</sup>⁄<sub>9</sub> =</td><td align='left'>10 ÷ <sup>1</sup>⁄<sub>10</sub> =</td><td align='left'>12 ÷ 1 =</td></tr> +<tr><td align='left'>16 ÷ <sup>1</sup>⁄<sub>2</sub> =</td><td align='left'></td><td align='left'></td><td align='left'>12 ÷ <sup>1</sup>⁄<sub>2</sub> =</td></tr> +<tr><td align='left'>16 ÷ <sup>1</sup>⁄<sub>4</sub> =</td><td align='left'></td><td align='left'></td><td align='left'>12 ÷ <sup>1</sup>⁄<sub>3</sub> =</td></tr> +<tr><td align='left'>16 ÷ <sup>1</sup>⁄<sub>8</sub> =</td><td align='left'></td><td align='left'></td><td align='left'>12 ÷ <sup>1</sup>⁄<sub>4</sub> =</td></tr> +<tr><td align='left'></td><td align='left'></td><td align='left'></td><td align='left'>12 ÷ <sup>1</sup>⁄<sub>6</sub> =</td></tr> +</table></div> + +<p>(8) <i>'% of' means 'hundredths times.'</i>—In the case of percentage +a series of bonds like the following should be +formed:—</p> + + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary=""> +<tr><td align='left'></td><td align='left'> 5</td><td align='left'>percent</td><td align='left'>of </td><td align='left'>= .05 times</td></tr> +<tr><td align='left'></td><td align='left'>20</td><td align='left'> " "</td><td align='left'>"</td><td align='left'>= .20 "</td></tr> +<tr><td align='left'></td><td align='left'> 6</td><td align='left'> " "</td><td align='left'>"</td><td align='left'>= .06 "</td></tr> +<tr><td align='left'></td><td align='left'>25</td><td align='left'>%</td><td align='left'>"</td><td align='left'>= .25 ×</td></tr> +<tr><td align='left'></td><td align='left'>12</td><td align='left'>%</td><td align='left'>"</td><td align='left'>= .12 ×</td></tr> +<tr><td align='left'></td><td align='left'> 3</td><td align='left'>%</td><td align='left'>"</td><td align='left'>= .03 ×</td></tr> +</table></div> + +<p>Four five-minute drills on such connections between +'<i>x</i> percent of' and 'its decimal equivalent times' are worth +an hour's study of verbal definitions of the meaning of +percent as per hundred or the like. The only use of the<span class='pagenum'><a name="Page_81" id="Page_81">[Pg 81]</a></span> +study of such definitions is to facilitate the later formation +of the bonds, and, with all save the brighter pupils, the +bonds are more needed for an understanding of the definitions +than the definitions are needed for the formation of the +bonds.</p> + +<p>(9) <i>Habits of verifying results.</i>—Bonds should early be +formed between certain manipulations of numbers and +certain means of checking, or verifying the correctness of, +the manipulation in question. The additions to 9 + 9 and +the subtractions to 18 − 9 should be verified by objective +addition and subtraction and counting until the pupil has +sure command; the multiplications to 9 × 9 should be +verified by objective multiplication and counting of the +result (in piles of tens and a pile of ones) eight or ten times,<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a> +and by addition eight or ten times;<a href="#Footnote_4_4" class="fnanchor">[4]</a> the divisions to 81 ÷ 9 +should be verified by multiplication and occasionally +objectively until the pupil has sure command; column +addition should be checked by adding the columns separately +and adding the sums so obtained, and by making two +shorter tasks of the given task and adding the two sums; +'short' multiplication should be verified eight or ten times by +addition; 'long' multiplication should be checked by +reversing multiplier and multiplicand and in other ways; +'short' and 'long' division should be verified by multiplication.</p> + +<p>These habits of testing an obtained result are of threefold +value. They enable the pupil to find his own errors, +and to maintain a standard of accuracy by himself. They +give him a sense of the relations of the processes and the +reasons why the right ways of adding, subtracting, multiplying, +and dividing are right, such as only the very bright +<span class='pagenum'><a name="Page_82" id="Page_82">[Pg 82]</a></span> +pupils can get from verbal explanations. They put his +acquisition of a certain power, say multiplication, to a +real and intelligible use, in checking the results of his practice +of a new power, and so instill a respect for arithmetical +power and skill in general. The time spent in such verification +produces these results at little cost; for the practice in +adding to verify multiplications, in multiplying to verify +divisions, and the like is nearly as good for general drill +and review of the addition and multiplication themselves +as practice devised for that special purpose.</p> + +<p>Early work in adding, subtracting, and reducing fractions +should be verified by objective aids in the shape of lines and +areas divided in suitable fractional parts. Early work with +decimal fractions should be verified by the use of the equivalent +common fractions for .25, .75, .125, .375, and the like. +Multiplication and division with fractions, both common and +decimal, should in the early stages be verified by objective +aids. The placing of the decimal point in multiplication +and division with decimal fractions should be verified by +such exercises as:—</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="1" summary=""> +<tr><td align='right'> +20<br /> +1.23 )<span class="overline"> 24.60</span><br /> +<span class="u"> 246</span> </td> +<td class='spacer'>It cannot be 200; for 200 × 1.23 is much more than 24.6.<br /> +It cannot be 2; for 2 × 1.23 is much less than 24.6.</td> +</tr> +</table></div> + + + +<p>The establishment of habits of verifying results and their +use is very greatly needed. The percentage of wrong +answers in arithmetical work in schools is now so high that +the pupils are often being practiced in error. In many +cases they can feel no genuine and effective confidence in the +processes, since their own use of the processes brings wrong +answers as often as right. In solving problems they often +cannot decide whether they have done the right thing or the +wrong, since even if they have done the right thing, they may<span class='pagenum'><a name="Page_83" id="Page_83">[Pg 83]</a></span> +have done it inaccurately. A wrong answer to a problem +is therefore too often ambiguous and uninstructive to them.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a></p> + +<p>These illustrations of the last few pages are samples of +the procedures recommended by a consideration of all the +bonds that one might form and of the contribution that +each would make toward the abilities that the study of +arithmetic should develop and improve. It is by doing more +or less at haphazard what psychology teaches us to do deliberately +and systematically in this respect that many of the +past advances in the teaching of arithmetic have been made.</p> + + +<h4>WASTEFUL AND HARMFUL BONDS</h4> + +<p>A scrutiny of the bonds now formed in the teaching of +arithmetic with questions concerning the exact service of +each, results in a list of bonds of small value or even no value, +so far as a psychologist can determine. I present here +samples of such psychologically unjustifiable bonds with +some of the reasons for their deficiencies.</p> + +<p>(1) <i>Arbitrary units.</i>—In drills intended to improve the +ability to see and use the meanings of numbers as names for +ratios or relative magnitudes, it is unwise to employ entirely +arbitrary units. The procedure in II (on page 84) is better +than that in I. Inches, half-inches, feet, and centimeters are +better as units of length than arbitrary As. Square inches, +square centimeters, and square feet are better for areas. +Ounces and pounds should be lifted rather than arbitrary +weights. Pints, quarts, glassfuls, cupfuls, handfuls, and +cubic inches are better for volume.</p> + +<p>All the real merit in the drills on relative magnitude +advocated by Speer, McLellan and Dewey, and others can +be secured without spending time in relating magnitudes +<span class='pagenum'><a name="Page_84" id="Page_84">[Pg 84]</a></span> +for the sake of relative magnitude alone. The use of units +of measure in drills which will never be used in <i>bona fide</i> +measuring is like the use of fractions like sevenths, elevenths, +and thirteenths. A very little of it is perhaps desirable to +test the appreciation of certain general principles, but for +regular training it should give place to the use of units of +practical significance.</p> + + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig3.jpg" width="640" height="240" alt="Fig. 3." title="" /> +<span class="caption smcap">Fig. 3.</span> +<p class="pblockquot">I. If <i>A</i> is 1 which line is 2? Which line is 4? Which line is 3? +<i>A</i> and <i>C</i> together equal what line? <i>A</i> and <i>B</i> together equal what +line? How much longer is <i>B</i> than <i>A</i>? How much longer is <i>B</i> +than <i>C</i>? How much longer is <i>D</i> than <i>A</i>?</p> +</div> + + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig4.jpg" width="800" height="300" alt="Fig. 4." title="" /> +<span class="caption smcap">Fig. 4.</span> +<p class="pblockquot">II. <i>A</i> is 1 inch long. Which line is 2 inches long? Which line +is 4 inches long? Which line is 3 inches long? <i>A</i> and <i>C</i> together +make ... inches? <i>A</i> and <i>B</i> together make ... inches? <i>B</i> is ... +... longer than <i>A</i>? <i>B</i> is ... ... longer than <i>C</i>? <i>D</i> is ... +... longer than <i>A</i>?</p> +</div> + + +<p><span class='pagenum'><a name="Page_85" id="Page_85">[Pg 85]</a></span></p> + +<p>(2) <i>Multiples of 11.</i>—The multiplications of 2 to 12 by +11 and 12 as single connections should be left for the pupil +to acquire by himself as he needs them. These connections +interfere with the process of learning two-place multiplication. +The manipulations of numbers there required can +be learned much more easily if 11 and 12 are used as multipliers +in just the same way that 78 or 96 would be. Later +the 12 × 2, 12 × 3, etc., may be taught. There is less reason +for knowing the multiples of 11 than for knowing the +multiples of 15, 16, or 25.</p> + +<p>(3) <i>Abstract and concrete numbers.</i>—The elaborate emphasis +of the supposed fact that we cannot multiply 726 +by 8 dollars and the still more elaborate explanations of +why nevertheless we find the cost of 726 articles at $8 each +by multiplying 726 by 8 and calling the answer dollars are +wasteful. The same holds of the corresponding pedantry +about division. These imaginary difficulties should not be +raised at all. The pupil should not think of multiplying +or dividing men or dollars, but simply of the necessary +equation and of the sort of thing that the missing number +represents. "8 × 726 = .... Answer is dollars," or "8, +726, multiply. Answer is dollars," is all that he needs to +think, and is in the best form for his thought. Concerning +the distinction between abstract and concrete numbers, both +logic and common sense as well as psychology support the +contention of McDougle ['14, p. 206f.], who writes:—</p> + +<p>"The most elementary counting, even that stage when the +counts were not carried in the mind, but merely in notches +on a stick or by DeMorgan's stones in a pot, requires some +thought; and the most advanced counting implies memory +of things. The terms, therefore, abstract and concrete number, +have long since ceased to be used by thinking people.</p> + +<p>"Recently the writer visited an arithmetic class in a<span class='pagenum'><a name="Page_86" id="Page_86">[Pg 86]</a></span> +State Normal School and saw a group of practically adult +students confused about this very question concerning abstract +and concrete numbers, according to their previous +training in the conventionalities of the textbook. Their +teacher diverted the work of the hour and she and the class +spent almost the whole period in reëstablishing the requirements +'that the product must always be the same kind of +unit as the multiplicand,' and 'addends must all be alike +to be added.' This is not an exceptional case. Throughout +the whole range of teaching arithmetic in the public schools +pupils are obfuscated by the philosophical encumbrances +which have been imposed upon the simplest processes of numerical +work. The time is surely ripe, now that we are readjusting +our ideas of the subject of arithmetic, to revise some +of these wasteful and disheartening practices. Algebra +historically grew out of arithmetic, yet it has not been +laden with this distinction. No pupil in algebra lets <i>x</i> +equal the horses; he lets <i>x</i> equal the <i>number</i> of horses, and +proceeds to drop the idea of horses out of his consideration. +He multiplies, divides, and extracts the root of the <i>number</i>, +sometimes handling fractions in the process, and finally +interprets the result according to the conditions of his +problem. Of course, in the early number work there have +been the sense-objects from which number has been perceived, +but the mind retreats naturally from objectivity to +the pure conception of number, and then to the number +symbol. The following is taken from the appendix to +Horn's thesis, where a seventh grade girl gets the population +of the United States in 1820:—</p> + +<div class='center'> +<table border="0" cellpadding="5" cellspacing="0" summary=""> +<tr><td align='right'> +7,862,166<br /> +233,634<br /> +1,538,022<br /> +<span class="overline">9,633,822</span> +</td> +<td align='left' valign='top'>whites<br /> +free negroes<br /> +slaves</td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_87" id="Page_87">[Pg 87]</a></span></p> + +<p class="noidt">In this problem three different kinds of addends are combined, +if we accept the usual distinction. Some may say +that this is a mistake,—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:—</p> + +<p>'In a garden on the Summit are as many cabbage-heads +as the total number of ladies and gentlemen in this class. +How many cabbage-heads in the garden?'</p> + +<p>And the blackboard solution looks like this each time:—</p> + +<div class='center'> +<table border="0" cellpadding="5" cellspacing="0" summary=""> +<tr><td align='right'> +29<br /> +15<br /> +<span class="overline">44</span> +</td> +<td align='left'>ladies<br /> +gentlemen<br /> +cabbage-heads</td></tr> +</table></div> + +<p class="noidt">So, also, one may say: I have 6 times as many sheep as you +have cows. If you have 5 cows, how many sheep have I? +Here we would multiply the number of cows, which is 5, +by 6 and call the result 30, which must be linked with the +idea of sheep because the conditions imposed by the problem +demand it. The mind naturally in this work separates the +pure number from its situation, as in algebra, handles it<span class='pagenum'><a name="Page_88" id="Page_88">[Pg 88]</a></span> +according to the laws governing arithmetical combinations, +and labels the result as the statement of the problem demands. +This is expressed in the following, which is tacitly accepted +in algebra, and should be accepted equally in arithmetic:</p> + +<p>'In all computations and operations in arithmetic, all +numbers are essentially abstract and should be so treated. +They are concrete only in the thought process that attends +the operation and interprets the result.'"</p> + +<p>(4) <i>Least common multiple.</i>—The whole set of bonds involved +in learning 'least common multiple' should be left +out. In adding and subtracting fractions the pupil should +<i>not</i> find the least common multiple of their denominators +but should find any common multiple that he can find +quickly and correctly. No intelligent person would ever +waste time in searching for the least common multiple of +sixths, thirds, and halves except for the unfortunate traditions +of an oversystematized arithmetic, but would think +of their equivalents in sixths or twelfths or twenty-fourths +or <i>any other convenient common multiple</i>. The process of +finding the least common multiple is of such exceedingly +rare application in science or business or life generally that +the textbooks have to resort to purely fantastic problems +to give drill in its use.</p> + +<p>(5) <i>Greatest common divisor.</i>—The whole set of bonds +involved in learning 'greatest common divisor' should +also be left out. In reducing fractions to lowest terms the +pupil should divide by anything that he sees that he can +divide by, favoring large divisors, and continue doing so +until he gets the fraction in terms suitable for the purpose +in hand. The reader probably never has had occasion to +compute a greatest common divisor since he left school. +If he has computed any, the chances are that he would have +saved time by solving the problem in some other way!<span class='pagenum'><a name="Page_89" id="Page_89">[Pg 89]</a></span></p> + +<p>The following problems are taken at random from those +given by one of the best of the textbooks that make the +attempt to apply the facts of Greatest Common Divisor +and Least Common Multiple to problems.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> Most of these +problems are fantastic. The others are trivial, or are better +solved by trial and adaptation.</p> + +<div class="pblockquot"><p><b>1.</b> A certain school consists of 132 pupils in the high school, +154 in the grammar, and 198 in the primary grades. If each group +is divided into sections of the same number containing as many +pupils as possible, how many pupils will there be in each section?</p> + +<p><b>2.</b> A farmer has 240 bu. of wheat and 920 bu. of oats, which +he desires to put into the least number of boxes of the same capacity, +without mixing the two kinds of grain. Find how many +bushels each box must hold.</p> + +<p><b>3.</b> Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively, +and begin to toll at the same instant. When will they +next toll together?</p> + +<p><b>4.</b> A, B, C, and D start together, and travel the same way +around an island which is 600 mi. in circuit. A goes 20 mi. per +day, B 30, C 25, and D 40. How long must their journeying continue, +in order that they may all come together again?</p> + +<p><b>5.</b> The periods of three planets which move uniformly in circular +orbits round the sun, are respectively 200, 250, and 300 da. +Supposing their positions relatively to each other and the sun to +be given at any moment, determine how many da. must elapse +before they again have exactly the same relative positions.</p></div> + +<p>(6) <i>Rare and unimportant words.</i>—The bonds between +rare or unimportant words and their meanings should not +be formed for the mere sake of verbal variety in the problems +of the textbook. A pupil should not be expected to solve +a problem that he cannot read. He should not be expected +in grades 2 and 3, or even in grade 4, to read words that he +has rarely or never seen before. He should not be given +elaborate drill in reading during the time devoted to the +treatment of quantitative facts and relations.</p> + +<p><span class='pagenum'><a name="Page_90" id="Page_90">[Pg 90]</a></span>All +this is so obvious that it may seem needless to relate. +It is not. With many textbooks it is now necessary to +give definite drill in reading the words in the printed problems +intended for grades 2, 3, and 4, or to replace them by +oral statements, or to leave the pupils in confusion concerning +what the problems are that they are to solve. Many +good teachers make a regular reading-lesson out of every +page of problems before having them solved. There should +be no such necessity.</p> + +<p>To define <i>rare</i> and <i>unimportant</i> concretely, I will say that +for pupils up to the middle of grade 3, such words as the +following are rare and unimportant (though each of them +occurs in the very first fifty pages of some well-known beginner's +book in arithmetic).</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="20" summary="" width="80%"> +<tr> +<td align='left'>absentees<br /> +account<br /> +Adele<br /> +admitted<br /> +Agnes<br /> +agreed<br /> +Albany<br /> +Allen<br /> +allowed<br /> +alternate<br /> +Andrew<br /> +Arkansas<br /> +arrived<br /> +assembly<br /> +automobile<br /> +baking powder<br /> +balance<br /> +barley<br /> +beggar<br /> +Bertie<br /> +Bessie<br /> +bin<br /> +Boston<br /> +bouquet<br /> +bronze<br /> +buckwheat<br /> +Byron<br /> +camphor<br /> +Carl<br /> +Carrie<br /> +Cecil<br /> +Charlotte<br /> +charity<br /> +Chicago<br /> +cinnamon<br /> +Clara<br /> +clothespins<br /> +collect<br /> +comma<br /> +committee<br /> +concert<br /> +confectioner<br /> +cranberries<br /> +crane<br /> +currants<br /> +dairyman<br /> +Daniel<br /> +David<br /> +dealer<br /> +debt<br /> +delivered<br /> +Denver<br /> +</td> +<td align='left'>department<br /> +deposited<br /> +dictation<br /> +discharged<br /> +discover<br /> +discovery<br /> +dish-water<br /> +drug<br /> +due<br /> +Edgar<br /> +Eddie<br /> +Edwin<br /> +election<br /> +electric<br /> +Ella<br /> +Emily<br /> +enrolled<br /> +entertainment<br /> +envelope<br /> +Esther<br /> +Ethel<br /> +exceeds<br /> +explanation<br /> +expression<br /> +generally<br /> +gentlemen<br /> +Gilbert<br /> +Grace<br /> +grading<br /> +Graham<br /> +grammar<br /> +Harold<br /> +hatchet<br /> +Heralds<br /> +hesitation<br /> +Horace Mann<br /> +impossible<br /> +income<br /> +indicated<br /> +inmost<br /> +inserts<br /> +installments<br /> +instantly<br /> +insurance<br /> +Iowa<br /> +Jack<br /> +Jennie<br /> +Johnny<br /> +Joseph<br /> +journey<br /> +Julia<br /> +Katherine<br /> +</td> +<td align='left'>lettuce-plant<br /> +library<br /> +Lottie<br /> +Lula<br /> +margin<br /> +Martha<br /> +Matthew<br /> +Maud<br /> +meadow<br /> +mentally<br /> +mercury<br /> +mineral<br /> +Missouri<br /> +molasses<br /> +Morton<br /> +movements<br /> +muslin<br /> +Nellie<br /> +nieces<br /> +Oakland<br /> +observing<br /> +obtained<br /> +offered<br /> +office<br /> +onions<br /> +opposite<br /> +original<br /> +package<br /> +packet<br /> +palm<br /> +Patrick<br /> +Paul<br /> +payments<br /> +peep<br /> +Peter<br /> +perch<br /> +phaeton<br /> +photograph<br /> +piano<br /> +pigeons<br /> +Pilgrims<br /> +preserving<br /> +proprietor<br /> +purchased<br /> +Rachel<br /> +Ralph<br /> +rapidity<br /> +rather<br /> +readily<br /> +receipts<br /> +register<br /> +remanded<br /> +</td> +<td align='left'>respectively<br /> +Robert<br /> +Roger<br /> +Ruth<br /> +rye<br /> +Samuel<br /> +San Francisco<br /> +seldom<br /> +sheared<br /> +shingles<br /> +skyrockets<br /> +sloop<br /> +solve<br /> +speckled<br /> +sponges<br /> +sprout<br /> +stack<br /> +Stephen<br /> +strap<br /> +successfully<br /> +suggested<br /> +sunny<br /> +supply<br /> +Susan<br /> +Susie's<br /> +syllable<br /> +talcum<br /> +term<br /> +test<br /> +thermometer<br /> +Thomas<br /> +torpedoes<br /> +trader<br /> +transaction<br /> +treasury<br /> +tricycle<br /> +tube<br /> +two-seated<br /> +united<br /> +usually<br /> +vacant<br /> +various<br /> +vase<br /> +velocipede<br /> +votes<br /> +walnuts<br /> +Walter<br /> +Washington<br /> +watched<br /> +whistle<br /> +woodland<br /> +worsted<br /> +</td> +</tr> +</table></div> + +<p><span class='pagenum'><a name="Page_91" id="Page_91">[Pg 91]</a></span></p> + +<p>(7) <i>Misleading facts and procedures.</i>—Bonds should not +be formed between articles of commerce and grossly inaccurate +prices therefor, between events and grossly improbable +consequences, or causes or accompaniments thereof, +nor between things, qualities, and events which have no +important connections one with another in the real world. +In general, things should not be put together in the pupil's +mind that do not belong together.</p> + +<p>If the reader doubts the need of this warning let him examine +problems 1 to 5, all from reputable books that are +in common use, or have been within a few years, and con<span class='pagenum'><a name="Page_92" id="Page_92">[Pg 92]</a></span>sider +how addition, subtraction, and the habits belonging +with each are confused by exercise 6.</p> + +<div class="pblockquot"> +<p><b>1.</b> If a duck flying <sup>3</sup>⁄<sub>5</sub> as fast as a hawk flies 90 miles in an hour, +how fast does the hawk fly?</p> + +<p><b>2.</b> At <sup>5</sup>⁄<sub>8</sub> of a cent apiece how many eggs can I buy for $60?</p> + +<p><b>3.</b> At $.68 a pair how many pairs of overshoes can you buy for +$816?</p> + +<p><b>4.</b> At $.13 a dozen how many dozen bananas can you buy for +$3.12?</p> + +<p><b>5.</b> How many pecks of beans can be put into a box that will +hold just 21 bushels?</p> + +<p><b>6.</b> Write answers:</p> + +<div class='center'> +<table border="0" cellpadding="20" cellspacing="0" summary=""> +<tr><td align='right'> +537<br /> +365<br /> +?<br /> +36<br /> +<span class="overline">1000</span> +</td> +<td align='left'>Beginning at the bottom say 11, 18, and 2 (writing it in +its place) are 20. 5, 11, 14, and 6 (writing it) are 20, 5, 10. The number, omitted, is 62. +</td></tr> +</table></div> + +<div class='center'> +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr> +<td align='left' valign='top'><i>a.</i></td><td align='right'>581<br />97<br />364<br />?<br /><span class="overline">1758</span></td> +<td align='left' valign='top'><i>b.</i></td><td align='right'>625<br />?<br />90<br />417<br /><span class="overline">2050</span></td> +<td align='left' valign='top'><i>c.</i></td><td align='right'>752<br />414<br />130<br />?<br /><span class="overline">2460</span></td> +<td align='left' valign='top'><i>d.</i></td><td align='right'>314<br />429<br />?<br />76<br /><span class="overline">1000</span></td> +<td align='left' valign='top'><i>e.</i></td><td align='right'>?<br />845<br />223<br />95<br /><span class="overline">2367</span></td> +</tr> +</table></div> +</div> + +<p>(8) <i>Trivialities and absurdities.</i>—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:—</p> + +<div class="pblockquot"><p>On one side of George's slate there are 32 words, and on the +other side 26 words. If he erases 6 words from one side, and 8 +from the other, how many words remain on his slate?</p> + +<p>A certain school has 14 rooms, and an average of 40 children +in a room. If every one in the school should make 500 straight +marks on each side of his slate, how many would be made in all?</p> +<p><span class='pagenum'><a name="Page_93" id="Page_93">[Pg 93]</a></span></p> +<p>8 times the number of stripes in our flag is the number of years +from 1800 until Roosevelt was elected President. In what year +was he elected President?</p> + +<p>From the Declaration of Independence to the World's Fair in +Chicago was 9 times as many years as there are stripes in the flag. +How many years was it?</p></div> + +<p>(9) <i>Useless methods.</i>—Bonds should not be formed between +a described situation and a method of treating the +situation which would not be a useful one to follow in the case +of the real situation. For example, "If I set 96 trees in +rows, sixteen trees in a row, how many rows will I have?" +forms the habit of treating by division a problem that in +reality would be solved by counting the rows. So also "I +wish to give 25 cents to each of a group of boys and find +that it will require $2.75. How many boys are in the +group?" forms the habit of answering a question by division +whose answer must already have been present to give the +data of the problem.</p> + +<p>(10) <i>Problems whose answers would, in real life, be already +known.</i>—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:—</p> + +<div class="pblockquot"><p>A clerk in an office addressed letters according to a given +list. After she had addressed 2500, <sup>4</sup>⁄<sub>9</sub> of the names on the list +had not been used; how many names were in the entire list?</p> + +<p>The Canadian power canal at Sault Ste. Marie furnished +<span class='pagenum'><a name="Page_94" id="Page_94">[Pg 94]</a></span>20,000 horse power. The canal on the Michigan side furnished +2½ times as much. How many horse power does the latter furnish?</p></div> + +<p>It may be asserted that the ideal of giving as described +problems only problems that might occur and demand the +same sort of process for solution with a real situation, is too +exacting. If a problem is comprehensible and serves to +illustrate a principle or give useful drill, that is enough, +teachers may say. For really scientific teaching it is not +enough. Moreover, if problems are given merely as tests +of knowledge of a principle or as means to make some fact +or principle clear or emphatic, and are not expected to be of +direct service in the quantitative work of life, it is better +to let the fact be known. For example, "I am thinking +of a number. Half of this number is twice six. What is +the number?" is better than "A man left his wife a certain +sum of money. Half of what he left her was twice as much +as he left to his son, who receives $6000. How much did +he leave his wife?" The former is better because it makes +no false pretenses.</p> + +<p>(11) <i>Needless linguistic difficulties.</i>—It should be unnecessary +to add that bonds should not be formed between +the pupil's general attitude toward arithmetic and needless, +useless difficulty in language or needless, useless, wrong +reasoning. Our teaching is, however, still tainted by both +of these unfortunate connections, which dispose the pupil +to think of arithmetic as a mystery and folly.</p> + +<p>Consider, for example, the profitless linguistic difficulty +of problems 1-6, whose quantitative difficulties are simply +those of:—</p> + +<div class="pblockquot"> +<div class="blockquot"><p class="noidt"> +<b>1.</b> 5 + 8 + 3 + 7<br /> +<b>2.</b> 64 ÷ 8, and knowledge that 1 peck = 8 quarts<br /> +<b>3.</b> 12 ÷ 4<br /> +<span class='pagenum'><a name="Page_95" id="Page_95">[Pg 95]</a></span><b>4.</b> 6 ÷ 2<br /> +<b>5.</b> 3 × 2<br /> +<b>6.</b> 4 × 4 +</p></div> + +<p><b>1.</b> What amount should you obtain by putting together 5 +cents, 8 cents, 3 cents, and 7 cents? Did you find this result by +adding or multiplying?</p> + +<p><b>2.</b> How many times must you empty a peck measure to fill a +basket holding 64 quarts of beans?</p> + +<p><b>3.</b> If a girl commits to memory 4 pages of history in one day, +in how many days will she commit to memory 12 pages?</p> + +<p><b>4.</b> If Fred had 6 chickens how many times could he give away +2 chickens to his companions?</p> + +<p><b>5.</b> If a croquet-player drove a ball through 2 arches at each +stroke, through how many arches will he drive it by 3 strokes?</p> + +<p><b>6.</b> If mamma cut the pie into 4 pieces and gave each person a +piece, how many persons did she have for dinner if she used 4 +whole pies for dessert?</p></div> + +<p>Arithmetically this work belongs in the first or second +years of learning. But children of grades 2 and 3, save a +few, would be utterly at a loss to understand the language.</p> + +<p>We are not yet free from the follies illustrated in the lessons +of pages 96 to 99, which mystified our parents.<span class='pagenum'><a name="Page_96" id="Page_96">[Pg 96]</a></span></p> + +<p> </p> +<div class="figcenter" style="width: 734px;"> +<img src="images/fig5.jpg" width="734" height="600" alt="Fig. 5." title="Fig. 5." /> +<span class="caption smcap">Fig. 5.</span> +</div> + +<p class="tabcap">LESSON I</p> +<div class="pblockquot"> + +<p><b>1.</b> In this picture, how many girls are in the swing?</p> + +<p><b>2.</b> How many girls are pulling the swing?</p> + +<p><b>3.</b> If you count both girls together, how many are they?<br /> + <b><i>One</i></b> girl and <b><i>one</i></b> other girl are how many?</p> + +<p><b>4.</b> How many kittens do you see on the stump?</p> + +<p><b>5.</b> How many on the ground?</p> + +<p><b>6.</b> How many kittens are in the picture? +One kitten and one other kitten are how many?</p> + +<p><b>7.</b> If you should ask me how many girls are in the swing, or +how many kittens are on the stump, I could answer aloud, <i>One</i>; +or I could write <i>One</i>; or thus, <b><i>1</i></b>.</p> + +<p><b>8.</b> If I write <i>One</i>, this is called the <i>word One</i>.</p> + +<p><b>9.</b> This, <b><i>1</i></b>, is named a <b><i>figure One</i></b>, because it means the same +as the word <i>One</i>, and stands for <i>One</i>.<span class='pagenum'><a name="Page_97" id="Page_97">[Pg 97]</a></span></p> + +<p><b>10.</b> Write 1. What is this named? Why?</p> + +<p><b>11.</b> A figure 1 may stand for <i>one</i> girl, <i>one</i> kitten, or <i>one</i> anything.</p> + +<p><b>12.</b> When children first attend school, what do they begin to +learn? <i>Ans.</i> Letters and words.</p> + +<p><b>13.</b> Could you read or write before you had learned either +letters or words?</p> + +<p><b>14.</b> If we have all the <i>letters</i> together, they are named the +Alphabet.</p> + +<p><b>15.</b> If we write or speak <i>words</i>, they are named Language.</p> + +<p><b>16.</b> You are commencing to study Arithmetic; and you can +read and write in Arithmetic only as you learn the Alphabet and +Language of Arithmetic. But little time will be required for this +purpose.</p> +</div> + +<p> </p> +<div class="figcenter" style="width: 727px;"> +<img src="images/fig6.jpg" width="727" height="600" alt="Fig. 6." title="Fig. 6." /> +<span class="caption smcap">Fig. 6.</span> +</div> + +<p class="tabcap">LESSON II</p> +<div class="pblockquot"> + +<p><b>1.</b> If we speak or write words, what do we name them, when +taken together?</p> + +<p><b>2.</b> What are you commencing to study? <i>Ans.</i> Arithmetic.</p> + +<p><b>3.</b> What Language must you now learn?</p> + +<p><b>4.</b> What do we name this, <b>1</b>? Why?</p> + +<p><b>5.</b> This figure, <b>1</b>, is part of the Language of Arithmetic.</p> + +<p><b>6.</b> If I should write something to stand for <b><i>Two</i></b>—<i>two</i> girls, +<i>two</i> kittens, or <i>two</i> things of any kind—what do you think we +would name it?</p> + +<p><b>7.</b> A <b><i>figure Two</i></b> is written thus: <b><i>2.</i></b> Make a <i>figure two</i>.</p> + +<p><b>8.</b> Why do we name this a <i>figure two</i>?</p> + +<p><b>9.</b> This figure two (<b>2</b>) is part of the Language of Arithmetic.<span class='pagenum'><a name="Page_98" id="Page_98">[Pg 98]</a></span></p> + +<p><b>10.</b> In this picture one boy is sitting, playing a flageolet. +What is the other boy doing? If the boy standing +should sit down by the other, how many boys would be +sitting together? One boy and one other boy are how many +boys?</p> + +<p><b>11.</b> You see a flageolet and a violin. They are musical +instruments. One musical instrument and one other musical +instrument are how many?</p> + +<p><b>12.</b> I will write thus: 1 1 2. We say that 1 boy and 1 other +boy, counted together, are 2 boys; or are equal to 2 boys. We +will now write something to show that the first 1 and the other 1 +are to be counted together.</p> + +<div class="figright" style="width: 336px;"> +<img src="images/image04.jpg" width="336" height="344" alt="Plus" title="" /> +</div> + +<p><b>13.</b> We name a line drawn thus,—, a <b><i>horizontal line</i></b>. Draw +such a line. Name it.</p> + +<p><b>14.</b> A line drawn thus, | , we name a <b><i>vertical line</i></b>. Draw such +a line. Name it.<span class='pagenum'><a name="Page_99" id="Page_99">[Pg 99]</a></span></p> + +<p><b>15.</b> Now I will put two such lines together; +thus, +. 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?</p> + +<p><b>16.</b> Each of you write thus: —, | , +.</p> + + +<p><b>17.</b> This, +, is named <b><i>Plus</i></b>. <i>Plus</i> means +<i>more</i>; and + also means <i>more</i>.</p> + +<p><b>18.</b> I will write.</p> + +<p class="center"> +<b><i>One and One More Equal Two</i></b>.<br /> +</p> + +<p><b>19.</b> Now I will write part of this in the Language of Arithmetic. +I write the first <i>One</i> thus, 1; then the other <i>One</i> thus, +1. Afterward I write, for the word <i>More</i>, thus, +, placing the + +between 1 and 1, so that the whole stands thus: 1 + 1. As I +write, I say, <i>One and One more</i>.</p> + +<p><b>20.</b> Each of you write 1 + 1. Read what you have written.</p> + +<p><b>21.</b> This +, when written between the 1s, shows that they are +to be put together, or counted together, so as to make 2.</p> + +<p><b>22.</b> Because + shows what is to be done, it is called a <i>Sign</i>. +If we take its name, <i>Plus</i>, and the word <i>Sign</i>, and put both words +together, we have <i>Sign Plus</i>, or <i>Plus Sign</i>. In speaking of this +we may call it <i>Sign Plus</i>, or <i>Plus Sign</i>, or <i>Plus</i>.</p> + +<p><b>23.</b> 1, 2, +, are part of the Language of Arithmetic.</p> + +<p class="center"> +<b><i>Write the following in the Language of Arithmetic</i></b>:<br /> +</p> + +<p><b>24.</b> One and one more.</p> + +<p><b>25.</b> One and two more.</p> + +<p><b>26.</b> Two and one more.</p> +</div> + +<p><span class='pagenum'><a name="Page_100" id="Page_100">[Pg 100]</a></span></p> + +<p>(12) <i>Ambiguities and falsities.</i>—Consider the ambiguities +and false reasoning of these problems.</p> + +<div class="pblockquot"> +<p><b>1.</b> If you can earn 4 cents a day, how much can you earn in +6 weeks? (Are Sundays counted? Should a child who earns 4 +cents some day expect to repeat the feat daily?)</p> + +<p><b>2.</b> How many lines must you make to draw ten triangles and +five squares? (I can do this with 8 lines, though the answer the +book requires is 50.)</p> + +<p><b>3.</b> A runner ran twice around an <sup>1</sup>⁄<sub>8</sub> mile track in two +minutes. What distance did he run in <sup>2</sup>⁄<sub>3</sub> of a minute? (I do not +know, but I do know that, save by chance, he did not run exactly +<sup>2</sup>⁄<sub>3</sub> of <sup>1</sup>⁄<sub>8</sub> mile.)</p> + +<p><b>4.</b> John earned $4.35 in a week, and Henry earned $1.93. +They put their money together and bought a gun. What did it +cost? (Maybe $5, maybe $10. Did they pay for the whole of it? +Did they use all their earnings, or less, or more?)</p> + +<p><b>5.</b> Richard has 12 nickels in his purse. How much more than +50 cents would you give him for them? (Would a wise child +give 60 cents to a boy who wanted to swap 12 nickels therefor, or +would he suspect a trick and hold on to his own coins?)</p> + +<p><b>6.</b> If a horse trots 10 miles in one hour how far will he travel +in 9 hours?</p> + +<p><b>7.</b> If a girl can pick 3 quarts of berries in 1 hour how many +quarts can she pick in 3 hours?</p> + +<div class="blockquot"><p>(These last two, with a teacher insisting on the 90 and 9, +might well deprive a matter-of-fact boy of respect for +arithmetic for weeks thereafter.)</p></div> + +<p>The economics and physics of the next four problems speak for +themselves.</p> + +<p><b>8.</b> I lost $15 by selling a horse for $85. What was the value +of the horse?</p> + +<p><b>9.</b> If floating ice has 7 times as much of it under the surface +of the water as above it, what part is above water? If an iceberg +is 50 ft. above water, what is the entire height of the iceberg? +How high above water would an iceberg 300 ft. high have to be?</p> + +<p><b>10.</b> A man's salary is $1000 a year and his expenses $625. +<span class='pagenum'><a name="Page_101" id="Page_101">[Pg 101]</a></span> +How many years will elapse before he is worth $10,000 if he is +worth $2500 at the present time?</p> + +<p><b>11.</b> Sound travels 1120 ft. a second. How long after a cannon +is fired in New York will the report be heard in Philadelphia, a +distance of 90 miles?</p> +</div> + + +<h4>GUIDING PRINCIPLES</h4> + +<p>The reader may be wearied of these special details concerning +bonds now neglected that should be formed and +useless or harmful bonds formed for no valid reason. Any +one of them by itself is perhaps a minor matter, but when we +have cured all our faults in this respect and found all the +possibilities for wiser selection of bonds, we shall have +enormously improved the teaching of arithmetic. The +ideal is such choice of bonds (and, as will be shown later, +such arrangement of them) as will most improve the functions +in question at the least cost of time and effort. The +guiding principles may be kept in mind in the form of seven +simple but golden rules:—</p> + +<p>1. Consider the situation the pupil faces.</p> + +<p>2. Consider the response you wish to connect with it.</p> + +<p>3. Form the bond; do not expect it to come by a miracle.</p> + +<p>4. Other things being equal, form no bond that will have +to be broken.</p> + +<p>5. Other things being equal, do not form two or three +bonds when one will serve.</p> + +<p>6. Other things being equal, form bonds in the way that +they are required later to act.</p> + +<p>7. Favor, therefore, the situations which life itself will +offer, and the responses which life itself will demand.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_102" id="Page_102">[Pg 102]</a></span></p> +<h2><a name="CHAPTER_V" id="CHAPTER_V"></a>CHAPTER V</h2> + +<h3>THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE +STRENGTH OF BONDS</h3> + + +<p>An inventory of the bonds to be formed in learning arithmetic +should be accompanied by a statement of how strong +each bond is to be made and kept year by year. Since, +however, the inventory itself has been presented here only +in samples, the detailed statement of desired strength for +each bond cannot be made. Only certain general facts +will be noted here.</p> + + +<h4>THE NEED OF STRONGER ELEMENTARY BONDS</h4> + +<p>The constituent bonds involved in the fundamental operations +with numbers need to be much stronger than they now +are. Inaccuracy in these operations means weakness of the +constituent bonds. Inaccuracy exists, and to a degree that +deprives the subject of much of its possible disciplinary +value, makes the pupil's achievements of slight value for +use in business or industry, and prevents the pupil from +verifying his work with new processes by some previously +acquired process.</p> + +<p>The inaccuracy that exists may be seen in the measurements +made by the many investigators who have used +arithmetical tasks as tests of fatigue, practice, individual +differences and the like, and in the special studies of arith<span class='pagenum'><a name="Page_103" id="Page_103">[Pg 103]</a></span>metical +achievements for their own sake made by Courtis +and others.</p> + +<p>Burgerstein ['91], using such examples as</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='right'>28704516938276546397<br /> +<span class="u">+ 35869427359163827263</span></td></tr> +</table></div> + +<p class="noidt">and similar long numbers to be multiplied by 2 or by 3 or +by 4 or by 5 or by 6, found 851 errors in 28,267 answer-figures, +or 3 per hundred answer-figures, or <sup>3</sup>⁄<sub>5</sub> of an error +per example. The children were 9½ 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 11½ 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 1½ per hundred. The +children were from all grades from the third to the eighth. +In Laser's work, 21, 19, 13, and 10 answer-figures were obtained +per minute. Friedrich ['97] with similar examples, +giving the very long time of 20 minutes for obtaining about +200 answer-figures, found from 1 to 2 per hundred wrong. +King ['07] had children in grade 5 do sums, each consisting +of 5 two-place numbers. In the most accurate work-period, +they made 1 error per 20 columns. In multiplying +a four-place by a four-place number they had less than one +total answer right out of three. In New York City Courtis +found ['11-'12] with his Test 7 that in 12 minutes the average +achievement of fourth-grade children is 8.8 units attempted +with 4.2 right. In grade 5 the facts are 10.9 attempts with +5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade +7, 15 attempts with 8.5 right; in grade 8, 15.7 attempts +with 10.1 right. These results are near enough to those +obtained from the country at large to serve as a text here.<span class='pagenum'><a name="Page_104" id="Page_104">[Pg 104]</a></span></p> + +<p>The following were set as official standards, in an excellent +school system, Courtis Series B being used:—</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="5" summary=""> +<tr><th></th><th class="smcap"> Grade. </th><th class="smcap">Speed<br /> Attempts. </th><th class="smcap"> Percent of <br />Correct Answers.</th></tr> +<tr><td align='left'>Addition</td><td align='center'>8</td><td align='center'>12</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>7</td><td align='center'>11</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>6</td><td align='center'>10</td><td align='center'>70</td></tr> +<tr><td align='center'></td><td align='center'>5</td><td align='center'>9</td><td align='center'>70</td></tr> +<tr><td align='center'></td><td align='center'>4</td><td align='center'>8</td><td align='center'>70</td></tr> +<tr><td colspan='3'> </td></tr> +<tr><td align='left'>Subtraction</td><td align='center'>8</td><td align='center'>12</td><td align='center'>90</td></tr> +<tr><td align='center'></td><td align='center'>7</td><td align='center'>11</td><td align='center'>90</td></tr> +<tr><td align='center'></td><td align='center'>6</td><td align='center'>10</td><td align='center'>90</td></tr> +<tr><td align='center'></td><td align='center'>5</td><td align='center'>9</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>4</td><td align='center'>7</td><td align='center'>80</td></tr> +<tr><td colspan='3'> </td></tr> +<tr><td align='left'>Multiplication</td><td align='center'>8</td><td align='center'>11</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>7</td><td align='center'>10</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>6</td><td align='center'>9</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>5</td><td align='center'>7</td><td align='center'>70</td></tr> +<tr><td align='center'></td><td align='center'>4</td><td align='center'>6</td><td align='center'>60</td></tr> +<tr><td colspan='3'> </td></tr> +<tr><td align='left'>Division</td><td align='center'>8</td><td align='center'>11</td><td align='center'>90</td></tr> +<tr><td align='center'></td><td align='center'>7</td><td align='center'>10</td><td align='center'>90</td></tr> +<tr><td align='center'></td><td align='center'>6</td><td align='center'>8</td><td align='center'>80</td></tr> +<tr><td align='center'></td><td align='center'>5</td><td align='center'>6</td><td align='center'>70</td></tr> +<tr><td align='center'></td><td align='center'>4</td><td align='center'>4</td><td align='center'>60</td></tr> +</table></div> + +<p>Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding +columns like those printed below, children in grade 4 got +on the average less than 80 percent of correct answers. +Their average speed was about 2 columns per minute. In +doing division of the sort printed below children of grades +3 <i>B</i> and 4 <i>A</i> got less than 95 percent of correct answers, the +average speed being 4 divisions per minute. In both cases +the slower computers were no more accurate than the faster +ones. Practice improved the speed very rapidly, but the +accuracy remained substantially unchanged. Brown ['11 +and '12] found a similar low status of ability and notable +improvement from a moderate amount of special practice.<span class='pagenum'><a name="Page_105" id="Page_105">[Pg 105]</a></span></p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="80%"> +<tr><td align='center'> 3</td><td align='center'>5</td><td align='center'>6</td><td align='center'>2</td><td align='center'>3</td><td align='center'>8</td><td align='center'>9</td><td align='center'>7</td><td align='center'>4</td><td align='center'>9</td></tr> +<tr><td align='center'> 7</td><td align='center'>9</td><td align='center'>6</td><td align='center'>5</td><td align='center'>5</td><td align='center'>6</td><td align='center'>4</td><td align='center'>5</td><td align='center'>8</td><td align='center'>2</td></tr> +<tr><td align='center'> 3</td><td align='center'>4</td><td align='center'>7</td><td align='center'>8</td><td align='center'>7</td><td align='center'>3</td><td align='center'>7</td><td align='center'>9</td><td align='center'>3</td><td align='center'>7</td></tr> +<tr><td align='center'> 8</td><td align='center'>8</td><td align='center'>4</td><td align='center'>8</td><td align='center'>2</td><td align='center'>6</td><td align='center'>8</td><td align='center'>2</td><td align='center'>9</td><td align='center'>8</td></tr> +<tr><td align='center'> 2</td><td align='center'>2</td><td align='center'>4</td><td align='center'>7</td><td align='center'>6</td><td align='center'>9</td><td align='center'>8</td><td align='center'>5</td><td align='center'>6</td><td align='center'>2</td></tr> +<tr><td align='center'> 6</td><td align='center'>9</td><td align='center'>5</td><td align='center'>7</td><td align='center'>8</td><td align='center'>5</td><td align='center'>2</td><td align='center'>3</td><td align='center'>2</td><td align='center'>4</td></tr> +<tr><td align='center'> 9</td><td align='center'>6</td><td align='center'>4</td><td align='center'>2</td><td align='center'>7</td><td align='center'>2</td><td align='center'>9</td><td align='center'>4</td><td align='center'>4</td><td align='center'>5</td></tr> +<tr><td align='center'> 3</td><td align='center'>3</td><td align='center'>7</td><td align='center'>9</td><td align='center'>9</td><td align='center'>9</td><td align='center'>2</td><td align='center'>8</td><td align='center'>9</td><td align='center'>7</td></tr> +<tr><td align='center'> 6</td><td align='center'>8</td><td align='center'>9</td><td align='center'>6</td><td align='center'>4</td><td align='center'>7</td><td align='center'>7</td><td align='center'>9</td><td align='center'>2</td><td align='center'>4</td></tr> +<tr><td align='center'> 8</td><td align='center'>4</td><td align='center'>6</td><td align='center'>9</td><td align='center'>9</td><td align='center'>2</td><td align='center'>6</td><td align='center'>9</td><td align='center'>8</td><td align='center'>9</td></tr> +<tr><td align='center'> —</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td><td align='center'>—</td></tr> +</table></div> + +<div class="blockquot"><p class="noidt"> +<span style="margin-left: 0.5em;">20 = .... 5s</span><br /> +<span style="margin-left: 0.5em;">56 = .... 9s and .... <i>r.</i></span><br /> +<span style="margin-left: 0.5em;">30 = .... 7s and .... <i>r.</i></span><br /> +<span style="margin-left: 0.5em;">89 = .... 9s and .... <i>r.</i></span><br /> +<span style="margin-left: 0.5em;">20 = .... 8s and .... <i>r.</i></span><br /> +<span style="margin-left: 0.5em;">56 = .... 6s and .... <i>r.</i></span><br /> +<span style="margin-left: 0.5em;">31 = .... 4s and .... <i>r.</i></span><br /> +<span style="margin-left: 0.5em;">86 = .... 9s and .... <i>r.</i></span><br /> +</p></div> + +<p>It is clear that numerical work as inaccurate as this has +little or no commercial or industrial value. If clerks got +only six answers out of ten right as in the Courtis tests, one +would need to have at least four clerks make each computation +and would even then have to check many of their discrepancies +by the work of still other clerks, if he wanted his +accounts to show less than one error per hundred accounting +units of the Courtis size.</p> + +<p>It is also clear that the "habits of ... absolute accuracy, +and satisfaction in truth as a result" which arithmetic +is supposed to further must be largely mythical in +pupils who get right answers only from three to nine times +out of ten!</p> + + +<h4>EARLY MASTERY</h4> + +<p>The bonds in question clearly must be made far stronger +than they now are. They should in fact be strong enough +to abolish errors in computation, except for those due to<span class='pagenum'><a name="Page_106" id="Page_106">[Pg 106]</a></span> +temporary lapses. It is much better for a child to know half +of the multiplication tables, and to know that he does not +know the rest, than to half-know them all; and this holds +good of all the elementary bonds required for computation. +Any bond should be made to work perfectly, though slowly, +very soon after its formation is begun. Speed can easily +be added by proper practice.</p> + +<p>The chief reasons why this is not done now seem to be the +following: (1) Certain important bonds (like the additions +with higher decades) are not given enough attention when +they are first used. (2) The special training necessary +when a bond is used in a different connection (as when the +multiplications to 9 × 9 are used in examples like</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='right'>729<br /> +<span class="u"> 8</span> +</td></tr> +</table></div> + +<p class="noidt">where the pupil has also to choose the right number to multiply, +keep in mind what is carried, use it properly, and write the +right figure in the right place, and carry a figure, or remember +that he carries none) is neglected. (3) The pupil is not +taught to check his work. (4) He is not made responsible +for substantially accurate results. Furthermore, the requirement +of (4) without the training of (1), (2), and (3) +will involve either a fruitless failure on the part of many +pupils, or an utterly unjust requirement of time. The +common error of supposing that the task of computation +with integers consists merely in learning the additions to +9 + 9, the subtractions to 18 − 9, the multiplications to 8 × 9, +and the divisions to 81 ÷ 9, and in applying this knowledge +in connection with the principles of decimal notation, has +had a large share in permitting the gross inaccuracy of +arithmetical work. The bonds involved in 'knowing the +tables' do not make up one fourth of the bonds involved +in real adding, subtracting, multiplying, and dividing (with +integers alone).<span class='pagenum'><a name="Page_107" id="Page_107">[Pg 107]</a></span></p> + +<p>It should be noted that if the training mentioned in (1) +and (2) is well cared for, the checking of results as recommended +in (3) becomes enormously more valuable than it +is under present conditions, though even now it is one of +our soundest practices. If a child knows the additions to +higher decades so that he can add a seen one-place number +to a thought-of two-place number in three seconds or less +with a correct answer 199 times out of 200, there is only +an infinitesimal chance that a ten-figure column twice added +(once up, once down) a few minutes apart with identical +answers will be wrong. Suppose that, in long multiplication, +a pupil can multiply to 9 × 9 while keeping +his place and keeping track of what he is 'carrying' +and of where to write the figure he writes, and can add +what he carries without losing track of what he is to add +it to, where he is to write the unit figure, what he is to +multiply next and by what, and what he will then have +to carry, in each case to a surety of 99 percent of +correct responses. Then two identical answers got by +multiplying one three-place number by another a few +minutes apart, and with reversal of the numbers, will not +be wrong more than twice in his entire school career. +Checks approach proofs when the constituent bonds are +strong.</p> + +<p>If, on the contrary, the fundamental bonds are so weak +that they do not work accurately, checking becomes much +less trustworthy and also very much more laborious. In +fact, it is possible to show that below a certain point of +strength of the fundamental bonds, the time required for +checking is so great that part of it might better be spent in +improving the fundamental bonds.</p> + +<p>For example, suppose that a pupil has to find the sum of +five numbers like $2.49, $5.25, $6.50, $7.89, and $3.75.<span class='pagenum'><a name="Page_108" id="Page_108">[Pg 108]</a></span> +Counting each act of holding in mind the number to be +carried and each writing of a column's result as equivalent +in difficulty to one addition, such a sum equals nineteen +single additions. On this basis and with certain additional +estimates<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> we can compute the practical consequences for +a pupil's use of addition in life according to the mastery of +it that he has gained in school.</p> + +<p>I have so computed the amount of checking a pupil will +have to do to reach two agreeing numbers (out of two, or +three, or four, or five, or whatever the number before he +gets two that are alike), according to his mastery of the elementary +processes. The facts appear in Table 1.</p> + +<p>It is obvious that a pupil whose mastery of the elements is +that denoted by getting them right 96 times out of 100 +will require so much time for checking that, even if he were +never to use this ability for anything save a few thousand +sums in addition, he would do well to improve this ability +before he tried to do the sums. An ability of 199 out of +200, or 995 out of 1000, seems likely to save much more +time than would be taken to acquire it, and a reasonable +defense could be made for requiring 996 or 997 out of +1000.</p> + +<p>A precision of from 995 to 997 out of 1000 being required, +and ordinary sagacity being used in the teaching, speed will +substantially take care of itself. Counting on the fingers +or in words will not give that precision. Slow recourse to +memory of serial addition tables will not give that precision. +Nothing save sure memory of the facts operating under the +conditions of actual examples will give it. And such memories +will operate with sufficient speed.</p> + +<p><span class='pagenum'><a name="Page_109" id="Page_109">[Pg 109]</a></span></p> + +<p class="tabcap">TABLE 1</p> +<p class="center"><span class="smcap">The Effect of Mastery of the Elementary Facts of Addition upon +the Labor Required to Secure Two Agreeing Answers When +Adding Five Three-figure Numbers</span></p> + +<div class='center'> +<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary=""> +<colgroup><col width="20%" /><col width="20%" /><col width="20%" /><col width="20%" /><col width="20%" /></colgroup> +<tr><th class="smcap bbox">Mastery of the Elementary Additions Times Right in 1000</th> +<th class="smcap bbox">Approximate Number of Wrong Answers in Sums of 5 Three-place Numbers per 1000</th> +<th class="smcap bbox">Approximate Number of Agreeing Answers, after One Checking, per 1000</th> +<th class="smcap bbox">Approximate Number of Agreeing Answers, after a Checking of the First Discrepancies</th> +<th class="smcap bbox">Approximate Number of Checkings Required (over and above the First General Checking of the 100 Sums) to Secure Two Agreeing Results</th></tr> +<tr><td align='center'> 960</td><td align='center'> 700</td><td align='center'> 90</td><td align='center'> 216</td><td align='center'> 4500</td></tr> +<tr><td align='center'> 980</td><td align='center'> 380</td><td align='center'> 384</td><td align='center'> 676</td><td align='center'> 1200</td></tr> +<tr><td align='center'> 990</td><td align='center'> 190</td><td align='center'> 656</td><td align='center'> 906</td><td align='center'> 470</td></tr> +<tr><td align='center'> 995</td><td align='center'> 95</td><td align='center'> 819</td><td align='center'> 975</td><td align='center'> 210</td></tr> +<tr><td align='center'> 996</td><td align='center'> 76</td><td align='center'> 854</td><td align='center'> 984</td><td align='center'> 165</td></tr> +<tr><td align='center'> 997</td><td align='center'> 54</td><td align='center'> 895</td><td align='center'> 992</td><td align='center'> 115</td></tr> +<tr><td align='center'> 998</td><td align='center'> 38</td><td align='center'> 925</td><td align='center'> 996</td><td align='center'> 80</td></tr> +<tr><td align='center'> 999</td><td align='center'> 19</td><td align='center'> 962</td><td align='center'> 999</td><td align='center'> 40</td></tr> +</table></div> + +<p>There is one intelligent objection to the special practice +necessary to establish arithmetical connections so fully as +to give the accuracy which both utilitarian and disciplinary +aims require. It may be said that the pupils in grades 3, +4, and 5 cannot appreciate the need and that consequently +the work will be dull, barren, and alien, without close personal +appropriation by the pupil's nature. It is true that no +vehement life-purpose is directly involved by the problem +of perfecting one's power to add 7 to 28 in grade 2, or by +the problem of multiplying 253 by 8 accurately in grade 3<span class='pagenum'><a name="Page_110" id="Page_110">[Pg 110]</a></span> +or by precise subtraction in long division in grade 4. It is +also true, however, that the most humanly interesting of +problems—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 <i>inaccurate</i> work of +the same sort. If we are to teach computation with +two- and three- and four-place numbers at all, it should +be taught as a reliable instrument, not as a combination +of vague memories and faith. The author is ready +to cut computation with numbers above 10 out of the +curriculum of grades 1-6 as soon as more valuable educational +instruments are offered in its place, but he is convinced +that nothing in child-nature makes a large variety of +inaccurate computing more interesting or educative or germane +to felt needs, than a smaller variety of accurate +computing!</p> +<p><span class='pagenum'><a name="Page_111" id="Page_111">[Pg 111]</a></span></p> + +<h4>THE STRENGTH OF BONDS FOR TEMPORARY SERVICE</h4> + +<p>The second general fact is that certain bonds are of service +for only a limited time and so need to be formed only to a +limited and slight degree of strength. The data of problems +set to illustrate a principle or improve some habit of computation +are, of course, the clearest cases. The pupil needs +to remember that John bought 3 loaves of bread and that +they were 5-cent loaves and that he gave 25 cents to the +baker only long enough to use the data to decide what change +John should receive. The connections between the total +described situation and the answer obtained, supposing +some considerable computation to intervene, is a bond that +we let expire almost as soon as it is born.</p> + +<p>It is sometimes assumed that the bond between a certain +group of features which make a problem a 'Buy <i>a</i> things at +<i>b</i> per thing, find total cost' problem or a 'Buy <i>a</i> things at <i>b</i> +per thing, what change from <i>c</i>' problem or a 'What gain +on buying for <i>a</i> and selling for <i>b</i>' problem or a 'How many +things at <i>a</i> each can I buy for <i>b</i> cents' problem—it is assumed +that the bond between these essential defining features +and the operation or operations required for solution +is as temporary as the bonds with the name of the buyer +or the price of the thing. It is assumed that all problems +are and should be solved by some pure act of reasoning +without help or hindrance from bonds with the particular +verbal structure and vocabulary of the problems. Whether +or not they <i>should</i> be, they <i>are not</i>. Every time that a pupil +solves a 'bought-sold' problem by subtraction he strengthens +the tendency to respond to any problem whatsoever that +contains the words 'bought for' and 'sold for' by subtraction; +and he will by no means surely stop and survey +every such problem in all its elements to make sure that no<span class='pagenum'><a name="Page_112" id="Page_112">[Pg 112]</a></span> +other feature makes inapplicable the tendency to subtract +which the 'bought sold' evokes.</p> + +<p>To prevent pupils from responding to the form of statement +rather than the essential facts, we should then not +teach them to forget the form of statement, but rather +give them all the common forms of statement to which +the response in question is an appropriate response, and +only such. If a certain form of statement does in life +always signify a certain arithmetical procedure, the bond +between it and that procedure may properly be made very +strong.</p> + +<p>Another case of the formation of bonds to only a slight +degree of strength concerns the use of so-called 'crutches' +such as writing +, −, and × in copying problems like those +below:—</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="60%"> +<tr><td align='center'>Add</td><td align='center'>Subtract</td><td align='center'>Multiply</td></tr> +<tr><td align='center'>23<br />61<br />—</td><td align='center'>79<br />24<br />—</td><td align='center'>32<br /> 3<br />—</td></tr> +</table></div> + +<p class="noidt">or altering the figures when 'borrowing' in subtraction, +and the like. Since it is undesirable that the pupil should +regard the 'crutch' response as essential to the total procedure, +or become so used to having it that he will be disturbed +by its absence later, it is supposed that the bond between +the situation and the crutch should not be fully +formed. There is a better way out of the difficulty, in case +crutches are used at all. This is to associate the crutch +with a special 'set,' and its non-use with the general set +which is to be the permanent one. For example, children +may be taught from the start never to write the crutch +sign or crutch figure unless the work is accompanied by +"Write ... to help you to...."<span class='pagenum'><a name="Page_113" id="Page_113">[Pg 113]</a></span></p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='justify' rowspan='2'>Write - to help you to remember that<br /> +you must subtract in this row.</td> +<td align='left' colspan='5'>Find the differences:—</td></tr> +<tr><td align='left'>39<br /><span class="u">23</span></td><td align='left'>67<br /><span class="u">44</span></td><td align='left'>78<br /><span class="u">36</span></td><td align='left'>56<br /><span class="u">26</span></td><td align='left'>45<br /><span class="u">24</span></td></tr> +<tr><td> </td></tr> +<tr><td align='justify' rowspan='2'>Remember that you must subtract<br /> +in this row.</td> +<td align='left' colspan='5'>Find the differences:—</td></tr> +<tr><td align='left'>85<br /><span class="u">63</span></td><td align='left'>27<br /><span class="u">14</span></td><td align='left'>96<br /><span class="u">51</span></td><td align='left'>38<br /><span class="u">45</span></td><td align='left'>78<br /><span class="u">32</span></td></tr> +</table></div> + +<p>The bond evoking the use of the crutch may then be +formed thoroughly enough so that there is no hesitation, +insecurity, or error, without interfering to any harmful extent +with the more general bond from the situation to work +without the crutch.</p> + + +<h4>THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS</h4> + +<p>Another instructive case concerns the bonds between certain +words and their meanings, and between certain situations +of commerce, industry, or agriculture and useful facts +about these situations. Illustrations of the former are the +bonds between <i>cube root</i>, <i>hectare</i>, <i>brokerage</i>, <i>commission</i>, +<i>indorsement</i>, <i>vertex</i>, <i>adjacent</i>, <i>nonagon</i>, <i>sector</i>, <i>draft</i>, <i>bill +of exchange</i>, and their meanings. Illustrations of the latter +are the bonds from "Money being lent 'with interest' at no +specified rate, what rate is charged?" to "The legal rate +of the state," from "$<i>X</i> per M as a rate for lumber" to +"Means $<i>X</i> per thousand board feet, a board foot being +1 ft. by 1 ft. by 1 in."</p> + +<p>It is argued by many that such bonds are valuable for a +short time; namely, while arithmetical procedures in connection +with which they serve are learned, but that their +value is only to serve as a means for learning these procedures +and that thereafter they may be forgotten. "They +are formed only as accessory means to certain more purely +arithmetical knowledge or discipline; after this is acquired<span class='pagenum'><a name="Page_114" id="Page_114">[Pg 114]</a></span> +they may be forgotten. Everybody does in fact forget +them, relearning them later if life requires." So runs the +argument.</p> + +<p>In some cases learning such words and facts only to use +them in solving a certain sort of problems and then forget +them may be profitable. The practice is, however, exceedingly +risky. It is true that everybody does in fact +forget many such meanings and facts, but this commonly +means either that they should not have been learned at all +at the time that they were learned, or that they should have +been learned more permanently, or that details should have +been learned with the expectation that they themselves +would be forgotten but that a general fact or attitude would +remain. For example, duodecagon should not be learned +at all in the elementary school; indorsement should either +not be learned at all there, or be learned for permanence of +a year or more; the details of the metric system should be +so taught as to leave for several years at least knowledge +of the facts that there is a system so named that is important, +whose tables go by tens, hundreds, or thousands, +and a tendency (not necessarily strong) to connect meter, +kilogram, and liter with measurement by the metric +system and with approximate estimates of their several +magnitudes.</p> + +<p>If an arithmetical procedure seems to require accessory +bonds which are to be forgotten, once the procedure +is mastered, we should be suspicious of the value +of the procedure itself. If pupils forget what compound +interest is, we may be sure that they will usually +also have forgotten how to compute it. Surely there +is waste if they have learned what it is only to learn +how to compute it only to forget how to compute +it!</p> +<p><span class='pagenum'><a name="Page_115" id="Page_115">[Pg 115]</a></span></p> + +<h4>THE STRENGTH OF BONDS CONCERNING THE REASONS FOR +ARITHMETICAL PROCESSES</h4> + +<p>The next case of the formation of bonds to slight strength +is the problematic one of forming the bonds involved in +understanding the reasons for certain processes only to +forget them after the process has become a habit. Should +a pupil, that is, learn why he inverts and multiplies, only to +forget it as soon as he can be trusted to divide by a fraction? +Should he learn why he puts the units figure of each +partial product in multiplication under the figure that he +multiplies by, only to forget the reason as soon as he has +command of the process? Should he learn why he gets +the number of square inches in a rectangle by multiplying +the length by the width, both being expressed in linear +inches, and forget why as soon as he is competent to make +computations of the areas of rectangles?</p> + +<p>On general psychological grounds we should be suspicious +of forming bonds only to let them die of starvation later, +and tend to expect that elaborate explanations learned only +to be forgotten either should not be learned at all, or should +be learned at such a time and in such a way that they would +not be forgotten. Especially we should expect that the +general principles of arithmetic, the whys and wherefores +of its fundamental ways of manipulating numbers, ought +to be the last bonds of all to be forgotten. Details of <i>how</i> +you arranged numbers to multiply might vanish, but the +general reasons for the placing would be expected to persist +and enable one to invent the detailed manipulations +that had been forgotten.</p> + +<p>This suspicion is, I think, justified by facts. The doctrine +that the customary deductive explanations of why we +invert and multiply, or place the partial products as we do<span class='pagenum'><a name="Page_116" id="Page_116">[Pg 116]</a></span> +before adding, may be allowed to be forgotten once the +actual habits are in working order, has a suspicious source. +It arose to meet the criticism that so much time and effort +were required to keep these deductive explanations in memory. +The fact was that the pupil learned to compute correctly +<i>irrespective of</i> the deductive explanations. They +were only an added burden. His inductive learning that +the procedure gave the right answer really taught him. So +he wisely shuffled off the extra burden of facts about the +consequences of the nature of a fraction or the place values +of our decimal notation. The bonds weakened because +they were not used. They were not used because they were +not useful in the shape and at the time that they were formed, +or because the pupil was unable to understand the explanations +so as to form them at all.</p> + +<p>The criticism was valid and should have been met in part +by replacing the deductive explanations by inductive verifications, +and in part by using the deductive reasoning as a +check after the process itself is mastered. The very same +discussions of place-value which are futile as proof that you +must do a certain thing before you have done it, often become +instructive as an explanation of why the thing that +you have learned to do and are familiar with and have +verified by other tests works as well as it does. The general +deductive theory of arithmetic should not be learned only +to be forgotten. Much of it should, by most pupils, not be +learned at all. What is learned should be learned much +later than now, as a synthesis and rationale of habits, +not as their creator. What is learned of such deductive +theory should rank among the most rather than least permanent +of a pupil's stock of arithmetical knowledge and +power. There are bonds which are formed only to be lost, +and bonds formed only to be lost <i>in their first form</i>, being<span class='pagenum'><a name="Page_117" id="Page_117">[Pg 117]</a></span> +used in a new organization as material for bonds of a higher +order; but the bonds involved in deductive explanations +of why certain processes are right are not such: they are +not to be formed just to be forgotten, nor as mere propædeutics +to routine manipulations.</p> + + +<h4>PROPÆDEUTIC BONDS</h4> + +<p>The formation of bonds to a limited strength because they +are to be lost in their first form, being worked over in different +ways in other bonds to which they are propædeutic or contributing +is the most important case of low strength, or +rather low permanence, in bonds.</p> + +<p>The bond between four 5s in a column to be added and +the response of thinking '10, 15, 20' is worth forming, but +it is displaced later by the multiplication bond or direct +connection of 'four 5s to be added' with '20.' Counting +by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc., forms +serial bonds which as series might well be left to disappear. +Their separate steps are kept as permanent bonds for use +in column addition, but their serial nature is changed from +2 (and 2) 4, (and 2) 6, (and 2) 8, etc., to two 2s = 4, three +2s = 6, four 2s = 8, etc.; after playing their part in producing +the bonds whereby any multiple of 2 by 2 to 9, can be got, +the original serial bonds are, as series, needed no longer. The +verbal response of saying 'and' in adding, after helping to establish +the bonds whereby the general set of the mind toward +adding coöperates with the numbers seen or thought of to +produce their sum, should disappear; or remain so slurred +in inner speech as to offer no bar to speed.</p> + +<p>The rule for such bonds is, of course, to form them strongly +enough so that they work quickly and accurately for the +time being and facilitate the bonds that are to replace them,<span class='pagenum'><a name="Page_118" id="Page_118">[Pg 118]</a></span> +but not to overlearn them. There is a difference between +learning something to be held for a short time, and the same +amount of energy spent in learning for long retention. The +former sort of learning is, of course, appropriate with many +of these propædeutic bonds.</p> + +<p>The bonds mentioned as illustrations are not <i>purely</i> +propædeutic, nor formed <i>only</i> to be transmuted into something +else. Even the saying of 'and' in addition has +some genuine, intrinsic value in distinguishing the process +of addition, and may perhaps be usefully reviewed for a +brief space during the first steps in adding common fractions. +Some such propædeutic bonds may be worth while +apart from their value in preparing for other bonds. Consider, +for example, exercises like those shown below which +are propædeutic to long division, giving the pupil some +basis in experience for his selection of the quotient figures. +These multiplications are intrinsically worth doing, especially +the 12s and 25s. Whatever the pupil remembers +of them will be to his advantage.</p> + +<div class="pblockquot"> +<p><b>1.</b> Count by 11s to 132, beginning 11, 22, 33.</p> + +<p><b>2.</b> Count by 12s to 144, beginning 12, 24, 36.</p> + +<p><b>3.</b> Count by 25s to 300, beginning 25, 50, 75.</p> + +<p><b>4.</b> State the missing numbers:—</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="10" summary=""> +<tr><th>A.</th><th> B.</th><th>C.</th><th>D.</th></tr> +<tr><td align='right'> 3 11s =</td><td align='right'>5 11s =</td><td align='right'>8 ft. = .... in.</td><td align='right'>2 dozen =</td></tr> +<tr><td align='right'> 4 12s =</td><td align='right'>3 12s =</td><td align='right'>10 ft. = .... in.</td><td align='right'>4 dozen =</td></tr> +<tr><td align='right'> 5 12s =</td><td align='right'>6 12s =</td><td align='right'>7 ft. = .... in.</td><td align='right'>10 dozen =</td></tr> +<tr><td align='right'> 6 11s =</td><td align='right'>12 11s =</td><td align='right'>4 ft. = .... in.</td><td align='right'>5 dozen =</td></tr> +<tr><td align='right'> 9 11s =</td><td align='right'>2 12s =</td><td align='right'>6 ft. = .... in.</td><td align='right'>7 dozen =</td></tr> +<tr><td align='right'> 7 12s =</td><td align='right'>9 12s =</td><td align='right'>9 ft. = .... in.</td><td align='right'>12 dozen =</td></tr> +<tr><td align='right'> 8 12s =</td><td align='right'>7 11s =</td><td align='right'>11 ft. = .... in.</td><td align='right'>9 dozen =</td></tr> +<tr><td align='right'>11 11s =</td><td align='right'> 12 12s =</td><td align='right'> 5 ft. = .... in.</td><td align='right'> 6 dozen =</td></tr> +</table></div> +<p><span class='pagenum'><a name="Page_119" id="Page_119">[Pg 119]</a></span></p> + +<p><b>5.</b> Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents, +one dollar," and so on.</p> + +<p><b>6.</b> Count by 15s to $1.50.</p> + +<p><b>7.</b> Find the products. Do not use pencil. Think what they are.</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="10" summary=""> +<tr><th>A.</th><th>B.</th><th>C.</th><th>D.</th><th>E.</th></tr> +<tr><td align='right'> 2 × 25</td><td align='right'>3 × 15</td><td align='right'>2 × 12</td><td align='right'>4 × 11</td><td align='right'>6 × 25</td></tr> +<tr><td align='right'> 3 × 25</td><td align='right'> 10 × 15</td><td align='right'>2 × 15</td><td align='right'>4 × 15</td><td align='right'>6 × 15</td></tr> +<tr><td align='right'> 5 × 25</td><td align='right'>4 × 15</td><td align='right'>2 × 25</td><td align='right'>4 × 12</td><td align='right'>6 × 12</td></tr> +<tr><td align='right'>10 × 25</td><td align='right'>2 × 15</td><td align='right'>2 × 11</td><td align='right'>4 × 25</td><td align='right'>6 × 11</td></tr> +<tr><td align='right'> 4 × 25</td><td align='right'>7 × 15</td><td align='right'>3 × 25</td><td align='right'>5 × 11</td><td align='right'>7 × 12</td></tr> +<tr><td align='right'> 6 × 25</td><td align='right'>9 × 15</td><td align='right'>3 × 15</td><td align='right'>5 × 12</td><td align='right'>7 × 15</td></tr> +<tr><td align='right'> 8 × 25</td><td align='right'>5 × 15</td><td align='right'>3 × 11</td><td align='right'>5 × 15</td><td align='right'>7 × 25</td></tr> +<tr><td align='right'> 7 × 25</td><td align='right'>8 × 15</td><td align='right'> 3 × 12</td><td align='right'>5 × 25</td><td align='right'>7 × 11</td></tr> +<tr><td align='right'> 9 × 25</td><td align='right'>6 × 15</td><td align='right'>8 × 12</td><td align='right'> 9 × 12</td><td align='right'> 8 × 25</td></tr> +</table></div> + +<p>State the missing numbers:—</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="10" summary=""> +<tr><td align='right'>A. 36 = .... 12s</td><td align='right'> B. 44 = .... 11s</td><td align='right'> C. 50 = .... 25s</td></tr> +<tr><td align='right'> 60 = .... 12s</td><td align='right'>88 = .... 11s</td><td align='right'>125 = .... 25s</td></tr> +<tr><td align='right'> 24 = .... 12s</td><td align='right'>77 = .... 11s</td><td align='right'>75 = .... 25s</td></tr> +<tr><td align='right'> 48 = .... 12s</td><td align='right'>55 = .... 11s</td><td align='right'>200 = .... 25s</td></tr> +<tr><td align='right'>144 = .... 12s</td><td align='right'>99 = .... 11s</td><td align='right'>250 = .... 25s</td></tr> +<tr><td align='right'>108 = .... 12s</td><td align='right'>110 = .... 11s</td><td align='right'>175 = .... 25s</td></tr> +<tr><td align='right'> 72 = .... 12s</td><td align='right'>33 = .... 11s</td><td align='right'>225 = .... 25s</td></tr> +<tr><td align='right'> 96 = .... 12s</td><td align='right'>66 = .... 11s</td><td align='right'>150 = .... 25s</td></tr> +<tr><td align='right'> 84 = .... 12s</td><td align='right'>22 = .... 11s</td><td align='right'>100 = .... 25s</td></tr> +</table></div> + +<p>Find the quotients and remainders. If you need to use paper +and pencil to find them, you may. But find as many as you can +without pencil and paper. Do Row A first. Then do Row B. +Then Row C, etc.</p> + + +<div class='center'> +<table border="0" cellpadding="6" cellspacing="20" summary=""> +<tr><td align='left'>Row A.</td><td align='left'>11|<span class="overline">45</span></td><td align='left'>12|<span class="overline">45</span></td><td align='left'>25|<span class="overline">45</span></td><td align='left'>15|<span class="overline">45</span></td><td align='left'>21|<span class="overline">45</span></td><td align='left'>22|<span class="overline">45</span></td></tr> +<tr><td align='left'>Row B.</td><td align='left'>25|<span class="overline">55</span></td><td align='left'>11|<span class="overline">55</span></td><td align='left'>12|<span class="overline">55</span></td><td align='left'>15|<span class="overline">55</span></td><td align='left'>22|<span class="overline">55</span></td><td align='left'>30|<span class="overline">55</span></td></tr> +<tr><td align='left'>Row C.</td><td align='left'>12|<span class="overline">60</span></td><td align='left'>25|<span class="overline">60</span></td><td align='left'>15|<span class="overline">60</span></td><td align='left'>11|<span class="overline">60</span></td><td align='left'>30|<span class="overline">60</span></td><td align='left'>21|<span class="overline">60</span></td></tr> +<tr><td align='left'>Row D.</td><td align='left'>12|<span class="overline">75</span></td><td align='left'>11|<span class="overline">75</span></td><td align='left'>15|<span class="overline">75</span></td><td align='left'>25|<span class="overline">75</span></td><td align='left'>30|<span class="overline">75</span></td><td align='left'>35|<span class="overline">75</span></td></tr> +<tr><td align='left'>Row E.</td><td align='left'>11|<span class="overline">100</span></td><td align='left'>12|<span class="overline">100</span></td><td align='left'>25|<span class="overline">100</span></td><td align='left'>15|<span class="overline">100</span></td><td align='left'>30|<span class="overline">100</span></td><td align='left'>22|<span class="overline">100</span></td></tr> +<tr><td align='left'>Row F.</td><td align='left'>11|<span class="overline">96</span></td><td align='left'>12|<span class="overline">96</span></td><td align='left'>25|<span class="overline">96</span></td><td align='left'>15|<span class="overline">96</span></td><td align='left'>30|<span class="overline">96</span></td><td align='left'>22|<span class="overline">96</span></td></tr> +<tr><td align='left'>Row G.</td><td align='left'>25|<span class="overline">105</span></td><td align='left'>11|<span class="overline">105</span></td><td align='left'>15|<span class="overline">105</span></td><td align='left'>12|<span class="overline">105</span></td><td align='left'>22|<span class="overline">105</span></td><td align='left'>35|<span class="overline">105</span></td></tr> +<tr><td align='left'>Row H.</td><td align='left'>12|<span class="overline">64</span></td><td align='left'>15|<span class="overline">64</span></td><td align='left'>25|<span class="overline">64</span></td><td align='left'>11|<span class="overline">64</span></td><td align='left'>22|<span class="overline">64</span></td><td align='left'>21|<span class="overline">64</span></td></tr> +<tr><td align='left'>Row I.</td><td align='left'>11|<span class="overline">80</span></td><td align='left'>12|<span class="overline">80</span></td><td align='left'>15|<span class="overline">80</span></td><td align='left'>25|<span class="overline">80</span></td><td align='left'>35|<span class="overline">80</span></td><td align='left'>21|<span class="overline">80</span></td></tr> +<tr><td align='left'>Row J.</td><td align='left'>25|<span class="overline">200</span></td><td align='left'>30|<span class="overline">200</span></td><td align='left'>75|<span class="overline">200</span></td><td align='left'>63|<span class="overline">200</span></td><td align='left'>65|<span class="overline">200</span></td><td align='left'>66|<span class="overline">200</span></td></tr> +</table></div> + +<p>Do this section again. Do all the first column first. Then do +the second column, then the third, and so on.</p></div> + +<p>Consider, from the same point of view, exercises like +(3 × 4) + 2, (7 × 6) + 5, (9 × 4) + 6, given as a preparation +for written multiplication. The work of</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr> +<td align='right'> 48<br /><span class="u"> 3</span></td> +<td align='right'> 68<br /><span class="u"> 7</span></td> +<td align='right'> 47<br /><span class="u"> 9</span></td> +</tr> +</table></div> + +<p class="noidt">and the like is facilitated if the pupil has easy control +of the process of getting a product, and keeping it in mind +while he adds a one-place number to it. The practice with +(3 × 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.</p> + +<p>In some cases the bonds are purely propædeutic or are +formed <i>only</i> for later reconstruction. They then differ +little from 'crutches.' The typical crutch forms a habit +which has actually to be broken, whereas the purely propædeutic +bond forms a habit which is left to rust out from +disuse.</p> + +<p>For example, as an introduction to long division, a pupil +may be given exercises using one-figure divisors in the long +form, as:—</p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='right'>773</td><td align='left'> and 5 remainder</td></tr> +<tr><td align='right'>7 )<span class="overline"> 5416</span></td></tr> +<tr><td align='right'><span class="u"> 49</span> </td></tr> +<tr><td align='right'>51 </td></tr> +<tr><td align='right'><span class="u"> 49</span> </td></tr> +<tr><td align='right'>26</td></tr> +<tr><td align='right'><span class="u">21</span></td></tr> +<tr><td align='right'>5</td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_121" id="Page_121">[Pg 121]</a></span></p> +<p>The important recommendation concerning these purely +propædeutic 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 propædeutic +bonds of dubious value (with letters, phonograms, +diacritical marks, and the like) often carried to demonstrably +wasteful extremes.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_122" id="Page_122">[Pg 122]</a></span></p> +<h2><a name="CHAPTER_VI" id="CHAPTER_VI"></a>CHAPTER VI</h2> + +<h3>THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE +AMOUNT OF PRACTICE AND THE ORGANIZATION OF +ABILITIES</h3> + +<h4>THE AMOUNT OF PRACTICE</h4> + + +<p>It will be instructive if the reader will perform the following +experiment as an introduction to the discussion of this +chapter, before reading any of the discussion.</p> + +<p>Suppose that a pupil does all the work, oral and written, +computation and problem-solving, presented for grades +1 to 6 inclusive (that is, in the first two books of a three-book +series) in the average textbook now used in the elementary +school. How many times will he have exercised +each of the various bonds involved in the four operations +with integers shown below? That is, how many times will +he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.? +Every case of the action of each bond is to be counted.</p> + + +<p class="center"><b>THE FUNDAMENTAL BONDS</b></p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary="" width="80%"> +<tr><td align='left'>1 + 1</td><td align='left'>2 − 1</td><td align='left'>1 × 1</td><td align='left'>2 ÷ 1</td></tr> +<tr><td align='left'>1 + 2</td><td align='left'>2 − 2</td><td align='left'>2 × 1</td><td align='left'>2 ÷ 2</td></tr> +<tr><td align='left'>1 + 3</td><td align='left'></td><td align='left'>3 × 1</td></tr> +<tr><td align='left'>1 + 4</td><td align='left'></td><td align='left'>4 × 1</td></tr> +<tr><td align='left'>1 + 5</td><td align='left'>3 − 1</td><td align='left'>5 × 1</td><td align='left'>3 ÷ 1</td></tr> +<tr><td align='left'>1 + 6</td><td align='left'>3 − 2</td><td align='left'>6 × 1</td><td align='left'>3 ÷ 2</td></tr> +<tr><td align='left'>1 + 7</td><td align='left'>3 − 3</td><td align='left'>7 × 1</td><td align='left'>3 ÷ 3</td></tr> +<tr><td align='left'>1 + 8</td><td align='left'></td><td align='left'>8 × 1</td></tr> +<tr><td align='left'>1 + 9</td><td align='left'></td><td align='left'>9 × 1</td></tr> +<tr><td align='left'><span class='pagenum'><a name="Page_123" id="Page_123">[Pg 123]</a></span></td><td align='left'>4 − 1</td><td align='left'></td><td align='left'>4 ÷ 1</td></tr> +<tr><td align='left'></td><td align='left'>4 − 2</td><td align='left'></td><td align='left'>4 ÷ 2</td></tr> +<tr><td align='left'>11 (or 21 or 31, etc.) + 1</td><td align='left'>4 − 3</td><td align='left'>1 × 2</td><td align='left'>4 ÷ 3</td></tr> +<tr><td align='left'>11 " + 2</td><td align='left'>4 − 4</td><td align='left'>2 × 2</td><td align='left'>4 ÷ 4</td></tr> +<tr><td align='left'>11 " + 3</td><td align='left'></td><td align='left'>3 × 2</td><td align='left'></td></tr> +<tr><td align='left'>11 " + 4</td><td align='left'></td><td align='left'>4 × 2</td><td align='left'></td></tr> +<tr><td align='left'>11 " + 5</td><td align='left'>5 − 1</td><td align='left'>5 × 2</td><td align='left'>5 ÷ 1</td></tr> +<tr><td align='left'>11 " + 6</td><td align='left'>5 − 2</td><td align='left'>6 × 2</td><td align='left'>5 ÷ 2</td></tr> +<tr><td align='left'>11 " + 7</td><td align='left'>5 − 3</td><td align='left'>7 × 2</td><td align='left'>5 ÷ 3</td></tr> +<tr><td align='left'>11 " + 8</td><td align='left'>5 − 4</td><td align='left'>8 × 2</td><td align='left'>5 ÷ 4</td></tr> +<tr><td align='left'>11 " + 9</td><td align='left'>5 − 5</td><td align='left'>9 × 2</td><td align='left'>5 ÷ 5</td></tr> +<tr><td> </td></tr> +<tr><td align='left'></td><td align='left'>6 − 1</td><td align='left'>1 × 3</td><td align='left'>6 ÷ 1</td></tr> +<tr><td align='left'>2 + 1</td><td align='left'>6 − 2</td><td align='left'>2 × 3</td><td align='left'>6 ÷ 2</td></tr> +<tr><td align='left'>2 + 2</td><td align='left'>6 − 3</td><td align='left'>3 × 3</td><td align='left'>6 ÷ 3</td></tr> +<tr><td align='left'>2 + 3</td><td align='left'>6 − 4</td><td align='left'>4 × 3</td><td align='left'>6 ÷ 4</td></tr> +<tr><td align='left'>2 + 4</td><td align='left'>6 − 5</td><td align='left'>5 × 3</td><td align='left'>6 ÷ 5</td></tr> +<tr><td align='left'>2 + 5</td><td align='left'>6 − 6</td><td align='left'>6 × 3</td><td align='left'>6 ÷ 6</td></tr> +<tr><td align='left'>2 + 6</td><td align='left'></td><td align='left'>7 × 3</td><td align='left'></td></tr> +<tr><td align='left'>2 + 7</td><td align='left'></td><td align='left'>8 × 3</td></tr> +<tr><td align='left'>2 + 8</td><td align='left'>7 − 1</td><td align='left'>9 × 3</td><td align='left'>7 ÷ 1</td></tr> +<tr><td align='left'>2 + 9</td><td align='left'>7 − 2</td><td align='left'></td><td align='left'>7 ÷ 2</td></tr> +<tr><td align='left'></td><td align='left'>7 − 3</td><td align='left'></td><td align='left'>7 ÷ 3</td></tr> +<tr><td align='left'></td><td align='left'>7 − 4</td><td align='left'>1 × 4</td><td align='left'>7 ÷ 4</td></tr> +<tr><td align='left'>12 (or 22 or 32, etc.) + 1</td><td align='left'>7 − 5</td><td align='left'>2 × 4</td><td align='left'>7 ÷ 5</td></tr> +<tr><td align='left'>12 " + 2</td><td align='left'>7 − 6</td><td align='left'>and so on</td><td align='left'>7 ÷ 6</td></tr> +<tr><td align='left'></td><td align='left'>7 − 7</td><td align='left'>to 9 × 9</td><td align='left'>7 ÷ 7</td></tr> +<tr><td align='left'>and so on to</td><td align='left'>and so on</td><td align='left'></td><td align='left'>and so on to</td></tr> +<tr><td align='left'>9 + 9</td><td align='left'>to 18 − 9</td><td align='left'></td><td align='left'>82 ÷ 9 83 ÷ 9, etc.</td></tr> +<tr><td align='left'>19 (or 29 or 39, etc.) + 9</td></tr> +</table></div> + +<p>If estimating for the entire series is too long a task, it will +be sufficient to use eight or ten from each, say:—</p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary="" width="80%"> +<colgroup><col width="25%" /><col width="25%" /><col width="25%" /><col width="25%" /></colgroup> +<tr><td align='right'>3 + 2</td><td align='right'>13, 23, etc. + 2</td><td align='right'>7 + 2</td><td align='right'>17, 27, etc. + 2</td></tr> +<tr><td align='right'> " 3</td><td align='right'> " 3</td> +<td align='right'> " 3</td><td align='right'> " 3</td></tr> +<tr><td align='right'> " 4</td><td align='right'> " 4</td> +<td align='right'> " 4</td><td align='right'> " 4</td></tr> +<tr><td align='right'> " 5</td><td align='right'> " 5</td> +<td align='right'> " 5</td><td align='right'> " 5</td></tr> +<tr><td align='right'> " 6</td><td align='right'> " 6</td> +<td align='right'> " 6</td><td align='right'> " 6</td></tr> +<tr><td align='right'> " 7</td><td align='right'> " 7</td> +<td align='right'> " 7</td><td align='right'> " 7</td></tr> +<tr><td align='right'> " 8</td><td align='right'> " 8</td> +<td align='right'> " 8</td><td align='right'> " 8</td></tr> +<tr><td align='right'> " 9</td><td align='right'> " 9</td> +<td align='right'> " 9</td><td align='right'> " 9</td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_124" id="Page_124">[Pg 124]</a></span></p> + + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary="" width="80%"> +<colgroup><col width="25%" /><col width="25%" /><col width="25%" /><col width="25%" /></colgroup> +<tr><td align='right'>3 − 3</td><td align='right'>7 − 7</td><td align='right'>9 × 7</td><td align='right'>63 ÷ 9</td></tr> +<tr><td align='right'>4 " </td><td align='right'>8 " </td><td align='right'>7 × 9</td><td align='right'>64 " </td></tr> +<tr><td align='right'>5 " </td><td align='right'>9 " </td><td align='right'>8 × 6</td><td align='right'>65 " </td></tr> +<tr><td align='right'>6 " </td><td align='right'>10 " </td><td align='right'>6 × 8</td><td align='right'>66 " </td></tr> +<tr><td align='right'>7 " </td><td align='right'>11 " </td><td align='right'></td><td align='right'>67 " </td></tr> +<tr><td align='right'>8 " </td><td align='right'>12 " </td><td align='right'></td><td align='right'>68 " </td></tr> +<tr><td align='right'>9 " </td><td align='right'>13 " </td><td align='right'></td><td align='right'>69 " </td></tr> +<tr><td align='right'>10 " </td><td align='right'>14 " </td><td align='right'></td><td align='right'>70 " </td></tr> +<tr><td align='right'>11 " </td><td align='right'>15 " </td><td align='right'></td><td align='right'>71 " </td></tr> +<tr><td align='right'>12 " </td><td align='right'>16 " </td><td align='right'></td></tr> +</table></div> + + +<p class="tabcap">TABLE 2</p> + +<p class="center"><span class="smcap">Estimates of the Amount of Practice Provided in Books I and II of +the Average Three-Book Text in Arithmetic; by 50 Experienced +Teachers</span></p> + +<div class='center'> +<table border="1" rules="cols" cellpadding="1" cellspacing="1" summary=""> +<colgroup><col width="30%" /><col width="15%" /><col width="15%" /><col width="15%" /><col width="25%" /></colgroup> +<tr><th class="smcap bbt">Arithmetical<br />Fact</th><th class="smcap bbt">Lowest<br />Estimate</th> +<th class="smcap bbt">Median<br />Estimate</th><th class="smcap bbt">Highest<br />Estimate</th> +<th class="smcap bbt">Range Required to<br />Include Half of<br />the Estimates</th></tr> +<tr><td align='right'>3 or 13 or 23, etc. + 2</td><td align='right'> 25</td><td align='right'> 1500</td><td align='right'>1,000,000</td><td align='right'> 800-5000</td></tr> +<tr><td align='right'>" " 3</td><td align='right'> 24</td><td align='right'> 1450</td><td align='right'> 80,000</td><td align='right'> 475-5000</td></tr> +<tr><td align='right'>" " 4</td><td align='right'> 23</td><td align='right'> 1150</td><td align='right'> 50,000</td><td align='right'> 750-5000</td></tr> +<tr><td align='right'>" " 5</td><td align='right'> 22</td><td align='right'> 1400</td><td align='right'> 44,000</td><td align='right'> 700-5000</td></tr> +<tr><td align='right'>" " 6</td><td align='right'> 21</td><td align='right'> 1350</td><td align='right'> 41,000</td><td align='right'> 700-4500</td></tr> +<tr><td align='right'>" " 7</td><td align='right'> 21</td><td align='right'> 1500</td><td align='right'> 37,000</td><td align='right'> 600-4000</td></tr> +<tr><td align='right'>" " 8</td><td align='right'> 20</td><td align='right'> 1400</td><td align='right'> 33,000</td><td align='right'> 550-4100</td></tr> +<tr><td align='right'>" " 9</td><td align='right'> 20</td><td align='right'> 1150</td><td align='right'> 28,000</td><td align='right'> 650-4500</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='right'>7 or 17 or 27, etc. + 2</td><td align='right'> 20</td><td align='right'> 1250</td><td align='right'>2,000,000</td><td align='right'> 600-5000</td></tr> +<tr><td align='right'>" " 3</td><td align='right'> 19</td><td align='right'> 1100</td><td align='right'>1,000,000</td><td align='right'> 650-4900</td></tr> +<tr><td align='right'>" " 4</td><td align='right'> 18</td><td align='right'> 1000</td><td align='right'> 80,000</td><td align='right'> 650-4900</td></tr> +<tr><td align='right'>" " 5</td><td align='right'> 17</td><td align='right'> 1300</td><td align='right'> 80,000</td><td align='right'> 650-4400</td></tr> +<tr><td align='right'>" " 6</td><td align='right'> 16</td><td align='right'> 1100</td><td align='right'> 29,000</td><td align='right'> 650-4500</td></tr> +<tr><td align='right'>" " 7</td><td align='right'> 15</td><td align='right'> 1100</td><td align='right'> 25,000</td><td align='right'> 500-4500</td></tr> +<tr><td align='right'>" " 8</td><td align='right'> 13</td><td align='right'> 1100</td><td align='right'> 21,000</td><td align='right'> 650-3800</td></tr> +<tr><td align='right'>" " 9</td><td align='right'> 10</td><td align='right'> 1275</td><td align='right'> 17,000</td><td align='right'> 500-4000</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='right'> 3 − 3</td><td align='right'> 25</td><td align='right'> 1000</td><td align='right'> 100,000</td><td align='right'> 500-4000</td></tr> +<tr><td align='right'> 4 − 3</td><td align='right'> 20</td><td align='right'> 1050</td><td align='right'> 500,000</td><td align='right'> 525-3000</td></tr> +<tr><td align='right'> 5 − 3</td><td align='right'> 20</td><td align='right'> 1100</td><td align='right'>2,500,000</td><td align='right'> 650-4200</td></tr> +<tr><td align='right'> 6 − 3</td><td align='right'> 10</td><td align='right'> 1050</td><td align='right'> 21,000</td><td align='right'> 650-3250</td></tr> +<tr><td align='right'> 7 − 3</td><td align='right'> 22</td><td align='right'> 1100</td><td align='right'> 15,000</td><td align='right'> 550-3050</td></tr> +<tr><td align='right'> 8 − 3</td><td align='right'> 21</td><td align='right'> 1075</td><td align='right'> 15,000</td><td align='right'> 650-3000</td></tr> +<tr><td align='right'> 9 − 3</td><td align='right'> 21</td><td align='right'> 1000</td><td align='right'> 15,000</td><td align='right'> 700-2600</td></tr> +<tr><td align='right'> 10 − 3</td><td align='right'> 20</td><td align='right'> 1000</td><td align='right'> 20,000</td><td align='right'> 600-2500</td></tr> +<tr><td align='right'> 11 − 3</td><td align='right'> 20</td><td align='right'> 1000</td><td align='right'> 15,000</td><td align='right'> 465-2550</td></tr> +<tr><td align='right'> 12 − 3</td><td align='right'> 18</td><td align='right'> 1000</td><td align='right'> 15,000</td><td align='right'> 650-2100</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='right'> 7 − 7</td><td align='right'> 10</td><td align='right'> 1000</td><td align='right'> 18,000</td><td align='right'> 425-3000</td></tr> +<tr><td align='right'> 8 − 7</td><td align='right'> 15</td><td align='right'> 1000</td><td align='right'> 18,000</td><td align='right'> 413-3100</td></tr> +<tr><td align='right'> 9 − 7</td><td align='right'> 15</td><td align='right'> 950</td><td align='right'> 18,000</td><td align='right'> 550-3000</td></tr> +<tr><td align='right'> 10 − 7</td><td align='right'> 15</td><td align='right'> 950</td><td align='right'> 18,000</td><td align='right'> 600-3950</td></tr> +<tr><td align='right'> 11 − 7</td><td align='right'> 10</td><td align='right'> 900</td><td align='right'> 18,000</td><td align='right'> 550-3000</td></tr> +<tr><td align='right'> 12 − 7</td><td align='right'> 10</td><td align='right'> 925</td><td align='right'> 18,000</td><td align='right'> 525-3100</td></tr> +<tr><td align='right'> 13 − 7</td><td align='right'> 10</td><td align='right'> 900</td><td align='right'> 18,000</td><td align='right'> 500-2600</td></tr> +<tr><td align='right'> 14 − 7</td><td align='right'> 10</td><td align='right'> 900</td><td align='right'> 18,000</td><td align='right'> 500-3100</td></tr> +<tr><td align='right'> 15 − 7</td><td align='right'> 10</td><td align='right'> 925</td><td align='right'> 18,000</td><td align='right'> 500-3000</td></tr> +<tr><td align='right'> 16 − 7</td><td align='right'> 10</td><td align='right'> 875</td><td align='right'> 18,000</td><td align='right'> 500-2500</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='right'> 9 × 7</td><td align='right'> 10</td><td align='right'> 700</td><td align='right'> 20,000</td><td align='right'> 500-2000</td></tr> +<tr><td align='right'> 7 × 9</td><td align='right'> 10</td><td align='right'> 700</td><td align='right'> 20,000</td><td align='right'> 500-1750</td></tr> +<tr><td align='right'> 8 × 6</td><td align='right'> 10</td><td align='right'> 750</td><td align='right'> 20,000</td><td align='right'> 500-2500</td></tr> +<tr><td align='right'> 6 × 8</td><td align='right'> 9</td><td align='right'> 700</td><td align='right'> 20,000</td><td align='right'> 500-2500</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='right'> 63 ÷ 9</td><td align='right'> 9</td><td align='right'> 500</td><td align='right'> 4,500</td><td align='right'> 300-2500</td></tr> +<tr><td align='right'> 64 ÷ 9</td><td align='right'> 9</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 100- 700</td></tr> +<tr><td align='right'> 65 ÷ 9</td><td align='right'> 8</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 100- 600</td></tr> +<tr><td align='right'> 66 ÷ 9</td><td align='right'> 7</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 100- 550</td></tr> +<tr><td align='right'> 67 ÷ 9</td><td align='right'> 7</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 75- 450</td></tr> +<tr><td align='right'> 68 ÷ 9</td><td align='right'> 6</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 87- 575</td></tr> +<tr><td align='right'> 69 ÷ 9</td><td align='right'> 6</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 87- 450</td></tr> +<tr><td align='right'> 70 ÷ 9</td><td align='right'> 5</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 75- 575</td></tr> +<tr><td align='right'> 71 ÷ 9</td><td align='right'> 5</td><td align='right'> 200</td><td align='right'> 4,000</td><td align='right'> 75- 700</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='right'> <i>XX</i></td><td align='right'> 40</td><td align='right'> 550</td><td align='right'>1,000,000</td><td align='right'> 300-2000</td></tr> +<tr><td align='right'> <i>XO</i></td><td align='right'> 20</td><td align='right'> 500</td><td align='right'> 11,500</td><td align='right'> 150-2000</td></tr> +<tr><td align='right'> <i>XXX</i></td><td align='right'> 15</td><td align='right'> 450</td><td align='right'> 12,000</td><td align='right'> 100-1000</td></tr> +<tr><td align='right'> <i>XXO</i></td><td align='right'> 25</td><td align='right'> 400</td><td align='right'> 15,000</td><td align='right'> 150-1000</td></tr> +<tr><td align='right'> <i>XOO</i></td><td align='right'> 15</td><td align='right'> 400</td><td align='right'> 5,000</td><td align='right'> 100-1000</td></tr> +<tr><td align='right'> <i>XOX</i></td><td align='right'> 10</td><td align='right'> 400</td><td align='right'> 10,000</td><td align='right'> 100- 975</td></tr> +</table></div> + +<p>Having made his estimates the reader should compare +them first with similar estimates made by experienced +teachers (shown on page 124 f.), and then with the results of +actual counts for representative textbooks in arithmetic +(shown on pages 126 to 132).</p> + +<p>It will be observed in Table 2 that even experienced +teachers vary enormously in their estimates of the amount +<span class='pagenum'><a name="Page_126" id="Page_126">[Pg 126]</a></span> +of practice given by an average textbook in arithmetic, +and that most of them are in serious error by overestimating +the amount of practice. In general it is the fact that +we use textbooks in arithmetic with very vague and erroneous +ideas of what is in them, and think they give much more +practice than they do.</p> + +<p>The authors of the textbooks as a rule also probably had +only very vague and erroneous ideas of what was in them. +If they had known, they would almost certainly have revised +their books. Surely no author would intentionally +provide nearly four times as much practice on 2 + 2 as on +8 + 8, or eight times as much practice on 2 × 2 as on 9 × 8, +or eleven times as much practice on 2 − 2 as on 17 − 8, or +over forty times as much practice on 2 ÷ 2 as on 75 ÷ 8 and +75 ÷ 9, both together. Surely no author would have provided +intentionally only twenty to thirty occurrences each +of 16 − 7, 16 − 8, 16 − 9, 17 − 8, 17 − 9, and 18 − 9 for the +entire course through grade 6; or have left the practice +on 60 ÷ 7, 60 ÷ 8, 60 ÷ 9, 61 ÷ 7, 61 ÷ 8, 61 ÷ 9, and the like +to occur only about once a year!</p> + + +<p class="tabcap">TABLE 3</p> + +<p class="nblockquot"><span class="smcap">Amount of Practice: Addition Bonds in a Recent Textbook (A) of +Excellent Repute. Books I and II, All Save Four Sections of +Supplementary Material, to be Used at the Teacher's Discretion</span></p> + +<p><small>The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times, +22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.</small></p> + + +<div class='center'> +<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary=""> +<tr><th class="bbt"></th><th class="bbt"> <b>2</b> </th><th class="bbt"> <b>3</b> </th><th class="bbt"> <b>4</b> </th><th class="bbt"> <b>5</b> </th><th class="bbt"> <b>6</b> </th><th class="bbt"> <b>7</b> </th><th class="bbt"> <b>8</b> </th><th class="bbt"> <b>9</b></th><th class="smcap bbt"><b>Total</b></th></tr> +<tr><td align='left'> <b>2</b></td><td align='right'> 226</td><td align='right'> 154</td><td align='right'> 162</td><td align='right'> 150</td><td align='right'> 97</td><td align='right'> 87</td><td align='right'> 66</td><td align='right'> 45</td><td></td></tr> +<tr><td align='left'><b>12</b></td><td align='right'> 74</td><td align='right'> 53</td><td align='right'> 76</td><td align='right'> 46</td><td align='right'> 51</td><td align='right'> 37</td><td align='right'> 36</td><td align='right'> 33</td><td></td></tr> +<tr><td align='left'><b>22</b>, etc.</td><td align='right'> 50</td><td align='right'> 60</td><td align='right'> 68</td><td align='right'> 63</td><td align='right'> 42</td><td align='right'> 50</td><td align='right'> 38</td><td align='right'> 26</td><td></td></tr> +<tr><td align='left'><span class='pagenum'><a name="Page_127" id="Page_127">[Pg 127]</a></span> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'> <b>3</b></td><td align='right'> 216</td><td align='right'> 141</td><td align='right'> 127</td><td align='right'> 89</td><td align='right'> 82</td><td align='right'> 54</td><td align='right'> 58</td><td align='right'> 40</td><td></td></tr> +<tr><td align='left'><b>13</b></td><td align='right'> 43</td><td align='right'> 43</td><td align='right'> 60</td><td align='right'> 70</td><td align='right'> 52</td><td align='right'> 30</td><td align='right'> 22</td><td align='right'> 18</td><td></td></tr> +<tr><td align='left'><b>23</b>, etc.</td><td align='right'> 15</td><td align='right'> 30</td><td align='right'> 51</td><td align='right'> 50</td><td align='right'> 42</td><td align='right'> 32</td><td align='right'> 29</td><td align='right'> 30</td><td></td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'> <b>7</b></td><td align='right'> 85</td><td align='right'> 90</td><td align='right'> 103</td><td align='right'> 103</td><td align='right'> 84</td><td align='right'> 81</td><td align='right'> 61</td><td align='right'> 47</td><td></td></tr> +<tr><td align='left'><b>17</b></td><td align='right'> 35</td><td align='right'> 25</td><td align='right'> 42</td><td align='right'> 32</td><td align='right'> 35</td><td align='right'> 21</td><td align='right'> 29</td><td align='right'> 16</td><td></td></tr> +<tr><td align='left'><b>27</b>, etc.</td><td align='right'> 30</td><td align='right'> 23</td><td align='right'> 32</td><td align='right'> 29</td><td align='right'> 24</td><td align='right'> 23</td><td align='right'> 25</td><td align='right'> 28</td><td></td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'> <b>8</b></td><td align='right'> 185</td><td align='right'> 112</td><td align='right'> 146</td><td align='right'> 99</td><td align='right'> 75</td><td align='right'> 71</td><td align='right'> 73</td><td align='right'> 61</td><td></td></tr> +<tr><td align='left'><b>18</b></td><td align='right'> 28</td><td align='right'> 35</td><td align='right'> 52</td><td align='right'> 46</td><td align='right'> 28</td><td align='right'> 29</td><td align='right'> 24</td><td align='right'> 14</td><td></td></tr> +<tr><td align='left'><b>28</b>, etc.</td><td align='right'> 53</td><td align='right'> 36</td><td align='right'> 34</td><td align='right'> 38</td><td align='right'> 23</td><td align='right'> 36</td><td align='right'> 27</td><td align='right'> 27</td><td></td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'> <b>9</b></td><td align='right'> 104</td><td align='right'> 81</td><td align='right'> 112</td><td align='right'> 96</td><td align='right'> 63</td><td align='right'> 74</td><td align='right'> 58</td><td align='right'> 57</td><td></td></tr> +<tr><td align='left'><b>19</b></td><td align='right'> 13</td><td align='right'> 11</td><td align='right'> 31</td><td align='right'> 38</td><td align='right'> 25</td><td align='right'> 14</td><td align='right'> 22</td><td align='right'> 11</td><td></td></tr> +<tr><td align='left'><b>29</b>, etc.</td><td align='right'> 19</td><td align='right'> 17</td><td align='right'> 27</td><td align='right'> 20</td><td align='right'> 32</td><td align='right'> 32</td><td align='right'> 19</td><td align='right'> 18</td><td></td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'><b>2, 12, 22</b>, etc.</td><td align='right'> 350</td><td align='right'> 277</td><td align='right'> 306</td><td align='right'> 260</td><td align='right'> 190</td><td align='right'> 174</td><td align='right'> 140</td><td align='right'> 104</td><td align='right'>1801</td></tr> +<tr><td align='left'><b>3, 13, 23</b>, etc.</td><td align='right'> 274</td><td align='right'> 214</td><td align='right'> 230</td><td align='right'> 209</td><td align='right'> 176</td><td align='right'> 116</td><td align='right'> 109</td><td align='right'> 88</td><td align='right'>1406</td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'><b>7, 17, 27</b>, etc.</td><td align='right'> 148</td><td align='right'> 138</td><td align='right'> 187</td><td align='right'> 164</td><td align='right'> 141</td><td align='right'> 125</td><td align='right'> 115</td><td align='right'> 91</td><td align='right'>1109</td></tr> +<tr><td align='left'><b>8, 18, 28</b>, etc.</td><td align='right'> 266</td><td align='right'> 183</td><td align='right'> 232</td><td align='right'> 185</td><td align='right'> 126</td><td align='right'> 136</td><td align='right'> 124</td><td align='right'> 102</td><td align='right'>1354</td></tr> +<tr><td align='left'><b>9, 19, 29</b>, etc.</td><td align='right'> 136</td><td align='right'> 109</td><td align='right'> 170</td><td align='right'> 154</td><td align='right'> 120</td><td align='right'> 120</td><td align='right'> 99</td><td align='right'> 86</td><td align='right'> 994</td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='center'>Totals</td><td align='right'>1164</td><td align='right'> 921</td><td align='right'>1125</td><td align='right'> 972</td><td align='right'> 753</td><td align='right'> 671</td><td align='right'> 687</td><td align='right'> 471</td><td></td></tr> +</table></div> + +<p> </p> +<p class="tabcap">TABLE 4</p> + +<p class="nblockquot"><span class="smcap">Amount of Practice: Subtraction Bonds in a Recent Textbook (A) +of Excellent Repute. Books I and II, All Save Four Sections of +Supplementary Material, to be Used at the Teacher's Discretion</span></p> + +<div class='center'> +<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary=""> +<tr><th class="bbt" rowspan='2'><span class="smcap">Minuends</span></th><th class="bbt" colspan='9'><span class="smcap">Subtrahends</span></th></tr> +<tr><th class="bbt"> <b>1</b> </th><th class="bbt"> <b>2</b> </th><th class="bbt"> <b>3</b> </th><th class="bbt"> <b>4</b> </th><th class="bbt"> <b>5</b> </th><th class="bbt"> <b>6</b> </th><th class="bbt"> <b>7</b> </th><th class="bbt"> <b>8</b> </th><th class="bbt"> <b>9</b></th></tr> +<tr><td align='right'><b>1</b></td><td align='right'> 372</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><b>2</b></td><td align='right'> 214</td><td align='right'> 311</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><b>3</b></td><td align='right'> 136</td><td align='right'> 149</td><td align='right'> 189</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><b>4</b></td><td align='right'> 146</td><td align='right'> 142</td><td align='right'> 103</td><td align='right'> 205</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><b>5</b></td><td align='right'> 171</td><td align='right'> 91</td><td align='right'> 92</td><td align='right'> 164</td><td align='right'> 136</td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><span class='pagenum'><a name="Page_128" id="Page_128">[Pg 128]</a></span><b>6</b></td><td align='right'> 80</td><td align='right'> 59</td><td align='right'> 69</td><td align='right'> 71</td><td align='right'> 81</td><td align='right'> 192</td><td align='right'></td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><b>7</b></td><td align='right'> 106</td><td align='right'> 57</td><td align='right'> 55</td><td align='right'> 67</td><td align='right'> 59</td><td align='right'> 156</td><td align='right'> 80</td><td align='right'></td><td align='right'></td></tr> +<tr><td align='right'><b>8</b></td><td align='right'> 73</td><td align='right'> 50</td><td align='right'> 50</td><td align='right'> 75</td><td align='right'> 50</td><td align='right'> 62</td><td align='right'> 48</td><td align='right'> 152</td><td align='right'></td></tr> +<tr><td align='right'><b>9</b></td><td align='right'> 71</td><td align='right'> 75</td><td align='right'> 54</td><td align='right'> 74</td><td align='right'> 48</td><td align='right'> 55</td><td align='right'> 55</td><td align='right'> 124</td><td align='right'> 133</td></tr> +<tr><td align='right'><b>10</b></td><td align='right'> 261</td><td align='right'> 84</td><td align='right'> 63</td><td align='right'> 100</td><td align='right'> 193</td><td align='right'> 83</td><td align='right'> 57</td><td align='right'> 124</td><td align='right'> 91</td></tr> +<tr><td align='right'></td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'></td></tr> +<tr><td align='right'><b>11</b></td><td align='right'></td><td align='right'> 48</td><td align='right'> 31</td><td align='right'> 50</td><td align='right'> 36</td><td align='right'> 41</td><td align='right'> 32</td><td align='right'> 46</td><td align='right'> 35</td></tr> +<tr><td align='right'><b>12</b></td><td align='right'></td><td align='right'></td><td align='right'> 48</td><td align='right'> 77</td><td align='right'> 57</td><td align='right'> 51</td><td align='right'> 35</td><td align='right'> 80</td><td align='right'> 30</td></tr> +<tr><td align='right'><b>13</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 35</td><td align='right'> 22</td><td align='right'> 40</td><td align='right'> 29</td><td align='right'> 35</td><td align='right'> 28</td></tr> +<tr><td align='right'><b>14</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 25</td><td align='right'> 37</td><td align='right'> 36</td><td align='right'> 49</td><td align='right'> 32</td></tr> +<tr><td align='right'><b>15</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 33</td><td align='right'> 19</td><td align='right'> 48</td><td align='right'> 20</td></tr> +<tr><td align='right'></td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'></td></tr> +<tr><td align='right'><b>16</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 16</td><td align='right'> 36</td><td align='right'> 26</td></tr> +<tr><td align='right'><b>17</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 27</td><td align='right'> 20</td></tr> +<tr><td align='right'><b>18</b></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'></td><td align='right'> 19</td></tr> +<tr><td align='right'></td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'> </td><td align='right'></td></tr> +<tr><td align='right'>Total excluding<br /><b>1 − 1</b>, <b>2 − 2,</b> etc.</td><td align='right'>1258</td><td align='right'> 755</td><td align='right'> 565</td><td align='right'> 713</td><td align='right'> 571</td><td align='right'> 558</td><td align='right'> 327</td><td align='right'> 569</td><td align='right'> 301</td></tr> +</table></div> + +<p> </p> +<p class="tabcap">TABLE 5</p> + +<p class="center"><span class="smcap">Frequencies of Subtractions not Included in Table 4</span></p> + +<p><small>These are cases where the pupil would, by reason of his stage of advancement, +probably operate 35 − 30, 46 − 46, etc., as one bond.</small></p> + + +<div class='center'> +<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary=""> +<tr><th class="bbt" rowspan='2'><span class="smcap">Minuends</span></th><th class="bbt" colspan='10'><span class="smcap">Subtrahends</span></th></tr> +<tr><td class="bbt"> <b>1</b><br /><b>11</b><br /><b>21</b><br />etc.</td> +<td class="bbt"> <b>2</b><br /><b>12</b><br /><b>22</b><br />etc.</td> +<td class="bbt"> <b>3</b><br /><b>13</b><br /><b>23</b><br />etc.</td> +<td class="bbt"> <b>4</b><br /><b>14</b><br /><b>24</b><br />etc.</td> +<td class="bbt"> <b>5</b><br /><b>15</b><br /><b>25</b><br />etc.</td> +<td class="bbt"> <b>6</b><br /><b>16</b><br /><b>26</b><br />etc.</td> +<td class="bbt"> <b>7</b><br /><b>17</b><br /><b>27</b><br />etc.</td> +<td class="bbt"> <b>8</b><br /><b>18</b><br /><b>28</b><br />etc.</td> +<td class="bbt"> <b>9</b><br /><b>19</b><br /><b>29</b><br />etc.</td> +<td class="bbt"> <br /><b>10</b><br /><b>20</b><br />etc.</td></tr> +<tr><td align='right'><b>10</b>, <b>20</b>, <b>30</b>, <b>40</b>, etc.</td><td align='right'> 11</td><td align='right'> 29</td><td align='right'> 16</td><td align='right'> 52</td><td align='right'> 32</td><td align='right'> 51</td><td align='right'> 7</td><td align='right'> 30</td><td align='right'> 22</td><td align='right'> 60</td></tr> +<tr><td align='right'><b>11</b>, <b>21</b>, <b>31</b>, <b>41</b>, etc.</td><td align='right'> 42</td><td align='right'> 14</td><td align='right'> 22</td><td align='right'> 32</td><td align='right'> 12</td><td align='right'> 26</td><td align='right'> 19</td><td align='right'> 52</td><td align='right'> 17</td><td align='right'> 10</td></tr> +<tr><td align='right'><b>12</b>, <b>22</b>, <b>32</b>, <b>42</b>, etc.</td><td align='right'> 47</td><td align='right'> 97</td><td align='right'> 5</td><td align='right'> 13</td><td align='right'> 9</td><td align='right'> 21</td><td align='right'> 11</td><td align='right'> 24</td><td align='right'> 19</td><td align='right'> 17</td></tr> +<tr><td align='right'><b>13</b>, <b>23</b>, <b>33</b>, <b>43</b>, etc.</td><td align='right'> 7</td><td align='right'> 40</td><td align='right'> 7</td><td align='right'> 14</td><td align='right'> 15</td><td align='right'> 13</td><td align='right'> 19</td><td align='right'> 19</td><td align='right'> 22</td><td align='right'> 3</td></tr> +<tr><td align='right'><b>14</b>, <b>24</b>, <b>34</b>, <b>44</b>, etc.</td><td align='right'> 8</td><td align='right'> 28</td><td align='right'> 14</td><td align='right'> 58</td><td align='right'> 13</td><td align='right'> 16</td><td align='right'> 14</td><td align='right'> 26</td><td align='right'> 19</td><td align='right'> 7<span class='pagenum'><a name="Page_129" id="Page_129">[Pg 129]</a></span></td></tr> +<tr><td align='right'><b>15, 25, 35, 45,</b> etc.</td><td align='right'> 21</td><td align='right'> 28</td><td align='right'> 29</td><td align='right'> 54</td><td align='right'> 51</td><td align='right'> 15</td><td align='right'> 21</td><td align='right'> 12</td><td align='right'> 24</td><td align='right'> 8</td></tr> +<tr><td align='right'><b>16, 26, 36, 46,</b> etc.</td><td align='right'> 5</td><td align='right'> 18</td><td align='right'> 12</td><td align='right'> 27</td><td align='right'> 35</td><td align='right'> 69</td><td align='right'> 13</td><td align='right'> 17</td><td align='right'> 19</td><td align='right'> 2</td></tr> +<tr><td align='right'><b>17, 27, 37, 47,</b> etc.</td><td align='right'> 5</td><td align='right'> 9</td><td align='right'> 12</td><td align='right'> 40</td><td align='right'> 32</td><td align='right'> 54</td><td align='right'> 24</td><td align='right'> 12</td><td align='right'> 12</td><td align='right'> 1</td></tr> +<tr><td align='right'><b>18, 28, 38, 48,</b> etc.</td><td align='right'> 2</td><td align='right'> 16</td><td align='right'> 10</td><td align='right'> 23</td><td align='right'> 22</td><td align='right'> 36</td><td align='right'> 18</td><td align='right'> 47</td><td align='right'> 16</td><td align='right'> 0</td></tr> +<tr><td align='left'><b>19, 29, 39,</b> etc.</td><td align='right'> 5</td><td align='right'> 7</td><td align='right'> 7</td><td align='right'> 10</td><td align='right'> 13</td><td align='right'> 28</td><td align='right'> 14</td><td align='right'> 23</td><td align='right'> 16</td><td align='right'> 0</td></tr> +<tr><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='left'>Totals</td><td align='right'>153</td><td align='right'>286</td><td align='right'>134</td><td align='right'>323</td><td align='right'>234</td><td align='right'>329</td><td align='right'>160</td><td align='right'>262</td><td align='right'>186</td><td align='right'>108</td></tr> +</table></div> + +<p> </p> +<p class="tabcap">TABLE 6</p> + +<p class="center"><span class="smcap">Amount of Practice: Multiplication Bonds in Another Recent +Textbook (B) of Excellent Repute. Books I and II</span></p> + + +<div class='center'> +<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary=""> +<tr><th class="bbt" rowspan='2'><span class="smcap">Multipliers</span></th><th class="bbt" colspan='11'><span class="smcap">Multiplicands</span></th></tr> +<tr><th class="bbt"> <b>0</b> </th><th class="bbt"> <b>1</b> </th><th class="bbt"> <b>2</b> </th><th class="bbt"> <b>3</b> </th><th class="bbt"> <b>4</b> </th><th class="bbt"> <b>5</b> </th><th class="bbt"> <b>6</b> </th><th class="bbt"> <b>7</b> </th><th class="bbt"> <b>8</b> </th><th class="bbt"> <b>9</b></th><th class="bbt">Totals</th></tr> +<tr><td align='center'><b>1</b></td><td align='right'> 299</td><td align='right'> 534</td><td align='right'> 472</td><td align='right'> 271</td><td align='right'> 310</td><td align='right'> 293</td><td align='right'> 261</td><td align='right'> 178</td><td align='right'> 195</td><td align='right'> 99</td><td align='right'> 2912</td></tr> +<tr><td align='center'><b>2</b></td><td align='right'> 350</td><td align='right'> 644</td><td align='right'> 668</td><td align='right'> 480</td><td align='right'> 458</td><td align='right'> 377</td><td align='right'> 332</td><td align='right'> 238</td><td align='right'> 239</td><td align='right'> 155</td><td align='right'> 3941</td></tr> +<tr><td align='center'><b>3</b></td><td align='right'> 280</td><td align='right'> 487</td><td align='right'> 509</td><td align='right'> 388</td><td align='right'> 318</td><td align='right'> 302</td><td align='right'> 247</td><td align='right'> 199</td><td align='right'> 227</td><td align='right'> 152</td><td align='right'> 3109</td></tr> +<tr><td align='center'><b>4</b></td><td align='right'> 186</td><td align='right'> 375</td><td align='right'> 398</td><td align='right'> 242</td><td align='right'> 203</td><td align='right'> 265</td><td align='right'> 197</td><td align='right'> 163</td><td align='right'> 159</td><td align='right'> 93</td><td align='right'> 2281</td></tr> +<tr><td align='center'><b>5</b></td><td align='right'> 268</td><td align='right'> 359</td><td align='right'> 393</td><td align='right'> 234</td><td align='right'> 263</td><td align='right'> 243</td><td align='right'> 217</td><td align='right'> 192</td><td align='right'> 197</td><td align='right'> 114</td><td align='right'> 2480</td></tr> +<tr><td align='center'><b>6</b></td><td align='right'> 180</td><td align='right'> 284</td><td align='right'> 265</td><td align='right'> 199</td><td align='right'> 196</td><td align='right'> 191</td><td align='right'> 168</td><td align='right'> 169</td><td align='right'> 165</td><td align='right'> 106</td><td align='right'> 1923</td></tr> +<tr><td align='center'><b>7</b></td><td align='right'> 135</td><td align='right'> 283</td><td align='right'> 277</td><td align='right'> 176</td><td align='right'> 187</td><td align='right'> 158</td><td align='right'> 155</td><td align='right'> 121</td><td align='right'> 145</td><td align='right'> 118</td><td align='right'> 1755</td></tr> +<tr><td align='center'><b>8</b></td><td align='right'> 137</td><td align='right'> 272</td><td align='right'> 292</td><td align='right'> 175</td><td align='right'> 192</td><td align='right'> 164</td><td align='right'> 158</td><td align='right'> 157</td><td align='right'> 126</td><td align='right'> 126</td><td align='right'> 1799</td></tr> +<tr><td align='center'><b>9</b></td><td align='right'> 71</td><td align='right'> 173</td><td align='right'> 140</td><td align='right'> 122</td><td align='right'> 97</td><td align='right'> 102</td><td align='right'> 101</td><td align='right'> 100</td><td align='right'> 82</td><td align='right'> 110</td><td align='right'> 1098</td></tr> +<tr><td align='center'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td><td align='left'> </td></tr> +<tr><td align='center'>Totals</td><td align='right'>1906</td><td align='right'>3411</td><td align='right'>3414</td><td align='right'>2287</td><td align='right'>2224</td><td align='right'>2095</td><td align='right'>1836</td><td align='right'>1517</td><td align='right'>1535</td><td align='right'>1073</td><td align='left'> </td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_130" id="Page_130">[Pg 130]</a></span></p> + +<p> </p> +<p class="tabcap">TABLE 7</p> +<p class="center"><span class="smcap">Amount Of Practice: Divisions Without Remainder In Textbook B, +Parts I And II</span></p> + +<div class='center'> +<table border="1" rules="cols" cellpadding="2" cellspacing="0" summary=""> +<tr><th class="bbt" rowspan='2'><span class="smcap">Dividends</span></th><th class="bbt" colspan='11'><span class="smcap">Divisors</span></th></tr> +<tr><th class="bbt"> <b>2</b> </th><th class="bbt"> <b>3</b> </th><th class="bbt"> <b>4</b> </th><th class="bbt"> <b>5</b> </th><th class="bbt"> <b>6</b> </th><th class="bbt"> <b>7</b> </th><th class="bbt"> <b>8</b> </th><th class="bbt"> <b>9</b></th><th class="bbt">Totals</th></tr> +<tr><td align='left' rowspan='6'>Integral multiples of <b>2</b> to <b>9</b><br /> +in sequence; <i>i.e.</i>,<br /> +<b>4 ÷ 2</b> occurred <b>397</b> times,<br /> +<b>6 ÷ 2</b> occurred <b>256</b> times,<br /> +<b>6 ÷ 3, 224</b> times,<br /> +<b>9 ÷ 3, 124</b> times.</td> +<td align='right'> 397</td><td align='right'> 224</td><td align='right'> 250</td><td align='right'> 130</td><td align='right'> 93</td><td align='right'> 44</td><td align='right'> 98</td><td align='right'> 23</td><td align='right'> 1259</td></tr> +<tr><td align='right'> 256</td><td align='right'> 124</td><td align='right'> 152</td><td align='right'> 79</td><td align='right'> 28</td><td align='right'> 43</td><td align='right'> 61</td><td align='right'> 25</td><td align='right'> 768</td></tr> +<tr><td align='right'> 318</td><td align='right'> 123</td><td align='right'> 130</td><td align='right'> 65</td><td align='right'> 50</td><td align='right'> 19</td><td align='right'> 39</td><td align='right'> 19</td><td align='right'> 763</td></tr> +<tr><td align='right'> 258</td><td align='right'> 98</td><td align='right'> 86</td><td align='right'> 105</td><td align='right'> 25</td><td align='right'> 24</td><td align='right'> 34</td><td align='right'> 20</td><td align='right'> 650</td></tr> +<tr><td align='right'> 198</td><td align='right'> 49</td><td align='right'> 76</td><td align='right'> 27</td><td align='right'> 22</td><td align='right'> 30</td><td align='right'> 33</td><td align='right'> 16</td><td align='right'> 451</td></tr> +<tr><td align='right'>77<br />180<br />69</td><td align='right'> 54<br />91<br />46</td><td align='right'> 36<br />50<br />37</td><td align='right'> 31<br />38<br />24</td><td align='right'> 28<br />17<br />12</td><td align='right'> 27<br />13<br />17</td><td align='right'> 16<br />22<br />16</td><td align='right'> 9<br />16<br />15</td><td align='right'> 278<br />427<br />236</td></tr> +<tr><td align='left'>Totals</td><td align='right'>1753</td><td align='right'> 809</td><td align='right'> 817</td><td align='right'> 499</td><td align='right'> 275</td><td align='right'> 217</td><td align='right'> 319</td><td align='right'> 142</td><td align='right'> </td></tr> +</table></div> + +<p> </p> +<p class="tabcap">TABLE 8</p> +<p class="center"><span class="smcap">Division Bonds, With And Without Remainders. Book B</span></p> + + + +<div class='pblockquot'> +<p>All work through grade 6, except estimates of quotient figures in long division.</p> + +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'>Dividend</td><td align='right'>2</td><td align='right' colspan='2'>3</td><td align='right' colspan='3'>4</td><td align='right' colspan='4'>5</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>1</td><td align='right'>2</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>41</td><td align='right'>386</td><td align='right'>27</td><td align='right'>189</td><td align='right'>240</td><td align='right'>26</td><td align='right'>397</td><td align='right'>66</td><td align='right'>185</td><td align='right'>23</td><td align='right'>136</td><td align='right'>43</td><td align='right'>53</td><td align='right'>135</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>6</td><td align='right' colspan='6'>7</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>21</td><td align='right'>256</td><td align='right'>224</td><td align='right'>68</td><td align='right'>43</td><td align='right'>83</td><td align='right'>23</td><td align='right'>72</td><td align='right'>55</td><td align='right'>38</td><td align='right'>46</td><td align='right'>32</td><td align='right'>54</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>8</td><td align='right' colspan='8'>9</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>17</td><td align='right'>318</td><td align='right'>30</td><td align='right'>250</td><td align='right'>22</td><td align='right'>28</td><td align='right'>39</td><td align='right'>91</td><td align='right'>19</td><td align='right'>50</td><td align='right'>124</td><td align='right'>49</td><td align='right'>25</td><td align='right'>15</td><td align='right'>18</td><td align='right'>30</td><td align='right'>38</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>10</td><td align='right' colspan='8'>11</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>258</td><td align='right'>38</td><td align='right'>46</td><td align='right'>120</td><td align='right'>19</td><td align='right'>9</td><td align='right'>24</td><td align='right'>24</td><td align='right'>32</td><td align='right'>21</td><td align='right'>16</td><td align='right'>3</td><td align='right'>7</td><td align='right'>11</td><td align='right'>14</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>12</td><td align='right' colspan='8'>13</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>198</td><td align='right'>123</td><td align='right'>152</td><td align='right'>29</td><td align='right'>93</td><td align='right'>9</td><td align='right'>16</td><td align='right'>7</td><td align='right'>45</td><td align='right'>16</td><td align='right'>15</td><td align='right'>11</td><td align='right'>7</td><td align='right'>4</td><td align='right'>5</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>14</td><td align='right' colspan='8'>15</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>77</td><td align='right'>20</td><td align='right'>13</td><td align='right'>5</td><td align='right'>8</td><td align='right'>44</td><td align='right'>8</td><td align='right'>6</td><td align='right'>69</td><td align='right'>98</td><td align='right'>16</td><td align='right'>79</td><td align='right'>8</td><td align='right'>8</td><td align='right'>4</td><td align='right'>6</td></tr> +<tr><td align='right'><span class='pagenum'><a name="Page_131" id="Page_131">[Pg 131]</a></span> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>16</td><td align='right' colspan='8'>17</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>180</td><td align='right'>19</td><td align='right'>130</td><td align='right'>14</td><td align='right'>6</td><td align='right'>9</td><td align='right'>98</td><td align='right'>3</td><td align='right'>61</td><td align='right'>9</td><td align='right'>15</td><td align='right'>14</td><td align='right'>6</td><td align='right'>6</td><td align='right'>12</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>18</td><td align='right' colspan='8'>19</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>2</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>69</td><td align='right'>49</td><td align='right'>13</td><td align='right'>6</td><td align='right'>28</td><td align='right'>7</td><td align='right'>7</td><td align='right'>23</td><td align='right'>21</td><td align='right'>6</td><td align='right'>10</td><td align='right'>5</td><td align='right'>3</td><td align='right'>4</td><td align='right'>10</td><td align='right'>4</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>20</td><td align='right' colspan='7'>21</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>24</td><td align='right'>86</td><td align='right'>65</td><td align='right'>11</td><td align='right'>3</td><td align='right'>23</td><td align='right'>5</td><td align='right'>54</td><td align='right'>12</td><td align='right'>8</td><td align='right'>5</td><td align='right'>43</td><td align='right'>10</td><td align='right'>5</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>22</td><td align='right' colspan='7'>23</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>17</td><td align='right'>16</td><td align='right'>15</td><td align='right'>8</td><td align='right'>13</td><td align='right'>6</td><td align='right'>15</td><td align='right'>7</td><td align='right'>8</td><td align='right'>11</td><td align='right'>8</td><td align='right'>6</td><td align='right'>3</td><td align='right'>2</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>24</td><td align='right' colspan='7'>25</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>91</td><td align='right'>76</td><td align='right'>18</td><td align='right'>50</td><td align='right'>5</td><td align='right'>61</td><td align='right'>1</td><td align='right'>11</td><td align='right'>13</td><td align='right'>105</td><td align='right'>5</td><td align='right'>6</td><td align='right'>5</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>26</td><td align='right' colspan='7'>27</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>5</td><td align='right'>6</td><td align='right'>3</td><td align='right'>3</td><td align='right'>4</td><td align='right'>6</td><td align='right'>3</td><td align='right'>46</td><td align='right'>8</td><td align='right'>10</td><td align='right'>4</td><td align='right'>2</td><td align='right'>6</td><td align='right'>25</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>28</td><td align='right' colspan='7'>29</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>3</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>36</td><td align='right'>8</td><td align='right'>3</td><td align='right'>19</td><td align='right'>3</td><td align='right'>7</td><td align='right'>6</td><td align='right'>8</td><td align='right'>0</td><td align='right'>5</td><td align='right'>11</td><td align='right'>2</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>30</td><td align='right' colspan='6'>31</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>21</td><td align='right'>27</td><td align='right'>25</td><td align='right'>6</td><td align='right'>7</td><td align='right'>13</td><td align='right'>4</td><td align='right'>3</td><td align='right'>1</td><td align='right'>1</td><td align='right'>4</td><td align='right'>2</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>32</td><td align='right' colspan='6'>33</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>50</td><td align='right'>11</td><td align='right'>3</td><td align='right'>6</td><td align='right'>39</td><td align='right'>5</td><td align='right'>8</td><td align='right'>7</td><td align='right'>7</td><td align='right'>2</td><td align='right'>6</td><td align='right'>1</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>34</td><td align='right' colspan='6'>35</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>8</td><td align='right'>3</td><td align='right'>5</td><td align='right'>2</td><td align='right'>1</td><td align='right'>1</td><td align='right'>10</td><td align='right'>31</td><td align='right'>5</td><td align='right'>24</td><td align='right'>5</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>36</td><td align='right' colspan='6'>37</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>37</td><td align='right'>16</td><td align='right'>22</td><td align='right'>2</td><td align='right'>6</td><td align='right'>19</td><td align='right'>12</td><td align='right'>8</td><td align='right'>7</td><td align='right'>5</td><td align='right'>3</td><td align='right'>9</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>38</td><td align='right' colspan='6'>39</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>4</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>7</td><td align='right'>8</td><td align='right'>7</td><td align='right'>1</td><td align='right'>1</td><td align='right'>5</td><td align='right'>4</td><td align='right'>3</td><td align='right'>7</td><td align='right'>4</td><td align='right'>3</td><td align='right'>1</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>40</td><td align='right' colspan='5'>41</td><td align='right' colspan='5'>42</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>38</td><td align='right'>9</td><td align='right'>2</td><td align='right'>34</td><td align='right'>2</td><td align='right'>6</td><td align='right'>6</td><td align='right'>3</td><td align='right'>7</td><td align='right'>5</td><td align='right'>7</td><td align='right'>28</td><td align='right'>30</td><td align='right'>10</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>43</td><td align='right' colspan='5'>44</td><td align='right' colspan='5'>45</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>7</td><td align='right'>5</td><td align='right'>10</td><td align='right'>3</td><td align='right'>2</td><td align='right'>7</td><td align='right'>6</td><td align='right'>4</td><td align='right'>5</td><td align='right'>0</td><td align='right'>24</td><td align='right'>6</td><td align='right'>7</td><td align='right'>10</td><td align='right'>20</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>46</td><td align='right' colspan='5'>47</td><td align='right' colspan='5'>48</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>3</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>2</td><td align='right'>6</td><td align='right'>2</td><td align='right'>2</td><td align='right'>0</td><td align='right'>3</td><td align='right'>7</td><td align='right'>17</td><td align='right'>4</td><td align='right'>33</td><td align='right'>2</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>49</td><td align='right' colspan='5'>50</td><td align='right' colspan='4'>51</td><td align='right' colspan='4'>52</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>5</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>7</td><td align='right'>27</td><td align='right'>9</td><td align='right'>2</td><td align='right'>4</td><td align='right'>6</td><td align='right'>3</td><td align='right'>8</td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>5</td><td align='right'>5</td><td align='right'>5</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>53</td><td align='right' colspan='4'>54</td><td align='right' colspan='4'>55</td><td align='right' colspan='4'>56</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>12</td><td align='right'>5</td><td align='right'>1</td><td align='right'>16</td><td align='right'>5</td><td align='right'>3</td><td align='right'>4</td><td align='right'>2</td><td align='right'>0</td><td align='right'>13</td><td align='right'>16</td><td align='right'>8</td></tr> +<tr><td align='right'> <span class='pagenum'><a name="Page_132" id="Page_132">[Pg 132]</a></span></td></tr> +<tr><td align='left'>Dividend</td><td align='right'>57</td><td align='right' colspan='4'>58</td><td align='right' colspan='4'>59</td><td align='right' colspan='4'>60</td><td align='right' colspan='3'>61</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>6</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>0</td><td align='right'>3</td><td align='right'>1</td><td align='right'>3</td><td align='right'>2</td><td align='right'>2</td><td align='right'>3</td><td align='right'>1</td><td align='right'>2</td><td align='right'>3</td><td align='right'>0</td><td align='right'>3</td><td align='right'>3</td><td align='right'>9</td><td align='right'>1</td><td align='right'>1</td><td align='right'>2</td><td align='right'>5</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>62</td><td align='right' colspan='3'>63</td><td align='right' colspan='3'>64</td><td align='right' colspan='3'>65</td><td align='right' colspan='3'>66</td><td align='right' colspan='3'>67</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>4</td><td align='right'>6</td><td align='right'>1</td><td align='right'>17</td><td align='right'>5</td><td align='right'>9</td><td align='right'>5</td><td align='right'>22</td><td align='right'>0</td><td align='right'>1</td><td align='right'>10</td><td align='right'>1</td><td align='right'>2</td><td align='right'>1</td><td align='right'>4</td><td align='right'>0</td><td align='right'>1</td><td align='right'>1</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>68</td><td align='right' colspan='3'>69</td><td align='right' colspan='3'>70</td><td align='right' colspan='2'>71</td><td align='right' colspan='2'>72</td><td align='right' colspan='2'>73</td><td align='right' colspan='2'>74</td><td align='right' colspan='2'>75</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>7</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>1</td><td align='right'>3</td><td align='right'>2</td><td align='right'>0</td><td align='right'>6</td><td align='right'>1</td><td align='right'>6</td><td align='right'>2</td><td align='right'>1</td><td align='right'>0</td><td align='right'>16</td><td align='right'>10</td><td align='right'>7</td><td align='right'>5</td><td align='right'>3</td><td align='right'>3</td><td align='right'>5</td><td align='right'>3</td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='left'>Dividend</td><td align='right'>76</td><td align='right' colspan='2'>77</td><td align='right' colspan='2'>78</td><td align='right' colspan='2'>79</td><td align='right' colspan='2'>80</td><td align='right'>81</td><td align='right'>82</td><td align='right'>83</td><td align='right'>84</td><td align='right'>85</td><td align='right'>86</td><td align='right'>87</td><td align='right'>88</td><td align='right'>89</td></tr> +<tr><td align='left'>Divisor</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>8</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td><td align='right'>9</td></tr> +<tr><td align='left'>Number of<br />Occurrences</td><td align='right'>3</td><td align='right'>2</td><td align='right'>3</td><td align='right'>0</td><td align='right'>4</td><td align='right'>1</td><td align='right'>0</td><td align='right'>2</td><td align='right'>4</td><td align='right'>15</td><td align='right'>2</td><td align='right'>4</td><td align='right'>1</td><td align='right'>2</td><td align='right'>0</td><td align='right'>3</td><td align='right'>2</td><td align='right'>7</td></tr> +</table></div> + +<p>Tables 3 to 8 show that even gifted authors make instruments +for instruction in arithmetic which contain much less +practice on certain elementary facts than teachers suppose; +and which contain relatively much more practice on the +more easily learned facts than on those which are harder +to learn.</p> + +<p>How much practice should be given in arithmetic? How +should it be divided among the different bonds to be +formed? Below a certain amount there is waste because, +as has been shown in Chapter VI, the pupil will need more +time to detect and correct his errors than would have been +required to give him mastery. Above a certain amount +there is waste because of unproductive overlearning. If +668 is just enough for 2 × 2, 82 is not enough for 9 × 8. If +82 is just enough for 9 × 8, 668 is too much for 2 × 2.</p> + +<p>It is possible to find the answers to these questions for +the pupil of median ability (or any stated ability) by suitable +experiments. The amount of practice will, of course, +<span class='pagenum'><a name="Page_133" id="Page_133">[Pg 133]</a></span> +vary according to the ability of the pupil. It will also vary +according to the interest aroused in him and the satisfaction +he feels in progress and mastery. It will also vary according +to the amount of practice of other related bonds; 7 + 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.</p> + +<p>Until suitable experiments are at hand we may estimate +for the fundamental bonds as follows, assuming that by +the end of grade 6 a strength of 199 correct out of 200 is +to be had, and that the teaching is by an intelligent person +working in accord with psychological principles as to both +ability and interest.</p> + +<p>For one of the easier bonds, most facilitated by other +bonds (such as 2 × 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.</p> + +<p>For bonds of ordinary difficulty, with average facilitation +from other bonds (such as 11 − 3, 4 × 7, or 48 ÷ 8 = 6) in the +case of the median or average pupil, we may estimate twenty +practices in the week of first learning, supported by thirty, +and maintained by fifty practices well spread over the later<span class='pagenum'><a name="Page_134" id="Page_134">[Pg 134]</a></span> +periods. Gifted pupils may gain and keep mastery with +twelve, fifteen, and twenty practices respectively. Pupils dull +at arithmetic may need up to twenty, sixty, and two hundred. +Here, again, it is to be doubted whether a pupil for whom +arithmetical facts, well taught and made interesting, are +so hard to acquire as this, should learn many of them.</p> + +<p>For bonds of greater difficulty, less facilitated by other +bonds (such as 17 − 9, 8 × 7, or 12½% of = <sup>1</sup>⁄<sub>8</sub> of), the practice +may be from ten to a hundred percent more than the above.</p> + + +<h4>UNDERLEARNING AND OVERLEARNING</h4> + +<p>If we accept the above provisional estimates as reasonable, +we may consider the harm done by giving less and by giving +more than these reasonable amounts. Giving less is indefensible. +The pupil's time is wasted in excessive checking +to find his errors. He is in danger of being practiced in error. +His attention is diverted from the learning of new facts +and processes by the necessity of thinking out these supposedly +mastered facts. All new bonds are harder to learn +than they should be because the bonds which should facilitate +them are not strong enough to do so. Giving more does +harm to some extent by using up time that could be spent +better for other purposes, and (though not necessarily) by +detracting from the pupil's interest in arithmetic. In +certain cases, however, such excess practice and overlearning +are actually desirable. Three cases are of special importance.</p> + +<p>The first is the case of a bond operating under a changed +mental set or adjustment. A pupil may know 7 × 8 adequately +as a thing by itself, but need more practice to operate +it in</p> + +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr> +<td align='right'>285<br /> +7<br /> +——</td> +</tr> +</table> + +<p class="noidt">where he has to remember that 3 is to be added to the<span class='pagenum'><a name="Page_135" id="Page_135">[Pg 135]</a></span> +56 when he obtains it, and that only the 9 is to be written +down, the 5 to be held in mind for later use. The practice +required to operate the bond efficiently in this new set is +desirable, even though it is excess from a narrower point of +view, and causes the straightforward 'seven eights are fifty-six' +to be overlearned. So also a pupil's work with 24, +34, 44, etc., + 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 <sup>5</sup>⁄<sub>12</sub> + <sup>7</sup>⁄<sub>12</sub> is +not quite the same task as adding 5 and 7 undisturbed by +the fact that they are twelfths. We know far too little +about the amount of practice needed to adapt arithmetical +bonds to efficient operation in these more complicated conditions +to estimate even approximately the allowances to +be made. But some allowance, and often a rather large +allowance, must be made.</p> + +<p>The second is the case where the computation in general +should be made very easy and sure for the pupil except for +some one new element that is being learned. For example, +in teaching the meaning and uses of 'Averages' and of +uneven division, we may deliberately use 2, 3, and 4 as divisors +rather than 7 and 9, so as to let all the pupil's energy be +spent in learning the new facts, and so that the fraction in +the quotient may be something easily understood, real, and +significant. In teaching the addition of mixed numbers, +we may use, in the early steps,</p> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr> +<td align='left'>11½<br /> +13½<br /> +24<br /> +——</td> +<td align='center'>rather than</td> +<td align='left'>79½<br /> +98½<br /> +67<br /> +——</td> +</tr> +</table></div> + +<p class="noidt">so as to save attention for the new process itself. In cancellation, +we may give excess practice to divisions by 2, 3, 4, +and 5 in order to make the transfer to the new habits of con<span class='pagenum'><a name="Page_136" id="Page_136">[Pg 136]</a></span>sidering +two numbers together from the point of view of their +divisibility by some number. In introducing trade discount, +we may give excess practice on '5% of' and '10% of' +deliberately, so that the meaning of discount may not be +obscured by difficulties in the computation itself. Excess +practice on, and overlearning of, certain bonds is thus very +often justifiable.</p> + +<p>The third case concerns bonds whose importance for +practical uses in life or as notable facilitators of other bonds +is so great that they may profitably be brought to a greater +strength than 199 correct out of 200 at a speed of 2 sec. or less, +or be brought to that degree of strength very early. Examples +of bonds of such special practical use are the subtractions +from 10, ½ + ½, ½ + ¼, ½ of 60, ¼ of 60, and the fractional parts +of 12 and of $1.00. Examples of notable facilitating bonds +are ten 10s = 100, ten 100s = 1000, additions like 2 + 2, +3 + 3, and 4 + 4, and all the multiplication tables to 9 × 9.</p> + +<p>In consideration of these three modifying cases or principles, +a volume could well be written concerning just how +much practice to give to each bond, in each of the types of +complex situations where it has to operate. There is evidently +need for much experimentation to expose the facts, +and for much sagacity and inventiveness in making sure of +effective learning without wasteful overlearning.</p> + +<p>The facts of primary importance are:—</p> + +<p class="nnblockquot">(1) The textbook or other instrument of instruction which +is a teacher's general guide may give far too little +practice on certain bonds.</p> + +<p class="nnblockquot">(2) It may divide the practice given in ways that are +apparently unjustifiable.</p> + +<p class="nnblockquot">(3) The teacher needs therefore to know how much practice +it does give, where to supplement it, and what to +omit.</p> + +<p><span class='pagenum'><a name="Page_137" id="Page_137">[Pg 137]</a></span></p> + +<p class="nnblockquot">(4) The omissions, on grounds of apparent excess practice, +should be made only after careful consideration +of the third principle described above.</p> + +<p class="nnblockquot">(5) The amount of practice should always be considered +in the light of its interest and appeal to the pupil's +tendency to work with full power and zeal. Mere +repetition of bonds when the learner does not care +whether he is improving is rarely justifiable on any +grounds.</p> + +<p class="nnblockquot">(6) Practice that is actually in excess is not a very grave +defect if it is enjoyed and improves the pupil's attitude +toward arithmetic. Not much time is lost; a +hundred practices for each of a thousand bonds after +mastery to 199 in 200 at 2 seconds will use up less than +60 hours, or 15 hours per year in grades 3 to 6.</p> + +<p class="nnblockquot">(7) By the proper division of practice among bonds, +the arrangement of learning so that each bond helps +the others, the adroit shifting of practice of a bond +to each new type of situation requiring it to operate +under changed conditions, and the elimination of +excess practice where nothing substantial is gained, +notable improvements over the past hit-and-miss +customs may be expected.</p> + +<p class="nnblockquot">(8) Unless the material for practice is adequate, well +balanced and sufficiently motivated, the teacher must +keep close account of the learning of pupils. Otherwise +disastrous underlearning of many bonds is +almost sure to occur and retard the pupil's development.</p> + + +<h4>THE ORGANIZATION OF ABILITIES</h4> + +<p>There is danger that the need of brevity and simplicity +which has made us speak so often of a bond or an ability,<span class='pagenum'><a name="Page_138" id="Page_138">[Pg 138]</a></span> +and of the amount of practice it requires, may mislead the +reader into thinking that these bonds and abilities are to be +formed each by itself alone and kept so. They should rarely +be formed so and never kept so. This we have indicated from +time to time by references to the importance of forming a +bond in the way in which it is to be used, to the action of +bonds in changed situations, to facilitation of one bond by +others, to the coöperation of abilities, and to their integration +into a total arithmetical ability.</p> + +<p>As a matter of fact, only a small part of drill work in +arithmetic should be the formation of isolated bonds. Even +the very young pupil learning 5 and 3 are 8 should learn it +with '5 and 5 = 10,' '5 and 2 = 7,' at the back of his mind, +so to speak. Even so early, 5 + 3 = 8 should be part of an +organized, coöperating 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.</p> + +<p>There are differences of course. Knowledge of square +root can be regarded somewhat as a separate tool to be +sharpened, polished, and used by itself, whereas knowledge +of the multiplication tables cannot. Yet even square root +is probably best made more closely a part of the total ability, +being taught as a special case of dividing where divisor is +to be the same as quotient, the process being one of estimating +and correcting.</p> + +<p>In general we do not wish the pupil to be a repository +of separated abilities, each of which may operate only if you +ask him the sort of questions which the teacher used to ask +him, or otherwise indicate to him which particular arithmetical +tool he is to use. Rather he is to be an effective +organization of abilities, coöperating in useful ways to meet<span class='pagenum'><a name="Page_139" id="Page_139">[Pg 139]</a></span> +the quantitative problems life offers. He should not as a +rule have to think in such fashion as: "Is this interest or +discount? Is it simple interest or compound interest? +What did I do in compound interest? How do I multiply +by 2 percent?" The situation that calls up interest should +also call up the kind of interest that is appropriate, and the +technique of operating with percents should be so welded +together with interest in his mind that the right coöperation +will occur almost without supervision by him.</p> + +<p>As each new ability is acquired, then, we seek to have it +take its place as an improvement of a thinking being, as a +coöperative member of a total organization, as a soldier +fighting together with others, as an element in an educated +personality. Such an organization of bonds will not form +itself any more than any one bond will create itself. If the +elements of arithmetical ability are to act together as a +total organized unified force they must be made to act together +in the course of learning. What we wish to have +work together we must put together and give practice in +teamwork.</p> + +<p>We can do much to secure such coöperative action when +and where and as it is needed by a very simple expedient; +namely, to give practice with computation and problems +such as life provides, instead of making up drills and problems +merely to apply each fact or principle by itself. Though a +pupil has solved scores of problems reading, "A triangle +has a base of <i>a</i> feet and an altitude of <i>b</i> feet, what is its +area?" he may still be practically helpless in finding the +area of a triangular plot of ground; still more helpless in +using the formula for a triangle which is one of two into which +a trapezoid is divided. Though a pupil has learned to solve +problems in trade discount, simple interest, compound +interest, and bank discount one at a time, stated in a few<span class='pagenum'><a name="Page_140" id="Page_140">[Pg 140]</a></span> +set forms, he may be practically helpless before the actual +series of problems confronting him in starting in business, +and may take money out of the savings bank when he ought +to borrow on a time loan, or delay payment on his bills when +by paying cash he could save money as well as improve his +standing with the wholesaler.</p> + +<p>Instead of making up problems to fit the abilities given +by school instruction, we should preferably modify school +instruction so that arithmetical abilities will be organized +into an effective total ability to meet the problems that life +will offer. Still more generally, <i>every bond formed should be +formed with due consideration of every other bond that has been +or will be formed; every ability should be practiced in the most +effective possible relations with other abilities</i>.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_141" id="Page_141">[Pg 141]</a></span></p> +<h2><a name="CHAPTER_VII" id="CHAPTER_VII"></a>CHAPTER VII</h2> + +<h3>THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION +OF BONDS</h3> + + +<p>The bonds to be formed having been chosen, the next +step is to arrange for their most economical order of formation—to +arrange to have each help the others as much as +possible—to arrange for the maximum of facilitation and +the minimum of inhibition.</p> + +<p>The principle is obvious enough and would probably be +admitted in theory by any intelligent teacher, but in practice +we are still wedded to conventional usages which arose +long before the psychology of arithmetic was studied. For +example, we inherit the convention of studying addition of +integers thoroughly, and then subtraction, and then multiplication, +and then division, and many of us follow it though +nobody has ever given a proof that this is the best order +for arithmetical learning. We inherit also the opposite +convention of studying in a so-called "spiral" plan, a little +addition, subtraction, multiplication, and division, and then +some more of each, and then some more, and many of us +follow this custom, with an unreasoned faith that changing +about from one process to another is <i>per se</i> helpful.</p> + +<p>Such conventions are very strong, illustrating our common +tendency to cherish most those customs which we cannot +justify! The reductions of denominate numbers ascending +and descending were, until recently, in most courses of study,<span class='pagenum'><a name="Page_142" id="Page_142">[Pg 142]</a></span> +kept until grade 4 or grade 5 was reached, although this material +is of far greater value for drills on the multiplication and +division tables than the customary problems about apples, +eggs, oranges, tablets, and penholders. By some historical +accident or for good reasons the general treatment of denominate +numbers was put late; by our naïve notions of +order and system we felt that any use of denominate numbers +before this time was heretical; we thus became blind to the +advantages of quarts and pints for the tables of 2s; yards +and feet for the tables of 3s; gallons and quarts for the tables +of 4s; nickels and cents for the 5s; weeks and days for the +7s; pecks and quarts for the 8s; and square yards and +square feet for the 9s. Problems like 5 yards = __ feet or +15 feet = __ yards have not only the advantages of brevity, +clearness, practical use, real reference, and ready variation, +but also the very great advantage that part of the data have +to be <i>thought of</i> in a useful way instead of <i>read off</i> from the +page. In life, when a person has twenty cents with which +to buy tablets of a certain sort, he <i>thinks of</i> the price +in making his purchase, asking it of the clerk only in +case he does not know it, and in planning his purchases +beforehand he <i>thinks of</i> prices as a rule. In spite of these +and other advantages, not one textbook in ten up to 1900 +made early use of these exercises with denominate numbers. +So strong is mere use and wont.</p> + +<p>Besides these conventional customs, there has been, in +those responsible for arithmetical instruction, an admiration +for an arrangement of topics that is easy for a person, after +he knows the subject, to use in thinking of its constituent +parts and their relations. Such arrangements are often +called 'logical' arrangements of subject matter, though +they are often far from logical in any useful sense. Now +the easiest order in which to think of a hierarchy of habits<span class='pagenum'><a name="Page_143" id="Page_143">[Pg 143]</a></span> +after you have formed them all may be an extremely difficult +order in which to form them. The criticism of other orders +as 'scrappy,' or 'unsystematic,' valid enough if the course +of study is thought of as an object of contemplation, may be +foolish if the course of study is regarded as a working instrument +for furthering arithmetical learning.</p> + +<p>We must remember that all our systematizing and labeling +is largely without meaning to the pupils. They cannot at +any point appreciate the system as a progression from that +point toward this and that, since they have no knowledge +of the 'this or that.' They do not as a rule think of their +work in grade 4 as an outcome of their work in grade +3 with extensions of <i>a</i> to <i>a</i><sub>1</sub>, and additions of <i>b</i><sub>2</sub> and <i>b</i><sub>3</sub> to +<i>b</i> and <i>b</i><sub>1</sub>, and refinements of <i>c</i> and <i>d</i> by <i>c</i><sub>4</sub> and <i>d</i><sub>5</sub>. They +could give only the vaguest account of what they did in +grade 3, much less of why it should have been done then. +They are not much disturbed by a lack of so-called 'system' +and 'logical' progression for the same reason that they +are not much helped by their presence. What they need +and can use is a <i>dynamically</i> effective system or order, one +that they can learn easily and retain long by, regardless of +how it would look in a museum of arithmetical systems. +Unless their actual arithmetical habits are usefully related +it does no good to see the so-called logical relations; and +if their habits are usefully related, it does not very much +matter whether or not they do see these; finally, they can be +brought to see them best by first acquiring the right habits +in a dynamically effective order.</p> + + +<h4>DECREASING INTERFERENCE AND INCREASING FACILITATION</h4> + +<p>Psychology offers no single, easy, royal road to discovering +this dynamically best order. It can only survey the +bonds, think what each demands as prerequisite and offers<span class='pagenum'><a name="Page_144" id="Page_144">[Pg 144]</a></span> +as future help, recommend certain orders for trial, and +measure the efficiency of each order as a means of attaining +the ends desired. The ingenious thought and careful experimentation +of many able workers will be required for +many years to come.</p> + +<p>Psychology can, however, even now, give solid constructive +help in many instances, either by recommending +orders that seem almost certainly better than those in +vogue, or by proposing orders for trial which can be justified +or rejected by crucial tests.</p> + +<p>Consider, for example, the situation, 'a column of one-place +numbers to be added, whose sum is over 9,' and the +response 'writing down the sum.' This bond is commonly +firmly fixed before addition with two-place numbers is +undertaken. As a result the pupil has fixed a habit that +he has to break when he learns two-place addition. If <i>oral</i> +answers only are given with such single columns until +two-place addition is well under way, the interference is +avoided.</p> + +<p>In many courses of study the order of systematic formation +of the multiplication table bonds is : 1 × 1, 2 × 1, etc., 1 × 2, +2 × 2, etc., 1 × 3, 2 × 3, etc., 1 × 9, 2 × 9, etc. This is probably +wrong in two respects. There is abundant reason to believe +that the × 5s should be learned first, since they are easier to +learn than the 1s or the 2s, and give the idea of multiplying +more emphatically and clearly. There is also abundant +reason to believe that the 1 × 5, 1 × 2, 1 × 3, etc., should +be put very late—after at least three or four tables are +learned, since the question "What is 1 times 2?" (or 3 or 5) +is unnecessary until we come to multiplication of two- and +three-place numbers, seems a foolish question until +then, and obscures the notion of multiplication if put +early. Also the facts are best learned once for all as the +<span class='pagenum'><a name="Page_145" id="Page_145">[Pg 145]</a></span> +habits "1 times <i>k</i> is the same as <i>k</i>," and "<i>k</i> times 1 is the +same as <i>k</i>."<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a></p> + +<p>In another connection it was recommended that the +divisions to 81 ÷ 9 be learned by selective thinking or reasoning +from the multiplications. This determines the order +of bonds so far as to place the formation of the division +bonds soon after the learning of the multiplications. For +other reasons it is well to make the proximity close.</p> + +<p>One of the arbitrary systematizations of the order of +formation of bonds restricts operations at first to the numbers +1 to 10, then to numbers under 100, then to numbers +under 1000, then to numbers under 10,000. Apart from the +avoidance of unreal and pedantic problems in applied +arithmetic to which work with large numbers in low grades +does somewhat predispose a teacher, there is little merit +in this restriction of the order of formation of bonds. Its +demerits are many. For example, when the pupil is learning +to 'carry' in addition he can be given better practice +by soon including tasks with sums above 100, and can get +a valuable sense of the general use of the process by being +given a few examples with three- and four-place numbers +to be added. The same holds for subtraction. Indeed, +there is something to be said in favor of using six- or seven-place +numbers in subtraction, enforcing the 'borrowing' +process by having it done again and again in the same example, +and putting it under control by having the decision +between 'borrowing' and 'not borrowing' made again +and again in the same example. When the multiplication +<span class='pagenum'><a name="Page_146" id="Page_146">[Pg 146]</a></span> +tables are learned the most important use for them is not +in tedious reviews or trivial problems with answers under +100, but in regular 'short' multiplication of two- and three- and +even four-place numbers. Just as the addition combinations +function mainly in the higher-decade modifications +of them, so the multiplication combinations function chiefly +in the cases where the bond has to operate while the added +tasks of keeping one's place, adding what has been carried, +writing down the right figure in the right place, and holding +the right number for later addition, are also taken care of. +It seems best to introduce such short multiplication as soon +as the × 5s, × 2s, × 3s, and × 4s are learned and to put the +× 6s, × 7s, and the rest to work in such short multiplication +as soon as each is learned.</p> + +<p>Still surer is the need for four-, five-, and six-place numbers +when two-place numbers are used in multiplying. When +the process with a two-place multiplier is learned, multiplications +by three-place numbers should soon follow. They +are not more difficult then than later. On the contrary, if +the pupil gets used to multiplying only as one does with +two-place multipliers, he will suffer more by the resulting +interference than he does from getting six- or seven-place +answers whose meaning he cannot exactly realize. They +teach the rationale and the manipulations of long multiplication +with especial economy because the principles and the +procedures are used two or three times over and the contrasts +between the values which the partial products have +in adding become three instead of one.</p> + +<p>The entire matter of long multiplication with integers +and United States money should be treated as a teaching +unit and the bonds formed in close organization, even though +numbers as large as 900,000 are occasionally involved. The +reason is not that it is more logical, or less scrappy, but +<span class='pagenum'><a name="Page_147" id="Page_147">[Pg 147]</a></span> +that each of the bonds in question thus gets much help from, +and gives much help to, the others.</p> + +<p>In sharp contrast to a topic like 'long multiplication' +stands a topic like denominate numbers. It most certainly +should not be treated as a large teaching unit, and all the +bonds involved in adding, subtracting, multiplying, and +dividing with all the ordinary sorts of measures should certainly +not be formed in close sequence. The reductions +ascending and descending for many of the measures should +be taught as drills on the appropriate multiplication and +division tables. The reduction of feet and inches to inches, +yards and feet to yards, gallons and quarts to quarts, and +the like are admirable exercises in connection with the +(<i>a</i> × <i>b</i>) + <i>c</i> = .... 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 <i>d</i> = (.... × <i>b</i>) + <i>c</i> or 'making change' +problem, which in its small-number forms is an excellent +preparatory step for short division. They are also of great +service in early work with fractions. The feet-mile, square-foot-square-inch, +and other simple relations give a genuine +and intelligible demand for multiplication with large numbers.</p> + +<p>Knowledge of the metric system for linear and square +measure would perhaps, as an introduction to decimal fractions, +more than save the time spent to learn it. It would +even perhaps be worth while to invent a measure (call it +the <i>twoqua</i>) midway between the quart and gallon and +teach carrying in addition and borrowing in subtraction +by teaching first the addition and subtraction of 'gallon, +twoqua, quart, and pint' series! Many of the bonds which +a system-made tradition huddled together uselessly in a +chapter on denominate numbers should thus be formed<span class='pagenum'><a name="Page_148" id="Page_148">[Pg 148]</a></span> +as helpful preparations for and applications of other bonds +all the way from the first to the eighth half-year of instruction +in arithmetic.</p> + +<p>The bonds involved in the ability to respond correctly to +the series:—</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='right'> 5 = .... 2s and .... remainder</td></tr> +<tr><td align='right'> 5 = .... 3s and .... remainder</td></tr> +<tr><td align='right'>88 = .... 9s and .... remainder</td></tr> +</table></div> + +<p class="noidt">should be formed before, not during, the training in short +division. They are admirable at that point as practice on +the division tables; are of practical service in the making-change +problems of the small purchase and the like; and +simplify the otherwise intricate task of keeping one's place, +choosing the quotient figure, multiplying by it, subtracting +and holding in mind the new number to be divided, which is +composed half of the remainder and half of a figure in the +written dividend. This change of order is a good illustration +of the nearly general rule that "<i>When the practice or review +required to perfect or hold certain bonds can, by an inexpensive +modification, be turned into a useful preparation for +new bonds, that modification should be made.</i>"</p> + +<p>The bonds involved in the four operations with United +States money should be formed in grades 3 and 4 +along with or very soon after the corresponding bonds with +three-place and four-place integers. This statement would +have seemed preposterous to the pedagogues of fifty years +ago. "United States money," they would have said, "is +an application of decimals. How can it be learned until the +essentials of decimal fractions are known? How will the +child understand when multiplying $.75 by 3 that 3 times +5 cents is 1 dime and 5 cents, or that 3 times 70 cents is +2 dollars and 1 dime? Why perplex the young pupils with +the difficulties of placing the decimal point? Why disturb +<span class='pagenum'><a name="Page_149" id="Page_149">[Pg 149]</a></span> +the learning of the four operations with integers by adding +at each step a second 'procedure with United States +money'?"</p> + +<p>The case illustrates very well the error of the older oversystematic +treatment of the order of topics and the still +more important error of confusing the logic of proof with +the psychology of learning. To prove that 3 × $.75 = $2.25 +to the satisfaction of certain arithmeticians, you may need +to know the theory of decimal fractions; but to do such +multiplication all a child needs is to do just what he has +been doing with integers and then "Put a $ before the answer +to show that it means dollars and cents, and put a decimal +point in the answer to show which figures mean dollars and +which figures mean cents." And this is general. The +ability to operate with integers plus the two habits of prefixing +$ and separating dollars from cents in the result will +enable him to operate with United States money.</p> + +<p>Consequently good practice came to use United States +money not as a consequence of decimal fractions, learned +by their aid, but as an introduction to decimal fractions +which aids the pupil to learn them. So it has gradually +pushed work with United States money further and further +back, though somewhat timidly.</p> + +<p>We need not be timid. The pupil will have no difficulty +in adding, subtracting, multiplying, and dividing with +United States money—unless we create it by our explanations! +If we simply form the two bonds described above +and show by proper verification that the procedure always +gives the right answer, the early teaching of the four operations +with United States money will in fact actually show a +learning profit! It will save more time in the work with +integers than was spent in teaching it! For, in the first +place, it will help to make work with four-place and five<span class='pagenum'><a name="Page_150" id="Page_150">[Pg 150]</a></span>-place +numbers more intelligible and vital. A pupil can +understand $16.75 or $28.79 more easily than 1675 or 2879. +The former may be the prices of a suit or sewing machine or +bicycle. In the second place, it permits the use of a large +stock of genuine problems about spending, saving, sharing, +and the like with advertisements and catalogues and school +enterprises. In the third place, it permits the use of common-sense +checks. A boy may find one fourth of 3000 as +7050 or 75 and not be disturbed, but he will much more +easily realize that one fourth of $30.00 is not over $70 or +less than $1. Even the decimal point of which we used to +be so afraid may actually help the eye to keep its place in +adding.</p> + + +<h4>INTEREST</h4> + +<p>So far, the illustrations of improvements in the order of +bonds so as to get less interference and more facilitation +than the customary orders secure have sought chiefly to +improve the mechanical organization of the bonds. Any +gain in interest which the changes described effected would +be largely due to the greater achievement itself. Dewey +and others have emphasized a very different principle of +improving the order of formation of bonds—the principle +of determination of the bonds to be formed by some vital, +engaging problem which arouses interest enough to lighten +the labor and which goes beyond or even against cut-and-dried +plans for sequences in order to get effective problems. +For example, the work of the first month in grade 2B might +sacrifice facilitations of the mechanical sort in order to put +arithmetic to use in deciding what dimensions a rabbit's +cage should have to give him 12 square feet of floor space, +how much bread he should have per meal to get 6 ounces +a day, how long a ten-cent loaf would last, how many loaves<span class='pagenum'><a name="Page_151" id="Page_151">[Pg 151]</a></span> +should be bought per week, how much it costs to feed the +rabbit, how much he has gained in weight since he was +brought to the school, and so on.</p> + +<p>Such sacrifices of the optimal order if interest were equal, +in order to get greater interest or a healthier interest, are +justifiable. Vital problems as nuclei around which to organize +arithmetical learning are of prime importance. It +is even safe probably to insist that some genuine problem-situation +requiring a new process, such as addition with +carrying, multiplication by two-place numbers, or division +with decimals, be provided in every case as a part of the +introduction to that process. The sacrifice should not +be too great, however; the search for vital problems that +fit an economical order of subject matter is as much needed +as the amendment of that order to fit known interests; +and the assurance that a problem helps the pupil to learn +arithmetic is as important as the assurance that arithmetic +is used to help the pupil solve his personal problems.</p> + +<p>Much ingenuity and experimentation will be required +to find the order that is satisfactory in both quality and +quantity of interest or motive and helpfulness of the bonds +one to another. The difficulty of organizing arithmetic +around attractive problems is much increased by the fact +of class instruction. For any one pupil vital, personal +problems or projects could be found to provide for many +arithmetical abilities; and any necessary knowledge and +technique which these projects did not develop could be +somehow fitted in along with them. But thirty children, +half boys and half girls, varying by five years in age, coming +from different homes, with different native capacities, will +not, in September, 1920, unanimously feel a vital need to +solve any one problem, and then conveniently feel another +on, say, October 15! In the mechanical laws of learning<span class='pagenum'><a name="Page_152" id="Page_152">[Pg 152]</a></span> +children are much alike, and the gain we may hope to make +from reducing inhibitions and increasing facilitations is, +for ordinary class-teaching, probably greater than that +to be made from the discovery of attractive central problems. +We should, however, get as much as possible of both.</p> + + +<h4>GENERAL PRINCIPLES</h4> + +<p>The reader may by now feel rather helpless before the +problem of the arrangement of arithmetical subject matter. +"Sometimes you complete a topic, sometimes you take it +piecemeal months or years apart, often you make queer +twists and shifts to get a strategic advantage over the +enemy," he may think, "but are there no guiding principles, +no general rules?" There is only one that is absolutely +general, to <i>take the order that works best for arithmetical learning</i>. +There are particular rules, but there are so many +and they are so limited by an 'other things being equal' +clause, that probably a general eagerness to think out the +<i>pros</i> and <i>cons</i> for any given proposal is better than a stiff +attempt to adhere to these rules. I will state and illustrate +some of them, and let the reader judge.</p> + +<p><i>Other things being equal, one new sort of bonds should not +be started until the previous set is fairly established, and two +different sets should not be started at once.</i> Thus, multiplication +of two- and three-place numbers by 2, 3, 4, and 5 will +first use numbers such that no carrying is required, and no +zero difficulties are encountered, then introduce carrying, +then introduce multiplicands like 206 and 320. If other +things were equal, the carrying would be split into two steps—first +drills with (4 × 6) + 2, (3 × 7) + 3, (5 × 4) + 1, and +the like, and second the actual use of these habits in the +multiplication. The objection to this separation of the +double habit is that the first part of it in isolation is too +<span class='pagenum'><a name="Page_153" id="Page_153">[Pg 153]</a></span> +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 (<i>a</i> × <i>b</i>) + <i>c</i> series. Experimental tests are +needed to decide this point.</p> + +<p><i>Other things being equal, bonds should be formed in such +order that none will have to be broken later.</i> For example, +there is a strong argument for teaching long division first, +or very early, with remainders, letting the case of zero remainder +come in as one of many. If the pupils have been +familiarized with the remainder notion by the drills recommended +as preparation for short division,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> the use of remainders +in long division will offer little difficulty. The +exclusive use of examples without remainders may form the +habit of not being exact in computation, of trusting to +'coming out even' as a sole check, and even of writing down +a number to fit the final number to be divided instead of +obtaining it by honest multiplication.</p> + +<p>For similar reasons additions with 2 and 3 as well as 1 +to be 'carried' have much to recommend them in the very +first stages of column addition with carrying. There is +here the added advantage that a pupil will be more likely to +remember to carry if he has to think <i>what</i> to carry. The +present common practice of using small numbers for ease +in the addition itself teaches many children to think of +carrying as adding one.</p> + +<p><i>Other things being equal, arrange to have variety.</i> Thus +it is probably, though not surely, wise to interrupt the +monotony of learning the multiplication and division tables, +by teaching the fundamentals of 'short' multiplication +and perhaps of division after the 5s, 2s, 3s, and 4s are learned. +This makes a break of several weeks. The facts for the 6s, +7s, 8s, and 9s can then be put to varied use as fast as learned. +<span class='pagenum'><a name="Page_154" id="Page_154">[Pg 154]</a></span> +It is almost certainly wise to interrupt the first half-year's +work with addition and subtraction, by teaching 2 × 2, 2 × 3, +3 × 2, 2 × 4, 4 × 2, 2 × 5, later by 2 × 10, 3 × 10, +4 × 10, 5 × 10, +later by ½ + ½, 1½ + ½, ½ of 2, ½ of 4, ½ of 6, and at some time +by certain profitable exercises wherein a pupil tells all he +knows about certain numbers which may be made nuclei of +important facts (say, 5, 8, 10, 12, 15, and 20).</p> + +<p><i>Other things being equal, use objective aids to verify an +arithmetical process or inference after it is made, as well as to +provoke it.</i> It is well at times to let pupils do everything +that they can with relations abstractly conceived, testing +their results by objective counting, measuring, adding, and +the like. For example, an early step in adding should be +to show three things, put them under a book, show two +more, put these under the book, and then ask how many +there are under the book, letting the objective counting +come later as the test of the correctness of the addition.</p> + +<p><i>Other things being equal, reserve all explanations of why a +process must be right until the pupils can use the process accurately, +and have verified the fact that it is right.</i> Except for +the very gifted pupils, the ordinary preliminary deductive +explanations of what must be done are probably useless as +means of teaching the pupils what to do. They use up +much time and are of so little permanent effect that, as we +have seen, the very arithmeticians who advocate making +them, admit that after a pupil has mastered the process he +may be allowed to forget the reasons for it. I am not sure +that the deductive proofs of why we place the decimal point +as we do in division by a decimal, or invert and multiply +in dividing by a fraction, and the like, are worth teaching at +all. If they are to be taught at all, the time to teach them +is (except for the very gifted) after the pupil has mastered +the process and has confidence in it. He then at least<span class='pagenum'><a name="Page_155" id="Page_155">[Pg 155]</a></span> +knows what process he is to prove is right, and that it is +right, and has had some chance of seeing <i>why</i> it is right from +his experience with it.</p> + +<p>One more principle may be mentioned without illustration. +<i>Arrange the order of bonds with due regard for the aims +of the other studies of the curriculum and the practical needs of +the pupil outside of school.</i> Arithmetic is not a book or a +closed system of exercises. It is the quantitative work of +the pupils in the elementary school. No narrower view of +it is adequate.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_156" id="Page_156">[Pg 156]</a></span></p> +<h2><a name="CHAPTER_VIII" id="CHAPTER_VIII"></a>CHAPTER VIII</h2> + +<h2>THE DISTRIBUTION OF PRACTICE</h2> + + +<h3>THE PROBLEM</h3> + +<p>The same amount of practice may be distributed in various +ways. Figures 7 to 10, for example, show 200 practices with +division by a fraction distributed over three and a half years +of 10 months in four different ways. In Fig. 7, practice is +somewhat equally distributed over the whole period. In +Fig. 8 the practice is distributed at haphazard. In Fig. 9 +there is a first main learning period, a review after about +ten weeks, a review at the beginning of the seventh grade, +another review at the beginning of the eighth grade, +and some casual practice rather at random. In Fig. 10 +there is a main learning period, with reviews diminishing +in length and separated by wider and wider intervals, with +occasional practice thereafter to keep the ability alive +and healthy.</p> + +<p>Plans I and II are obviously inferior to Plans III and IV; +and Plan IV gives promise of being more effective than +Plan III, since there seems danger that the pupil working +by Plan III might in the ten weeks lose too much of what +he had gained in the initial practice, and so again in the next +ten weeks.</p> + +<p><span class='pagenum'><a name="Page_157" id="Page_157">[Pg 157]</a></span></p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig07.jpg" width="800" height="109" alt="Fig. 7." title="" /> +<p class="nblockquot"><b><span class="smcap">Fig. 7.</span>—Plan I. 200 practices distributed somewhat evenly over 3½ years of 10 +months. In Figs. 7, 8, 9, and 10, each tenth of an inch along the base line +represents one month. Each hundredth of a square inch represents four practices, +a little square <sup>1</sup>⁄<sub>20</sub> of an inch wide and <sup>1</sup>⁄<sub>20</sub> inch high representing one practice.</b></p> +</div> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig08.jpg" width="800" height="134" alt="Fig. 8." title="" /> +<span class="caption"><span class="smcap">Fig. 8.</span>—Plan II. 200 practices distributed haphazard over 3½ years of 10 months.</span> +</div> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig09.jpg" width="800" height="489" alt="Fig. 9." title="" /> +<span class="caption"><span class="smcap">Fig. 9.</span>—Plan III. A learning period, three reviews, and incidental practice.</span> +</div> + + +<p>It is not wise, however, to try now to make close decisions +in the case of practice with division by a fraction; or to +determine what the best distribution of practice is for that +or any other ability to be improved. The facts of psychology +are as yet not adequate for very close decisions, nor are the<span class='pagenum'><a name="Page_158" id="Page_158">[Pg 158]</a></span> +types of distribution of practice that are best adapted to +different abilities even approximately worked out.</p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig10.jpg" width="800" height="466" alt="Fig. 10." title="" /> +<span class="caption"><span class="smcap">Fig. 10.</span>—Plan IV. A learning period with reviews of decreasing length at increasing intervals.</span> +</div> + + + +<h3>SAMPLE DISTRIBUTIONS</h3> + +<p>Let us rather examine some actual cases of distribution +of practice found in school work and consider, not the +attainment of the best possible distribution, but simply +the avoidance of gross blunders and the attainment of +reasonable, defensible procedures in this regard.</p> + +<p>Figures 11 to 18 show the distribution of examples in multiplication +with multipliers of various sorts. <i>X</i> stands for any +digit except zero. <i>O</i> stands for 0. <i>XXO</i> thus means a multiplier +like 350 or 270 or 160; <i>XOX</i> means multipliers like 407, +905, or 206; <i>XX</i> means multipliers like 25, 17, 38. Each of +these diagrams covers approximately 3½ years of school work, +or from about the middle of grade 3 to the end of grade 6. +They are made from counts of four textbooks (A, B, C, and +D), the count being taken for each successive 8 pages.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a> +<span class='pagenum'><a name="Page_159" id="Page_159">[Pg 159]</a></span> +Each tenth of an inch along the base line equals 8 pages of the +text in question. Each .01 sq. in. equals one example. The +books, it will be observed, differ in the amount of practice +given, as well as in the way in which it is distributed.</p> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig11.jpg" width="800" height="380" alt="Fig. 11." title="" /> +<span class="caption"><span class="smcap">Fig. 11.</span>—Distribution of practise with multipliers of the <i>XX</i> type in the first two books of the three-book text A.</span> +</div> + + +<p><span class='pagenum'><a name="Page_160" id="Page_160">[Pg 160]</a></span></p> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig12.jpg" width="800" height="427" alt="Fig. 12." title="" /> +<p class="nblockquot"><b><span class="smcap">Fig. 12.</span>—Same as Fig. 11, but for text B. Following this period come certain pages +of computation to be used by the teacher at her discretion, containing 24 <i>XX</i> +multiplications.</b></p> +</div> + + +<p>These distributions are worthy of careful study; we shall +note only a few salient facts about them here. Of the distributions +of multiplications with multipliers of the <i>XX</i> +type, that of book D (Fig. 14) is perhaps the best. A +(Fig. 11) has too much of the practice too late; B (Fig. 12) +gives too little practice in the first learning; C (Fig. 13) +gives too much in the first learning and in grade 6. Among +the distributions of multiplication with multipliers of the +<i>XOX</i> type, that of book D (Fig. 18) is again probably the +best. A, B, and C (Figs. 15, 16, and 17) have too much +practice early and too long intervals between reviews. +Book C (Fig. 17) by a careless oversight has one case of<span class='pagenum'><a name="Page_161" id="Page_161">[Pg 161]</a></span> +this very difficult process, without any explanation, weeks +before the process is taught!</p> + +<div class="figcenter" style="width: 768px;"> +<img src="images/fig13.jpg" width="768" height="795" alt="Fig. 13." title="" /> +<span class="caption"><span class="smcap">Fig. 13.</span>—Same as Fig. 11, but for text C.</span> +</div><p><span class='pagenum'><a name="Page_162" id="Page_162">[Pg 162]</a></span></p> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig14.jpg" width="800" height="333" alt="Fig. 14." title="" /> +<span class="caption"><span class="smcap">Fig. 14.</span>—Same as Fig. 11, but for text D.</span> +</div> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig15.jpg" width="800" height="412" alt="Fig. 15." title="" /> +<span class="caption"><span class="smcap">Fig. 15.</span>—Distribution of practice with multipliers of the <i>XOX</i> type in the first +two books of the three-book text A.</span> +</div><p><span class='pagenum'><a name="Page_163" id="Page_163">[Pg 163]</a></span></p> + +<div class="figcenter" style="width: 768px;"> +<img src="images/fig16.jpg" width="768" height="804" alt="Fig. 16." title="" /> +<p class="nblockquot"><b><span class="smcap">Fig. 16.</span>—Same as Fig. 15, but for text B. Following this period come certain +pages of computation to be used by the teacher at her discretion, containing +17 <i>XOX</i> multiplications.</b></p> +</div><p><span class='pagenum'><a name="Page_164" id="Page_164">[Pg 164]</a></span></p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig17.jpg" width="800" height="584" alt="Fig. 17." title="" /> +<span class="caption"><span class="smcap">Fig. 17.</span>—Same as Fig. 16, but for text C.</span> +</div> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig18.jpg" width="800" height="220" alt="Fig. 18." title="" /> +<span class="caption"><span class="smcap">Fig. 18.</span>—Same as Fig. 16, but for text D.</span> +</div> + + + +<p>Figures 19, 20, 21, 22, and 23 all concern the first two books +of the three-book text E.</p> + +<p>Figure 19 shows the distribution of practice on 5 × 5 in +the first two books of text E. The plan is the same as in +Figs. 11 to 18, except that each tenth of an inch along the +base line represents ten pages. Figure 20 shows the dis<span class='pagenum'><a name="Page_165" id="Page_165">[Pg 165]</a></span>tribution +of practice on 7 × 7; Fig. 21 shows it for 6 × 7 and +7 × 6 together. In Figs. 20 and 21 also, 0.1 inch along the +base line equals ten pages.</p> + +<p>Figures 22 and 23 show the distribution of practice on the +divisions of 72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or +9, and on the divisions of 81, 82 ... 89 by 9. Each tenth +of an inch along the base line represents ten pages here +also.</p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig19.jpg" width="800" height="559" alt="Fig. 19." title="" /> +<span class="caption"><span class="smcap">Fig. 19.</span>—Distribution of practice with 5 × 5 in the first two books of the three-book +text E.</span> +</div> + +<p>Figures 19 to 23 show no consistent plan for distributing +practice. With 5 × 5 (Fig. 19) the amount of practice increases +from the first treatment in grade 3 to the end of +grade 6, so that the distribution would be better if the pupil +began at the end and went backward! With 7 × 7 (Fig. 20)<span class='pagenum'><a name="Page_166" id="Page_166">[Pg 166]</a></span> +the practice is distributed rather evenly and in small doses. +With 6 × 7 and 7 × 6 (Fig. 21) much of it is in very large +doses. With the divisions (Figs. 22 and 23) the practice is +distributed more suitably, though in Fig. 23 there is too +much of it given at one time in the middle of the period.</p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig20.jpg" width="800" height="239" alt="Fig. 20." title="" /> +<span class="caption"><span class="smcap">Fig. 20.</span>—Distribution of practice with 7 × 7 in the first two books of text E.</span> +</div> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig21.jpg" width="800" height="527" alt="Fig. 21." title="" /> +<span class="caption"><span class="smcap">Fig. 21.</span>—Distribution of practice with 6 × 7 or 7 × 6 in the first two books of text E.</span> +</div><p><span class='pagenum'><a name="Page_167" id="Page_167">[Pg 167]</a></span></p> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig22.jpg" width="800" height="374" alt="Fig. 22." title="" /> +<span class="caption"><span class="smcap">Fig. 22.</span>—Distribution of practice with 72, 73 ... 79 ÷ 8 or 9 in the first two +books of text E.</span> +</div> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig23.jpg" width="800" height="233" alt="Fig. 23." title="" /> +<span class="caption"><span class="smcap">Fig. 23.</span>—Distribution of practice with 81, 82 ... 89 ÷ 9 in the first two books +of text E.</span> +</div> + + +<h4>POSSIBLE IMPROVEMENTS</h4> + +<p>Even if we knew what the best distribution of practice +was for each ability of the many to be inculcated by arithmetical +instruction, we could perhaps not provide it for all +of them. For, in the first place, the allotments for some of<span class='pagenum'><a name="Page_168" id="Page_168">[Pg 168]</a></span> +them might interfere with those for others. In the second +place, there are many other considerations of importance +in the ordering of topics besides giving the optimal distribution +of practice to each ability. Such are considerations of +interest, of welding separate abilities into an integrated +total ability, and of the limitations due to the school schedule +with its Saturdays, Sundays, holidays, and vacations.</p> + +<p>Improvement can, however, be made over present practice +in many respects. A scientific examination of the teaching +of almost any class for a year, or of many of our standard +instruments of instruction, will reveal opportunities for +improving the distribution of practice with no sacrifice of +interest, and with an actual gain in integrated functioning +arithmetical power. In particular it will reveal cases where +an ability is given practice and then, never being used again, +left to die of inactivity. It will reveal cases where an ability +is given practice and then left so long without practice that +the first effect is nearly lost. There will be cases where +practice is given and reviews are given, but all in such isolation +from everything else in arithmetic that the ability, though +existent, does not become a part of the pupil's general working +equipment. There will be cases where more practice +is given in the late than the earlier periods for no apparent +extrinsic advantage; and cases where the practice is put +where it is for no reason that is observable save that the +teacher or author in question has decided to have some drill +work at that time!</p> + +<p>Each ability has its peculiar needs in this matter, and no +set rules are at present of much value. It will be enough +for the present if we are aroused to the problem of distribution, +avoid obvious follies like those just noted, and exercise +what ingenuity we have.</p> + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_169" id="Page_169">[Pg 169]</a></span></p> +<h2><a name="CHAPTER_IX" id="CHAPTER_IX"></a>CHAPTER IX</h2> + +<h3>THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND +GENERAL NOTIONS IN ARITHMETIC<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a></h3> + + +<h4>RESPONSES TO ELEMENTS AND CLASSES</h4> + +<p>The plate which you see, the egg before you at the breakfast +table, and this page are concrete things, but whiteness, whether +of plate, egg, or paper, is, we say, an abstract quality. To be able +to think of whiteness irrespective of any concrete white object is +to be able to have an abstract idea or notion of white; to be able +to respond to whiteness, irrespective of whether it is a part of +china, eggshell, paper or whatever object, is to be able to respond +to the abstract element of whiteness.</p> + +<p>Learning arithmetic involves the formation of very many such +ideas, the acquisition of very many such powers of response to +elements regardless of the gross total situations in which they +appear. To appreciate the fiveness of five boys, five pencils, five +inches, five rings of a bell; to understand the division into eight +equal parts of 40 cents, 32 feet, 64 minutes, or 16 ones; to respond +correctly to the fraction relation in <sup>2</sup>⁄<sub>3</sub>, <sup>5</sup>⁄<sub>6</sub>, +<sup>3</sup>⁄<sub>4</sub>, <sup>7</sup>⁄<sub>12</sub>, <sup>1</sup>⁄<sub>8</sub>, or any other; to +be sensitive to the common element of 9 = 3 × 3, 16 = 4 × 4, +625 = 25 × 25, .04 = .2 × .2, ¼ = ½ × ½,—these are obvious illustrations. +All the numbers which the pupil learns to understand and manipulate +are in fact abstractions; all the operations are abstractions; +percent, discount, interest, height, length, area, volume, are abstractions; +sum, difference, product, quotient, remainder, average, are +facts that concern elements or aspects which may appear with +countless different concrete surroundings or concomitants.</p> + +<p>Towser is a particular dog; your house lot on Elm Street is a +particular rectangle; Mr. and Mrs. I.S. Peterson and their +daughter Louise are a particular family of three. In contrast to +<span class='pagenum'><a name="Page_170" id="Page_170">[Pg 170]</a></span> +these particulars, we mean by a dog, a rectangle, and a family of +three, <i>any</i> specimens of these classes of facts. The idea of a dog, +of rectangles in general, of any family of three is a general notion, +a concept or idea of a class or species. The ability to respond to +any dog, or rectangle, or family of three, regardless of which +particular one it may be, is the general notion in action.</p> + +<p>Learning arithmetic involves the formation of very many such +general notions, such powers of response to any member of a +certain class. Thus a hundred different sized lots may all be +responded to as rectangles; <sup>9</sup>⁄<sub>18</sub>, <sup>12</sup>⁄<sub>27</sub>, +<sup>15</sup>⁄<sub>24</sub>, and <sup>27</sup>⁄<sub>36</sub> may all be responded +to as members of the class, 'both members divisible by 3.' The +same fact may be responded to in different ways according to the +class to which it is assigned. Thus 4 in <sup>3</sup>⁄<sub>4</sub>, <sup>4</sup>⁄<sub>5</sub>, 45, 54, and 405 is +classed respectively as 'a certain sized part of unity,' 'a certain +number of parts of the size shown by the 5,' 'a certain number +of tens,' 'a certain number of ones,' and 'a certain number of +hundreds.' Each abstract quality may become the basis of a +class of facts. So fourness as a quality corresponds to the class +'things four in number or size'; the fractional quality or relation +corresponds to the class 'fractions.' The bonds formed with +classes of facts and with elements or features by which one whole +class of facts is distinguished from another, are in fact, a chief concern +of arithmetical learning.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a></p> + + +<h4>FACILITATING THE ANALYSIS OF ELEMENTS</h4> + +<p>Abstractions and generalizations then depend upon analysis +and upon bonds formed with more or less subtle elements rather +than with gross total concrete situations. The process involved is +most easily understood by considering the means employed to +facilitate it.</p> + +<p>The first of these is having the learner respond to the total +situations containing the element in question with the attitude +of piecemeal examination, and with attentiveness to one element +after another, especially to so near an approximation to the +element in question as he can already select for attentive examination. +This attentiveness to one element after another serves to +<span class='pagenum'><a name="Page_171" id="Page_171">[Pg 171]</a></span> +emphasize whatever appropriate minor bonds from the element +in question the learner already possesses. Thus, in teaching +children to respond to the 'fiveness' of various collections, we +show five boys or five girls or five pencils, and say, "See how +many boys are standing up. Is Jack the only boy that is standing +here? Are there more than two boys standing? Name the boys +while I point at them and count them. (Jack) is one, and (Fred) +is one more, and (Henry) is one more. Jack and Fred make (two) +boys. Jack and Fred and Henry make (three) boys." (And so on +with the attentive counting.) The mental set or attitude is +directed toward favoring the partial and predominant activity +of 'how-many-ness' as far as may be; and the useful bonds that +the 'fiveness,' the 'one and one and one and one and one-ness,' +already have, are emphasized as far as may be.</p> + +<p>The second of the means used to facilitate analysis is having +the learner respond to many situations each containing the element +in question (call it A), but with varying concomitants (call these +V. C.) his response being so directed as, so far as may be, to separate +each total response into an element bound to the A and an +element bound to the V. C.</p> + +<p>Thus the child is led to associate the responses—'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 <i>per se</i> tends strongly to call up the +'Boys' element of the response). The other elements of the +responses (boys, girls, pencils, etc.) have each only a slight connection +with the 'fiveness' element of the situations. These slight +connections also in large part<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a> counteract each other, leaving the +field clear for whatever uninhibited bond the 'fiveness' has.</p> + +<p>The third means used to facilitate analysis is having the learner +respond to situations which, pair by pair, present the element +in a certain context and present that same context with <i>the opposite +of the element in question</i>, or with something at least very unlike the +element. Thus, a child who is being taught to respond to 'one +fifth' is not only led to respond to 'one fifth of a cake,' 'one +<span class='pagenum'><a name="Page_172" id="Page_172">[Pg 172]</a></span> +fifth of a pie,' 'one fifth of an apple,' 'one fifth of ten inches,' +'one fifth of an army of twenty soldiers,' and the like; he is also +led to respond to each of these <i>in contrast with</i> 'five cakes,' 'five +pies,' 'five apples,' 'five times ten inches,' 'five armies of +twenty soldiers.' Similarly the 'place values' of tenths, +hundredths, and the rest are taught by contrast with the tens, +hundreds, and thousands.</p> + +<p>These means utilize the laws of connection-forming to disengage +a response element from gross total responses and attach +it to some situation element. The forces of use, disuse, satisfaction, +and discomfort are so maneuvered that an element which +never exists by itself in nature can influence man almost as if it +did so exist, bonds being formed with it that act almost or quite +irrespective of the gross total situation in which it inheres. What +happens can be most conveniently put in a general statement by +using symbols.</p> + +<p>Denote by <i>a + b</i>, <i>a + g</i>, <i>a + l</i>, <i>a + q</i>, <i>a + v</i>, and <i>a + B</i> certain +situations alike in the element <i>a</i> and different in all else. Suppose +that, by original nature or training, a child responds to these +situations respectively by <i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>7</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>12</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>17</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>22</sub>, +<i>r</i><sub>1</sub> + <i>r</i><sub>27</sub>. Suppose that man's neurones are capable of such action +that <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>7</sub>, <i>r</i><sub>12</sub>, <i>r</i><sub>22</sub>, and <i>r</i><sub>27</sub>, can each be made singly.</p> + + +<p class="center">Case I. Varying Concomitants</p> + +<p>Suppose that <i>a</i> + <i>b</i>, <i>a</i> + <i>g</i>, <i>a</i> + <i>l</i>, etc., occur once each.</p> + + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'>We have</td><td align='left'><i>a</i> + <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>g</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>7</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>l</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>12</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>q</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>17</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>v</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>22</sub>, and</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>B</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>27</sub>, as shown in Scheme I.</td></tr> +</table></div> + +<p class="center">Scheme I</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='center'></td><td align='center'><i>a</i></td><td align='center'><i>b</i></td><td align='center'><i>g</i></td><td align='center'><i>l</i></td><td align='center'><i>q</i></td><td align='center'><i>v</i></td><td align='center'><i>B</i></td></tr> +<tr><td align='center'><i>r</i><sub>1</sub></td><td align='center'>6</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr> +<tr><td align='center'><i>r</i><sub>2</sub></td><td align='center'>1</td><td align='center'>1</td></tr> +<tr><td align='center'><i>r</i><sub>7</sub></td><td align='center'>1</td><td></td><td align='center'>1</td></tr> +<tr><td align='center'><i>r</i><sub>12</sub></td><td align='center'>1</td><td></td><td></td><td align='center'>1</td></tr> +<tr><td align='center'><i>r</i><sub>17</sub></td><td align='center'>1</td><td></td><td></td><td></td><td align='center'>1</td></tr> +<tr><td align='center'><i>r</i><sub>22</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr> +<tr><td align='center'><i>r</i><sub>27</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr> +</table></div> +<p><span class='pagenum'><a name="Page_173" id="Page_173">[Pg 173]</a></span></p> + +<p class="noidt"><i>a</i> is thus responded to by <i>r</i><sub>1</sub> (that is, connected with <i>r</i><sub>1</sub>) each time, +or six in all, but only once each with <i>b</i>, <i>g</i>, <i>l</i>, <i>q</i>, <i>v</i>, and <i>B</i>. <i>b</i>, <i>g</i>, <i>l</i>, <i>q</i>, <i>v</i>, +and <i>B</i> are connected once each with <i>r</i><sub>1</sub> and once respectively with +<i>r</i><sub>2</sub>, <i>r</i><sub>7</sub>, <i>r</i><sub>12</sub>, etc. The bond from <i>a</i> to <i>r</i><sub>1</sub>, has had six times as much +exercise as the bond from <i>a</i> to <i>r</i><sub>2</sub>, or from <i>a</i> to <i>r</i><sub>7</sub>, etc. In any new +gross situation, <i>a</i> 0, <i>a</i> will be more predominant in determining +response than it would otherwise have been; and <i>r</i><sub>1</sub> will be more +likely to be made than <i>r</i><sub>2</sub>, <i>r</i><sub>7</sub>, <i>r</i><sub>12</sub>, etc., the other previous associates +in the response to a situation containing <i>a</i>. That is, the bond +from the element <i>a</i> to the response <i>r</i><sub>1</sub> has been notably strengthened.</p> + + +<p class="center">Case II. Contrasting Concomitants</p> + +<p>Now suppose that <i>b</i> and <i>g</i> are very dissimilar elements (<i>e.g.</i>, +white and black), that <i>l</i> and <i>q</i> are very dissimilar (<i>e.g.</i>, long and +short), and that <i>v</i> and <i>B</i> are also very dissimilar. To be very +dissimilar means to be responded to very differently, so that <i>r</i><sub>7</sub>, +the response to <i>g</i>, will be very unlike <i>r</i><sub>2</sub>, the response to <i>b</i>. So <i>r</i><sub>7</sub> +may be thought of as <i>r</i><sub>not 2</sub> or <i>r</i><sub>-2</sub>. In the same way <i>r</i><sub>12</sub> may be +thought of as <i>r</i><sub>not 12</sub> or <i>r</i><sub>-12</sub>, and <i>r</i><sub>27</sub> may be called <i>r</i><sub>not 22</sub> or <i>r</i><sub>-22</sub>.</p> + +<p>Then, if the situations <i>a b</i>, <i>a g</i>, <i>a l</i>, <i>a q</i>, <i>a v</i>, and <i>a B</i> are responded +to, each once, we have:—</p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>g</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>not 2</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>l</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>12</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>q</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>not 12</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>v</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>22</sub>, and</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>B</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>not 22</sub>, as shown in Scheme II.</td></tr> +</table></div> + +<p class="center">Scheme II</p> + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='center'></td><td align='center'><i>a</i></td><td align='center'><i>b</i></td><td align='center'><i>g</i></td><td align='center'><i>l</i></td><td align='center'><i>q</i></td><td align='center'><i>v</i></td><td align='center'><i>B</i></td></tr> +<tr><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'>(opp. of <i>b</i>)</td><td align='center'></td><td align='center'>(opp. of <i>l</i>)</td><td align='center'></td><td align='center'>(opp. of <i>v</i>)</td></tr> +<tr><td align='left'><i>r</i><sub>1</sub></td><td align='center'>6</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>not 1</sub></td></tr> +<tr><td align='left'><i>r</i><sub>2</sub></td><td align='center'>1</td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>not 2</sub></td><td align='center'>1</td><td></td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>12</sub></td><td align='center'>1</td><td></td><td></td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>not 12</sub></td><td align='center'>1</td><td></td><td></td><td></td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>22</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>not 22</sub></td><td align='center'>1</td><td></td><td></td><td></td><td></td><td></td><td align='center'>1</td></tr> +</table></div> + + +<p class="noidt"><i>r</i><sub>1</sub> is connected to <i>a</i> by 6 repetitions. <i>r</i><sub>2</sub> and <i>r</i><sub>not 2</sub> are each connected +to <i>a</i> by 1 repetition, but since they interfere, canceling each<span class='pagenum'><a name="Page_174" id="Page_174">[Pg 174]</a></span> +other so to speak, the net result is for <i>a</i> to have zero tendency to +call up <i>r</i><sub>2</sub> or <i>r</i><sub>not 2</sub>. <i>r</i><sub>12</sub> and <i>r</i><sub>not 12</sub> are each connected to <i>a</i> by 1 +repetition, but they interfere with or cancel each other with the +net result that <i>a</i> has zero tendency to call up <i>r</i><sub>12</sub> or <i>r</i><sub>not 12</sub>. So +with <i>r</i><sub>22</sub> and <i>r</i><sub>not 22</sub>. Here then the net result of the six connections +of <i>a b</i>, <i>a g</i>, <i>a l</i>, <i>a q</i>, <i>a v</i>, and <i>a B</i> is to connect <i>a</i> with <i>r</i>, and with +nothing else.</p> + + +<p class="center">Case III. Contrasting Concomitants and Contrasting Element</p> + +<p>Suppose now that the facts are as in Case II, but with the +addition of six experiences where a certain element which is the +opposite of, or very dissimilar to, <i>a</i> is connected with the response +<i>r</i><sub>not 1</sub>, or <i>r</i><sub>-1</sub> which is opposite to, or very dissimilar to <i>r</i><sub>1</sub>. Call +this opposite of <i>a</i>, − <i>a</i>.</p> + +<p>That is, we have not only</p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>g</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>not 2</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>l</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>12</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>q</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>not 12</sub>,</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>v</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>22</sub>, and</td></tr> +<tr><td align='left'></td><td align='left'><i>a</i> + <i>B</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>1</sub> + <i>r</i><sub>not 22</sub>,</td></tr> +</table></div> + +<p class="noidt">but also</p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'>− <i>a</i> + <i>b</i></td><td align='left'>responded to by</td><td align='left'><i>r</i><sub>not 1</sub> + <i>r</i><sub>2</sub>,</td></tr> +<tr><td align='left'>− <i>a</i> + <i>g</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>not 1</sub> + <i>r</i><sub>not 2</sub>,</td></tr> +<tr><td align='left'>− <i>a</i> + <i>l</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>not 1</sub> + <i>r</i><sub>12</sub>,</td></tr> +<tr><td align='left'>− <i>a</i> + <i>q</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>not 1</sub> + <i>r</i><sub>not 12</sub>,</td></tr> +<tr><td align='left'>− <i>a</i> + <i>v</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>not 1</sub> + <i>r</i><sub>22</sub>, and</td></tr> +<tr><td align='left'>− <i>a</i> + <i>B</i></td><td align='center'>" "</td><td align='left'><i>r</i><sub>not 1</sub> + <i>r</i><sub>not 22</sub>, as shown in Scheme III.</td></tr> +</table></div> + + +<p class="center">Scheme III</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='center'></td><td align='center'><i>a</i></td><td align='center'><i>opp.</i></td><td align='center'><i>b</i></td><td align='center'><i>g</i></td><td align='center'><i>l</i></td><td align='center'><i>q</i></td><td align='center'><i>v</i></td><td align='center'><i>B</i></td></tr> +<tr><td align='center'></td><td align='center'></td><td align='center'><i>of a</i></td><td align='center' colspan='2'>(opp. of <i>b</i>)</td><td align='center' colspan='2'>(opp. of <i>l</i>)</td><td align='center' colspan='2'>(opp. of <i>v</i>)</td></tr> +<tr><td align='left'><i>r</i><sub>1</sub></td><td align='center'>6</td><td></td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>not 1</sub></td><td></td><td align='center'>6</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td><td align='center'>1</td></tr> +<tr><td align='left'><i>r</i><sub>2</sub></td><td align='center'>1</td><td align='center'>1</td><td align='center'>2</td></tr> +<tr><td align='left'><i>r</i><sub>not 2</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td align='center'>2</td></tr> +<tr><td align='left'><i>r</i><sub>12</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td align='center'>2<span class='pagenum'><a name="Page_175" id="Page_175">[Pg 175]</a></span></td></tr> +<tr><td align='left'><i>r</i><sub>not 12</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td></td><td align='center'>2</td></tr> +<tr><td align='left'><i>r</i><sub>22</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td></td><td></td><td align='center'>2</td></tr> +<tr><td align='left'><i>r</i><sub>not 22</sub></td><td align='center'>1</td><td align='center'>1</td><td></td><td></td><td></td><td></td><td></td><td align='center'>2</td></tr> +</table></div> + + + +<p>In this series of twelve experiences <i>a</i> connects with <i>r</i><sub>1</sub> six times +and the opposite of <i>a</i> connects with <i>r</i><sub>not 1</sub> six times. <i>a</i> connects +equally often with three pairs of mutual destructives <i>r</i><sub>2</sub> and <i>r</i><sub>not 2</sub>, +<i>r</i><sub>12</sub> and <i>r</i><sub>not 12</sub>, <i>r</i><sub>22</sub> and <i>r</i><sub>not 22</sub>, and so has zero tendency to call +them up. − <i>a</i> has also zero tendency to call up any of these +responses except its opposite, <i>r</i><sub>not 1</sub>. <i>b</i>, <i>g</i>, <i>l</i>, <i>q</i>, <i>v</i>, and <i>B</i> are made +to connect equally often with <i>r</i><sub>1</sub> and <i>r</i><sub>not 1</sub>. So, of these elements, +<i>a</i> is the only one left with a tendency to call up <i>r</i><sub>1</sub>.</p> + +<p>Thus, by the mere action of frequency of connection, <i>r</i><sub>1</sub> is connected +with <i>a</i>; the bonds from <i>a</i> to anything except <i>r</i><sub>1</sub> are being +counteracted, and the slight bonds from anything except <i>a</i> to <i>r</i><sub>1</sub> +are being counteracted. The element <i>a</i> becomes predominant +in situations containing it; and its bond toward <i>r</i><sub>1</sub> becomes +relatively enormously strengthened and freed from competition.</p> + +<p>These three processes occur in a similar, but more complicated, +form if the situations <i>a</i> + <i>b</i>, <i>a</i> + <i>g</i>, etc., are replaced by <i>a</i> + <i>b</i> + <i>c</i> + <i>d</i> + <i>e</i> + <i>f</i>, +<i>a</i> + <i>g</i> + <i>h</i> + <i>i</i> + <i>j</i> + <i>k</i>, etc., and the responses <i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>7</sub>, +<i>r</i><sub>1</sub> + <i>r</i><sub>12</sub>, etc., are replaced by <i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> + <i>r</i><sub>3</sub> + <i>r</i><sub>4</sub> + <i>r</i><sub>5</sub> + <i>r</i><sub>6</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>7</sub> + <i>r</i><sub>8</sub> + <i>r</i><sub>9</sub> + <i>r</i><sub>10</sub> + <i>r</i><sub>11</sub>, +etc.—<i>provided the r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub>, etc., <i>can be made singly</i>. +In so far as any one of the responses is necessarily co-active with +any one of the others (so that, for example, <i>r</i><sub>13</sub> always brings <i>r</i><sub>26</sub> +with it and <i>vice versa</i>), the exact relations of the numbers recorded +in schemes like schemes I, II, and III on pages 172 to 174 will +change; but, unless <i>r</i><sub>1</sub> has such an inevitable co-actor, the general +results of schemes I, II, and III will hold good. If <i>r</i><sub>1</sub> does have +such an inseparable co-actor, say <i>r</i><sub>2</sub>, then, of course, <i>a</i> can never +acquire bonds with <i>r</i><sub>1</sub> alone, but everywhere that <i>r</i><sub>1</sub> or <i>r</i><sub>2</sub> appears +in the preceding schemes the other element must appear also. +<i>r</i><sub>1</sub> <i>r</i><sub>2</sub> would then have to be used as a unit in analysis.</p> + +<p>The '<i>a</i> + <i>b</i>,' '<i>a</i> + <i>g</i>,' '<i>a</i> + <i>l</i>,' ... '<i>a</i> + <i>B</i>' situations may occur +unequal numbers of times, altering the exact numerical relations +of the connections formed and presented in schemes I, II, and III; +but the process in general remains the same.</p> + +<p>So much for the effect of use and disuse in attaching appropriate +response elements to certain subtle elements of situations. There +are three main series of effects of satisfaction and discomfort.<span class='pagenum'><a name="Page_176" id="Page_176">[Pg 176]</a></span> +They serve, first, to emphasize, from the start, the desired bonds +leading to the responses <i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>, <i>r</i><sub>1</sub> + <i>r</i><sub>7</sub>, etc., to the total situations, +and to weed out the undesirable ones. They also act to emphasize, +in such comparisons and contrasts as have been described, every +action of the bond from <i>a</i> to <i>r</i><sub>1</sub>; and to eliminate every tendency +of <i>a</i> to connect with aught save <i>r</i><sub>1</sub>, and of aught save <i>a</i> to +connect with <i>r</i><sub>1</sub>. Their third service is to strengthen the bonds +produced of appropriate responses to <i>a</i> wherever it occurs, +whether or not any formal comparisons and contrasts take place.</p> + +<p>The process of learning to respond to the difference of pitch +in tones from whatever instrument, to the 'square-root-ness' of +whatever number, to triangularity in whatever size or combination +of lines, to equality of whatever pairs, or to honesty in whatever +person or instance, is thus a consequence of associative learning, +requiring no other forces than those of use, disuse, satisfaction, +and discomfort. "What happens in such cases is that the response, +by being connected with many situations alike in the +presence of the element in question and different in other respects, +is bound firmly to that element and loosely to each of its concomitants. +Conversely any element is bound firmly to any one +response that is made to all situations containing it and very, very +loosely to each of those responses that are made to only a few +of the situations containing it. The element of triangularity, for +example, is bound firmly to the response of saying or thinking +'triangle' but only very loosely to the response of saying or +thinking white, red, blue, large, small, iron, steel, wood, paper, +and the like. A situation thus acquires bonds not only with some +response to it as a gross total, but also with responses to any of +its elements that have appeared in any other gross totals. Appropriate +response to an element regardless of its concomitants is a +necessary consequence of the laws of exercise and effect if an animal +learns to make that response to the gross total situations that +contain the element and not to make it to those that do not. +Such prepotent determination of the response by one or another +element of the situation is no transcendental mystery, but, given +the circumstances, a general rule of all learning." Such are at +bottom only extreme cases of the same learning as a cat exhibits +that depresses a platform in a certain box whether it faces north +or south, whether the temperature is 50 or 80 degrees, whether one +or two persons are in sight, whether she is exceedingly or moderately +hungry, whether fish or milk is outside the box. All learning is +analytic, representing the activity of elements within a total +situation. In man, by virtue of certain instincts and the course<span class='pagenum'><a name="Page_177" id="Page_177">[Pg 177]</a></span> +of his training, very subtle elements of situations can so operate.</p> + +<hr style='width: 45%;' /> + +<p>Learning by analysis does not often proceed in the carefully +organized way represented by the most ingenious +marshaling of comparing and contrasting activities. The +associations with gross totals, whereby in the end an element +is elevated to independent power to determine response, +may come in a haphazard order over a long interval +of time. Thus a gifted three-year-old boy will have the +response element of 'saying or thinking <i>two</i>,' bound to the +'two-ness' element of very many situations in connection +with the 'how-many' mental set; and he will have made +this analysis without any formal, systematic training. An +imperfect and inadequate analysis already made is indeed +usually the starting point for whatever systematic abstraction +the schools direct. Thus the kindergarten exercises +in analyzing out number, color, size, and shape commonly +assume that 'one-ness' <i>versus</i> 'more-than-one-ness,' black +and white, big and little, round and not round are, at least +vaguely, active as elements responded to in some independence +of their contexts. Moreover, the tests of actual +trial and success in further undirected exercises usually +coöperate to confirm and extend and refine what the systematic +drills have given. Thus the ordinary child in school +is left, by the drills on decimal notation, with only imperfect +power of response to the 'place-values.' He continues to +learn to respond properly to them by finding that 4 × 40 += 160, 4 × 400 = 1600, 800 − 80 = 720, 800 − 8 = 792, 800 − 800 += 0, 42 × 48 = 2016, 24 × 48 = 1152, and the like, are +satisfying; while 4 × 40 = 16, 23 × 48 = 832, 800 − 8 = 0, +and the like, are not. The process of analysis is +the same in such casual, unsystematized formation of +connections with elements as in the deliberately man<span class='pagenum'><a name="Page_178" id="Page_178">[Pg 178]</a></span>aged, +piecemeal inspection, comparison, and contrast described +above.</p> + + +<h4>SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS</h4> + +<p>The arrangement of a pupil's experiences so as to direct +his attention to an element, vary its concomitants instructively, +stimulate comparison, and throw the element into relief +by contrast may be by fixed, formal, systematic exercises. +Or it may be by much less formal exercises, spread +over a longer time, and done more or less incidentally in +other connections. We may call these two extremes the +'systematic' and 'opportunistic,' since the chief feature of +the former is that it systematically provides experiences designed +to build up the power of correct response to the element, +whereas the chief feature of the latter is that it uses +especially such opportunities as occur by reason of the +pupil's activities and interests.</p> + +<p>Each method has its advantages and disadvantages. +The systematic method chooses experiences that are specially +designed to stimulate the analysis; it provides these at a +certain fixed time so that they may work together; it can +then and there test the pupils to ascertain whether they +really have the power to respond to the element or aspect +or feature in question. Its disadvantages are, first, that +many of the pupils will feel no need for and attach no interest +or motive to these formal exercises; second, that some of the +pupils may memorize the answers as a verbal task instead +of acquiring insight into the facts; third, that the ability +to respond to the element may remain restricted to the +special cases devised for the systematic training, and not be +available for the genuine uses of arithmetic.</p> + +<p>The opportunistic method is strong just where the systematic +is weak. Since it seizes upon opportunities created<span class='pagenum'><a name="Page_179" id="Page_179">[Pg 179]</a></span> +by the pupil's abilities and interests, it has the attitude of +interest more often. Since it builds up the experiences +less formally and over a wider space of time, the pupils are +less likely to learn verbal answers. Since its material comes +more from the genuine uses of life, the power acquired is +more likely to be applicable to life.</p> + +<p>Its disadvantage is that it is harder to manage. More +thought and experimentation are required to find the best +experiences; greater care is required to keep track of the +development of an abstraction which is taught not in two +days, but over two months; and one may forget to test +the pupils at the end. In so far as the textbook and teacher +are able to overcome these disadvantages by ingenuity and +care, the opportunistic method is better.</p> + + +<h4>ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS</h4> + +<p>We may expect much improvement in the formation of +abstract and general ideas in arithmetic from the application +of three principles in addition to those already described. +They are: (1) Provide enough actual experiences before +asking the pupil to understand and use an abstract or general +idea. (2) Develop such ideas gradually, not attempting to +give complete and perfect ideas all at once. (3) Develop +such ideas so far as possible from experiences which will be +valuable to the pupil in and of themselves, quite apart from +their merit as aids in developing the abstraction or general +notion. Consider these three principles in order.</p> + +<p>Children, especially the less gifted intellectually, need +more experiences as a basis for and as applications of an +arithmetical abstraction or concept than are usually given +them. For example, in paving the way for the principle, +"Any number times 0 equals 0," it is not safe to say, "John +worked 8 days for 0 minutes per day. How many minutes<span class='pagenum'><a name="Page_180" id="Page_180">[Pg 180]</a></span> +did he work?" and "How much is 0 times 4 cents?" It +will be much better to spend ten or fifteen minutes as follows:<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a> +"What does zero mean? (Not any. No.) How +many feet are there in eight yards? In 5 yards? In 3 +yards? In 2 yards? In 1 yard? In 0 yard? How many +inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... +qt. 5 pk. = .... qt. 0 pk. = .... qt. A boy receives +60 cents an hour when he works. How much does he receive +when he works 3 hr.? 8 hr.? 6 hr.? 0 hr.? A +boy received 60 cents a day for 0 days. How much did he +receive? How much is 0 times $600? How much is 0 +times $5000? How much is 0 times a million dollars? 0 +times any number equals....</p> + + + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'> 232</td><td align='left'>(At the blackboard.) 0 time 232 equals what?</td></tr> +<tr><td align='left'> 30</td><td align='left'>I write 0 under the 0.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> 3 times 232 equals what?</td></tr> +<tr><td align='left'>——</td></tr> +<tr><td align='left'>6960</td><td align='left'>Continue at the blackboard with</td></tr> +</table></div> + + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='right'></td><td align='right'>734</td><td align='right'>321</td><td align='right'>312</td><td align='right'>41</td></tr> +<tr><td align='right'></td><td align='right'>20</td><td align='right'>40</td><td align='right'>30</td><td align='right'>60</td><td align='right'>etc."</td></tr> +<tr><td align='right'></td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td><td align='right'>——</td></tr> +</table></div> + +<p>Pupils in the elementary school, except the most gifted, +should not be expected to gain mastery over such concepts +as <i>common fraction</i>, <i>decimal fraction</i>, <i>factor</i>, and <i>root</i> quickly. +They can learn a definition quickly and learn to use it in +very easy cases, where even a vague and imperfect understanding +of it will guide response correctly. But complete +<span class='pagenum'><a name="Page_181" id="Page_181">[Pg 181]</a></span> +and exact understanding commonly requires them to take, +not one intellectual step, but many; and mastery in use +commonly comes only as a slow growth. For example, +suppose that pupils are taught that .1, .2, .3, etc., mean <sup>1</sup>⁄<sub>10</sub>, +<sup>2</sup>⁄<sub>10</sub>, <sup>3</sup>⁄<sub>10</sub>, etc., that .01, .02, .03, etc., mean <sup>1</sup>⁄<sub>100</sub>, +<sup>2</sup>⁄<sub>100</sub>, <sup>3</sup>⁄<sub>100</sub>, +etc., that .001, .002, .003, etc., mean <sup>1</sup>⁄<sub>1000</sub>, +<sup>2</sup>⁄<sub>1000</sub>, <sup>3</sup>⁄<sub>1000</sub>, etc., +and that .1, .02, .001, etc., are decimal fractions. They +may then respond correctly when asked to write a decimal +fraction, or to state which of these,—<sup>1</sup>⁄<sub>4</sub>, .4, <sup>3</sup>⁄<sub>8</sub>, .07, .002, +<sup>5</sup>⁄<sub>6</sub>,—are +common fractions and which are decimal fractions. +They may be able, though by no means all of them will be, +to write decimal fractions which equal <sup>1</sup>⁄<sub>2</sub> and <sup>1</sup>⁄<sub>5</sub>, and the +common fractions which equal .1 and .09. Most of them +will not, however, be able to respond correctly to "Write +a decimal mixed number"; or to state which of these,—<sup>1</sup>⁄<sub>100</sub> +.4½, <sup>.007</sup>⁄<sub>350</sub>, $.25,—are common fractions, and which are +decimals; or to write the decimal fractions which equal <sup>3</sup>⁄<sub>4</sub> +and <sup>1</sup>⁄<sub>3</sub>.</p> + +<p>If now the teacher had given all at once the additional +experiences needed to provide the ability to handle these +more intricate and subtle features of decimal-fraction-ness, +the result would have been confusion for most pupils. The +general meaning of .32, .14, .99, and the like requires some +understanding of .30, .10, .90, and .02, .04, .08; but it is +not desirable to disturb the child with .30 while he is trying +to master 2.3, 4.3, 6.3, and the like. Decimals in general +require connection with place value and the contrasts of +.41 with 41, 410, 4.1, and the like, but if the relation to place +values in general is taught in the same lesson with the relation +to ⁄<sub>10</sub>s, ⁄<sub>100</sub>s, ⁄<sub>1000</sub>s, the mind will suffer from +violent indigestion.</p> + +<p>A wise pedagogy in fact will break up the process of learn<span class='pagenum'><a name="Page_182" id="Page_182">[Pg 182]</a></span>ing +the meaning and use of decimal fractions into many +teaching units, for example, as follows:—</p> + +<p>(1) Such familiarity with fractions with large denominators +as is desirable for pupils to have, as by an exercise +in reducing to lowest terms, +<sup>8</sup>⁄<sub>10</sub>, <sup>36</sup>⁄<sub>64</sub>, +<sup>20</sup>⁄<sub>25</sub>, <sup>18</sup>⁄<sub>24</sub>, +<sup>24</sup>⁄<sub>32</sub>, <sup>21</sup>⁄<sub>30</sub>, +<sup>25</sup>⁄<sub>100</sub>, <sup>40</sup>⁄<sub>100</sub>, +and the like. This is good as a review of cancellation, and +as an extension of the idea of a fraction.</p> + +<p>(2) Objective work, showing <sup>1</sup>⁄<sub>10</sub> sq. ft., <sup>1</sup>⁄<sub>50</sub> sq. ft., <sup>1</sup>⁄<sub>100</sub> +sq. ft., and <sup>1</sup>⁄<sub>1000</sub> sq. ft., and having these identified and the +forms <sup>1</sup>⁄<sub>10</sub> sq. ft., <sup>1</sup>⁄<sub>100</sub> sq. ft., and <sup>1</sup>⁄<sub>1000</sub> sq. ft. learned. Finding +how many feet = <sup>1</sup>⁄<sub>10</sub> mile and <sup>1</sup>⁄<sub>100</sub> mile.</p> + +<p>(3) Familiarity with ⁄<sub>100</sub>s and ⁄<sub>1000</sub>s by reductions of +<sup>750</sup>⁄<sub>1000</sub>, <sup>50</sup>⁄<sub>100</sub>, etc., to lowest terms and by writing the missing +numerators in <sup>500</sup>⁄<sub>1000</sub> = ⁄<sub>100</sub> = ⁄<sub>10</sub> and the like, and by finding +<sup>1</sup>⁄<sub>10</sub>, <sup>1</sup>⁄<sub>100</sub>, and <sup>1</sup>⁄<sub>1000</sub> of 3000, 6000, 9000, etc.</p> + +<p>(4) Writing <sup>1</sup>⁄<sub>10</sub> as .1 and <sup>1</sup>⁄<sub>100</sub> as .01, +<sup>11</sup>⁄<sub>100</sub>, <sup>12</sup>⁄<sub>100</sub>, <sup>13</sup>⁄<sub>100</sub>, etc., as +.11, .12, .13. United States money is used as the introduction. +Application is made to miles.</p> + +<p>(5) Mixed numbers with a first decimal place. The +cyclometer or speedometer. Adding numbers like 9.1, +14.7, 11.4, etc.</p> + +<p>(6) Place value in general from thousands to hundredths.</p> + +<p>(7) Review of (1) to (6).</p> + +<p>(8) Tenths and hundredths of a mile, subtraction when +both numbers extend to hundredths, using a railroad table +of distances.</p> + +<p>(9) Thousandths. The names 'decimal fractions or +decimals,' and 'decimal mixed numbers or decimals.' +Drill in reading any number to thousandths. The work +will continue with gradual extension and refinement of the +understanding of decimals by learning how to operate with +them in various ways.</p> + +<p>Such may seem a slow progress, but in fact it is not, and<span class='pagenum'><a name="Page_183" id="Page_183">[Pg 183]</a></span> +many of these exercises whereby the pupil acquires his +mastery of decimals are useful as organizations and applications +of other arithmetical facts.</p> + +<p>That, it will be remembered, was the third principle:—"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:—</p> + +<p>Is <sup>2.75</sup>⁄<sub>25</sub> a common fraction?</p> + +<p>Is $.25 a decimal fraction?</p> + +<p>Is one <i>x</i>th of <i>y</i> a fraction?</p> + +<p>Can the same words mean both a common fraction and a +decimal fraction?</p> + +<p>Express 1 as a common fraction.</p> + +<p>Express 1 as a decimal fraction.</p> + +<p>These same children can, however, be taught to operate +correctly with fractions in the ordinary uses thereof. And +that is the chief value of arithmetic to them. They should +not be deprived of it because they cannot master its subtler +principles. So we seek to provide experiences that will +teach all pupils something of value, while stimulating in +those who have the ability the growth of abstract ideas and +general principles.</p> + +<p>Finally, we should bear in mind that working with qualities +and relations that are only partly understood or even +misunderstood does under certain conditions give control +over them. The general process of analytic learning in<span class='pagenum'><a name="Page_184" id="Page_184">[Pg 184]</a></span> +life is to respond as well as one can; to get a clearer idea +thereby; to respond better the next time; and so on. For +instance, one gets some sort of notion of what <sup>1</sup>⁄<sub>5</sub> means; +he then answers such questions as <sup>1</sup>⁄<sub>5</sub> of 10 = ? <sup>1</sup>⁄<sub>5</sub> of 5 = ? <sup>1</sup>⁄<sub>5</sub> +of 20 = ?; by being told when he is right and when he is +wrong, he gets from these experiences a better idea of <sup>1</sup>⁄<sub>5</sub>; +again he does his best with <sup>1</sup>⁄<sub>5</sub> = ⁄<sub>10</sub>, <sup>1</sup>⁄<sub>5</sub> = ⁄<sub>15</sub>, etc., and as before +refines and enlarges his concept of <sup>1</sup>⁄<sub>5</sub>. He adds <sup>1</sup>⁄<sub>5</sub> to <sup>2</sup>⁄<sub>5</sub>, etc., +<sup>1</sup>⁄<sub>5</sub> to <sup>3</sup>⁄<sub>10</sub>, etc., <sup>1</sup>⁄<sub>5</sub> to <sup>1</sup>⁄<sub>2</sub>, etc., and thereby gains still further, +and so on.</p> + +<p>What begins as a blind habit of manipulation started by +imitation may thus grow into the power of correct response +to the essential element. The pupil who has at the start +no notion at all of 'multiplying' may learn what multiplying +is by his experience that '4 6 multiplying gives 24'; +'3 9 multiplying gives 27,' etc. If the pupil keeps on +doing something with numbers and differentiates right +results, he will often reach in the end the abstractions which +he is supposed to need in the beginning. It may even be +the case with some of the abstractions required in arithmetic +that elaborate provision for comprehension beforehand is +not so efficient as the same amount of energy devoted partly +to provision for analysis itself beforehand and partly to +practice in response to the element in question without full +comprehension.</p> + +<p>It certainly is not the best psychology and not the best +educational theory to think that the pupil first masters a +principle and then merely applies it—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.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_185" id="Page_185">[Pg 185]</a></span></p> +<h2><a name="CHAPTER_X" id="CHAPTER_X"></a>CHAPTER X</h2> + +<h3>THE PSYCHOLOGY OF THINKING: REASONING IN +ARITHMETIC</h3> + + +<h4>THE ESSENTIALS OF ARITHMETICAL REASONING</h4> + +<p>We distinguish aimless reverie, as when a child dreams +of a vacation trip, from purposive thinking, as when he tries +to work out the answer to "How many weeks of vacation +can a family have for $120 if the cost is $22 a week for board, +$2.25 a week for laundry, and $1.75 a week for incidental +expenses, and if the railroad fares for the round trip are +$12?" We distinguish the process of response to familiar +situations, such as five integral numbers to be added, from +the process of response to novel situations, such as (for a +child who has not been trained with similar problems):—"A +man has four pieces of wire. The lengths are 120 yd., +132 meters, 160 feet, and <sup>1</sup>⁄<sub>8</sub> mile. How much more does +he need to have 1000 yd. in all?" We distinguish 'thinking +things together,' as when a diagram or problem or proof +is understood, from thinking of one thing after another as +when a number of words are spelled or a poem in an unknown +tongue is learned. In proportion as thinking is purposive, +with selection from the ideas that come up, and in +proportion as it deals with novel problems for which no +ready-made habitual response is available, and in proportion +as many bonds act together in an organized way to +produce response, we call it reasoning.<span class='pagenum'><a name="Page_186" id="Page_186">[Pg 186]</a></span></p> + +<p>When the conclusion is reached as the effect of many +particular experiences, the reasoning is called inductive. +When some principle already established leads to another +principle or to a conclusion about some particular fact, the +reasoning is called deductive. In both cases the process +involves the analysis of facts into their elements, the selection +of the elements that are deemed significant for the +question at hand, the attachment of a certain amount of +importance or weight to each of them, and their use in the +right relations. Thought may fail because it has not suitable +facts, or does not select from them the right ones, or +does not attach the right amount of weight to each, or does +not put them together properly.</p> + +<p>In the world at large, many of our failures in thinking +are due to not having suitable facts. Some of my readers, +for example, cannot solve the problem—"What are the +chances that in drawing a card from an ordinary pack of +playing-cards four times in succession, the same card will be +drawn each time?" And it will be probably because they do +not know certain facts about the theory of probabilities. +The good thinkers among such would look the matter up +in a suitable book. Similarly, if a person did not happen +to know that there were fifty-two cards in all and that no +two were alike, he could not reason out the answer, no matter +what his mastery of the theory of probabilities. If a competent +thinker, he would first ask about the size and nature of +the pack. In the actual practice of reasoning, that is, we have +to survey our facts to see if we lack any that are necessary. +If we do, the first task of reasoning is to acquire those +facts.</p> + +<p>This is specially true of the reasoning about arithmetical +facts in life. "Will 3½ yards of this be enough for a dress?" +Reason directs you to learn how wide it is, what style of<span class='pagenum'><a name="Page_187" id="Page_187">[Pg 187]</a></span> +dress you intend to make of it, how much material that +style normally calls for, whether you are a careful or a +wasteful cutter, and how big the person is for whom the +dress is to be made. "How much cheaper as a diet is bread +alone, than bread with butter added to the extent of 10% of +the weight of the bread?" Reason directs you to learn the +cost of bread, the cost of butter, the nutritive value of bread, +and the nutritive value of butter.</p> + +<p>In the arithmetic of the school this feature of reasoning +appears in cases where some fact about common measures +must be brought to bear, or some table of prices or discounts +must be consulted, or some business custom must be remembered +or looked up.</p> + +<p>Thus "How many badges, each 9 inches long, can be +made from 2½ yd. ribbon?" cannot be solved without +getting into mind 1 yd. = 36 inches. "At Jones' prices, +which costs more, 3¾ lb. butter or 6½ lb. lard? How much +more?" is a problem which directs the thinker to ascertain +Jones' prices.</p> + +<p>It may be noted that such problems are, other things +being equal, somewhat better training in thinking than +problems where all the data are given in the problem itself +(<i>e.g.</i>, "Which costs more, 3¾ lb. butter at 48¢ per lb. or +6½ lb. lard at 27¢ per lb.? How much more?"). At least +it is unwise to have so many problems of the latter sort that +the pupil may come to think of a problem in applied arithmetic +as a problem where everything is given and he has +only to manipulate the data. Life does not present its +problems so.</p> + +<p>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,<span class='pagenum'><a name="Page_188" id="Page_188">[Pg 188]</a></span> +can save a little over $20 per month, and can borrow from +your father at 6% interest?"</p> + +<div class="pblockquot"> +<p class="center">A</p> + +<p>A Reliable Piano. The Famous D.C.K. Upright. You pay +$50 cash down and $21 a month for only a year and a half. <i>No +interest</i> to pay. We ask you to pay only for the piano and allow +you plenty of time.</p> + +<p class="center">B</p> + +<p>We offer the well-known D.C.K. Piano for $390. $50 cash +and $20 a month thereafter. Regular interest at 6%. The +interest soon is reduced to less than $1 a month.</p> + +<p class="center">C</p> + +<p>The D.C.K. Piano. Special Offer, $375, cash. Compare our +prices with those of any reliable firm.</p></div> + +<p>If you consider chiefly the "only," "No interest to pay," +"only," and "plenty of time" in offer A, attaching much +weight to them and little to the thought, "How much will +$50 plus <span class="inline_eqn">(18 × $21)</span> be?", you will probably decide wrongly.</p> + +<p>The situations of life are often complicated by many elements +of little or even of no relevance to the correct solution. +The offerer of A may belong to your church; your dearest +friend may urge you to accept offer B; you may dislike +to talk with the dealer who makes offer C; you may have +a prejudice against owing money to a relative; that prejudice +may be wise or foolish; you may have a suspicion that the +B piano is shopworn; that suspicion may be well-founded +or groundless; the salesman for C says, "You don't want +your friends to say that you bought on the installment plan. +Only low-class persons do that," etc. The statement of +arithmetical problems in school usually assists the pupil to the +extent of ruling out all save definitely quantitative elements,<span class='pagenum'><a name="Page_189" id="Page_189">[Pg 189]</a></span> +and of ruling out all quantitative elements except those +which should be considered. The first of the two simplifications +is very beneficial, on the whole, since otherwise there +might be different correct solutions to a problem according +to the nature and circumstances of the persons involved. +The second simplification is often desirable, since it will often +produce greater improvement in the pupils, per hour of +time spent, than would be produced by the problems requiring +more selection. It should not, however, be a universal +custom; for in that case the pupils are tempted to +think that in every problem they must use all the quantities +given, as one must use all the pieces in a puzzle picture.</p> + +<p>It is obvious that the elements selected must not only be +right but also be in the right relations to one another. For +example, in the problems below, the 6 must be thought of +in relation to a dozen and as being half of a dozen, and also +as being 6 times 1. 1 must be mentally tied to "each." +The 6 as half of a dozen must be related to the $1.00, $1.60, +etc. The 6 as 6 times 1 must be related to the $.09, $.14, +etc.</p> + +<div class="pblockquot"> +<p class="center"><b>Buying in Quantity</b></p> + +<p>These are a grocer's prices for +certain things by the dozen +and for a single one. He sells +a half dozen at half the price +of a dozen. Find out how +much you save by buying 6 +all at one time instead of buying +them one at a time.</p></div> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><th> </th><th>Doz.</th><th>Each</th></tr> +<tr><td align='left'><b>1.</b> Evaporated Milk</td><td align='right'>$1.00</td><td align='right'>$.09</td></tr> +<tr><td align='left'><b>2.</b> Puffed Rice</td><td align='right'>1.60</td><td align='right'>.14</td></tr> +<tr><td align='left'><b>3.</b> Puffed Wheat</td><td align='right'>1.10</td><td align='right'>.10</td></tr> +<tr><td align='left'><b>4.</b> Canned Soup</td><td align='right'>1.90</td><td align='right'>.17</td></tr> +<tr><td align='left'><b>5.</b> Sardines</td><td align='right'>1.80</td><td align='right'>.16</td></tr> +<tr><td align='left'><b>6.</b> Beans (No. 2 cans)</td><td align='right'>1.50</td><td align='right'>.13</td></tr> +<tr><td align='left'><b>7.</b> Pork and Beans</td><td align='right'>1.70</td><td align='right'>.15</td></tr> +<tr><td align='left'><b>8.</b> Peas (No. 2 cans)</td><td align='right'>1.40</td><td align='right'>.12</td></tr> +<tr><td align='left'><b>9.</b> Tomatoes (extra cans)</td><td align='right'>3.20</td><td align='right'>.28</td></tr> +<tr><td align='left'><b>10.</b> Ripe olives (qt. cans)</td><td align='right'>7.20</td><td align='right'>.65</td></tr> +</table></div> + +<p>It is obvious also that in such arithmetical work as we<span class='pagenum'><a name="Page_190" id="Page_190">[Pg 190]</a></span> +have been describing, the pupil, to be successful, must +'think things together.' Many bonds must coöperate to +determine his final response.</p> + +<p>As a preface to reasoning about a problem we often have +the discovery of the problem and the classification of just +what it is, and as a postscript we have the critical inspection +of the answer obtained to make sure that it is verified by +experiment or is consistent with known facts. During the +process of searching for, selecting, and weighting facts, there +may be similar inspection and validation, item by item.</p> + + +<h4>REASONING AS THE COÖPERATION OF ORGANIZED HABITS</h4> + +<p>The pedagogy of the past made two notable errors in +practice based on two errors about the psychology of reasoning. +It considered reasoning as a somewhat magical power +or essence which acted to counteract and overrule the ordinary +laws of habit in man; and it separated too sharply +the 'understanding of principles' by reasoning from the +'mechanical' work of computation, reading problems, remembering +facts and the like, done by 'mere' habit and +memory.</p> + +<p>Reasoning or selective, inferential thinking is not at all +opposed to, or independent of, the laws of habit, but really +is their necessary result under the conditions imposed by +man's nature and training. A closer examination of selective +thinking will show that no principles beyond the +laws of readiness, exercise, and effect are needed to explain +it; that it is only an extreme case of what goes on in associative +learning as described under the 'piecemeal' +activity of situations; and that attributing certain features +of learning to mysterious faculties of abstraction or reasoning +gives no real help toward understanding or controlling +them.<span class='pagenum'><a name="Page_191" id="Page_191">[Pg 191]</a></span></p> + +<p>It is true that man's behavior in meeting novel problems +goes beyond, or even against, the habits represented by +bonds leading from gross total situations and customarily +abstracted elements thereof. One of the two reasons therefor, +however, is simply that the finer, subtle, preferential +bonds with subtler and less often abstracted elements go +beyond, and at times against, the grosser and more usual +bonds. One set is as much due to exercise and effect as the +other. The other reason is that in meeting novel problems +the mental set or attitude is likely to be one which rejects +one after another response as their unfitness to satisfy a +certain desideratum appears. What remains as the apparent +course of thought includes only a few of the many +bonds which did operate, but which, for the most part, were +unsatisfying to the ruling attitude or adjustment.</p> + +<p>Successful responses to novel data, associations by similarity +and purposive behavior are in only apparent opposition +to the fundamental laws of associative learning. Really +they are beautiful examples of it. Man's successful responses +to novel data—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.</p> + +<p>Nothing is less like the mysterious operations of a faculty +of reasoning transcending the laws of connection-forming, +than the behavior of men in response to novel situations. +Let children who have hitherto confronted only such arithmetical +tasks, in addition and subtraction with one- and +two-place numbers and multiplication with one-place numbers, +as those exemplified in the first line below, be told to +do the examples shown in the second line.<span class='pagenum'><a name="Page_192" id="Page_192">[Pg 192]</a></span></p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" width="90%"> +<tr><td align='center'><span class="smcap">Add</span></td><td align='center'><span class="smcap">Add</span></td><td align='center'><span class="smcap">Add</span></td><td align='center'><span class="smcap">Subt.</span></td><td align='center'><span class="smcap">Subt.</span></td><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td></tr> +<tr> + <td align='center'>8<br />5<br />—</td> + <td align='center'>37<br />24<br />—</td> + <td align='center'>35<br />68<br />23<br />19<br />—</td> + <td align='center'>8<br />5<br />—</td> + <td align='center'>37<br />24<br />—</td> + <td align='center'>8<br />5<br />—</td> + <td align='center'>9<br />7<br />—</td> + <td align='center'>6<br />3<br />—</td> +</tr> +<tr><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td><td align='center'><span class="smcap">Multiply</span></td></tr> +<tr> + <td align='center'>32<br />23<br />—</td> + <td align='center'>43<br />22<br />—</td> + <td align='center'>34<br />26<br />—</td> +</tr> +</table></div> + +<p>They will add the numbers, or subtract the lower from the +upper number, or multiply <span class="inline_eqn">3 × 2</span> and <span class="inline_eqn">2 × 3,</span> etc., getting 66, +86, and 624, or respond to the element of 'Multiply' attached +to the two-place numbers by "I can't" or "I don't +know what to do," or the like; or, if one is a child of great +ability, he may consider the 'Multiply' element and the +bigness of the numbers, be reminded by these two aspects +of the situation of the fact that</p> + +<p class="noidt"> +'9<br /> + 9 multiply'<br /> +— +</p> + +<p class="noidt">gave only 81, and that</p> + +<p class="noidt"> +'10<br /> + 10 multiply'<br /> +—— +</p> + +<p class="noidt">gave only 100, or the like; and +so may report an intelligent and justified "I can't," or reject +the plan of <span class="inline_eqn">3 × 2</span> and <span class="inline_eqn">2 × 3,</span> +with 66, 86, and 624 for answers, +as unsatisfactory. What the children will do will, in every +case, be a product of the elements in the situation that are +potent with them, the responses which these evoke, and the +further associates which these responses in turn evoke. If +the child were one of sufficient genius, he might infer the +procedure to be followed as a result of his knowledge of the +principles of decimal notation and the meaning of 'Multiply,' +responding correctly to the 'place-value' element +of each digit and adding his 6 tens and 9 tens, 20 twos and +3 thirties; but if he did thus invent the shorthand addition +of a collection of twenty-three collections, each of 32 units, +he would still do it by the operation of bonds, subtle but real.<span class='pagenum'><a name="Page_193" id="Page_193">[Pg 193]</a></span></p> + +<p>Association by similarity is, as James showed long ago, +simply the tendency of an element to provoke the responses +which have been bound to it. <i>abcde</i> leads to <i>vwxyz</i> because +<i>a</i> has been bound to <i>vwxyz</i> by original nature, exercise, or +effect.</p> + +<p>Purposive behavior is the most important case of the +influence of the attitude or set or adjustment of an organism +in determining (1) which bonds shall act, and (2) which +results shall satisfy. James early described the former fact, +showing that the mechanism of habit can give the directedness +or purposefulness in thought's products, provided that +mechanism includes something paralleling the problem, the +aim, or need, in question.</p> + +<p>The second fact, that the set or attitude of the man helps +to determine which bonds shall satisfy, and which shall +annoy, has commonly been somewhat obscured by vague +assertions that the selection and retention is of what is +"in point," or is "the right one," or is "appropriate," or the +like. It is thus asserted, or at least hinted, that "the will," +"the voluntary attention," "the consciousness of the +problem," and other such entities are endowed with magic +power to decide what is the "right" or "useful" bond and +to kill off the others. The facts are that in purposive thinking +and action, as everywhere else, bonds are selected and +retained by the satisfyingness, and are killed off by the discomfort, +which they produce; and that the potency of the +man's set or attitude to make this satisfy and that annoy—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.</p> + +<p>Reasoning is not a radically different sort of force operating +against habit but the organization and coöperation of +many habits, thinking facts together. Reasoning is not<span class='pagenum'><a name="Page_194" id="Page_194">[Pg 194]</a></span> +the negation of ordinary bonds, but the action of many of +them, especially of bonds with subtle elements of the situation. +Some outside power does not enter to select and +criticize; the pupil's own total repertory of bonds relevant +to the problem is what selects and rejects. An unsuitable +idea is not killed off by some <i>actus purus</i> of intellect, but by +the ideas which it itself calls up, in connection with the total +set of mind of the pupil, and which show it to be inadequate.</p> + +<p>Almost nothing in arithmetic need be taught as a matter +of mere unreasoning habit or memory, nor need anything, +first taught as a principle, ever become a matter of mere +habit or memory. <span class="inline_eqn">5 × 4 = 20</span> should not be learned as an +isolated fact, nor remembered as we remember that Jones' +telephone number is 648 J 2. Almost everything in arithmetic +should be taught as a habit that has connections with +habits already acquired and will work in an organization +with other habits to come. The use of this organized +hierarchy of habits to solve novel problems is reasoning.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_195" id="Page_195">[Pg 195]</a></span></p> +<h2><a name="CHAPTER_XI" id="CHAPTER_XI"></a>CHAPTER XI</h2> + +<h3>ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE +SCHOOL</h3> + + +<h4>THE UTILIZATION OF INSTINCTIVE INTERESTS</h4> + +<p>The activities essential to acquiring ability in arithmetic +can rely on little in man's instinctive equipment beyond the +purely intellectual tendencies of curiosity and the satisfyingness +of thought for thought's sake, and the general enjoyment +of success rather than failure in an enterprise to +which one sets oneself. It is only by a certain amount +of artifice that we can enlist other vehement inborn interests +of childhood in the service of arithmetical knowledge +and skill. When this can be done at no cost the +gain is great. For example, marching in files of two, in files +of three, in files of four, etc., raising the arms once, two +times, three times, showing a foot, a yard, an inch with +the hands, and the like are admirable because learning +the meanings of numbers thus acquires some of the zest +of the passion for physical action. Even in late grades +chances to make pictures showing the relations of fractional +parts, to cut strips, to fold paper, and the like will +be useful.</p> + +<p>Various social instincts can be utilized in matches after +the pattern of the spelling match, contests between rows, +certain number games, and the like. The scoring of both<span class='pagenum'><a name="Page_196" id="Page_196">[Pg 196]</a></span> +the play and the work of the classroom is a useful field for +control by the teacher of arithmetic.</p> + +<p>Hunt ['12] has noted the more important games which +have some considerable amount of arithmetical training as +a by-product and which are more or less suitable for class +use. Flynn ['12] has described games, most of them for home +use, which give very definite arithmetical drill, though in +many cases the drills are rather behind the needs of children +old enough to understand and like the game itself.</p> + +<p>It is possible to utilize the interests in mystery, tricks, +and puzzles so as to arouse a certain form of respect for +arithmetic and also to get computational work done. I +quote one simple case from Miss Selkin's admirable collection +['12, p. 69 f.]:—</p> + + +<p class="tabcap">I. <span class="smcap">ADDITION</span></p> +<div class="pblockquot"> + +<p>"We must admit that there is nothing particularly interesting +in a long column of numbers to be added. Let the teacher, however, +suggest that he can write the answer at sight, and the task +will assume a totally different aspect.</p> + +<p>"A very simple number trick of this kind can be performed by +making use of the principle of complementary addition. The +arithmetical complement of a number with respect to a larger +number is the difference between these two numbers. Most +interesting results can be obtained by using complements with +respect to 9.</p> + +<p>"The children may be called upon to suggest several numbers of +two, three, or more digits. Below these write an equal number of +addends and immediately announce the answer. The children, +impressed by this apparently rapid addition, will set to work +enthusiastically to test the results of this lightning calculation.</p> + +<p>"Example:—</p></div> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='right'>357 </td><td align='left' rowspan='3' valign='middle'><span class="sz30">}</span></td><td></td><td align='right'>999</td></tr> +<tr><td align='right'>682 </td><td align='left'>A </td><td align='right'>× 3</td></tr> +<tr><td align='right'>793 </td><td align='left'></td><td align='right'><span class="overline">2997</span></td></tr> +<tr><td align='right'> </td></tr> +<tr><td align='right'>642 </td><td align='left' rowspan='3' valign='middle'><span class="sz30">}</span></td></tr> +<tr><td align='right'>317 </td><td align='left'>B</td></tr> +<tr><td align='right'>206 </td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_197" id="Page_197">[Pg 197]</a></span></p> +<div class="pblockquot"> +<p>"Explanation:—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."</p></div> + +<p>Respect for arithmetic as a source of tricks and magic is +very much less important than respect for its everyday +services; and computation to test such tricks is likely to be +undertaken zealously only by the abler pupils. Consequently +this source of interest should probably be used only +sparingly, and perhaps the teacher should give such exhibitions +only as a reward for efficiency in the regular work. +For example, if the work for a week is well done in four +days the fifth day might be given up to some semi-arithmetical +entertainment, such as the demonstration of an +adding machine, the story of primitive methods of counting, +team races in computation, an exhibition of lightning +calculation and intellectual sleight-of-hand by the teacher, +or the voluntary study of arithmetical puzzles.</p> + +<p>The interest in achievement, in success, mentioned above +is stronger in children than is often realized and makes advisable +the systematic use of the practice experiment as a +method of teaching much of arithmetic. Children who thus +compete with their own past records, keeping an exact score +from week to week, make notable progress and enjoy hard +work in making it.</p> + + +<h4>THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES</h4> + +<p>Negatively the difficulty of the work that pupils should +be expected to do is conditioned by the gradual maturing<span class='pagenum'><a name="Page_198" id="Page_198">[Pg 198]</a></span> +of their capacities. Other things being equal, the common +custom of reserving hard things for late in the elementary +school course is, of course, sound. It seems probable that +little is gained by using any of the child's time for arithmetic +before grade 2, though there are many arithmetical facts +that he can learn in grade 1. Postponement of systematic +work in arithmetic to grade 3 or even grade 4 is allowable +if better things are offered. With proper textbooks and +oral and written exercises, however, a child in grades 2 and +3 can spend time profitably on arithmetical work. When +all children can be held in school through the eighth grade +it does not much matter whether arithmetic is begun early +or late. If, however, many children are to leave in grades 5 +and 6 as now, we may think it wise to provide somehow that +certain minima of arithmetical ability be given them.</p> + +<p>There are, so far as is known, no special times and seasons +at which the human animal by inner growth is specially ripe +for one or another section or aspect of arithmetic, except +in so far as the general inner growth of intellectual powers +makes the more abstruse and complex tasks suitable to +later and later years.</p> + +<p>Indeed, very few of even the most enthusiastic devotees +of the recapitulation theory or culture-epoch theory have +attempted to apply either to the learning of arithmetic, and +Branford is the only mathematician, so far as I know, who +has advocated such application, even tempered by elaborate +shiftings and reversals of the racial order. He says:—</p> + +<div class="pblockquot"><p>"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 Ar<span class='pagenum'><a name="Page_199" id="Page_199">[Pg 199]</a></span>chæan +mathematics passing on through Greek and Hindu to mediæval +European mathematics; while the student is become sufficiently +mature to begin the assimilation of modern and highly +abstract European thought. The filling in of details must necessarily +be left to the individual teacher, and also, within some such +broadly marked limits, the precise order of the marshalling of the +material for each age. For, though, on the whole, mathematical +development has gone forward, yet there have been lapses from +advances already made. Witness the practical world-loss of much +valuable Hindu thought, and, for long centuries, the neglect of +Greek thought: witness the world-loss of the invention by the +Babylonians of the Zero, until re-invented by the Hindus, passed +on by them to the Arabs, and by these to Europe.</p> + +<p>"Moreover, many blunders and false starts and false principles +have marked the whole course of development. In a phrase, +rivers have their backwaters. But it is precisely the teacher's +function to avoid such racial mistakes, to take short cuts ultimately +discovered, and to guide the young along the road ultimately +found most accessible with such halts and retracings—returns +up side-cuts—as the mental peculiarities of the pupils +demand.</p> + +<p>"All this, the practical realization of the spirit of the principle, +is to be wisely left to the mathematical teacher, familiar with the +history of mathematical science and with the particular limitations +of his pupils and himself." ['08, p. 245.]</p></div> + +<p>The latitude of modification suggested by Branford reduces +the guidance to be derived from racial history to +almost <i>nil</i>. Also it is apparent that the racial history in +the case of arithmetical achievement is entirely a matter of +acquisition and social transmission. Man's original nature +is destitute of all arithmetical ideas. The human germs +do not know even that one and one make two!</p> + + +<h4>INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL</h4> + +<p>A scientific plan for teaching arithmetic would begin with +an exact inventory of the knowledge and skill which the +pupils already possessed. Our ordinary notions of what a +child knows at entrance to grade 1, or grade 2, or grade 3,<span class='pagenum'><a name="Page_200" id="Page_200">[Pg 200]</a></span> +and of what a first-grade child or second-grade child can do, +are not adequate. If they were, we should not find reputable +textbooks arranging to teach elaborately facts already +sufficiently well known to over three quarters of the pupils +when they enter school. Nor should we find other textbooks +presupposing in their first fifty pages a knowledge +of words which not half of the children can read even +at the end of the 2 B grade.</p> + +<p>We do find just such evidence that ordinary ideas about +the abilities of children at the beginning of systematic school +training in arithmetic may be in gross error. For example, a +reputable and in many ways admirable recent book has +fourteen pages of exercises to teach the meaning of two and +the fact that one and one make two! As an example of the +reverse error, consider putting all these words in the first +twenty-five pages of a beginner's book:—<i>absentees, attendance, +blanks, continue, copy, during, examples, grouped, memorize, +perfect, similar, splints, therefore, total</i>!</p> + +<p>Little, almost nothing, has been done toward providing +an exact inventory compared with what needs to be done. +We may note here (1) the facts relevant to arithmetic found +by Stanley Hall, Hartmann, and others in their general investigations +of the knowledge possessed by children at entrance +to school, (2) the facts concerning the power of children +to perceive differences in length, area, size of collection, +and organization within a collection such as is shown in +Fig. 24, and certain facts and theories about early awareness +of number.</p> + +<p>In the Berlin inquiry of 1869, knowledge of the meaning +of two, three, and four appeared in 74, 74, and 73 percent of +the children upon entrance to school. Some of those recorded +as ignorant probably really knew, but failed to understand +that they were expected to reply or were shy. Only<span class='pagenum'><a name="Page_201" id="Page_201">[Pg 201]</a></span> +85 percent were recorded as knowing their fathers' names. +Seven eighths as many children knew the meanings of two, +three, and four as knew their fathers' names. In a similar +but more careful experiment with Boston children in September, +1880, Stanley Hall found that 92 percent knew three, +83 percent knew four, and 71½ percent knew five. Three +was known about as well as the color red; four was known +about as well as the color blue or yellow or green. Hartmann +['90] found that two thirds of the children entering school +in Annaberg could count from one to ten. This is about as +many as knew money, or the familiar objects of the town, +or could repeat words spoken to them.</p> + +<div class="figcenter" style="width: 661px;"> +<img src="images/fig24.jpg" width="661" height="600" alt="Fig. 24." title="Fig. 24." /> +<span class="caption"><span class="smcap">Fig. 24.</span>—Objective presentation.</span> +</div> + +<p>In the Stanford form of the Binet tests counting four +pennies is given as an ability of the typical four-year-old. +Counting 13 pennies correctly in at least one out of two<span class='pagenum'><a name="Page_202" id="Page_202">[Pg 202]</a></span> +trials, and knowing three of the four coins,—penny, nickel, +dime, and quarter,—are given as abilities of the typical +six-year-old.</p> + + +<h4>THE PERCEPTION OF NUMBER AND QUANTITY</h4> + +<p>We know that educated adults can tell how many lines or +dots, etc., they see in a single glance (with an exposure too +short for the eye to move) up to four or more, according to +the clearness of the objects and their grouping. For example, +Nanu ['04] reports that when a number of bright circles on +a dark background are shown to educated adults for only +.033 second, ten can be counted when arranged to form a +parallelogram, but only five when arranged in a row. With +certain groupings, of course, their 'perception' involves +much inference, even conscious addition and multiplication. +Similarly they can tell, up to twenty and beyond, the +number of taps, notes, or other sounds in a series too rapid +for single counting if the sounds are grouped in a convenient +rhythm.</p> + +<p>These abilities are, however, the product of a long and +elaborate learning, including the learning of arithmetic +itself. Elementary psychology and common experience +teach us that the mere observation of groups or quantities, +no matter how clear their number quality appears to the +person who already knows the meanings of numbers, does +not of itself create the knowledge of the meanings of numbers +in one who does not. The experiments of Messenger ['03] +and Burnett ['06] showed that there is no direct intuitive +apprehension even of two as distinct from one. We have to +<i>learn</i> to feel the two touches or see the two dots or lines as two.</p> + +<div class="figcenter" style="width: 640px;"> +<img src="images/img220.jpg" width="640" height="445" alt="" title="" /> +</div> + +<p>We do not know by exact measurements the growth in +children of this ability to count or infer the number of elements +in a collection seen or series heard. Still less do we<span class='pagenum'><a name="Page_203" id="Page_203">[Pg 203]</a></span> +know what the growth would be without the influence of +school training in counting, grouping, adding, and multiplying. +Many textbooks and teachers seem to overestimate +it greatly. Not all educated adults can, apart from measurement, +decide with surety which of these lines is the longer, +or which of these areas is the larger, or whether this is a +ninth or a tenth or an eleventh of a circle.</p> + + +<div class="figleft" style="width: 480px;"> +<img src="images/img221.jpg" width="480" height="594" alt="" title="" /> +</div> + +<p>Children upon entering school have not been tested carefully +in respect to judgments of length and area, but we +know from such studies as Gilbert's ['94] that the difference +required in their case is probably over twice that required +for children of 13 or 14. In judging weights, for example, +a difference of 6 is perceived as easily by children 13 to +15 years of age as a difference of 15 by six-year-olds.</p> + +<p>A teacher who has adult powers of estimating length or +area or weight and who also knows already which of the two<span class='pagenum'><a name="Page_204" id="Page_204">[Pg 204]</a></span> +is longer or larger or heavier, may use two lines to illustrate +a difference which they really hide from the child. +It is unlikely, for example, that the first of these lines +______________ ________________ would be recognized as +shorter than the second by every child in a fourth-grade class, +and it is extremely unlikely that it would be recognized as +being <sup>7</sup>⁄<sub>8</sub> of the length of the latter, rather than <sup>3</sup>⁄<sub>4</sub> of it or <sup>5</sup>⁄<sub>6</sub> of +it or <sup>9</sup>⁄<sub>10</sub> of it or <sup>11</sup>⁄<sub>12</sub> of it. If the two were shown to a second +grade, with the question, "The first line is 7. How long +is the other line?" there would be very many answers +of 7 or 9; and these might be entirely correct arithmetically, +the pupils' errors being all due to their inability +to compare the lengths accurately.</p> + +<p>The quantities used +should be such that their +mere discrimination offers +no difficulty even +to a child of blunted +sense powers. If <sup>7</sup>⁄<sub>8</sub> and +1 are to be compared, +<i>A</i> and <i>B</i> are not allowable. +<i>C</i>, <i>D</i>, and <i>E</i> are +much better.</p> + + +<p>Teachers probably often +underestimate or +neglect the sensory +difficulties of the tasks +they assign and of the +material they use to +illustrate absolute +and relative magnitudes. +The result may be more pernicious when the pupils +answer correctly than when they fail. For their correct +<span class='pagenum'><a name="Page_205" id="Page_205">[Pg 205]</a></span> +answering may be due to their divination of what the +teacher wants; and they may call a thing an inch larger +to suit her which does not really seem larger to them +at all. This, of course, is utterly destructive of their respect +for arithmetic as an exact and matter-of-fact instrument. +For example, if a teacher drew a series of lines 20, 21, 22, 23, +24, and 25 inches long on the blackboard in this form— +_____ ________ and asked, "This is 20 inches long, how long +is this?" she might, after some errors and correction thereof, +finally secure successful response to all the lines by all the +children. But their appreciation of the numbers 20, 21, 22, +23, 24, and 25 would be actually damaged by the exercise.</p> + + +<h4>THE EARLY AWARENESS OF NUMBER</h4> + +<p>There has been some disagreement concerning the origin +of awareness of number in the individual, in particular concerning +the relative importance of the perception of how-many-ness +and that of how-much-ness, of the perception +of a defined aggregate and the perception of a defined ratio. +(See McLellan and Dewey ['95], Phillips ['97 and '98], and +Decroly and Degand ['12].)</p> + +<p>The chief facts of significance for practice seem to be these: +(1) Children with rare exceptions hear the names <i>one</i>, <i>two</i>, +<i>three</i>, <i>four</i>, <i>half</i>, <i>twice</i>, <i>two times</i>, <i>more</i>, <i>less</i>, <i>as many as</i>, <i>again</i>, +<i>first</i>, <i>second</i>, and <i>third</i>, long before they have analyzed out +the qualities and relations to which these words refer so as +to feel them at all clearly. (2) Their knowledge of the qualities +and relations is developed in the main in close association +with the use of these words to the child and by the child. +(3) The ordinary experiences of the first five years so develop +in the child awareness of the 'how many somethings' in +various groups, of the relative magnitudes of two groups or +quantities of any sort, and of groups and magnitudes as<span class='pagenum'><a name="Page_206" id="Page_206">[Pg 206]</a></span> +related to others in a series. For instance, if fairly gifted, +a child comes, by the age of five, to see that a row of four +cakes is an aggregate of four, seeing each cake as a part of +the four and the four as the sum of its parts, to know that +two of them are as many as the other two, that half of them +would be two, and to think, when it is useful for him to do +so, of four as a step beyond three on the way to five, or to +think of hot as a step from warm on the way to very hot. +The degree of development of these abilities depends upon +the activity of the law of analysis in the individual and the +character of his experiences.</p> + +<p>(4) He gets certain bad habits of response from the +ambiguity of common usage of 2, 3, 4, etc., for second, third, +fourth. Thus he sees or hears his parents or older children +or others count pennies or rolls or eggs by saying one, two, +three, four, and so on. He himself is perhaps misled into so +counting. Thus the names properly belonging to a series +of aggregations varying in amount come to be to him the +names of the positions of the parts in a counted whole. +This happens especially with numbers above 3 or 4, where the +correct experience of the number as a name for the group +has rarely been present. This attaching to the cardinal +numbers above three or four the meanings of the ordinal +numbers seems to affect many children on entrance to +school. The numbering of pages in books, houses, streets, +etc., and bad teaching of counting often prolong this error.</p> + +<p>(5) He also gets the habit, not necessarily bad, but often +indirectly so, of using many names such as eight, nine, ten, +eleven, fifteen, a hundred, a million, without any meaning.</p> + +<p>(6) The experiences of half, twice, three times as many, +three times as long, etc., are rarer; even if they were not, +they would still be less easily productive of the analysis of +the proper abstract element than are the experiences of<span class='pagenum'><a name="Page_207" id="Page_207">[Pg 207]</a></span> +two, three, four, etc., in connection with aggregates of things +each of which is usually called one, such as boys, girls, balls, +apples. Experiences of the names, two, three, and four, in +connection with two twos, two threes, two fours, are very +rare.</p> + +<p>Hence, the names, two, three, etc., mean to these children +in the main, "one something and one something," "one +something usually called one, and one something usually +called one, and another something usually called one," and +more rarely and imperfectly "two times anything," "three +times anything," etc.</p> + +<p>With respect to Mr. Phillips' emphasis of the importance +of the series-idea in children's minds, the matters of importance +are: first, that the knowledge of a series of number +names in order is of very little consequence to the teaching +of arithmetic and of still less to the origin of awareness of +number. Second, the habit of applying this series of words +in counting in such a way that 8 is associated with the eighth +thing, 9 with the ninth thing, etc., is of consequence because +it does so much mischief. Third, the really valuable idea of +the number series, the idea of a series of groups or of magnitudes +varying by steps, is acquired later, as a result, not a +cause, of awareness of numbers.</p> + +<p>With respect to the McLellan-Dewey doctrine, the ratio +aspect of numbers should be emphasized in schools, not +because it is the main origin of the child's awareness of +number, but because it is <i>not</i>, and because the ordinary practical +issues of child life do <i>not</i> adequately stimulate its action. +It also seems both more economical and more scientific to +introduce it through multiplication, division, and fractions +rather than to insist that 4 and 5 shall from the start mean +4 or 5 times anything that is called 1, for instance, that 8 +inches shall be called 4 two-inches, or 10 cents, 5 two-cents. +<span class='pagenum'><a name="Page_208" id="Page_208">[Pg 208]</a></span> +If I interpret Professor Dewey's writings correctly, he would +agree that the use of inch, foot, yard, pint, quart, ounce, +pound, glassful, cupful, handful, spoonful, cent, nickel, +dime, and dollar gives a sufficient range of units for the +first two school years. Teaching the meanings of ½ of 4, +½ of 6, ½ of 8, ½ of 10, ½ of 20, +<sup>1</sup>⁄<sub>3</sub> of 6, <sup>1</sup>⁄<sub>3</sub> of 9, <sup>1</sup>⁄<sub>3</sub> of 30, ¼ of 8, +two 2s, five 2s, and the like, in early grades, each in connection +with many different units of measure, provides a +sufficient assurance that numbers will connect with relationships +as well as with collections.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_209" id="Page_209">[Pg 209]</a></span></p> +<h2><a name="CHAPTER_XII" id="CHAPTER_XII"></a>CHAPTER XII</h2> + +<h3>INTEREST IN ARITHMETIC</h3> + + +<h4>CENSUSES OF PUPILS' INTERESTS</h4> + +<p>Arithmetic, although it makes little or no appeal to +collecting, muscular manipulation, sensory curiosity, or +the potent original interests in things and their mechanisms +and people and their passions, is fairly well liked by children. +The censuses of pupils' likes and dislikes that have been +made are not models of scientific investigation, and the +resulting percentages should not be used uncritically. +They are, however, probably not on the average over-favorable +to arithmetic in any unfair way. Some of their +results are summarized below. In general they show arithmetic +to be surpassed in interest clearly by only the manual +arts (shopwork and manual training for boys, cooking and +sewing for girls), drawing, certain forms of gymnastics, and +history. It is about on a level with reading and science. +It clearly surpasses grammar, language, spelling, geography, +and religion.</p> + +<p>Lobsien ['03], who asked one hundred children in +each of the first five grades (<i>Stufen</i>) of the elementary +schools of Kiel, "Which part of the school work (literally, +'which instruction period') do you like best?" found +arithmetic led only by drawing and gymnastics in the +case of the boys, and only by handwork in the case of the +girls.<span class='pagenum'><a name="Page_210" id="Page_210">[Pg 210]</a></span></p> + +<p>This is an exaggerated picture of the facts, since no count +is made of those who especially dislike arithmetic. Arithmetic +is as unpopular with some as it is popular with others. +When full allowance is made for this, arithmetic still has +popularity above the average. Stern ['05] asked, "Which +subject do you like most?" and "Which subject do you +like least?" The balance was greatly in favor of gymnastics +for boys (28-1), handwork for girls (32-1½), and drawing +for both (16½-6). Writing (6½-4), arithmetic (14½-13), +history (9-6½), reading (8½-8), and singing (6-7½) come +next. Religion, nature study, physiology, geography, geometry, +chemistry, language, and grammar are low.</p> + +<p>McKnight ['07] found with boys and girls in grades 7 and +8 of certain American cities that arithmetic was liked better +than any of the school subjects except gymnastics and +manual training. The vote as compared with history +was:—</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="4" summary=""> +<tr><td align='left'>Arithmetic</td><td align='left'>327 liked greatly,</td><td align='left'> 96 disliked greatly.</td></tr> +<tr><td align='left'>History</td><td align='left'>164 liked greatly,</td><td align='left'>113 disliked greatly.</td></tr> +</table></div> + +<p>In a later study Lobsien ['09] had 6248 pupils from 9 to +15 years old representing all grades of the elementary school +report, so far as they could, the subject most disliked, the +subject most liked, the subject next most liked, and the +subject next in order. No child was forced to report all of +these four judgments, or even any of them. Lobsien counts +the likes and the dislikes for each subject. Gymnastics, +handwork, and cooking are by far the most popular. History +and drawing are next, followed by arithmetic and reading. +Below these are geography, writing, singing, nature study, +biblical history, catechism, and three minor subjects.</p> + +<p>Lewis ['13] secured records from English children in elementary +schools of the order of preference of all the studies<span class='pagenum'><a name="Page_211" id="Page_211">[Pg 211]</a></span> +listed below. He reports the results in the following +table of percents:</p> + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="5" summary="" rules="cols"> +<tr><th class="bbt"> </th> +<th class="bbt"><span class="smcap">Top Third<br />of Studies for Interest</span></th> +<th class="bbt"><span class="smcap">Middle Third<br />of Studies for Interest</span></th> +<th class="bbt"><span class="smcap">Lowest Third<br />of Studies for Interest</span></th> +</tr> +<tr><td align='left'>Drawing</td><td align='center'>78</td><td align='center'>20</td><td align='center'> 2</td></tr> +<tr><td align='left'>Manual Subjects</td><td align='center'>66</td><td align='center'>26</td><td align='center'> 8</td></tr> +<tr><td align='left'>History</td><td align='center'>64</td><td align='center'>24</td><td align='center'>12</td></tr> +<tr><td align='left'>Reading</td><td align='center'>53</td><td align='center'>38</td><td align='center'> 9</td></tr> +<tr><td align='left'>Singing</td><td align='center'>32</td><td align='center'>48</td><td align='center'>20</td></tr> +<tr><td></td></tr> +<tr><td align='left'>Drill</td><td align='center'>20</td><td align='center'>55</td><td align='center'>25</td></tr> +<tr><td align='left'>Arithmetic</td><td align='center'>16</td><td align='center'>53</td><td align='center'>31</td></tr> +<tr><td align='left'>Science</td><td align='center'>23</td><td align='center'>37</td><td align='center'>40</td></tr> +<tr><td align='left'>Nature Study</td><td align='center'>16</td><td align='center'>36</td><td align='center'>48</td></tr> +<tr><td align='left'>Dictation</td><td align='center'> 4</td><td align='center'>57</td><td align='center'>39</td></tr> +<tr><td></td></tr> +<tr><td align='left'>Composition</td><td align='center'>18</td><td align='center'>28</td><td align='center'>54</td></tr> +<tr><td align='left'>Scripture</td><td align='center'> 4</td><td align='center'>38</td><td align='center'>58</td></tr> +<tr><td align='left'>Recitation</td><td align='center'> 9</td><td align='center'>23</td><td align='center'>68</td></tr> +<tr><td align='left'>Geography</td><td align='center'> 4</td><td align='center'>24</td><td align='center'>72</td></tr> +<tr><td align='left'>Grammar</td><td align='center'>—</td><td align='center'> 6</td><td align='center'>94</td></tr> +<tr><td class="bb"></td><td class="bb"></td><td class="bb"></td><td class="bb"></td></tr> +</table></div> + +<p>Brandell ['13] obtained data from 2137 Swedish children +in Stockholm (327), Norrköping (870), and Gothenburg +(940).</p> + +<p>In general he found, as others have, that handwork, shopwork +for boys and household work for girls, and drawing +were reported as much better liked than arithmetic. So also +was history, and (in this he differs from most students of +this matter) so were reading and nature study. Gymnastics +he finds less liked than arithmetic. Religion, geography, +language, spelling, and writing are, as in other studies, much +less popular than arithmetic.<span class='pagenum'><a name="Page_212" id="Page_212">[Pg 212]</a></span></p> + +<p>Other studies are by Lilius ['11] in Finland, Walsemann +['07], Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern +['13 and '14], in Germany. They confirm the general results +stated.</p> + +<p>The reasons for the good showing that arithmetic makes +are probably the strength of its appeal to the interest in +definite achievement, success, doing what one attempts to +do; and of its appeal, in grades 5 to 8, to the practical +interest of getting on in the world, acquiring abilities that +the world pays for. Of these, the former is in my opinion +much the more potent interest. Arithmetic satisfies it especially +well, because, more than any other of the 'intellectual' +studies of the elementary school, it permits the +pupil to see his own progress and determine his own success +or failure.</p> + +<p>The most important applications of the psychology of +satisfiers and annoyers to arithmetic will therefore be in +the direction of utilizing still more effectively this interest +in achievement. Next in importance come the plans to +attach to arithmetical learning the satisfyingness of bodily +action, play, sociability, cheerfulness, and the like, and of +significance as a means of securing other desired ends than +arithmetical abilities themselves. Next come plans to relieve +arithmetical learning from certain discomforts such as +the eyestrain of some computations and excessive copying +of figures. These will be discussed here in the inverse +order.</p> + + +<h4>RELIEVING EYESTRAIN</h4> + +<p>At present arithmetical work is, hour for hour, probably +more of a tax upon the eyes than reading. The task of +copying numbers from a book to a sheet of paper is one of +the very hardest tasks that the eyes of a pupil in the ele<span class='pagenum'><a name="Page_213" id="Page_213">[Pg 213]</a></span>mentary +schools have to perform. A certain amount of +such work is desirable to teach a child to write numbers, to +copy exactly, and to organize material in shape for computation. +But beyond that, there is no more reason for a pupil +to copy every number with which he is to compute than for +him to copy every word he is to read. The meaningless +drudgery of copying figures should be mitigated by arranging +much work in the form of exercises like those shown on pages +216, 217, and 218, and by having many of the textbook +examples in addition, subtraction, and multiplication done +with a slip of paper laid below the numbers, the answers +being written on it. There is not only a resulting gain in +interest, but also a very great saving of time for the pupil +(very often copying an example more than quadruples the +time required to get its answer), and a much greater efficiency +in supervision. Arithmetical errors are not confused with +errors of copying,<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a> and the teacher's task of following a +pupil's work on the page is reduced to a minimum, each +pupil having put the same part of the day's work in just the +same place. The use of well-printed and well-spaced pages +of exercises relieves the eyestrain of working with badly +made gray figures, unevenly and too closely or too widely +spaced. I reproduce in Fig. 25 specimens taken at random +from one hundred random samples of arithmetical work by +pupils in grade 8. Contrast the task of the eyes in working +with these and their task in working with pages 216 to 218. +The customary method of always copying the numbers +to be used in computation from blackboard or book to +a sheet of paper is an utterly unjustifiable cruelty and +waste.<span class='pagenum'><a name="Page_214" id="Page_214">[Pg 214]</a></span></p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/fig25a.jpg" width="600" height="794" alt="Fig. 25a." title="Fig. 25a." /> +<p class="nblockquot"><b><span class="smcap">Fig.</span> 25<i>a</i>.—Specimens taken at random from the computation work of eighth-grade +pupils. This computation occurred in a genuine test. In the original gray of +the pencil marks the work is still harder to make out.</b></p> +</div> + + +<p> <span class='pagenum'><a name="Page_215" id="Page_215">[Pg 215]</a></span></p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/fig25b.jpg" width="577" height="800" alt="Fig. 25b." title="Fig. 25b." /><br /> +<p class="nblockquot"><b><span class="smcap">Fig.</span> 25<i>b</i>.—Specimens taken at random from the computation work of eighth-grade +pupils. This computation occurred in a genuine test. In the original gray of +the pencil marks the work is still harder to make out.</b></p> +</div> + +<p><span class='pagenum'><a name="Page_216" id="Page_216">[Pg 216]</a></span></p> + +<p>Write the products:—</p> + +<div class="dblockquot"> + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='right'><small>A.</small> 3 4s =</td><td align='right'><small>B.</small> 5 7s =</td><td align='right'><small>C.</small> 9 2s =</td></tr> +<tr><td align='right'>5 2s =</td><td align='right'>8 3s =</td><td align='right'>4 4s =</td></tr> +<tr><td align='right'>7 2s =</td><td align='right'>4 2s =</td><td align='right'>2 7s =</td></tr> +<tr><td align='right'>1 6 =</td><td align='right'>4 5s =</td><td align='right'>6 4s =</td></tr> +<tr><td align='right'>1 3 =</td><td align='right'>4 7s =</td><td align='right'>5 5s =</td></tr> +<tr><td align='right'>3 7s =</td><td align='right'>5 9s =</td><td align='right'>3 6s =</td></tr> +<tr><td align='right'>4 1s =</td><td align='right'>7 5s =</td><td align='right'>3 2s =</td></tr> +<tr><td align='right'>6 8s =</td><td align='right'>7 1s =</td><td align='right'>3 9s =</td></tr> +<tr><td align='right'>9 8s =</td><td align='right'>6 3s =</td><td align='right'>5 1s =</td></tr> +<tr><td align='right'>4 3s =</td><td align='right'>4 9s =</td><td align='right'>8 6s =</td></tr> +<tr><td align='right'>2 4s =</td><td align='right'>3 5s =</td><td align='right'>8 4s =</td></tr> +<tr><td align='right'>2 2s =</td><td align='right'>9 6s =</td><td align='right'>8 5s =</td></tr> +<tr><td align='right'>8 7s =</td><td align='right'>2 5s =</td><td align='right'>7 9s =</td></tr> +<tr><td align='right'>5 8s =</td><td align='right'>5 4s =</td><td align='right'>6 2s =</td></tr> +<tr><td align='right'>7 6s =</td><td align='right'>8 2s =</td><td align='right'>7 4s =</td></tr> +<tr><td align='right'>7 3s =</td><td align='right'>8 9s =</td><td align='right'>9 3s =</td></tr> +</table></div> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr><td align='right'><small>D.</small> 4 20s =</td><td align='right'><small>E.</small> 9 60s =</td><td align='left'><small>F.</small> 40 × 2 = 80</td></tr> +<tr><td align='right'> 4 200s =</td><td align='right'>9 600s =</td><td align='left'> 20 × 2 =</td></tr> +<tr><td align='right'> 6 30s =</td><td align='right'>5 30s =</td><td align='left'> 30 × 2 =</td></tr> +<tr><td align='right'> 6 300s =</td><td align='right'>5 300s =</td><td align='left'> 40 × 2 =</td></tr> +<tr><td align='right'> 7 × 50 =</td><td align='right'>8 × 20 =</td><td align='left'> 20 × 3 =</td></tr> +<tr><td align='right'> 7 × 500 =</td><td align='right'>8 × 200 =</td><td align='left'> 30 × 3 =</td></tr> +<tr><td align='right'> 3 × 40 =</td><td align='right'>2 × 70 =</td><td align='left'> 300 × 3 = 900</td></tr> +<tr><td align='right'> 3 × 400 =</td><td align='right'>2 × 700 =</td><td align='left'> 300 × 2 =</td></tr> +</table></div> +</div> + +<p><span class='pagenum'><a name="Page_217" id="Page_217">[Pg 217]</a></span></p> + +<p>Write the missing numbers: (<i>r</i> stands for remainder.)</p> + + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='left'>25 = .... 3s and .... <i>r</i>.</td><td align='left'>30 = .... 4s and .... <i>r</i>.</td></tr> +<tr><td align='left'>25 = .... 4s " .... <i>r</i>.</td><td align='left'>30 = .... 5s " .... <i>r</i>.</td></tr> +<tr><td align='left'>25 = .... 5s " .... <i>r</i>.</td><td align='left'>30 = .... 6s " .... <i>r</i>.</td></tr> +<tr><td align='left'>25 = .... 6s " .... <i>r</i>.</td><td align='left'>30 = .... 7s " .... <i>r</i>.</td></tr> +<tr><td align='left'>25 = .... 7s " .... <i>r</i>.</td><td align='left'>30 = .... 8s " .... <i>r</i>.</td></tr> +<tr><td align='left'>25 = .... 8s " .... <i>r</i>.</td><td align='left'>30 = .... 9s " .... <i>r</i>.</td></tr> +<tr><td align='left'>25 = .... 9s " .... <i>r</i>.</td></tr> +<tr><td> </td></tr> +<tr><td align='left'>26 = .... 3s and .... <i>r</i>.</td><td align='left'>31 = .... 4s and .... <i>r</i>.</td></tr> +<tr><td align='left'>26 = .... 4s " .... <i>r</i>.</td><td align='left'>31 = .... 5s " .... <i>r</i>.</td></tr> +<tr><td align='left'>26 = .... 5s " .... <i>r</i>.</td><td align='left'>31 = .... 6s " .... <i>r</i>.</td></tr> +<tr><td align='left'>26 = .... 6s " .... <i>r</i>.</td><td align='left'>31 = .... 7s " .... <i>r</i>.</td></tr> +<tr><td align='left'>26 = .... 7s " .... <i>r</i>.</td><td align='left'>31 = .... 8s " .... <i>r</i>.</td></tr> +<tr><td align='left'>26 = .... 8s " .... <i>r</i>.</td><td align='left'>31 = .... 9s " .... <i>r</i>.</td></tr> +<tr><td align='left'>26 = .... 9s " .... <i>r</i>.</td></tr> +</table></div> + +<p>Write the whole numbers or mixed numbers which these +fractions equal:—</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr> + <td align='center'>5<br /><span class="overline">4</span></td> + <td align='center'>4<br /><span class="overline">3</span></td> + <td align='center'>9<br /><span class="overline">5</span></td> + <td align='center'>4<br /><span class="overline">2</span></td> + <td align='center'>7<br /><span class="overline">3</span></td> +</tr> +<tr> + <td align='center'>7<br /><span class="overline">4</span></td> + <td align='center'>5<br /><span class="overline">3</span></td> + <td align='center'>11<br /><span class="overline"> 8 </span></td> + <td align='center'>3<br /><span class="overline">2</span></td> + <td align='center'>8<br /><span class="overline">8</span></td> +</tr> +<tr> + <td align='center'>8<br /><span class="overline">4</span></td> + <td align='center'>6<br /><span class="overline">3</span></td> + <td align='center'>9<br /><span class="overline">8</span></td> + <td align='center'>9<br /><span class="overline">4</span></td> + <td align='center'>16<br /><span class="overline"> 8 </span></td> +</tr> +<tr> + <td align='center'>11<br /><span class="overline"> 4 </span></td> + <td align='center'>7<br /><span class="overline">5</span></td> + <td align='center'>13<br /><span class="overline"> 8 </span></td> + <td align='center'>8<br /><span class="overline">5</span></td> + <td align='center'>6<br /><span class="overline">6</span></td> +</tr> +</table></div> + +<p>Write the missing figures:—</p> + +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr> + <td align='right'>6<br /><span class="overline">8</span></td><td>=</td><td align='left'> <br /><span class="overline">4</span></td> + <td> </td> + <td align='right'>2<br /><span class="overline">4</span></td><td>=</td><td align='left'> <br /><span class="overline">2</span></td> + <td> </td> + <td align='right'>8<br /><span class="overline">10</span></td><td>=</td><td align='left'> <br /><span class="overline">5</span></td> + <td> </td> + <td align='right'>1<br /><span class="overline">5</span></td><td>=</td><td align='left'> <br /><span class="overline">10</span></td> + <td> </td> + <td align='right'>2<br /><span class="overline">3</span></td><td>=</td><td align='left'> <br /><span class="overline">6</span></td> +</tr> +</table></div> + +<p><span class='pagenum'><a name="Page_218" id="Page_218">[Pg 218]</a></span></p> + +<p>Write the missing numerators:—</p> + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr> + <td align='right'>1<br /><span class="overline">2</span></td><td>=</td> + <td align='left'> <br /><span class="overline">12</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">8</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">10</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">4</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">16</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">6</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">14</span></td> +</tr> +<tr> + <td align='right'>1<br /><span class="overline">3</span></td><td>=</td> + <td align='left'> <br /><span class="overline">12</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">9</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">18</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">6</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">15</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">24</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">21</span></td> +</tr> +<tr> + <td align='right'>1<br /><span class="overline">4</span></td><td>=</td> + <td align='left'> <br /><span class="overline">12</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">16</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">8</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">24</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">20</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">28</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">32</span></td> +</tr> +<tr> + <td align='right'>1<br /><span class="overline">5</span></td><td>=</td> + <td align='left'> <br /><span class="overline">10</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">20</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">15</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">25</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">40</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">35</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">30</span></td> +</tr> +<tr> + <td align='right'>2<br /><span class="overline">3</span></td><td>=</td> + <td align='left'> <br /><span class="overline">12</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">18</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">21</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">6</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">15</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">24</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">9</span></td> +</tr> +<tr> + <td align='right'>3<br /><span class="overline">4</span></td><td>=</td> + <td align='left'> <br /><span class="overline">8</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">16</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">12</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">20</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">24</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">32</span></td> + <td> </td> + <td align='left'> <br /><span class="overline">28</span></td> +</tr> +</table></div> + +<p>Find the products. Cancel when you can:—</p> + + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr> + <td align='center'> 5 <br /><span class="overline">16</span></td><td>× 4 =</td> + <td> </td> + <td align='center'>11<br /><span class="overline">12</span></td><td>× 3 =</td> + <td> </td> + <td align='center'>2<br /><span class="overline">3</span></td><td>× 5 =</td> +</tr> +<tr> + <td align='center'> 7 <br /><span class="overline">12</span></td><td>× 8 =</td> + <td> </td> + <td align='center'>8<br /><span class="overline">5</span></td><td>× 15 =</td> + <td> </td> + <td align='center'>1<br /><span class="overline">6</span></td><td>× 8 =</td> +</tr> +</table></div> + + +<p><span class='pagenum'><a name="Page_219" id="Page_219">[Pg 219]</a></span></p> + +<h4>SIGNIFICANCE FOR RELATED ACTIVITIES</h4> + +<p>The use of bodily action, social games, and the like was discussed +in the section on original tendencies. "Significance as a +means of securing other desired ends than arithmetical learning +itself" is therefore our next topic. Such significance can +be given to arithmetical work by using that work as a means +to present and future success in problems of sports, housekeeping, +shopwork, dressmaking, self-management, other +school studies than arithmetic, and general school life and +affairs. Significance as a means to future ends alone can also +be more clearly and extensively attached to it than it now is.</p> + +<p>Whatever is done to supply greater strength of motive +in studying arithmetic must be carefully devised so as not +to get a strong but wrong motive, so as not to get abundant +interest but in something other than arithmetic, and so as +not to kill the goose that after all lays the golden eggs—the +interest in intellectual activity and achievement itself. +It is easy to secure an interest in laying out a baseball +diamond, measuring ingredients for a cake, making a balloon +of a certain capacity, or deciding the added cost of an extra +trimming of ribbon for one's dress. The problem is to +<i>attach</i> that interest to arithmetical learning. Nor should a +teacher be satisfied with attaching the interest as a mere +tail that steers the kite, so long as it stays on, or as a sugar-coating +that deceives the pupil into swallowing the pill, or +as an anodyne whose dose must be increased and increased +if it is to retain its power. Until the interest permeates the +arithmetical activity itself our task is only partly done, and +perhaps is made harder for the next time.</p> + +<p>One important means of really interfusing the arithmetical +learning itself with these derived interests is to lead +the pupil to seek the help of arithmetic himself—to lead +him, in Dewey's phrase, to 'feel the need'—to take the<span class='pagenum'><a name="Page_220" id="Page_220">[Pg 220]</a></span> +'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.</p> + +<p>Even if we do not make the most skillful use possible of +these interests derived from the quantitative problems of +sports, housekeeping, shopwork, dressmaking, self-management, +other school studies, and school life and affairs, the +gain will still be considerable. To have them in mind will +certainly preserve us from giving to children of grades 3 and +4 problems so devoid of relation to their interests as those +shown below, all found (in 1910) in thirty successive pages +of a book of excellent repute:—</p> + +<div class="pblockquot"> +<p>A chair has 4 legs. How many legs have 8 chairs? 5 chairs?</p> + +<p>A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?</p> + +<p class="center">(Eight more of the same sort.)</p> + +<p>In 1890 New York had 1,513,501 inhabitants, Milwaukee had +206,308, Boston had 447,720, San Francisco 297,990. How many +had these cities together?</p> + +<p class="center">(Five more of the same sort.)</p> + +<p>Milton was born in 1608 and died in 1674. How many years +did he live?</p> + +<p class="center">(Several others of the same sort.)</p> + +<p>The population of a certain city was 35,629 in 1880 and 106,670 +in 1890. Find the increase.</p> + +<p class="center">(Several others of this sort.)</p> + +<p>A number of others about the words in various inaugural addresses +and the Psalms in the Bible.</p></div> + +<p>It also seems probable that with enough care other systematic +plans of textbooks can be much improved in this respect. +From every point of view, for example, the early work in arithmetic +should be adapted to some extent to the healthy childish +interests in home affairs, the behavior of other children, +and the activities of material things, animals, and plants.<span class='pagenum'><a name="Page_221" id="Page_221">[Pg 221]</a></span></p> + +<p class="tabcap">TABLE 9</p> + +<p class="center"><span class="smcap">Frequency of Appearance of Certain Words about Family Life, +Play, and Action in Eight Elementary Textbooks in Arithmetic</span>, +pp. 1-50.</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" width="80%" rules="cols"> +<tr><th class='bbt'></th><th class='bbt'> A</th><th class='bbt'> B</th><th class='bbt'> C</th><th class='bbt'> D</th><th class='bbt'> E</th><th class='bbt'> F</th><th class='bbt'> G</th><th class='bbt'> H</th></tr> +<tr><td align='left'>baby</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 4</td><td align='center'></td><td></td></tr> +<tr><td align='left'>brother</td><td align='center'> 2</td><td align='center'></td><td align='center'> 6</td><td align='center'> 1</td><td align='center'> 1</td><td align='center'></td><td align='center'> 1</td><td></td></tr> +<tr><td align='left'>family</td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 4</td><td></td></tr> +<tr><td align='left'>father</td><td align='center'> 1</td><td align='center'></td><td align='center'> 3</td><td align='center'> 5</td><td align='center'></td><td align='center'> 2</td><td align='center'> 1</td><td></td></tr> +<tr><td align='left'>help</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>home</td><td align='center'> 2</td><td align='center'></td><td align='center'> 4</td><td align='center'> 4</td><td align='center'> 2</td><td align='center'> 2</td><td align='center'> 7</td><td align='center'> 1</td></tr> +<tr><td align='left'>mother</td><td align='center'> 4</td><td align='center'> 2</td><td align='center'> 9</td><td align='center'> 5</td><td align='center'></td><td align='center'> 5</td><td align='center'> 1</td><td align='center'> 7</td></tr> +<tr><td align='left'>sister</td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'> 2</td><td align='center'> 2</td><td align='center'> 9</td><td align='center'> 1</td><td align='center'> 1</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='left'>fork</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>knife</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>plate</td><td align='center'> 4</td><td align='center'> 2</td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td></td></tr> +<tr><td align='left'>spoon</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='left'>doll</td><td align='center'> 10</td><td align='center'> 1</td><td align='center'> 10</td><td align='center'> 6</td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td align='center'> 9</td></tr> +<tr><td align='left'>game</td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'> 3</td><td align='center'></td><td align='center'></td><td align='center'> 5</td><td align='center'> 5</td></tr> +<tr><td align='left'>jump</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 4</td></tr> +<tr><td align='left'>marbles</td><td align='center'> 10</td><td align='center'> 4</td><td align='center'> 10</td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td align='center'> 1</td><td></td></tr> +<tr><td align='left'>play</td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'> 3</td><td align='center'></td><td></td></tr> +<tr><td align='left'>run</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td align='center'> 3</td></tr> +<tr><td align='left'>sing</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>tag</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>toy</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td></tr> +<tr><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td></tr> +<tr><td align='left'>car</td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'> 4</td><td align='center'></td><td align='center'> 2</td><td align='center'> 3</td><td align='center'> 1</td></tr> +<tr><td align='left'>cut</td><td align='center'></td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td align='center'> 6</td><td align='center'> 2</td><td align='center'></td><td align='center'> 8</td></tr> +<tr><td align='left'>dig</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td></td></tr> +<tr><td align='left'>flower</td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'> 4</td><td align='center'> 1</td><td align='center'> 1</td><td align='center'> 2</td><td></td></tr> +<tr><td align='left'>grow</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>plant</td><td align='center'></td><td align='center'></td><td align='center'> 2</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td></td></tr> +<tr><td align='left'>seed</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 3</td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td></td></tr> +<tr><td align='left'>string</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 1</td><td align='center'> 10</td><td align='center'> 1</td><td align='center'> 1</td></tr> +<tr><td align='left'>wheel</td><td align='center'> 5</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> 10</td><td align='center'></td><td></td></tr> +<tr><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td></tr> +</table></div> + +<p><span class='pagenum'><a name="Page_222" id="Page_222">[Pg 222]</a></span></p> + +<p>The words used by textbooks give some indication of how +far this aim is being realized, or rather of how far short we +are of realizing it. Consider, for example, the words home, +mother, father, brother, sister, help, plate, knife, fork, +spoon, play, game, toy, tag, marbles, doll, run, jump, sing, +plant, seed, grow, flower, car, wheel, string, cut, dig. The +frequency of appearance in the first fifty pages of eight beginners' +arithmetics was as shown in Table 9. The eight +columns refer to the eight books (the first fifty pages of +each). The numbers refer to the number of times the word +in question appeared, the number 10 meaning 10 <i>or more</i> +times in the fifty pages. Plurals, past tenses, and the like +were counted. <i>Help</i>, <i>fork</i>, <i>knife</i>, <i>spoon</i>, <i>jump</i>, <i>sing</i>, and <i>tag</i> +did not appear at all! <i>Toy</i> and <i>grow</i> appeared each once +in the 400 pages! <i>Play</i>, <i>run</i>, <i>dig</i>, <i>plant</i>, and <i>seed</i> appeared +once in a hundred or more pages. <i>Baby</i> did not appear as +often as <i>buggy</i>. <i>Family</i> appeared no oftener than <i>fence</i> or +<i>Friday</i>. <i>Father</i> appears about a third as often as <i>farmer</i>.</p> + +<p>Book A shows only 10 of these thirty words in the fifty +pages; book B only 4; book C only 12; and books D, E, F, +G, and H only 13, 8, 14, 13, 10, respectively. The total number +of appearances (counting the 10s as only 10 in each case) +is 40 for A, 9 for B, 60 for C, 42 for D, 25 for E, 62 for F, 30 +for G, and 37 for H. The five words—apple, egg, Mary, +milk, and orange—are used oftener than all these thirty +together.</p> + +<p>If it appeared that this apparent neglect of childish affairs +and interests was deliberate to provide for a more systematic +treatment of pure arithmetic, a better gradation of problems, +and a better preparation for later genuine use than could be +attained if the author of the textbook were tied to the child's +apron strings, the neglect could be defended. It is not at all +certain that children in grade 2 get much more enjoyment<span class='pagenum'><a name="Page_223" id="Page_223">[Pg 223]</a></span> +or ability from adding the costs of purchases for Christmas +or Fourth of July, or multiplying the number of cakes each +child is to have at a party by the number of children who +are to be there, than from adding gravestones or multiplying +the number of hairs of bald-headed men. When, however, +there is nothing gained by substituting remote facts for +those of familiar concern to children, the safe policy is surely +to favor the latter. In general, the neglect of childish +data does not seem to be due to provision for some other +end, but to the same inertia of tradition which has carried +over the problems of laying walls and digging wells into city +schools whose children never saw a stone wall or dug well.</p> + +<hr style='width: 45%;' /> + +<p>I shall not go into details concerning the arrangement of +courses of study, textbooks, and lesson-plans to make desirable +connections between arithmetical learning and sports, +housework, shopwork, and the rest. It may be worth while, +however, to explain the term <i>self-management</i>, since this +source of genuine problems of real concern to the pupils +has been overlooked by most writers.</p> + +<p>By self-management is meant the pupil's use of his time, +his abilities, his knowledge, and the like. By the time he +reaches grade 5, and to some extent before then, a boy +should keep some account of himself, of how long it takes +him to do specified tasks, of how much he gets done in a +specified time at a certain sort of work and with how many +errors, of how much improvement he makes month by +month, of which things he can do best, and the like. Such +objective, matter-of-fact, quantitative study of one's behavior +is not a stimulus to morbid introspection or egotism; +it is one of the best preventives of these. To treat oneself +impersonally is one of the essential elements of mental +balance and health. It need not, and should not, encourage<span class='pagenum'><a name="Page_224" id="Page_224">[Pg 224]</a></span> +priggishness. On the contrary, this matter-of-fact study of +what one is and does may well replace a certain amount of +the exhortations and admonitions concerning what one ought +to do and be. All this is still truer for a girl.</p> + +<p>The demands which such an accounting of one's own +activities make of arithmetic have the special value of connecting +directly with the advanced work in computation. +They involve the use of large numbers, decimals, averaging, +percentages, approximations, and other facts and processes +which the pupil has to learn for later life, but to which his +childish activities as wage-earner, buyer and seller, or shopworker +from 10 to 14 do not lead. Children have little +money, but they have time in thousands of units! They do +not get discounts or bonuses from commercial houses, but +they can discount their quantity of examples done for the +errors made, and credit themselves with bonuses of all sorts +for extra achievements.</p> + + +<h4>INTRINSIC INTEREST IN ARITHMETICAL LEARNING</h4> + +<p>There remains the most important increase of interest in +arithmetical learning—an increase in the interest directly +bound to achievement and success in arithmetic itself. +"Arithmetic," says David Eugene Smith, "is a game and +all boys and girls are players." It should not be a <i>mere</i> +game for them and they should not <i>merely</i> play, but their +unpractical interest in doing it because they can do it and can +see how well they do do it is one of the school's most precious +assets. Any healthy means to give this interest more and +better stimulus should therefore be eagerly sought and +cherished.</p> + +<p>Two such means have been suggested in other connections. +The first is the extension of training in checking and verifying +work so that the pupil may work to a standard of ap<span class='pagenum'><a name="Page_225" id="Page_225">[Pg 225]</a></span>proximately +100% success, and may know how nearly he is +attaining it. The second is the use of standardized practice +material and tests, whereby the pupil may measure himself +against his own past, and have a clear, vivid, and trustworthy +idea of just how much better or faster he can do the +same tasks than he could do a month or a year ago, and of +just how much harder things he can do now than then.</p> + +<p>Another means of stimulating the essential interest in +quantitative thinking itself is the arrangement of the work +so that real arithmetical thinking is encouraged more than +mere imitation and assiduity. This means the avoidance +of long series of applied problems all of one type to be +solved in the same way, the avoidance of miscellaneous series +and review series which are almost verbatim repetitions of +past problems, and in general the avoidance of excessive +repetition of any one problem-situation. Stimulation to +real arithmetical thinking is weak when a whole day's +problem work requires no choice of methods, or when a +review simply repeats without any step of organization or +progress, or when a pupil meets a situation (say the 'buy <i>x</i> +things at <i>y</i> per thing, how much pay' situation) for the five-hundredth +time.</p> + +<p>Another matter worthy of attention in this connection is +the unwise tendency to omit or present in diluted form some +of the topics that appeal most to real intellectual interests, +just because they are hard. The best illustration, perhaps, +is the problem of ratio or "How many times as large (long, +heavy, expensive, etc.) as <i>x</i> is <i>y</i>?" Mastery of the 'times +as' relation is hard to acquire, but it is well worth acquiring, +not only because of its strong intellectual appeal, but also +because of its prime importance in the applications of +arithmetic to science. In the older arithmetics it was confused +by pedantries and verbal difficulties and penalized<span class='pagenum'><a name="Page_226" id="Page_226">[Pg 226]</a></span> +by unreal problems about fractions of men doing parts of a +job in strange and devious times. Freed from these, it +should be reinstated, beginning as early as grade 5 with such +simple exercises as those shown below and progressing to the +problems of food values, nutritive ratios, gears, speeds, and +the like in grade 8.</p> + + +<div class="blockquot"><p><br /> +John is 4 years old.<br /> +Fred is 6 years old.<br /> +Mary is 8 years old.<br /> +Nell is 10 years old.<br /> +Alice is 12 years old.<br /> +Bert is 15 years old.<br /> +<br /> +Who is twice as old as John?<br /> +Who is half as old as Alice?<br /> +Who is three times as old as John?<br /> +Who is one and one half times as old as Nell?<br /> +Who is two thirds as old as Fred?<br /> +<span style="margin-left: 3em;">etc., etc., etc.</span><br /> +<br /> +Alice is .... times as old as John.<br /> +John is .... as old as Mary.<br /> +Fred is .... times as old as John.<br /> +Alice is .... times as old as Fred.<br /> +Fred is .... as old as Mary.<br /> +<span style="margin-left: 3em;">etc., etc., etc.</span><br /> +</p></div> + +<p>Finally it should be remembered that all improvements +in making arithmetic worth learning and helping the pupil +to learn it will in the long run add to its interest. Pupils +like to learn, to achieve, to gain mastery. Success is interesting. +If the measures recommended in the previous chapters +are carried out, there will be little need to entice pupils to +take arithmetic or to sugar-coat it with illegitimate attractions.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_227" id="Page_227">[Pg 227]</a></span></p> +<h2><a name="CHAPTER_XIII" id="CHAPTER_XIII"></a>CHAPTER XIII</h2> + +<h3>THE CONDITIONS OF LEARNING</h3> + + +<p>We shall consider in this chapter the influence of time of +day, size of class, and amount of time devoted to arithmetic +in the school program, the hygiene of the eyes in arithmetical +work, the use of concrete objects, and the use of +sounds, sights, and thoughts as situations and of speech and +writing and thought as responses.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a></p> + + +<h4>EXTERNAL CONDITIONS</h4> + +<p>Computation of one or another sort has been used by +several investigators as a test of efficiency at different times +in the day. When freed from the effects of practice on the +one hand and lack of interest due to repetition on the other, +the results uniformly show an increase in speed late in the +school session with a falling off in accuracy that about +balances it.<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a> There is no wisdom in putting arithmetic +early in the session because of its <i>difficulty</i>. Lively and +sociable exercises in mental arithmetic with oral answers in +fact seem to be admirably fitted for use late in the session. +Except for the general principles (1) of starting the day with +work that will set a good standard of cheerful, efficient +<span class='pagenum'><a name="Page_228" id="Page_228">[Pg 228]</a></span> +production and (2) of getting the least interesting features of +the day's work done fairly early in the day, psychology +permits practical exigencies to rule the program, so far as +present knowledge extends. Adequate measurements of the +effect of time of day on <i>improvement</i> have not been made, +but there is no reason to believe that any one time between +9 <span class="smcap">a.m.</span> and 4 <span class="smcap">p.m.</span> is appreciably more favorable to arithmetical +learning than to learning geography, history, +spelling, and the like.</p> + +<p>The influence of size of class upon progress in school +studies is very difficult to measure because (1) within the +same city system the average of the six (or more) sizes of +class that a pupil has experienced will tend to approximate +closely to the corresponding average for any other child; +because further (2) there may be a tendency of supervisory +officers to assign more pupils to the better teachers; and +because (3) separate systems which differ in respect to size +of class probably differ in other respects also so that their +differences in achievement may be referable to totally +different differences.</p> + +<p>Elliott ['14] has made a beginning by noting size of class +during the year of test in connection with his own measures +of the achievements of seventeen hundred pupils, supplemented +by records from over four hundred other classes. +As might be expected from the facts just stated, he finds no +appreciable difference between classes of different sizes +within the same school system, the effect of the few months +in a small class being swamped by the antecedents or concomitants +thereof.</p> + +<p>The effect of the amount of time devoted to arithmetic +in the school program has been studied extensively by Rice +['02 and '03] and Stone ['08].</p> + +<p>Dr. Rice ['02] measured the arithmetical ability of some<span class='pagenum'><a name="Page_229" id="Page_229">[Pg 229]</a></span> +6000 children in 18 different schools in 7 different cities. +The results of these measurements are summarized in Table +10. This table "gives two averages for each grade as well +as for each school as a whole. Thus, the school at the top +shows averages of 80.0 and 83.1, and the one at the bottom, +25.3 and 31.5. The first represents the percentage of +answers which were absolutely correct; the second shows +what per cent of the problems were correct in principle, <i>i.e.</i> +the average that would have been received if no mechanical +errors had been made."</p> + +<p>The facts of Dr. Rice's table show that there is a positive +relation between the general standing of a school system +in the tests and the amount of time devoted to arithmetic +by its program. The relation is not close, however, being +that expressed by a correlation coefficient of .36½. Within +any one school system there is no relation between the +standing of a particular school and the amount of time devoted +to arithmetic in that school's program. It must be +kept in mind that the amount of time given in the school +program may be counterbalanced by emphasizing work at +home and during study periods, or, on the other hand, +may be a symptom of correspondingly small or great emphasis +on arithmetic in work set for the study periods at +home.</p> + +<p>A still more elaborate investigation of this same topic was +made by Stone ['08]. I quote somewhat fully from it, since +it is an instructive sample of the sort of studies that will +doubtless soon be made in the case of every elementary +school subject. He found that school systems differed notably +in the achievements made by their sixth-grade pupils +in his tests of computation (the so-called 'fundamentals') +and of the solution of verbally described problems (the +so-called 'reasoning'). The facts were as shown in Table 11.<span class='pagenum'><a name="Page_230" id="Page_230">[Pg 230]</a></span></p> + +<p class="tabcap">TABLE 10</p> + +<p class="tabcap"><span class="smcap">Averages for Individual Schools in Arithmetic</span></p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary="" rules="cols"> +<tr><th class="bbox1" rowspan='2'><span class="smcap">City</span></th><th class="bbox1" rowspan='2'><span class="smcap">School</span></th><th colspan='2' class='bbox1'><span class="smcap">6th Year</span></th><th colspan='2' class='bbox1'><span class="smcap">7th Year</span></th><th colspan='2' class='bbox1'><span class="smcap">8th Year</span></th><th colspan='3' class='bbox1'><span class="smcap">School Average</span></th><th class='bbox1'></th></tr> +<tr><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class="bbox1">Result</th><th class="bbox1">Principle</th><th class='bbox1'>Percent of<br />Mechanical Errors</th><th class='bbox1'>Minutes<br />Daily</th></tr> +<tr><td align='right'>III</td><td align='center'> 1</td><td align='center'>79.3</td><td align='center'>80.3</td><td align='center'>81.1</td><td align='center'>82.3</td><td align='center'>91.7</td><td align='center'>93.9</td><td align='center'>80.0</td><td align='center'>83.1</td><td align='center'> 3.7</td><td align='center'> 53</td></tr> +<tr><td align='right'> I</td><td align='center'> 1</td><td align='center'>80.4</td><td align='center'>81.5</td><td align='center'>64.2</td><td align='center'>67.2</td><td align='center'>80.9</td><td align='center'>82.8</td><td align='center'>76.6</td><td align='center'>80.3</td><td align='center'> 4.6</td><td align='center'> 60</td></tr> +<tr><td align='right'> I</td><td align='center'> 2</td><td align='center'>80.9</td><td align='center'>83.4</td><td align='center'>43.5</td><td align='center'>50.9</td><td align='center'>72.7</td><td align='center'>79.1</td><td align='center'>69.3</td><td align='center'>75.1</td><td align='center'> 7.7</td><td align='center'> 25</td></tr> +<tr><td align='right'> I</td><td align='center'> 3</td><td align='center'>72.2</td><td align='center'>74.0</td><td align='center'>63.5</td><td align='center'>66.2</td><td align='center'>74.5</td><td align='center'>76.6</td><td align='center'>67.8</td><td align='center'>72.2</td><td align='center'> 6.1</td><td align='center'> 45</td></tr> +<tr><td align='right'> I</td><td align='center'> 4</td><td align='center'>69.9</td><td align='center'>72.2</td><td align='center'>54.6</td><td align='center'>57.8</td><td align='center'>66.5</td><td align='center'>69.1</td><td align='center'>64.3</td><td align='center'>70.3</td><td align='center'> 8.5</td><td align='center'> 45</td></tr> +<tr><td align='right'> II</td><td align='center'> 1</td><td align='center'>71.2</td><td align='center'>75.3</td><td align='center'>33.6</td><td align='center'>35.7</td><td align='center'>36.8</td><td align='center'>40.0</td><td align='center'>60.2</td><td align='center'>64.8</td><td align='center'> 7.1</td><td align='center'> 60</td></tr> +<tr><td align='right'>III</td><td align='center'> 2</td><td align='center'>43.7</td><td align='center'>45.0</td><td align='center'>53.9</td><td align='center'>56.7</td><td align='center'>51.1</td><td align='center'>53.1</td><td align='center'>54.5</td><td align='center'>58.9</td><td align='center'> 7.4</td><td align='center'> 60</td></tr> +<tr><td align='right'> IV</td><td align='center'> 1</td><td align='center'>58.9</td><td align='center'>60.4</td><td align='center'>31.2</td><td align='center'>34.1</td><td align='center'>41.6</td><td align='center'>43.5</td><td align='center'>55.1</td><td align='center'>58.4</td><td align='center'> 5.6</td><td align='center'> 60</td></tr> +<tr><td align='right'> IV</td><td align='center'> 2</td><td align='center'>59.8</td><td align='center'>63.1</td><td align='center'> —</td><td align='center'> —</td><td align='center'>22.5</td><td align='center'>22.5</td><td align='center'>53.9</td><td align='center'>58.8</td><td align='center'> 8.3</td><td align='center'> —</td></tr> +<tr><td align='right'> IV</td><td align='center'> 3</td><td align='center'>54.9</td><td align='center'>58.1</td><td align='center'>35.2</td><td align='center'>38.6</td><td align='center'>43.5</td><td align='center'>45.0</td><td align='center'>51.5</td><td align='center'>57.6</td><td align='center'>10.5</td><td align='center'> 60</td></tr> +<tr><td align='right'> IV</td><td align='center'> 4</td><td align='center'>42.3</td><td align='center'>45.1</td><td align='center'>16.1</td><td align='center'>19.2</td><td align='center'>48.7</td><td align='center'>48.7</td><td align='center'>42.8</td><td align='center'>48.2</td><td align='center'>11.2</td><td align='center'> —</td></tr> +<tr><td align='right'> V</td><td align='center'> 1</td><td align='center'>44.1</td><td align='center'>48.7</td><td align='center'>29.2</td><td align='center'>32.5</td><td align='center'>51.1</td><td align='center'>58.3</td><td align='center'>45.9</td><td align='center'>51.3</td><td align='center'>10.5</td><td align='center'> 40</td></tr> +<tr><td align='right'> VI</td><td align='center'> 1</td><td align='center'>68.3</td><td align='center'>71.3</td><td align='center'>33.5</td><td align='center'>36.6</td><td align='center'>26.9</td><td align='center'>30.7</td><td align='center'>39.0</td><td align='center'>42.9</td><td align='center'> 9.0</td><td align='center'> 33</td></tr> +<tr><td align='right'> VI</td><td align='center'> 2</td><td align='center'>46.1</td><td align='center'>49.5</td><td align='center'>19.5</td><td align='center'>24.2</td><td align='center'>30.2</td><td align='center'>40.6</td><td align='center'>36.5</td><td align='center'>43.6</td><td align='center'>16.2</td><td align='center'> 30</td></tr> +<tr><td align='right'> VI</td><td align='center'> 3</td><td align='center'>34.5</td><td align='center'>36.4</td><td align='center'>30.5</td><td align='center'>35.1</td><td align='center'>23.3</td><td align='center'>24.1</td><td align='center'>36.0</td><td align='center'>42.5</td><td align='center'>15.2</td><td align='center'> 48</td></tr> +<tr><td align='right'>VII</td><td align='center'> 1</td><td align='center'>35.2</td><td align='center'>37.7</td><td align='center'>29.1</td><td align='center'>32.5</td><td align='center'>25.1</td><td align='center'>27.2</td><td align='center'>40.5</td><td align='center'>45.9</td><td align='center'>11.7</td><td align='center'> 42</td></tr> +<tr><td align='right'>VII</td><td align='center'> 2</td><td align='center'>35.2</td><td align='center'>38.7</td><td align='center'>15.0</td><td align='center'>16.4</td><td align='center'>19.6</td><td align='center'>21.2</td><td align='center'>36.5</td><td align='center'>40.6</td><td align='center'>10.1</td><td align='center'> 75</td></tr> +<tr><td align='right'>VII</td><td align='center'> 3</td><td align='center'>27.6</td><td align='center'>33.7</td><td align='center'> 8.9</td><td align='center'>10.1</td><td align='center'>11.3</td><td align='center'>11.3</td><td align='center'>25.3</td><td align='center'>31.5</td><td align='center'>19.6</td><td align='center'> 45</td></tr> +<tr><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td><td class='bb'></td></tr> +</table></div> + +<p>High achievement by a system in computation went with +high achievement in solving the problems, the correlation +being about .50; and the system that scored high in addition +or subtraction or multiplication or division usually showed +closely similar excellence in the other three, the correlations +being about .90.</p> + +<p class="tabcap">TABLE 11</p> + +<p class="center"><span class="smcap">Scores Made by the Sixth-Grade Pupils of Each of Twenty-Six School +Systems</span></p> + + + +<div class='center'> +<table border="0" cellpadding="1" cellspacing="0" summary="" rules="cols"> +<tr><th class="bbt smcap">System</th><th class="bbt smcap">Score in Tests<br />with Problems</th><th class="bbt smcap">Score in Tests<br />in Computing</th></tr> +<tr><td align='center'>23</td><td align='center'> 356</td><td align='center'> 1841</td></tr> +<tr><td align='center'>24</td><td align='center'> 429</td><td align='center'> 3513</td></tr> +<tr><td align='center'>17</td><td align='center'> 444</td><td align='center'> 3042</td></tr> +<tr><td align='center'> 4</td><td align='center'> 464</td><td align='center'> 3563</td></tr> +<tr><td align='center'>25</td><td align='center'> 464</td><td align='center'> 2167</td></tr> +<tr><td align='center'>22</td><td align='center'> 468</td><td align='center'> 2311</td></tr> +<tr><td align='center'>16</td><td align='center'> 469</td><td align='center'> 3707</td></tr> +<tr><td align='center'>20</td><td align='center'> 491</td><td align='center'> 2168</td></tr> +<tr><td align='center'>18</td><td align='center'> 509</td><td align='center'> 3758</td></tr> +<tr><td align='center'>15</td><td align='center'> 532</td><td align='center'> 2779</td></tr> +<tr><td align='center'> 3</td><td align='center'> 533</td><td align='center'> 2845</td></tr> +<tr><td align='center'> 8</td><td align='center'> 538</td><td align='center'> 2747</td></tr> +<tr><td align='center'> 6</td><td align='center'> 550</td><td align='center'> 3173</td></tr> +<tr><td align='center'> 1</td><td align='center'> 552</td><td align='center'> 2935</td></tr> +<tr><td align='center'>10</td><td align='center'> 601</td><td align='center'> 2749</td></tr> +<tr><td align='center'> 2</td><td align='center'> 615</td><td align='center'> 2958</td></tr> +<tr><td align='center'>21</td><td align='center'> 627</td><td align='center'> 2951</td></tr> +<tr><td align='center'>13</td><td align='center'> 636</td><td align='center'> 3049</td></tr> +<tr><td align='center'>14</td><td align='center'> 661</td><td align='center'> 3561</td></tr> +<tr><td align='center'> 9</td><td align='center'> 691</td><td align='center'> 3404</td></tr> +<tr><td align='center'> 7</td><td align='center'> 734</td><td align='center'> 3782</td></tr> +<tr><td align='center'>12</td><td align='center'> 736</td><td align='center'> 3410</td></tr> +<tr><td align='center'>11</td><td align='center'> 759</td><td align='center'> 3261</td></tr> +<tr><td align='center'>26</td><td align='center'> 791</td><td align='center'> 3682</td></tr> +<tr><td align='center'>19</td><td align='center'> 848</td><td align='center'> 4099</td></tr> +<tr><td align='center'> 5</td><td align='center'> 914</td><td align='center'> 3569</td></tr> +<tr><td class="bb"></td><td class="bb"></td><td class="bb"></td></tr> +</table></div> + +<p>Of the conditions under which arithmetical learning took +place, the one most elaborately studied was the amount of +time devoted to arithmetic. On the basis of replies by +principals of schools to certain questions, he gave each of<span class='pagenum'><a name="Page_231" id="Page_231">[Pg 231]</a></span> +the twenty-six school systems a measure for the probable +time spent on arithmetic up through grade 6. Leaving +home study out of account, there seems to be little or no +correlation between the amount of time a system devotes +to arithmetic and its score in problem-solving, and not +much more between time expenditure and score in computation. +With home study included there is little relation to<span class='pagenum'><a name="Page_232" id="Page_232">[Pg 232]</a></span> +the achievement of the system in solving problems, but +there is a clear effect on achievement in computation. +The facts as given by Stone are:—</p> + +<p class="tabcap">TABLE 12</p> + +<p class="center"><span class="smcap">Correlation of Time Expenditures with Abilities</span></p> + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td rowspan='2'>Without Home Study <span class="sz30">{</span></td><td align='left'> Reasoning and Time Expenditure</td><td align='right'>−.01</td></tr> +<tr><td align='left'> Fundamentals and Time Expenditure</td><td align='right'>.09</td></tr> +<tr><td colspan='3'> </td></tr> +<tr><td rowspan='2'>Including Home Study <span class="sz30">{</span></td><td align='left'> Reasoning and Time Expenditure</td><td align='right'>.13</td></tr> +<tr><td align='left'> Fundamentals and Time Expenditure</td><td align='right'>.49</td></tr> +</table></div> + + +<p>These correlations, it should be borne in mind, are for +school systems, not for individual pupils. It might be that, +though the system which devoted the most time to arithmetic +did not show corresponding superiority in the product over +the system devoting only half as much time, the pupils +within the system did achieve in exact proportion to the +time they gave to study. Neither correlation would permit +inference concerning the effect of different amounts of time +spent by the same pupil.</p> + +<p>Stone considered also the printed announcements of the +courses of study in arithmetic in these twenty-six systems. +Nineteen judges rated these announced courses of study for +excellence according to the instructions quoted below:—</p> + + +<h4>CONCERNING THE RATING OF COURSES OF STUDY</h4> + +<p class="center">Judges please read before scoring</p> + +<p class="noidt">I. Some Factors Determining Relative Excellence.</p> + +<p>(N. B. The following enumeration is meant to be suggestive +rather than complete or exclusive. And each scorer is urged to +rely primarily on his own judgment.)</p> + +<div class="sblockquot"> + +<p class="nblockquot">1. Helpfulness to the teacher in teaching the subject matter +outlined.</p> +<p><span class='pagenum'><a name="Page_233" id="Page_233">[Pg 233]</a></span></p> +<p class="nblockquot">2. Social value or concreteness of sources of problems.</p> + +<p class="nblockquot">3. The arrangement of subject matter.</p> + +<p class="nblockquot">4. The provision made for adequate drill.</p> + +<p class="nblockquot">5. A reasonable minimum requirement with suggestions for +valuable additional work.</p> + +<p class="nblockquot">6. The relative values of any predominating so-called methods—such +as Speer, Grube, etc.</p> + +<p class="nblockquot">7. The place of oral or so-called mental arithmetic.</p> + +<p class="nblockquot">8. The merit of textbook references.</p> +</div> + +<p class="noidt">II. Cautions and Directions.</p> + +<p>(Judges please follow as implicitly as possible.)</p> + +<div class="sblockquot"> +<p class="nblockquot">1. Include references to textbooks as parts of the Course of Study.</p> + +<p>This necessitates judging the parts of the texts referred to.</p> + +<p class="nblockquot">2. As far as possible become equally familiar with all courses +before scoring any.</p> + +<p class="nblockquot">3. When you are ready to begin to score, (1) arrange in serial +order according to excellence, (2) starting with the +middle one score it 50, then score above and below 50 +according as courses are better or poorer, indicating relative +differences in excellence by relative differences in +scores, <i>i.e.</i> in so far as you find that the courses differ +by about equal steps, score those better than the middle +one 51, 52, etc., and those poorer 49, 48, etc., but if you +find that the courses differ by unequal steps show these +inequalities by omitting numbers.</p> + +<p class="nblockquot">4. Write ratings on the slip of paper attached to each course.</p> +</div> + +<p>The systems whose courses of study were thus rated highest +did not manifest any greater achievement in Stone's tests +than the rest. The thirteen with the most approved announcements +of courses of study were in fact a little inferior +in achievement to the other thirteen, and the correlation +coefficients were slightly negative.</p> + +<p>Stone also compared eighteen systems where there was +supervision of the work by superintendents or supervisors +as well as by principals with four systems where the principals +and teachers had no such help. The scores in his tests +were very much lower in the four latter cities.</p> +<p><span class='pagenum'><a name="Page_234" id="Page_234">[Pg 234]</a></span></p> + +<h4>THE HYGIENE OF THE EYES IN ARITHMETIC</h4> + + +<div class="figcenter" style="width: 336px;"> +<img src="images/fig26.jpg" width="336" height="388" alt="Fig. 26." title="Fig. 26." /> +<span class="caption"><span class="smcap">Fig. 26.</span>—Type too large.</span> +</div> + +<p>We have already noted that the task of reading and copying +numbers is one of the hardest that the eyes have to perform +in the elementary school, and that it should be alleviated +by arranging much of the work so that only answers need +be written by the pupil. The figures to be read and copied<span class='pagenum'><a name="Page_235" id="Page_235">[Pg 235]</a></span> +should obviously be in type of suitable size and style, so +arranged and spaced on the page or blackboard as to cause +a minimum of effort and strain.</p> + +<div class="figcenter" style="width: 448px;"> +<img src="images/fig27.jpg" width="448" height="160" alt="Fig. 27." title="Fig. 27." /> +<span class="caption"><span class="smcap">Fig. 27.</span>—12-point, 11-point, and 10-point type.</span> +</div> + +<p><i>Size.</i>—Type may be too large as well as too small, though +the latter is the commoner error. If it is too large, as in +Fig. 26, which is a duplicate of type actually used in a form +of practice pad, the eye has to make too many fixations to +take in a given content. All things considered, 12-point +type in grades 3 and 4, 11-point in grades 5 and 6, and +10-point in grades 7 and 8 seem the most desirable +sizes. These are shown in Fig. 27. Too small type occurs +oftenest in fractions and in the dimension-numbers or scale +numbers of drawings. Figures 28, 29, and 30 are samples +from actual school practice. Samples of the desirable size +are shown in Figs. 31 and 32. The technique of modern +typesetting makes it very difficult and expensive to make +fractions of the horizontal type</p> + + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="0" summary=""> +<tr> + <td>(</td> + <td class="ft05">1<br /><span class="overline">4</span></td> + <td>,</td> + <td class="ft05">3<br /><span class="overline">8</span></td> + <td>,</td> + <td class="ft05">5<br /><span class="overline">6</span></td> + <td>)</td> +</tr> +</table></div> + +<p class="noidt">large enough without +making the whole-number figures with which they are +mingled too large or giving an uncouth appearance to the +total. Consequently fractions somewhat smaller than are +desirable may have to be used occasionally in textbooks.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a> +There is no valid excuse, however, for the excessively small +fractions which often are made in blackboard work.</p> + +<p><span class='pagenum'><a name="Page_236" id="Page_236">[Pg 236]</a></span></p> +<div class="figcenter" style="width: 600px;"> +<div class="figleft" style="width: 300px;"> +<img src="images/fig28.jpg" width="225" height="200" alt="Fig. 28." title="Fig. 28." /><br /> +<span class="caption"><span class="smcap">Fig. 28.</span>—Type of measurements too small.</span><br /> +<br /> +</div> + +<p> </p> +<p> </p> + +<p>This is a picture of Mary's garden.<br /> +How many feet is it around the garden?</p> +</div> + + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig29.png" width="640" height="296" alt="Fig. 29." title="Fig. 29." /> +<span class="caption"><span class="smcap">Fig. 29.</span>—Type too small.</span> +</div> + +<p> <span class='pagenum'><a name="Page_237" id="Page_237">[Pg 237]</a></span></p> + +<div class="figcenter" style="width: 725px;"> +<img src="images/fig30.png" width="725" height="600" alt="Fig. 30." title="Fig. 30." /> +<span class="caption"><span class="smcap">Fig. 30.</span>—Numbers too small and badly designed.</span> +</div> + +<div class="figcenter" style="width: 350px;"> +<img src="images/fig31.jpg" width="255" height="235" alt="Fig. 31." title="Fig. 31." /><br /> +<span class="caption"><span class="smcap">Fig. 31.</span>—Figure 28 with suitable numbers.</span> +</div> + +<p> <span class='pagenum'><a name="Page_238" id="Page_238">[Pg 238]</a></span></p> + +<div class="figcenter" style="width: 751px;"> +<img src="images/fig32.png" width="751" height="600" alt="Fig. 32." title="Fig. 32." /> +<span class="caption"><span class="smcap">Fig. 32.</span>—Figure 30 with suitable numbers.</span> +</div> + +<p><i>Style.</i>—The ordinary type forms often have 3 and 8 so +made as to require strain to distinguish them. 5 is sometimes +easily confused with 3 and even with 8. 1, 4, and 7 +may be less easily distinguishable than is desirable. Figure +33 shows a specially good type in which each figure is represented +by its essential<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a> features without any distracting +shading or knobs or turns. Figure 34 shows some of the types +in common use. There are no demonstrably great differences +amongst these. In fractions there is a notable gain +from using the slant form (<sup>2</sup>⁄<sub>3</sub>, <sup>3</sup>⁄<sub>4</sub>) for exercises in addition +<span class='pagenum'><a name="Page_239" id="Page_239">[Pg 239]</a></span> +and subtraction, and for almost all mixed numbers. This +appears clearly to the eye in the comparison of Fig. 35 below, +where the same fractions all in 10-point type are displayed +in horizontal and in slant form. The figures in the +slant form are in general larger and the space between them +and the fraction-line is wider. Also the slant form makes +it easier for the eye to examine the denominators to see +whether reductions are necessary. Except for a few cases +to show that the operations can be done just as truly with +the horizontal forms, the book and the blackboard should +display mixed numbers and fractions to be added or subtracted +in the slant form. The slant line should be at an +angle of approximately 45 degrees. Pupils should be taught +to use this form in their own work of this sort.</p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/fig33.jpg" width="448" height="200" alt="Fig. 33." title="Fig. 33." /><br /> +<span class="caption"><span class="smcap">Fig. 33.</span>—Block type; a very desirable type except that it is somewhat too +heavy.</span> +</div> + +<p> </p> +<div class="figcenter" style="width: 400px;"> +<img src="images/fig34.png" width="294" height="448" alt="Fig. 34." title="Fig. 34." /><br /> +<span class="caption"><span class="smcap">Fig. 34.</span>—Common styles of printed numbers.</span> +</div> + +<p><span class='pagenum'><a name="Page_240" id="Page_240">[Pg 240]</a></span></p> + + +<p>When script figures are presented they should be of simple +design, showing clearly the essential features of the figure, +the line being everywhere of equal or nearly equal width +(that is, without shading, and without ornamentation or +eccentricity of any sort). The opening of the 3 should be +wide to prevent confusion with 8; the top of the 3 should be +curved to aid its differentiation from 5; the down stroke +of the 9 should be almost or quite straight; the 1, 4, 7, and 9 +should be clearly distinguishable. There are many ways of +distinguishing them clearly, the best probably being to use +the straight line for 1, the open 4 with clear angularity, a +wide top to the 7, and a clearly closed curve for the top of the 9.</p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig35.jpg" width="448" height="140" alt="Fig. 35." title="Fig. 35." /><br /> +<span class="caption"><span class="smcap">Fig. 35.</span>—Diagonal and horizontal fractions compared.</span> +</div> + +<p> <span class='pagenum'><a name="Page_241" id="Page_241">[Pg 241]</a></span></p> + +<div class="figcenter" style="width: 640px;"> +<img src="images/p241.png" width="480" height="598" alt="Figs. 36, 37." title="Fig. 36, 37." /><br /> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td align='left'><b><span class="smcap">Fig. 36.</span>—Good vertical spacing.</b></td> +<td align='right'><b><span class="smcap">Fig. 37.</span>—Bad vertical spacing.</b></td> +</tr> +</table> +</div> + +<p> <span class='pagenum'><a name="Page_242" id="Page_242">[Pg 242]</a></span></p> + +<div class="figcenter" style="width: 480px;"> +<img src="images/p242.png" width="480" height="612" alt="Figs. 38, 39." title="Figs. 38, 39." /> +<span class="caption"><span class="smcap">Figs.</span> 38 (above) and 39 (below).—Good and bad left-right spacing.</span> +</div> + +<p><span class='pagenum'><a name="Page_243" id="Page_243">[Pg 243]</a></span></p> + +<p>The pupil's writing of figures should be clear. He will +thereby be saved eyestrain and errors in his school work as +well as given a valuable ability for life. Handwriting of +figures is used enormously in spite of the development of +typewriters; illegible figures are commonly more harmful +than illegible letters or words, since the context far less often +tells what the figure is intended to be; the habit of making +clear figures is not so hard to acquire, since they are written +unjoined and require only the automatic action of ten minor +acts of skill. The schools have missed a great opportunity in +this respect. Whereas the hand writing of words is often better +than it needs to be for life's purposes, the writing of figures is +usually much worse. The figures presented in books on penmanship +are also commonly bad, showing neglect or misunderstanding +of the matter on the part of leaders in penmanship.</p> + +<p><i>Spacing.</i>—Spacing up and down the column is rarely too +wide, but very often too narrow. The specimens shown in +Figs. 36 and 37 show good practice contrasted with the +common fault.</p> + +<p>Spacing from right to left is generally fairly satisfactory +in books, though there is a bad tendency to adopt some one +routine throughout and so to miss chances to use reductions +and increases of spacing so as to help the eye and the mind +in special cases. Specimens of good and bad spacing are +shown in Figs. 38 and 39. In the work of the pupils, the +spacing from right to left is often too narrow. This crowding +of letters, together with unevenness of spacing, adds +notably to the task of eye and mind.</p> + +<p><i>The composition or make-up of the page.</i>—Other things +being equal, that arrangement of the page is best which +helps a child most to keep his place on a page and to find +it after having looked away to work on the paper on which +he computes, or for other good reasons. A good page and +a bad page in this respect are shown in Figs. 40 and 41.<span class='pagenum'><a name="Page_244" id="Page_244">[Pg 244]</a></span></p> + + +<div class="figcenter" style="width: 480px;"> +<img src="images/fig40.png" width="480" height="635" alt="Fig. 40." title="Fig. 40." /> +<span class="caption"><span class="smcap">Fig. 40.</span>—A page well made up to suit the action of the eye.</span> +</div> + +<p> <span class='pagenum'><a name="Page_245" id="Page_245">[Pg 245]</a></span></p> + +<div class="figcenter" style="width: 480px;"> +<img src="images/fig41.png" width="477" height="439" alt="Fig. 41." title="Fig. 41." /><br /> +<span class="caption"><span class="smcap">Fig. 41.</span>—The same matter as in Fig. 40, much +less well made up.</span> +</div> + +<p><span class='pagenum'><a name="Page_246" id="Page_246">[Pg 246]</a></span></p> + +<p><i>Objective presentations.</i>—Pictures, diagrams, maps, and +other presentations should not tax the eye unduly,</p> + +<div class="blockquot"> +<p class="nblockquot">(<i>a</i>) by requiring too fine distinctions, or</p> + +<p class="nblockquot">(<i>b</i>) by inconvenient arrangement of the data, preventing +easy counting, measuring, comparison, or whatever +the task is, or</p> + +<p class="nblockquot">(<i>c</i>) by putting too many facts in one picture so that the +eye and mind, when trying to make out any one, are +confused by the others.</p></div> + +<p>Illustrations of bad practices in these respects are shown +in Figs. 42 to 52. A few specimens of work well arranged for +the eye are shown in Figs. 53 to 56.</p> + +<p>Good rules to remember are:—</p> + +<p>Other things being equal, make distinctions by the clearest +method, fit material to the tendency of the eye to see an +'eyeful' at a time (roughly 1½ inch by ½ inch in a book; +1½ ft. by ½ ft. on the blackboard), and let one picture teach +only one fact or relation, or such facts and relations as do +not interfere in perception.</p> + +<p>The general conditions of seating, illumination, paper, and +the like are even more important when the eyes are used +with numbers than when they are used with words.</p> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig42.jpg" width="800" height="321" alt="Fig. 42." title="Fig. 42." /> +<span class="caption"><span class="smcap">Fig. 42.</span>—Try to count the rungs on the ladder, or the shocks in the wagon.</span> +</div> + +<p> <span class='pagenum'><a name="Page_247" id="Page_247">[Pg 247]</a></span></p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig43.jpg" width="800" height="386" alt="Fig. 43." title="Fig. 43." /> +<span class="caption"><span class="smcap">Fig. 43.</span>—How many oars do you see? How many birds? How many fish?</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig44.jpg" width="800" height="271" alt="Fig. 44." title="Fig. 44." /> +<span class="caption"><span class="smcap">Fig. 44.</span>—Count the birds in each of the three flocks of birds.</span> +</div> + + +<p><span class='pagenum'><a name="Page_248" id="Page_248">[Pg 248]</a></span></p> + +<div class="figcenter" style="width: 720px;"> +<img src="images/fig45.png" width="720" height="660" alt="Fig. 45." title="Fig. 45." /> +<span class="caption"><span class="smcap">Fig. 45.</span>—Note the lack of clear division of the hundreds. Consider the difficulty +of counting one of these columns of dots.</span> +</div> + +<p> <span class='pagenum'><a name="Page_249" id="Page_249">[Pg 249]</a></span></p> + + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig46.png" width="640" height="360" alt="Fig. 46." title="Fig. 46." /> +<span class="caption"><span class="smcap">Fig. 46.</span>—What do you suppose these pictures are intended to show?</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig47.png" width="800" height="500" alt="Fig. 47." title="Fig. 47." /> +<p class="nblockquot"><b><span class="smcap">Fig. 47.</span>—Would a beginner know that after THIRTEEN he was to switch around +and begin at the other end? Could you read the SIX of TWENTY-SIX if you +did not already know what it ought to be? What meaning would all the +brackets have for a little child in grade 2? Does this picture illustrate or +obfuscate?</b></p> +</div> + +<p> <span class='pagenum'><a name="Page_250" id="Page_250">[Pg 250]</a></span></p> + + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig48.jpg" width="800" height="700" alt="Fig. 48." title="Fig. 48." /> +<span class="caption"><span class="smcap">Fig. 48.</span>—How long did it take you to find out what these pictures mean?</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig49.png" width="640" height="250" alt="Fig. 49." title="Fig. 49." /><br /> +<p class="nblockquot"><b><span class="smcap">Fig. 49.</span>—Count the figures in the first row, using your eyes alone; have some one +make lines of 10, 11, 12, 13, and more repetitions of this figure spaced closely as +here. Count 20 or 30 such lines, using the eye unaided by fingers, pencil, etc.</b></p> +</div> + +<p> <span class='pagenum'><a name="Page_251" id="Page_251">[Pg 251]</a></span></p> + + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig50.png" width="640" height="147" alt="Fig. 50." title="Fig. 50." /> +<span class="caption"><span class="smcap">Fig. 50.</span>—Can you answer the question without measuring? Could a child of +seven or eight?</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig51.png" width="640" height="320" alt="Fig. 51." title="Fig. 51." /><br /> +<span class="caption"><span class="smcap">Fig. 51.</span>—What are these drawings intended to show? Why do they show the +facts only obscurely and dubiously?</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 579px;"> +<img src="images/fig52.png" width="579" height="480" alt="Fig. 52." title="Fig. 52." /> +<span class="caption"><span class="smcap">Fig. 52.</span>—What are these drawings intended to show? What simple change would +make them show the facts much more clearly?</span> +</div> + +<p> <span class='pagenum'><a name="Page_252" id="Page_252">[Pg 252]</a></span></p> + +<div class="figcenter" style="width: 683px;"> +<img src="images/fig53.png" width="683" height="600" alt="Fig. 53." title="Fig. 53." /> +<span class="caption"><span class="smcap">Fig. 53.</span>—Arranged in convenient "eye-fulls."</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 600px;"> +<div class="figleft" style="width: 336px;"> +<img src="images/fig54.png" width="235" height="235" alt="Fig. 54." title="Fig. 54." /><br /> +<span class="caption"><span class="smcap">Fig. 54.</span>—Clear, simple, and easy of comparison.</span><br /> +<br /> +</div> + +<p> </p> +<p> </p> +<p class="nblockquot"> Tell which bar has—<br /> +<small> +<b>1.</b> About 5 percent of its length black.<br /> +<b>2.</b> About 10 percent of its length black.<br /> +<b>3.</b> About 25 percent of its length black.<br /> +<b>4.</b> About 75 percent of its length black.<br /> +<b>5.</b> About 90 percent of its length black.<br /> +<b>6.</b> About 95 percent of its length black.<br /> +</small></p> +</div> +<p> <span class='pagenum'><a name="Page_253" id="Page_253">[Pg 253]</a></span></p> + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig55.png" width="640" height="155" alt="Fig. 55." title="Fig. 55." /> +<span class="caption"><span class="smcap">Fig. 55.</span>—Clear, simple, and well spaced.</span> +</div> + +<p> </p> + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig56.png" width="640" height="206" alt="Fig. 56." title="Fig. 56." /> +<span class="caption"><span class="smcap">Fig. 56.</span>—Well arranged, though a little wider spacing between the squares would +make it even better.</span> +</div> + + +<h4>THE USE OF CONCRETE OBJECTS IN ARITHMETIC</h4> + +<p>We mean by concrete objects actual things, events, and +relations presented to sense, in contrast to words and numbers +and symbols which mean or stand for these objects or +for more abstract qualities and relations. Blocks, tooth-picks, +coins, foot rules, squared paper, quart measures, +bank books, and checks are such concrete things. A foot +rule put successively along the three thirds of a yard rule, +a bell rung five times, and a pound weight balancing sixteen +ounce weights are such concrete events. A pint beside +a quart, an inch beside a foot, an apple shown cut in halves +display such concrete relations to a pupil who is attentive +to the issue.</p> + +<p>Concrete presentations are obviously useful in arithmetic +to teach meanings under the general law that a word or +number or sign or symbol acquires meaning by being connected +with actual things, events, qualities, and relations.<span class='pagenum'><a name="Page_254" id="Page_254">[Pg 254]</a></span> +We have also noted their usefulness as means to verifying +the results of thinking and computing, as when a pupil, +having solved, "How many badges each 5 inches long can +be made from 3<small><sup>1</sup>⁄<sub>3</sub></small> yd. of ribbon?" by using 10 × <sup>12</sup>⁄<sub>5</sub>, draws a +line 3<small><sup>1</sup>⁄<sub>3</sub></small> yd. long and divides it into 5-inch lengths.</p> + +<p>Concrete experiences are useful whenever the meaning of a +number, like 9 or <sup>7</sup>⁄<sub>8</sub> or .004, or of an operation, like multiplying +or dividing or cubing, or of some term, like rectangle or +hypothenuse or discount, or some procedure, like voting or +insuring property against fire or borrowing money from a +bank, is absent or incomplete or faulty. Concrete work thus +is by no means confined to the primary grades but may be +appropriate at all stages when new facts, relations, and procedures +are to be taught.</p> + +<p>How much concrete material shall be presented will depend +upon the fact or relation or procedure which is to be +made intelligible, and the ability and knowledge of the +pupil. Thus 'one half' will in general require less concrete +illustration than 'five sixths'; and five sixths will require +less in the case of a bright child who already knows +<sup>2</sup>⁄<sub>3</sub>, <sup>3</sup>⁄<sub>4</sub>, <sup>3</sup>⁄<sub>8</sub>, <sup>5</sup>⁄<sub>8</sub>, <sup>7</sup>⁄<sub>8</sub>, <sup>2</sup>⁄<sub>5</sub>, +<sup>3</sup>⁄<sub>5</sub>, and <sup>4</sup>⁄<sub>5</sub> +than in the case of a dull child or one who +only knows <sup>2</sup>⁄<sub>3</sub> and <sup>3</sup>⁄<sub>4</sub>. As a general rule the same topic will +require less concrete material the later it appears in the school +course. If the meanings of the numbers are taught in +grade 2 instead of grade 1, there will be less need of blocks, +counters, splints, beans, and the like. If 1½ + ½ = 2 is +taught early in grade 3, there will be more gain from the +use of 1½ inches and ½ inch on the foot rule than if the same +relations were taught in connection with the general addition +of like fractions late in grade 4. Sometimes the understanding +can be had either by connecting the idea with the +reality directly, or by connecting the two indirectly <i>via</i> +some other idea. The amount of concrete material to be<span class='pagenum'><a name="Page_255" id="Page_255">[Pg 255]</a></span> +used will depend on its relative advantage per unit of time +spent. Thus it might be more economical to connect <sup>5</sup>⁄<sub>12</sub>, +<sup>7</sup>⁄<sub>12</sub>, and <sup>11</sup>⁄<sub>12</sub> with real meanings indirectly by calling up the +resemblance to the <sup>2</sup>⁄<sub>3</sub>, <sup>3</sup>⁄<sub>4</sub>, <sup>3</sup>⁄<sub>8</sub>, <sup>5</sup>⁄<sub>8</sub>, <sup>7</sup>⁄<sub>8</sub>, <sup>2</sup>⁄<sub>5</sub>, +<sup>3</sup>⁄<sub>5</sub>, <sup>4</sup>⁄<sub>5</sub>, and <sup>5</sup>⁄<sub>6</sub> already studied, +than by showing <sup>5</sup>⁄<sub>12</sub> of an apple, <sup>7</sup>⁄<sub>12</sub> of a yard, <sup>11</sup>⁄<sub>12</sub> of a foot, +and the like.</p> + +<p>In general the economical course is to test the understanding +of the matter from time to time, using more concrete +material if it is needed, but being careful to encourage +pupils to proceed to the abstract ideas and general principles +as fast as they can. It is wearisome and debauching to +pupils' intellects for them to be put through elaborate concrete +experiences to get a meaning which they could have +got themselves by pure thought. We should also remember +that the new idea, say of the meaning of decimal fractions, +will be improved and clarified by using it (see page 183 f.), so +that the attainment of a <i>perfect</i> conception of decimal fractions +before doing anything with them is unnecessary and +probably very wasteful.</p> + +<p>A few illustrations may make these principles more instructive.</p> + +<p>(<i>a</i>) Very large numbers, such as 1000, 10,000, 100,000, and +1,000,000, need more concrete aids than are commonly given. +Guessing contests about the value in dollars of the school +building and other buildings, the area of the schoolroom +floor and other surfaces in square inches, the number of +minutes in a week, and year, and the like, together with +proper computations and measurements, are very useful to +reënforce the concrete presentations and supply genuine problems +in multiplication and subtraction with large numbers.</p> + +<p>(<i>b</i>) Numbers very much smaller than one, such as <sup>1</sup>⁄<sub>32</sub>, <sup>1</sup>⁄<sub>64</sub>, +.04, and .002, also need some concrete aids. A diagram like +that of Fig. 57 is useful.<span class='pagenum'><a name="Page_256" id="Page_256">[Pg 256]</a></span></p> + +<p>(<i>c</i>) <i>Majority</i> and <i>plurality</i> should be understood by every +citizen. They can be understood without concrete aid, but +an actual vote is well worth while for the gain in vividness +and surety.</p> + +<div class="figcenter" style="width: 640px;"> +<img src="images/fig57.png" width="480" height="480" alt="Fig. 57." title="Fig. 57." /><br /> +<span class="caption"><span class="smcap">Fig. 57.</span>—Concrete aid to understanding fractions with large denominators.<br /> +A = <sup>1</sup>⁄<sub>1000</sub> sq. ft.; B = <sup>1</sup>⁄<sub>100</sub> sq. ft.; C = <sup>1</sup>⁄<sub>50</sub> sq. ft.; D = <sup>1</sup>⁄<sub>10</sub> sq. ft.</span> +</div> + +<p>(<i>d</i>) Insurance against loss by fire can be taught by explanation +and analogy alone, but it will be economical to +have some actual insuring and payment of premiums and +a genuine loss which is reimbursed.<span class='pagenum'><a name="Page_257" id="Page_257">[Pg 257]</a></span></p> + +<p>(<i>e</i>) Four play banks in the corners of the room, receiving +deposits, cashing checks, and later discounting notes will +give good educational value for the time spent.</p> + +<p>(<i>f</i>) Trade discount, on the contrary, hardly requires more +concrete illustration than is found in the very problems to +which it is applied.</p> + +<p>(<i>g</i>) The process of finding the number of square units in a +rectangle by multiplying with the appropriate numbers +representing length and width is probably rather hindered +than helped by the ordinary objective presentation as an +introduction. The usual form of objective introduction is +as follows:—</p> + +<div class="figcenter" style="width: 448px;"> +<img src="images/fig58.png" width="448" height="336" alt="Fig. 58." title="Fig. 58." /><br /> +<span class="caption"><span class="smcap">Fig. 58.</span></span> +</div> + +<div class="pblockquot"><p>How long is this rectangle? How large is each square? How +many square inches are there in the top row? How many rows are<span class='pagenum'><a name="Page_258" id="Page_258">[Pg 258]</a></span> +there? How many square inches are there in the whole rectangle? +Since there are three rows each containing 4 square inches, +we have 3 × 4 square inches = 12 square inches.</p> + +<p>Draw a rectangle 7 inches long and 2 inches wide. If you divide +it into inch squares how many rows will there be? How many +inch squares will there be in each row? How many square inches +are there in the rectangle?</p></div> + +<div class="figcenter" style="width: 525px;"> +<img src="images/fig59.png" width="525" height="403" alt="Fig. 59." title="Fig. 59." /><br /> +<span class="caption"><span class="smcap">Fig. 59.</span></span> +</div> + +<p>It is better actually to hide the individual square units as +in Fig. 59. There are four reasons: (1) The concrete rows +and columns rather distract attention from the essential +thing to be learned. This is not that "<i>x</i> rows one square +wide, <i>y</i> squares in a row will make <i>xy</i> squares in all," but +that "by using proper units and the proper operation the +area of any rectangle can be found from its length and +width." (2) Children have little difficulty in learning to<span class='pagenum'><a name="Page_259" id="Page_259">[Pg 259]</a></span> +multiply rather than add, subtract, or divide when computing +area. (3) The habit so formed holds good for areas like +1<small><sup>2</sup>⁄<sub>3</sub></small> by 4½, 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 ½' by 2' or <sup>1</sup>⁄<sub>3</sub> in. by 3 in. or 1½ in. by <sup>2</sup>⁄<sub>3</sub> in. is +likely to be formed too emphatically if much time is spent +upon the sort of concrete presentation shown above. It is +then better to use concrete counting of rows of small areas +as a means of <i>verification after</i> the procedure is learned, than +as a means of deriving it.</p> + +<p>There has been, especially in Germany, much argument +concerning what sort of number-pictures (that is, arrangement +of dots, lines, or the like, as shown in Fig. 60) is best +for use in connection with the number names in the early +years of the teaching of arithmetic.</p> + +<p>Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell +['14], and others have measured the accuracy of children in +estimating the number of dots in arrangements of one or +more of these different types.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a> Many writers interpret a +difference in favor of estimating, say, the square arrangements +of Born or Lay as meaning that such is the best +arrangement to use in teaching. The inference is, however, +unjustified. That certain number-pictures are easier to +estimate numerically does not necessarily mean that they +are more instructive in learning. One set may be easier +to estimate just because they are more familiar, having +been oftener experienced. Even if the favored set was so +after equal experience with all sets, accuracy of estimation +would be a sign of superiority for use in instruction only +if all other things were equal (or in favor of the arrangement +<span class='pagenum'><a name="Page_260" id="Page_260">[Pg 260]</a></span> +in question). Obviously the way to decide which of these +is best to use in teaching is by using them in teaching and +measuring all relevant results, not by merely recording which +of them are most accurately estimated in certain time exposures.</p> + +<div class="figcenter" style="width: 1024px;"> +<img src="images/fig60.png" width="1024" height="565" alt="Fig. 60." title="Fig. 60." /> +<span class="caption"><span class="smcap">Fig. 60.</span>—Various proposed arrangements of dots for use in teaching the meanings of the numbers 1 to 10.</span> +</div> + +<p><span class='pagenum'><a name="Page_261" id="Page_261">[Pg 261]</a></span></p> + + +<p>It may be noted that the Born, Lay, and Freeman pictures +have claims for special consideration on grounds of probable +instructiveness. Since they are also superior in the tests in +respect to accuracy of estimate, choice should probably be +made from these three by any teacher who wishes to connect +one set of number-pictures systematically with the number +names, as by drills with the blackboard or with cards.</p> + +<p>Such drills are probably useful if undertaken with zeal, +and if kept as supplementary to more realistic objective +work with play money, children marching, material to be +distributed, garden-plot lengths to be measured, and the +like, and if so administered that the pupils soon get the +generalized abstract meaning of the numbers freed from +dependence on an inner picture of any sort. This freedom +is so important that it may make the use of many types of +number-pictures advisable rather than the use of the one +which in and of itself is best.</p> + +<p>As Meumann says: "Perceptual reckoning can be overdone. +It had its chief significance for the surety and clearness +of the first foundation of arithmetical instruction. If, +however, it is continued after the first operations become +familiar to the child, and extended to operations which develop +from these elementary ones, it necessarily works as a +retarding force and holds back the natural development of +arithmetic. This moves on to work with abstract number +and with mechanical association and reproduction." ['07, +Vol. 2, p. 357.]</p> + +<p>Such drills are commonly overdone by those who make<span class='pagenum'><a name="Page_262" id="Page_262">[Pg 262]</a></span> +use of them, being given too often, and continued after their +instructiveness has waned, and used instead of more significant, +interesting, and varied work in counting and estimating +and measuring real things. Consequently, there is now +rather a prejudice against them in our better schools. They +should probably be reinstated but to a moderate and judicious +use.</p> + + +<h4>ORAL, MENTAL, AND WRITTEN ARITHMETIC</h4> + +<p>There has been much dispute over the relative merits of +oral and written work in arithmetic—a question which is +much confused by the different meanings of 'oral' and +'written.' <i>Oral</i> has meant (1) work where the situations +are presented orally and the pupil's final responses are given +orally, or (2) work where the situations are presented orally +and the pupils' final responses are written or partly written +and partly oral, or (3) work where the situations are presented +in writing or print and the final responses are oral. <i>Written</i> +has meant (1) work where the situations are presented in +writing or print and the final responses are made in writing, +or (2) work where also many of the intermediate responses +are written, or (3) work where the situations are presented +orally but the final responses and a large percentage of the +intermediate computational responses are written. There +are other meanings than these.</p> + +<p>It is better to drop these very ambiguous terms and ask +clearly what are the merits and demerits, in the case of any +specified arithmetical work, of auditory and of visual presentation +of the situations, and of saying and of writing each +specified step in the response.</p> + +<p>The disputes over mental <i>versus</i> written arithmetic are +also confused by ambiguities in the use of 'mental.' Mental +has been used to mean "done without pencil and paper"<span class='pagenum'><a name="Page_263" id="Page_263">[Pg 263]</a></span> +and also "done with few overt responses, either written or +spoken, between the setting of the task and the announcement +of the answer." In neither case is the word <i>mental</i> +specially appropriate as a description of the total fact. As +before, we should ask clearly, "What are the merits and +demerits of making certain specified intermediate responses in +inner speech or imaged sounds or visual images or imageless +thought—that is, <i>without</i> actual writing or overt speech?"</p> + +<p>It may be said at the outset that oral, written, and inner +presentations of initial situations, oral, written, and inner +announcements of final responses, and oral, written, and +inner management of intermediate processes have varying +degrees of merit according to the particular arithmetical +exercise, pupil, and context. Devotion to oralness or mentalness +as such is simply fanatical. Various combinations, such +as the written presentation of the situation with inner +management of the intermediate responses and oral announcement +of the final response have their special merits +for particular cases.</p> + +<p>These merits the reader can evaluate for himself for any +given sort of work for a given class by considering: (1) The +amount of practice received by the class per hour spent; +(2) the ease of correction of the work; (3) the ease of understanding +the tasks; (4) the prevention of cheating; (5) the +cheerfulness and sociability of the work; (6) the freedom +from eyestrain, and other less important desiderata.</p> + +<p>It should be noted that the stock schemes A, B, C, and D +below are only a few of the many that are possible and that +schemes E, F, G, and H have special merits.</p> + +<p>The common practice of either having no use made of +pencil and paper or having all computations and even much +verbal analysis written out elaborately for examination is +unfavorable for learning. The demands which life itself<span class='pagenum'><a name="Page_264" id="Page_264">[Pg 264]</a></span> +will make of arithmetical knowledge and skill will range from +tasks done with every percentage of written work from zero +up to the case where every main result obtained by thought +is recorded for later use by further thought. In school the +best way is that which, for the pupils in question, has the best +total effect upon quality of product, speed, and ease of production, +reënforcement of training already given, and preparation +for training to be given. There is nothing intellectually +criminal about using a pencil as well as inner thought; on the +other hand there is no magical value in writing out for the +teacher's inspection figures that the pupil does not need in +order to attain, preserve, verify, or correct his result.</p> + + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><th><span class="smcap">Presentation of Initial Situation</span></th> +<th><span class="smcap">Management of Intermediate Processes</span></th> +<th><span class="smcap">Announcement of Final Response</span></th></tr> +<tr><td align='left'>A. Printed or written</td><td align='left'>Written</td><td align='left'>Written</td></tr> +<tr><td align='left'>B. " "</td><td align='left'>Inner</td><td align='left'>Oral by one pupil, inner by the rest</td></tr> +<tr><td align='left'>C. Oral (by teacher)</td><td align='left'>Written</td><td align='left'>Written</td></tr> +<tr><td align='left'>D. " "</td><td align='left'>Inner</td><td align='left'>Oral by one pupil, inner by the rest</td></tr> +<tr><td align='left'>E. As in A or C</td><td align='left'>A mixture, the pupil writing what he needs</td><td align='left'>As in A or B or H</td></tr> +<tr><td align='left'>F. The real situation itself, in part at least</td><td align='left'>As in E</td><td align='left'>As in A or B or H</td></tr> +<tr><td align='left'>G. Both read by the pupil and put orally by the teacher</td><td align='left'>As in E</td><td align='left'>As in A or B or H</td></tr> +<tr><td align='left'>H. As in A or C or G</td><td align='left'>As in E</td><td align='left'>Written by all pupils, announced orally by one pupil</td></tr> +</table></div> + + + + + +<p><span class='pagenum'><a name="Page_265" id="Page_265">[Pg 265]</a></span></p> + +<p>The common practice of having the final responses of all +<i>easy</i> tasks given orally has no sure justification. On the +contrary, the great advantage of having all pupils really +do the work should be secured in the easy work more than +anywhere else. If the time cost of copying the figures is +eliminated by the simple plan of having them printed, and +if the supervision cost of examining the papers is eliminated +by having the pupils correct each other's work in these easy +tasks, written answers are often superior to oral except for +the elements of sociability and 'go' and freedom from eyestrain +of the oral exercise. Such written work provides the +gifted pupils with from two to ten times as much practice +as they would get in an oral drill on the same material, supposing +them to give inner answers to every exercise done +by the class as a whole; it makes sure that the dull pupils +who would rarely get an inner answer at the rate demanded +by the oral exercise, do as much as they are able to do.</p> + +<p>Two arguments often made for the oral statement of +problems by the teacher are that problems so put are better +understood, especially in the grades up through the fifth, +and that the problems are more likely to be genuine and +related to the life the pupils know. When these statements +are true, the first is a still better argument for having +the pupils read the problems <i>aided by the teacher's oral statement +of them</i>. For the difficulty is largely that the pupils +cannot read well enough; and it is better to help them to +surmount the difficulty rather than simply evade it. The +second is not an argument for oralness <i>versus</i> writtenness, +but for good problems <i>versus</i> bad; the teacher who makes +up such good problems should, in fact, take special care to +write them down for later use, which may be by voice or by +the blackboard or by printed sheet, as is best.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_266" id="Page_266">[Pg 266]</a></span></p> +<h2><a name="CHAPTER_XIV" id="CHAPTER_XIV"></a>CHAPTER XIV</h2> + +<h3>THE CONDITIONS OF LEARNING: THE PROBLEM +ATTITUDE</h3> + + +<p>Dewey, and others following him, have emphasized the +desirability of having pupils do their work as active seekers, +conscious of problems whose solution satisfies some real +need of their own natures. Other things being equal, it is +unwise, they argue, for pupils to be led along blindfold as it +were by the teacher and textbook, not knowing where they +are going or why they are going there. They ought rather +to have some living purpose, and be zealous for its attainment.</p> + +<p>This doctrine is in general sound, as we shall see, but it is +often misused as a defense of practices which neglect the +formation of fundamental habits, or as a recommendation +to practices which are quite unworkable under ordinary +classroom conditions. So it seems probable that its nature +and limitations are not thoroughly known, even to its followers, +and that a rather detailed treatment of it should be +given here.</p> + + +<h4>ILLUSTRATIVE CASES</h4> + +<p>Consider first some cases where time spent in making +pupils understand the end to be attained before attacking +the task by which it is attained, or care about attaining the +end (well or ill understood) is well spent.<span class='pagenum'><a name="Page_267" id="Page_267">[Pg 267]</a></span></p> + +<p>It is well for a pupil who has learned (1) the meanings of +the numbers one to ten, (2) how to count a collection of ten +or less, and (3) how to measure in inches a magnitude of +ten, nine, eight inches, etc., to be confronted with the +problem of true adding without counting or measuring, as +in 'hidden' addition and measurement by inference. For +example, the teacher has three pencils counted and put +under a book; has two more counted and put under the +book; and asks, "How many pencils are there under the +book?" Answers, when obtained, are verified or refuted +by actual counting and measuring.</p> + +<p>The time here is well spent because the children can do the +necessary thinking if the tasks are well chosen; because they +are thereby prevented from beginning their study of addition +by the bad habit of pseudo-adding by looking at the two +groups of objects and counting their number instead of +real adding, that is, thinking of the two numbers and inferring +their sum; and further, because facing the problem of +adding as a real problem is in the end more economical for +learning arithmetic and for intellectual training in general +than being enticed into adding by objective or other processes +which conceal the difficulty while helping the pupil to master +it.</p> + +<p>The manipulation of short multiplication may be introduced +by confronting the pupils with such problems as, +"How to tell how many Uneeda biscuit there are in four +boxes, by opening only one box." Correct solutions by +addition should be accepted. Correct solutions by multiplication, +if any gifted children think of this way, should be +accepted, even if the children cannot justify their procedure. +(Inferring the manipulation from the place-values of numbers +is beyond all save the most gifted and probably beyond +them.) Correct solution by multiplication by some child<span class='pagenum'><a name="Page_268" id="Page_268">[Pg 268]</a></span> +who happens to have learned it elsewhere should be accepted. +Let the main proof of the trustworthiness of the +manipulation be by measurement and by addition. Proof +by the stock arguments from the place-values of numbers +may also be used. If no child hits on the manipulation in +question, the problem of finding the length <i>without</i> adding +may be set. If they still fail, the problem may be made +easier by being put as "4 times 22 gives the answer. Write +down what you think 4 times 22 will be." Other reductions +of the difficulty of the problem may be made, or the teacher +may give the answer without very great harm being done. +The important requirement is that the pupils should be +aware of the problem and treat the manipulation as a solution +of it, not as a form of educational ceremonial which +they learn to satisfy the whims of parents and teachers. +In the case of any particular class a situation that is more +appealing to the pupils' practical interests than the situation +used here can probably be devised.</p> + +<p>The time spent in this way is well spent (1) because all +but the very dull pupils can solve the problem in some way, +(2) because the significance of the manipulation as an +economy over addition is worth bringing out, and (3) because +there is no way of beginning training in short multiplication +that is much better.</p> + +<p>In the same fashion multiplication by two-place numbers +may be introduced by confronting pupils with the problem +of the number of sheets of paper in 72 pads, or pieces of chalk +in 24 boxes, or square inches in 35 square feet, or the number +of days in 32 years, or whatever similar problem can be +brought up so as to be felt as a problem.</p> + +<p>Suppose that it is the 35 square feet. Solutions by +(5 × 144) + (30 × 144), however arranged, or by (10 × 144) + +(10 × 144) + (10 × 144) + (5 × 144), or by 3500 + (35 × 40) + +<span class='pagenum'><a name="Page_269" id="Page_269">[Pg 269]</a></span> +(35 × 4), or by 7 × (5 × 144), however arranged, should all be +listed for verification or rejection. The pupils need not be +required to justify their procedures by a verbal statement. +Answers like 432,720, or 720,432, or 1152, or 4220, or 3220 +should be listed for verification or rejection. Verification +may be by a mixture of short multiplication and objective +work, or by a mixture of short multiplication and addition, +or by addition abbreviated by taking ten 144s as +1440, or even (for very stupid pupils) by the authority +of the teacher. Or the manipulation in cases like 53 × 9 +or 84 × 7 may be verified by the reverse short multiplication. +The deductive proof of the correctness of the manipulation +may be given in whole or in part in connection with +exercises like</p> + + +<div class='center'> +<table border="0" cellpadding="2" cellspacing="5" width="60%" summary=""> +<tr><td align='left'>10 × 2 =</td><td align='left'>30 × 14 =</td></tr> +<tr><td align='left'>10 × 3 =</td><td align='left'> 3 × 44 =</td></tr> +<tr><td align='left'>10 × 4 =</td><td align='left'>30 × 44 =</td></tr> +<tr><td align='left'>10 × 14 =</td><td align='left'> 3 × 144 =</td></tr> +<tr><td align='left'>10 × 44 =</td><td align='left'>20 × 144 =</td></tr> +<tr><td align='left'>10 × 144 =</td><td align='left'>40 × 144 =</td></tr> +<tr><td align='left'>20 × 2 =</td><td align='left'>30 × 144 =</td></tr> +<tr><td align='left'>20 × 3 =</td><td align='left'> 5 × 144 =</td></tr> +<tr><td align='left'>30 × 3 =</td><td align='left'>35 = 30 + ....</td></tr> +<tr><td align='left'>30 × 4 =</td><td align='left'>30 × 144 added to 5 × 144 =</td></tr> +<tr><td align='left'> 3 × 14 =</td></tr> +</table></div> + +<p>Certain wrong answers may be shown to be wrong +in many ways; <i>e.g.</i>, 432,720 is too big, for 35 times a +thousand square inches is only 35,000; 1152 is too small, +for 35 times a hundred square inches would be 3500, or +more than 1152.</p> + +<p>The time spent in realizing the problem here is fairly well +spent because (1) any successful original manipulation in +<span class='pagenum'><a name="Page_270" id="Page_270">[Pg 270]</a></span> +this case represents an excellent exercise of thought, because +(2) failures show that it is useless to juggle the figures +at random, and because (3) the previous experience with +short multiplication makes it possible for the pupils to realize +the problem in a very few minutes. It may, however, be +still better to give the pupils the right method just as soon +as the problem is realized, without having them spend more +time in trying to solve it. Thus:—</p> + +<p>1 square foot has 144 square inches. How many square +inches are there in 35 square feet (marked out in chalk on +the floor as a piece 10 ft. × 3 ft. plus a piece 5 ft. × 1 ft.)?</p> + +<p>1 yard = 36 inches. How many inches long is this wall +(found by measure to be 13 yards)?</p> + +<p>Here is a quick way to find the answers:—</p> + +<div class="blockquot"> +<p class="noidt"> +<span style="margin-left: 0.5em;">144</span><br /> +<span style="margin-left: 1em;">35</span><br /> +<span style="margin-left: 0em;">——</span><br /> +<span style="margin-left: 0.5em;">720</span><br /> +432<br /> +——<br /> +5040 sq. inches in 35 sq. ft.<br /> +<br /> + +<br /> +<span style="margin-left: 0.5em;">36</span><br /> +<span style="margin-left: 0.5em;">13</span><br /> +——<br /> +108<br /> +36<br /> +——<br /> +468 inches in 13 yd.<br /> +</p> +</div> + +<p>Consider now the following introduction to dividing by a +decimal:—</p> + +<p class="center"><b>Dividing by a Decimal</b></p> + +<div class="pblockquot"> +<p><b>1.</b> How many minutes will it take a motorcycle, to go +12.675 miles at the rate of .75 mi. per minute?</p> +</div> + +<div class="iblockquot"> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='right'>16.9</td></tr> +<tr><td align='right'>.75 |<span class="overline"> 12.675</span></td></tr> +<tr><td align='right'><span class="u"> 7 5</span> </td></tr> +<tr><td align='right'>5 17 </td></tr> +<tr><td align='right'><span class="u"> 4 50</span> </td></tr> +<tr><td align='right'>675</td></tr> +<tr><td align='right'><span class="u">675</span></td></tr> +</table></div> + + +<p><span class='pagenum'><a name="Page_271" id="Page_271">[Pg 271]</a></span></p> + +<div class="pblockquot"> +<p><b>2.</b> Check by multiplying 16.9 by .75.</p> + +<p><b>3.</b> How do you know that the quotient cannot be as little as 1.69?</p> + +<p><b>4.</b> How do you know that the quotient cannot be as large as 169?</p> + +<p><b>5.</b> Find the quotient for 3.75 ÷ 1.5.</p> + +<p><b>6.</b> Check your result by multiplying the quotient by the divisor.</p> + +<p><b>7.</b> How do you know that the quotient cannot be .25 or 25 ?</p> + +<p><b>8.</b> Look at this problem. .25<span class="overline">|7.5</span></p> + +<p> How do you know that 3.0 is wrong for the quotient?</p> + +<p> How do you know that 300 is wrong for the quotient?</p> + +<p>State which quotient is right for each of these:—</p> + +<p><b>9.</b> <small>.021 or .21 or 2.1 or 21 or 210</small><br /> + 1.8<span class="overline">|3.78</span></p> + +<p><b>10.</b> <small>.021 or .21 or 21 or 210</small><br /> + 1.8<span class="overline">|37.8</span></p> + +<p><b>11.</b> <small>.03 or .3 or 3 or 30 or 300</small><br /> + 1.25<span class="overline">|37.5</span></p> + +<p><b>12.</b> <small>.03 or .3 or 3 or 30 or 300</small><br /> + 12.5<span class="overline">|37.5</span></p> + +<p><b>13.</b> <small>.05 or .5 or 5 or 50 or 500</small><br /> + 1.25<span class="overline">|6.25</span></p> + +<p><b>14.</b> <small>.05 or .5 or 5 or 50 or 500</small><br /> + 12.5<span class="overline">|6.25</span></p> + +<p><b>15.</b> Is this rule true? If it is true, learn it.</p> + +<div class="blockquot"> +<p class="noidt"><b>In a correct result, the number of decimal places in +the divisor and quotient together equals the number +of decimal places in the dividend.</b></p> +</div> +</div> + +<p>These and similar exercises excite the problem attitude +in children <i>who have a general interest in getting right answers</i>. +Such a series carefully arranged is a desirable introduction +to a statement of the rule for placing the decimal point in +division with decimals. For it attracts attention to the +general principle (divisor × quotient should equal dividend), +which is more important than the rule for convenient location +of the decimal point, and it gives training in placing +the point by inspection of the divisor, quotient, and dividend, +which suffices for nineteen out of twenty cases that the pupil +will ever encounter outside of school. He is likely to remember +this method by inspection long after he has forgotten +the fixed rule.</p> + +<p>It is well for the pupil to be introduced to many arith<span class='pagenum'><a name="Page_272" id="Page_272">[Pg 272]</a></span>metical +facts by way of problems about their common uses. +The clockface, the railroad distance table in hundredths +of a mile, the cyclometer and speedometer, the recipe, and +the like offer problems which enlist his interest and energy +and also connect the resulting arithmetical learning with +the activities where it is needed. There is no time cost, +but a time-saving, for the learning as a means to the solution +of the problems is quicker than the mere learning of the +arithmetical facts by themselves alone. A few samples of +such procedure are shown below:—</p> + +<div class="blockquot"> + +<div class="pblockquot"> +<p class="tabcap">GRADE 3</p> + +<p class="center"><b>To be Done at Home</b></p> + +<p>Look at a watch. Has it any hands besides the hour hand and +the minute hand? Find out all that you can about how a watch +tells seconds, how long a second is, and how many seconds make a +minute.</p> + +<p class="tabcap">GRADE 5</p> + +<p class="center"><b>Measuring Rainfall</b></p> + + +<div class='figleft'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='center' colspan='3'><b>Rainfall per Week</b><br /> +(<b>cu. in. per sq. in. of area</b>)</td></tr> +<tr><td align='center'>June</td><td align='left'> 1-7</td><td align='center'>1.056</td></tr> +<tr><td align='center'></td><td align='left'> 8-14</td><td align='center'>1.103</td></tr> +<tr><td align='center'></td><td align='left'>15-21</td><td align='center'>1.040</td></tr> +<tr><td align='center'></td><td align='left'>22-28</td><td align='center'> .960</td></tr> +<tr><td align='center'></td><td align='left'>29-July 5</td><td align='center'> .915</td></tr> +<tr><td align='center'>July</td><td align='left'> 6-12</td><td align='center'> .782</td></tr> +<tr><td align='center'></td><td align='left'>13-19</td><td align='center'> .790</td></tr> +<tr><td align='center'></td><td align='left'>20-26</td><td align='center'> .670</td></tr> +<tr><td align='center'></td><td align='left'>27-Aug. 2</td><td align='center'> .503</td></tr> +<tr><td align='center'>Aug.</td><td align='left'> 3-9</td><td align='center'> .512</td></tr> +<tr><td align='center'></td><td align='left'>10-16</td><td align='center'> .240</td></tr> +<tr><td align='center'></td><td align='left'>17-23</td><td align='center'> .215</td></tr> +<tr><td align='center'></td><td align='left'>24-30</td><td align='center'> .811</td></tr> +</table></div> +<p><b>1.</b> In which weeks was the rainfall 1 +or more?</p> + +<p><b>2.</b> Which week of August had the +largest rainfall for that month?</p> + +<p><b>3.</b> Which was the driest week of the +summer? (Driest means with +the least rainfall.)</p> + +<p><b>4.</b> Which week was the next to the +driest?</p> + +<p><b>5.</b> In which weeks was the rainfall +between .800 and 1.000?</p> + +<p><b>6.</b> Look down the table and estimate +whether the average rainfall for +one week was about .5, or about +.6, or about .7, or about .8, or +about .9.</p> +<div style="clear: both;"></div> +<p><span class='pagenum'><a name="Page_273" id="Page_273">[Pg 273]</a></span></p> + +<p class="center"><b>Dairy Records</b></p> + + +<div class='figleft'> +<table border="0" cellpadding="1" cellspacing="0" summary=""> +<tr><td align='center' colspan='3'><b>Record of Star Elsie</b></td></tr> +<tr><th></th><th><small>Pounds of<br />Milk</small></th><th><small>Butter-Fat<br />per Pound<br />of Milk</small></th></tr> +<tr><td align='left'>Jan.</td><td align='left'><b>1742</b></td><td align='left'><b>.0461</b></td></tr> +<tr><td align='left'>Feb.</td><td align='left'><b>1690</b></td><td align='left'><b>.0485</b></td></tr> +<tr><td align='left'>Mar.</td><td align='left'><b>1574</b></td><td align='left'><b>.0504</b></td></tr> +<tr><td align='left'>Apr.</td><td align='left'><b>1226</b></td><td align='left'><b>.0490</b></td></tr> +<tr><td align='left'>May</td><td align='left'><b>1202</b></td><td align='left'><b>.0466</b></td></tr> +<tr><td align='left'>June</td><td align='left'><b>1251</b></td><td align='left'><b>.0481</b></td></tr> +</table></div> + +<p>Read this record of the +milk given by the cow Star +Elsie. The first column tells +the number of pounds of +milk given by Star Elsie each +month. The second column +tells what fraction of a pound +of butter-fat each pound of +milk contained.</p> + +<p><b>1.</b> Read the first line, saying, "In January this cow gave 1742 +pounds of milk. There were 461 ten thousandths of a +pound of butter-fat per pound of milk." Read the other +lines in the same way.</p> + +<p><b>2.</b> How many pounds of butter-fat did the cow produce in Jan.? +<b>3.</b> In Feb.? <b>4.</b> In Mar.? <b>5.</b> In Apr.? <b>6.</b> In May? +<b>7.</b> In June?</p> +<div style="clear: both;"></div> + +<p class="tabcap">GRADE 5 OR LATER</p> + +<p class="center"><b>Using Recipes to Make Larger or Smaller Quantities</b></p> + +<p>I. State how much you would use of each material in the +following recipes: (<i>a</i>) To make double the quantity. (<i>b</i>) To +make half the quantity. (<i>c</i>) To make 1½ times the quantity. +You may use pencil and paper when you cannot find the right +amount mentally.</p> + +<div class='center sblockquot'> +<table border="0" cellpadding="4" cellspacing="0" width="80%" summary=""> +<tr><td align='left'><b>1.</b> <span class="smcap">Peanut Penuche</span></td><td align='left'><b>2</b>. <span class="smcap">Molasses Candy</span></td></tr> +<tr><td align='left'>1 tablespoon butter</td><td align='left'>½ cup butter</td></tr> +<tr><td align='left'>2 cups brown sugar</td><td align='left'>2 cups sugar</td></tr> +<tr><td align='left'><sup>1</sup>⁄<sub>3</sub> cup milk or cream</td><td align='left'>1 cup molasses</td></tr> +<tr><td align='left'>¾ cup chopped peanuts</td><td align='left'>1½ cups boiling water</td></tr> +<tr><td align='left'><sup>1</sup>⁄<sub>3</sub> teaspoon salt</td></tr> +<tr><td colspan='2'> </td></tr> +<tr><td align='left'><b>3.</b> <span class="smcap">Raisin Opera Caramels</span></td><td align='left'><b>4.</b> <span class="smcap">Walnut Molasses Squares</span></td></tr> +<tr><td align='left'>2 cups light brown sugar</td><td align='left'>2 tablespoons butter</td></tr> +<tr><td align='left'><sup>7</sup>⁄<sub>8</sub> cup thin cream</td><td align='left'>1 cup molasses</td></tr> +<tr><td align='left'>½ cup raisins</td><td align='left'><sup>1</sup>⁄<sub>3</sub> cup sugar</td></tr> +<tr><td align='left'></td><td align='left'>½ cup walnut meats</td></tr> +<tr><td colspan='2'> <span class='pagenum'><a name="Page_274" id="Page_274">[Pg 274]</a></span></td></tr> +<tr><td align='left'><b>5.</b> <span class="smcap">Reception Rolls</span></td><td align='left'><b>6.</b> <span class="smcap">Graham Raised Loaf</span></td></tr> +<tr><td align='left'>1 cup scalded milk</td><td align='left'>2 cups milk</td></tr> +<tr><td align='left'>1½ tablespoons sugar</td><td align='left'>6 tablespoons molasses</td></tr> +<tr><td align='left'>1 teaspoon salt</td><td align='left'>1½ teaspoons salt</td></tr> +<tr><td align='left'>¼ cup lard</td><td align='left'><sup>1</sup>⁄<sub>3</sub> yeast cake</td></tr> +<tr><td align='left'>1 yeast cake</td><td align='left'>¼ cup lukewarm water</td></tr> +<tr><td align='left'>¼ cup lukewarm water</td><td align='left'>2 cups sifted Graham flour</td></tr> +<tr><td align='left'>White of 1 egg</td><td align='left'>½ cup Graham bran</td></tr> +<tr><td align='left'>3½ cups flour</td><td align='left'>¾ cup flour (to knead)</td></tr> +</table></div> + + +<p>II. How much would you use of each material in the following +recipes: (<i>a</i>) To make <sup>2</sup>⁄<sub>3</sub> as large a quantity? (<i>b</i>) To make 1<small><sup>1</sup>⁄<sub>3</sub></small> +times as much? (<i>c</i>) To make 2½ times as much?</p> + +<div class='center sblockquot'> +<table border="0" cellpadding="4" cellspacing="0" width="80%" summary=""> +<tr><td align='left'><b>1.</b> <span class="smcap">English Dumplings</span></td><td align='left'><b>2.</b> <span class="smcap">White Mountain Angel Cake</span></td></tr> +<tr><td align='left'>½ pound beef suet</td><td align='left'>1½ cups egg whites</td></tr> +<tr><td align='left'>1¼ cups flour</td><td align='left'>1½ cups sugar</td></tr> +<tr><td align='left'>3 teaspoons baking powder</td><td align='left'>1 teaspoon cream of tartar</td></tr> +<tr><td align='left'>1 teaspoon salt</td><td align='left'>1 cup bread flour</td></tr> +<tr><td align='left'>½ teaspoon pepper</td><td align='left'>¼ teaspoon salt</td></tr> +<tr><td align='left'>1 teaspoon minced parsley</td><td align='left'>1 teaspoon vanilla</td></tr> +<tr><td align='left'>¼ cup cold water</td></tr> +</table></div> +</div> +</div> + +<p>In many cases arithmetical facts and principles can be +well taught in connection with some problem or project +which is not arithmetical, but which has special potency to +arouse an intellectual activity in the pupil which by some +ingenuity can be directed to arithmetical learning. Playing +store is the most fundamental case. Planning for a +party, seeing who wins a game of bean bag, understanding +the calendar for a month, selecting Christmas presents, +planning a picnic, arranging a garden, the clock, the watch +with second hand, and drawing very simple maps are situations +suggesting problems which may bring a living purpose +into arithmetical learning in grade 2. These are all available +under ordinary conditions of class instruction. A +sample of such problems for a higher grade (6) is shown +below.</p> + +<div class="pblockquot"> +<p class="center"><b>Estimating Areas</b></p> + +<p>The children in the geography class had a contest in estimating +the areas of different surfaces. Each child wrote his estimates +<span class='pagenum'><a name="Page_275" id="Page_275">[Pg 275]</a></span> +for each of these maps, A, B, C, D, and E. (Only C and D are +shown here.) In the arithmetic class they learned how to find +the exact areas. Then they compared their estimates with the +exact areas to find who came nearest.</p> + +<div class="figcenter" style="width: 512px;"> +<img src="images/img292.jpg" width="512" height="202" alt="Estimating Areas" title="Estimating Areas" /> +</div> + +<p>Write your estimates for A, B, C, D, and E. Then study the +next 6 pages and learn how to find the exact areas.</p> + +<p>(The next 6 pages comprise training in the mensuration of +parallelograms and triangles.)</p></div> + +<p>In some cases the affairs of individual pupils include problems +which may be used to guide the individual in question +to a zealous study of arithmetic as a means of achieving +his purpose—of making a canoe, surveying an island, keeping +the accounts of a Girls' Canning Club, or the like. It +requires much time and very great skill to direct the work +of thirty or more pupils each busy with a special type of his +own, so as to make the work instructive for each, but in +some cases the expense of time and skill is justified.</p> + + +<h4>GENERAL PRINCIPLES</h4> + +<p>In general what should be meant when one says that it is +desirable to have pupils in the problem-attitude when they +are studying arithmetic is substantially as follows:—</p> + +<p><i>First.</i>—Information that comes as an answer to questions<span class='pagenum'><a name="Page_276" id="Page_276">[Pg 276]</a></span> +is better attended to, understood, and remembered than +information that just comes.</p> + +<p><i>Second.</i>—Similarly, movements that come as a step +toward achieving an end that the pupil has in view are +better connected with their appropriate situations, and such +connections are longer retained, than is the case with movements +that just happen.</p> + +<p><i>Third.</i>—The more the pupil is set toward getting the +question answered or getting the end achieved, the greater +is the satisfyingness attached to the bonds of knowledge +or skill which mean progress thereto.</p> + +<p><i>Fourth.</i>—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.</p> + +<p><i>Fifth.</i>—It would be still worse policy to rely exclusively<span class='pagenum'><a name="Page_277" id="Page_277">[Pg 277]</a></span> +on problems arising outside arithmetic. To learn arithmetic +is itself a series of problems of intrinsic interest and worth +to healthy-minded children. The need for ability to +multiply with United States money or to add fractions or +to compute percents may be as truly vital and engaging as +the need for skill to make a party dress or for money to buy +it or for time to play baseball. The intellectualistic needs +and problems should be considered along with all others, and +given whatever weight their educational value deserves.</p> + + +<h4>DIFFICULTY AND SUCCESS AS STIMULI</h4> + +<p>There are certain misconceptions of the doctrine of the +problem-attitude. The most noteworthy is that difficulty—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.<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a></p> + +<p>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.</p> + +<p>Mere difficulty, mere failure, mere inadequacy of existing +bonds, does nothing. It is hard for me to add three eight-place +numbers at a glance; I have failed to find as effective +illustrations for pages 276 to 277 as I wished; my existing +sensori-motor connections are inadequate to playing a golf +course in 65. But these events and conditions have done +nothing to stimulate me in respect to the behavior in question. +In the first of the three there is no annoying lack and +no dynamic influence at all; in the second there was to some +<span class='pagenum'><a name="Page_278" id="Page_278">[Pg 278]</a></span> +degree an annoying lack—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.</p> + +<p>The real issue in all this concerns the relative strength, +in the pupil's intellectual life, of the "negative reaction" of +behavior in general. An animal whose life processes are +interfered with so that an annoying state of affairs is set +up, changes his behavior, making one after another responses +as his instincts and learned tendencies prescribe, until the +annoying state of affairs is terminated, or the animal dies, +or suffers the annoyance as less than the alternatives which +his responses have produced. When the annoying state of +affairs is characterized by the failure of things as they are +to minister to a craving—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.</p> + +<p>Such is in some measure true of man's intellectual life. +In recalling a forgotten name, in solving certain puzzles, or +in simplifying an algebraic complex, there is an annoying<span class='pagenum'><a name="Page_279" id="Page_279">[Pg 279]</a></span> +lack of the name, solution, or factor, a trial of one after +another response, until the annoyance is relieved by success +or made less potent by fatigue or distraction. Even here +the <i>difficulty</i> does not do anything—but only the annoying +interference with our intellectual peace by the problem. +Further, although for the particular problem, the annoying +lack stimulates, and the successful attainment stops thinking, +the later and more important general effect on thinking is +the reverse. Successful attainment stops our thinking <i>on +that problem</i> but makes us more predisposed later to thinking +<i>in general</i>.</p> + +<p>Overt negative reaction, however, plays a relatively small +part in man's intellectual life. Filling intellectual voids or +relieving intellectual strains in this way is much less frequent +than being stimulated positively by things seen, words read, +and past connections acting under modified circumstances. +The notion of thinking as coming to a lack, filling it, meeting +an obstacle, dodging it, being held up by a difficulty and +overcoming it, is so one-sided as to verge on phantasy. The +overt lacks, strains, and difficulties come perhaps once in +five hours of smooth straightforward use and adaptation +of existing connections, and they might as truly be called +hindrances to thought—barriers which past successes help +the thinker to surmount. Problems themselves come more +often as cherished issues which new facts reveal, and whose +contemplation the thinker enjoys, than as strains or lacks +or 'problems which I need to solve.' It is just as true that +the thinker gets many of his problems as results from, or +bonuses along with, his information, as that he gets much of +his information as results of his efforts to solve problems.</p> + +<p>As between difficulty and success, success is in the long +run more productive of thinking. Necessity is not the +mother of invention. Knowledge of previous inventions is<span class='pagenum'><a name="Page_280" id="Page_280">[Pg 280]</a></span> +the mother; original ability is the father. The solutions +of previous problems are more potent in producing both new +problems and their solutions than is the mere awareness of +problems and desire to have them solved.</p> + +<p>In the case of arithmetic, learning to cancel instead of +getting the product of the dividends and the product of the +divisors and dividing the former by the latter, is a clear +case of very valuable learning, with ease emphasized rather +than difficulty, with the adequacy of existing bonds (when +slightly redirected) as the prime feature of the process +rather than their inadequacy, and with no sense of failure +or lack or conflict. It would be absurd to spend time in +arousing in the pupil, before beginning cancellation, a sense +of a difficulty—viz., that the full multiplying and dividing +takes longer than one would like. A pupil in grade 4 or +5 might well contemplate that difficulty for years to no advantage. +He should at once begin to cancel and prove by +checking that errorless cancellation always gives the right +answer. To emphasize before teaching cancellation the +inadequacy of the old full multiplying and dividing would, +moreover, not only be uneconomical as a means to teaching +cancellation; it would amount to casting needless slurs +on valuable past acquisitions, and it would, scientifically, +be false. For, until a pupil has learned to cancel, the old +full multiplying is not inadequate; it is admirable in every +respect. The issue of its inadequacy does not truly appear +until the new method is found. It is the best way until +the better way is mastered.</p> + +<p>In the same way it is unwise to spend time in making +pupils aware of the annoying lacks to be supplied by the +multiplication tables, the division tables, the casting out +of nines, or the use of the product of the length and breadth +of a rectangle as its area, the unit being changed to the<span class='pagenum'><a name="Page_281" id="Page_281">[Pg 281]</a></span> +square erected on the linear unit as base. The annoying +lack will be unproductive, while the learning takes place +readily as a modification of existing habits, and is sufficiently +appreciated as soon as it does take place. The +multiplication tables come when instead of merely counting +by 7s from 0 up saying "7, 14, 21," etc., the pupil counts +by 7s from 0 up saying "Two sevens make 14, three sevens +make 21, four sevens make 28," etc. The division tables +come as easy selections from the known multiplications; +the casting out of nines comes as an easy device. The +computation of the area of a rectangle is best facilitated, +not by awareness of the lack of a process for doing it, but by +awareness of the success of the process as verified objectively.</p> + +<p>In all these cases, too, the pupil would be misled if we +aroused first a sense of the inadequacy of counting, adding, +and objective division, an awareness of the difficulties which +the multiplication and division tables and nines device and +area theorem relieve. The displaced processes are admirable +and no unnecessary fault should be found with them, and they +are <i>not</i> inadequate until the shorter ways have been learned.</p> + + +<h4>FALSE INFERENCES</h4> + +<p>One false inference about the problem-attitude is that the +pupil should always understand the aim or issue before beginning +to form the bonds which give the method or process +that provides the solution. On the contrary, he will often +get the process more easily and value it more highly if he +is taught what it is <i>for</i> gradually while he is learning it. +The system of decimal notation, for example, may better +be taken first as a mere fact, just as we teach a child to talk +without trying first to have him understand the value of verbal +intercourse, or to keep clean without trying first to have +him understand the bacteriological consequences of filth.<span class='pagenum'><a name="Page_282" id="Page_282">[Pg 282]</a></span></p> + +<p>A second inference—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<sup>2</sup>⁄<sub>3</sub> × = divide by 6," ".333<sup>1</sup>⁄<sub>3</sub> × = divide by 3," "multiply +by .875 = divide the number by 8 and subtract the +quotient from the number."</p> + +<p>A third unwise tendency is to degrade the mere giving of +information—to belittle the value of facts acquired in any +other way than in the course of deliberate effort by the pupil +to relieve a problem or conflict or difficulty. As a protest +against merely verbal knowledge, and merely memoriter +knowledge, and neglect of the active, questioning search for +knowledge, this tendency to belittle mere facts has been +healthy, but as a general doctrine it is itself equally one-sided. +Mere facts not got by the pupil's thinking are often +of enormous value. They may stimulate to active thinking +just as truly as that may stimulate to the reception of facts. +In arithmetic, for example, the names of the numbers, the +use of the fractional form to signify that the upper number is +divided by the lower number, the early use of the decimal +point in U. S. money to distinguish dollars from cents, and +the meanings of "each," "whole," "part," "together," +"in all," "sum," "difference," "product," "quotient," and +the like are self-justifying facts.</p> + +<p>A fourth false inference is that whatever teaching makes +the pupil face a question and think out its answer is thereby +justified. This is not necessarily so unless the question is a +worthy one and the answer that is thought out an intrinsically +valuable one and the process of thinking used one that is<span class='pagenum'><a name="Page_283" id="Page_283">[Pg 283]</a></span> +appropriate for that pupil for that question. Merely to +think may be of little value. To rely much on formal discipline +is just as pernicious here as elsewhere. The tendency +to emphasize the methods of learning arithmetic at the expense +of what is learned is likely to lead to abuses different +in nature but as bad in effect as that to which the emphasis +on disciplinary rather than content value has led in the +study of languages and grammar, or in the old puzzle problems +of arithmetic.</p> + +<p>The last false inference that I shall discuss here is the +inference that most of the problems by which arithmetical +learning is stimulated had better be external to arithmetic +itself—problems about Noah's Ark or Easter Flowers or +the Merry Go Round or A Trip down the Rhine.</p> + +<p>Outside interests should be kept in mind, as has been +abundantly illustrated in this volume, but it is folly to +neglect the power, even for very young or for very stupid +children, of the problem "How can I get the right answer?" +Children do have intellectual interests. They do like +dominoes, checkers, anagrams, and riddles as truly as playing +tag, picking flowers, and baking cake. With carefully +graded work that is within their powers they like to learn +to add, subtract, multiply, and divide with integers, fractions, +and decimals, and to work out quantitative relations.</p> + +<p>In some measure, learning arithmetic is like learning to +typewrite. The learner of the latter has little desire to +present attractive-looking excuses for being late, or to save +expense for paper. He has no desire to hoard copies of such +and such literary gems. He may gain zeal from the fact +that a school party is to be given and invitations are to be +sent out, but the problem "To typewrite better" is after +all his main problem. Learning arithmetic is in some +measure a game whose moves are motivated by the general<span class='pagenum'><a name="Page_284" id="Page_284">[Pg 284]</a></span> +set of the mind toward victory—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.</p> + + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_285" id="Page_285">[Pg 285]</a></span></p> +<h2><a name="CHAPTER_XV" id="CHAPTER_XV"></a>CHAPTER XV</h2> + +<h3>INDIVIDUAL DIFFERENCES</h3> + + +<p>The general facts concerning individual variations in +abilities—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.</p> + + +<h4>NATURE AND AMOUNT</h4> + +<p>The surfaces of frequency shown in Figs. 61, 62, and 63 +are samples. In these diagrams each space along the baseline +represents a certain score or degree of ability, and the +height of the surface above it represents the number of +individuals obtaining that score. Thus in Fig. 61, 63 out of +1000 soldiers had no correct answer, 36 out of 1000 had one +correct answer, 49 had two, 55 had three, 67 had four, and +so on, in a test with problems (stated in words).</p> + +<p>Figure 61 shows that these adults varied from no problems +solved correctly to eighteen, around eight as a central +tendency. Figure 62 shows that children of the same +year-age (they were also from the same neighborhood and +in the same school) varied from under 40 to over 200 figures +correct. Figure 63 shows that even among children who +have all reached the same school grade and so had rather +<span class='pagenum'><a name="Page_286" id="Page_286">[Pg 286]</a></span> +similar educational opportunities in arithmetic, the variation +is still very great. It requires a range from 15 to over +30 examples right to include even nine tenths of them.</p> + + +<div class="figcenter" style="width: 1024px;"> +<img src="images/fig61.jpg" width="1024" height="297" alt="Fig. 61." title="Fig. 61." /> +<p class="nblockquot"><b><span class="smcap">Fig. 61.</span>—The scores of 1000 soldiers in the National Army born in English-speaking +countries, in Test 2 of the Army Alpha. The score is the number of +correct answers obtained in five minutes. Probably 10 to 15 percent of these +men were unable to read or able to read only very easy sentences at a very slow +rate. Data furnished by the Division of Psychology in the office of the Surgeon +General.</b></p> +</div> + +<p>It should, however, be noted that if each individual had +been scored by the average of his work on eight or ten different +days instead of by his work in just one test, the variability +would have been somewhat less than appears in +Figs. 61, 62, and 63.</p> + + +<div class="figcenter" style="width: 1024px;"> +<img src="images/fig62.jpg" width="1024" height="257" alt="Fig. 62." title="Fig. 62." /> +<p class="nblockquot"><b><span class="smcap">Fig. 62.</span>—The scores of 100 11-year-old pupils in a test of computation. Estimated +from the data given by Burt ['17, p. 68] for 10-, 11-, and 12-year-olds. The score +equals the number of correct figures.</b></p> +</div> + +<p>It is also the case that if each individual had been scored, +not in problem-solving alone or division alone, but in an +elaborate examination on the whole field of arithmetic, the +variability would have been somewhat less than appears in +Figs. 61, 62, and 63. On the other hand, if the officers and +<span class='pagenum'><a name="Page_287" id="Page_287">[Pg 287]</a></span> +the soldiers rejected for feeblemindedness had been included +in Fig. 61, if the 11-year-olds in special classes for the very +dull had been included in Fig. 62, and if all children who had +been to school six years had been included in Fig. 63, no +matter what grade they had reached, the effect would have +been to <i>increase</i> the variability.</p> + +<div class="figcenter" style="width: 1024px;"> +<img src="images/fig63.jpg" width="1024" height="439" alt="Fig. 63." title="Fig. 63." /> +<p class="nblockquot"><b><span class="smcap">Fig. 63.</span>—The scores of pupils in grade 6 in city schools in the Woody Division +Test A. The score is the number of correct answers obtained in 20 minutes. +From Woody ['16, p. 61].</b></p> +</div> + +<p>In spite of the effort by school officers to collect in any +one school grade those somewhat equal in ability or in +achievement or in a mixture of the two, the population of +the same grades in the same school system shows a very +wide range in any arithmetical ability. This is partly because +promotion is on a more general basis than arithmetical +ability so that some very able arithmeticians are deliberately +held back on account of other deficiencies, and some very +incompetent arithmeticians are advanced on account of +other excellencies. It is partly because of general inaccuracy +in classifying and promoting pupils.</p> + +<p>In a composite score made up of the sum of the scores in +Woody tests,—Add. A, Subt. A, Mult. A, and Div. A, and +two tests in problem-solving (ten and six graded problems, +<span class='pagenum'><a name="Page_288" id="Page_288">[Pg 288]</a></span> +with maximum attainable credits of 30 and 18), Kruse +['18] found facts from which I compute those of Table 13, +and Figs. 64 to 66, for pupils all having the training of the +same city system, one which sought to grade its pupils very +carefully.</p> + +<div class="figcenter" style="width: 768px;"> +<img src="images/figs64_66.jpg" width="768" height="797" alt="Figs. 64-66." title="Figs. 64-66." /> +<p class="nblockquot"><b><span class="smcap">Figs.</span> 64, 65, and 66.—The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65), +and grade 8 (Fig. 66) in a composite of tests in computation and problem-solving. +The time was about 120 minutes. The maximum score attainable was 196.</b></p> +</div> + +<p><span class='pagenum'><a name="Page_289" id="Page_289">[Pg 289]</a></span></p> + +<p>The overlapping of grade upon grade should be noted. +Of the pupils in grade 6 about 18 percent do better than the +average pupil in grade 7, and about 7 percent do better than +the average pupil in grade 8. Of the pupils in grade 8 about +33 percent do worse than the average pupil in grade 7 and +about 12 percent do worse than the average pupil in grade 6.</p> + + +<p class="tabcap">TABLE 13</p> + +<p class="center"><span class="smcap">Relative Frequencies of Scores in an Extensive Team of +Arithmetical Tests.<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> In Percents</span></p> + + +<table border="1" frame="hsides" rules="cols" cellpadding="2" cellspacing="0" width="60%" summary="Table 13"> +<thead class="bb bt"> +<tr><th><span class="smcap">Score</span></th><th><span class="smcap">Grade 6</span></th><th><span class="smcap">Grade 7</span></th><th><span class="smcap">Grade 8</span></th></tr> +</thead> +<tbody> +<tr><td align='center'>70 to 79</td><td align='center'> 1.3</td><td align='center'> .9</td><td align='center'> .4</td></tr> +<tr><td align='center'>80 " 89</td><td align='center'> 5.5</td><td align='center'> 2.3</td><td align='center'> .4</td></tr> +<tr><td align='center'>90 " 99</td><td align='center'> 10.6</td><td align='center'> 4.3</td><td align='center'> 2.9</td></tr> +<tr><td align='center'>100 " 109</td><td align='center'> 19.4</td><td align='center'> 5.2</td><td align='center'> 4.4</td></tr> +<tr><td align='center'>110 " 119</td><td align='center'> 19.8</td><td align='center'> 18.5</td><td align='center'> 5.8</td></tr> +<tr><td align='center'>120 " 129</td><td align='center'> 23.5</td><td align='center'> 16.2</td><td align='center'> 16.8</td></tr> +<tr><td align='center'>130 " 139</td><td align='center'> 12.6</td><td align='center'> 17.5</td><td align='center'> 16.8</td></tr> +<tr><td align='center'>140 " 149</td><td align='center'> 4.6</td><td align='center'> 13.9</td><td align='center'> 22.9</td></tr> +<tr><td align='center'>150 " 159</td><td align='center'> 1.7</td><td align='center'> 13.6</td><td align='center'> 17.1</td></tr> +<tr><td align='center'>160 " 169</td><td align='center'> 1.2</td><td align='center'> 4.8</td><td align='center'> 9.4</td></tr> +<tr><td align='center'>170 " 179</td><td align='center'></td><td align='center'> 2.5</td><td align='center'> 3.3</td></tr> +</tbody> +</table> + +<h4>DIFFERENCES WITHIN ONE CLASS</h4> + +<p>The variation within a single class for which a single +teacher has to provide is great. Even when teaching is +departmental and promotion is by subjects, and when also +the school is a large one and classification within a grade is +by ability—there may be a wide range for any given special +component ability. Under ordinary circumstances the +range is so great as to be one of the chief limiting conditions +for the teaching of arithmetic. Many methods appropriate +<span class='pagenum'><a name="Page_290" id="Page_290">[Pg 290]</a></span> +to the top quarter of the class will be almost useless for the +bottom quarter, and <i>vice versa</i>.</p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/fig67.jpg" width="600" height="766" alt="Fig. 67." title="Fig. 67." /> +<span class="caption"><span class="smcap">Fig. 67.</span></span> +</div> + +<p> </p> +<p><span class='pagenum'><a name="Page_291" id="Page_291">[Pg 291]</a></span></p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/fig68.jpg" width="600" height="735" alt="Fig. 68." title="Fig. 68." /> +<span class="caption"><span class="smcap">Fig. 68.</span><br /> +</span> +<p class="nblockquot"><b><span class="smcap">Figs.</span> 67 and 68.—The scores of ten 6 B classes in a 12-minute test in computation +with integers (the Courtis Test 7). The score is the number of units done. +Certain long tasks are counted as two units.</b></p> + +</div> + +<p>Figures 67 and 68 show the scores of ten classes taken at +random from ninety 6 B classes in one city by Courtis ['13, +p. 64] in amount of computation done in 12 minutes. Observe +the very wide variation present in the case of every +<span class='pagenum'><a name="Page_292" id="Page_292">[Pg 292]</a></span> +class. The variation within a class would be somewhat +reduced if each pupil were measured by his average in eight +or ten such tests given on different days. If a rather generous +allowance is made for this we still have a variation in +speed as great as that shown in Fig. 69, as the fact to be +expected for a class of thirty-two 6 B pupils.</p> + +<div class="figcenter" style="width: 800px;"> +<img src="images/fig69.jpg" width="800" height="274" alt="Fig. 69." title="Fig. 69." /> +<p class="nblockquot"><b><span class="smcap">Fig. 69.</span>—A conservative estimate of the amount of variation to be expected +within a single class of 32 pupils in grade 6, in the number of units done in +Courtis Test 7 when all chance variations are eliminated.</b></p> +</div> + +<p>The variations within a class in respect to what processes +are understood so as to be done with only occasional errors +may be illustrated further as follows:—A teacher in grade +4 at or near the middle of the year in a city doing the customary +work in arithmetic will probably find some pupil +in her class who cannot do column addition even without +carrying, or the easiest written subtraction</p> + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td rowspan='2'><span class="sz30">(</span></td><td align='center'>8</td><td align='center'>9</td><td align='center'> </td><td align='center'>78</td><td rowspan='2'><span class="sz30">)</span>,</td></tr> +<tr><td align='center'><span class="u">5</span></td><td align='center'><span class="u">3</span></td><td align='center'>or</td><td align='center'><span class="u">37</span></td></tr> +</table></div> + +<p class="noidt">who does not know his multiplication tables or how to derive +them, or understand the meanings of + − × and ÷, or +have any useful ideas whatever about division.</p> + +<p>There will probably be some child in the class who can do +such work as that shown below, and with very few errors.<span class='pagenum'><a name="Page_293" id="Page_293">[Pg 293]</a></span></p> + +<p class="noidt">Add</p> +<div class='center'> +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr><td align='center'></td> +<td align='center'><sup>3</sup>⁄<sub>8</sub> + <sup>5</sup>⁄<sub>8</sub> + <sup>7</sup>⁄<sub>8</sub> + <sup>1</sup>⁄<sub>8</sub> </td> +<td align='center'>2½<br />6<small><sup>3</sup>⁄<sub>8</sub></small><br /><span class="u">3¾</span></td> +<td align='center'> <sup>1</sup>⁄<sub>6</sub> + <sup>3</sup>⁄<sub>8</sub></td></tr> +</table></div> + + +<p class="noidt">Subtract</p> +<div class='center'> +<table border="0" cellpadding="10" cellspacing="0" summary=""> +<tr><td align='right'>10.00<br /><span class="u">3.49</span></td><td align='right'>4 yd. 1 ft. 6 in.<br /><span class="u">2 yd. 2 ft. 3 in.</span></td></tr> +</table></div> + +<p class="noidt">Multiply</p> +<div class='center'> +<table border="0" cellpadding="0" cellspacing="0" summary=""> +<tr><td align='center'>1¼ × 8</td> +<td align='center'>16<br />2<small><sup>5</sup>⁄<sub>8</sub></small></td><td align='center'>145<br />206</td></tr> +<tr><td align='center'></td><td align='center'>——</td><td align='center'>——</td></tr> +</table></div> + + +<p class="noidt">Divide</p> +<p class="center">2 )<span class="overline"> 13.50</span> 25 )<span class="overline"> 9750</span> +</p> + +<p>The invention of means of teaching thirty so different +children at once with the maximum help and minimum +hindrance from their different capacities and acquisitions +is one of the great opportunities for applied science.</p> + +<p>Courtis, emphasizing the social demand for a certain +moderate arithmetical attainment in the case of nearly all +elementary school children of, say, grade 6, has urged that +definite special means be taken to bring the deficient children +up to certain standards, without causing undesirable 'overlearning' +by the more gifted children. Certain experimental +work to this end has been carried out by him and +others, but probably much more must be done before an +authoritative program for securing certain minimum standards +for all or nearly all pupils can be arranged.</p> + + +<h4>THE CAUSES OF INDIVIDUAL DIFFERENCES</h4> + +<p>The differences found among children of the same grade +in the same city are due in large measure to inborn differences +in their original natures. If, by a miracle, the children +studied by Courtis, or by Woody, or by Kruse had all re<span class='pagenum'><a name="Page_294" id="Page_294">[Pg 294]</a></span>ceived +exactly the same nurture from birth to date, they +would still have varied greatly in arithmetical ability, perhaps +almost as much as they now do vary.</p> + +<p>The evidence for this is the general evidence that variation +in original nature is responsible for much of the eventual +variation found in intellectual and moral traits, plus certain +special evidence in the case of arithmetical abilities themselves.</p> + +<p>Thorndike found ['05] that in tests with addition and +multiplication twins were very much more alike than +siblings<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a> two or three years apart in age, though the resemblance +in home and school training in arithmetic should +be nearly as great for the latter as for the former. Also +the young twins (9-11) showed as close a resemblance in +addition and multiplication as the older twins (12-15), +although the similarities of training in arithmetic have had +twice as long to operate in the latter case.</p> + +<p>If the differences found, say among children in grade 6 in +addition, were due to differences in the quantity and quality +of training in addition which they have had, then by giving +each of them 200 minutes of additional identical training +the differences should be reduced. For the 200 minutes of +identical training is a step toward equalizing training. It +has been found in many investigations of the matter that +when we make training in arithmetic more nearly equal for +any group the variation within the group is not reduced.</p> + +<p>On the contrary, equalizing training seems rather to increase +differences. The superior individual seems to have +attained his superiority by his own superiority of nature +rather than by superior past training, for, during a period +of equal training for all, he increases his lead. For example, +compare the gains of different individuals due to +<span class='pagenum'><a name="Page_295" id="Page_295">[Pg 295]</a></span> +about 300 minutes of practice in mental multiplication of a +three-place number by a three-place number shown in +Table 14 below, from data obtained by the author ['08].<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a></p> + +<p class="tabcap">TABLE 14</p> + +<p class="center"><span class="smcap">The Effect of Equal Amounts of Practice upon Individual Difference +in the Multiplication Of Three-Place Numbers</span></p> + +<div class='center'> +<table border="1" cellpadding="4" cellspacing="0" width="80%" summary="Table 14"> +<thead class="bb bt"> +<tr><th rowspan='2'> </th><th colspan='2'><span class="smcap">Amount</span></th><th colspan='2'><span class="smcap">Percentage of Correct Figures</span></th></tr> +<tr><th>Initial Score</th><th>Gain</th><th>Initial Score</th><th>Gain</th></tr> +</thead> +<tbody> +<tr><td align='center'>Initially highest five individuals</td><td align='center'> 85</td><td align='center'> 61</td><td align='center'> 70</td><td align='center'> 18</td></tr> +<tr><td align='center'>next five "</td><td align='center'> 56</td><td align='center'> 51</td><td align='center'> 68</td><td align='center'> 10</td></tr> +<tr><td align='center'>next six "</td><td align='center'> 46</td><td align='center'> 22</td><td align='center'> 74</td><td align='center'> 8</td></tr> +<tr><td align='center'>next six "</td><td align='center'> 38</td><td align='center'> 8</td><td align='center'> 58</td><td align='center'> 12</td></tr> +<tr><td align='center'>next six "</td><td align='center'> 29</td><td align='center'> 24</td><td align='center'> 56</td><td align='center'> 14</td></tr> +</tbody> +</table></div> + +<h4>THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES</h4> + +<p>Achievement in arithmetic depends upon a number of +different abilities. For example, accuracy in copying numbers +depends upon eyesight, ability to perceive visual details, +and short-term memory for these. Long column +addition depends chiefly upon great strength of the addition +combinations especially in higher decades, 'carrying,' and +keeping one's place in the column. The solution of problems +framed in words requires understanding of language, +the analysis of the situation described into its elements, the +selection of the right elements for use at each step and their +use in the right relations.</p> + +<p><span class='pagenum'><a name="Page_296" id="Page_296">[Pg 296]</a></span></p> +<p>Since the abilities which together constitute arithmetic +ability are thus specialized, the individual who is the best of +a thousand of his age or grade in respect to, say, adding +integers, may occupy different stations, perhaps from 1st +to 600th, in multiplying with integers, placing the decimal +point in division with decimals, solving novel problems, +copying figures, etc., etc. Such specialization is in part +due to his having had, relatively to the others in the thousand, +more or better training in certain of these abilities than in +others, and to various circumstances of life which have +caused him to have, relatively to the others in the thousand, +greater interest in certain of these achievements than in +others. The specialization is not wholly due thereto, however. +Certain inborn characteristics of an individual predispose +him to different degrees of superiority or inferiority +to other men in different features of arithmetic.</p> + +<p>We measure the extent to which ability of one sort goes +with or fails to go with ability of some other sort by the +coefficient of correlation between the two. If every individual +keeps the same rank in the second ability—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.</p> + +<p>The meanings of coefficients of correlation of .90, .70, .50, +and 0 are shown by Tables 15, 16, 17 and 18.<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a></p> + +<p><span class='pagenum'><a name="Page_297" id="Page_297">[Pg 297]</a></span></p> + +<p class="tabcap">TABLE 15</p> + +<p class="center"><span class="smcap">Distribution of Arrays in Successive Tenths of the Group +When</span> <i>r</i> = .90</p> + +<div class='center'> +<table border="1" frame="hsides" rules="cols" cellpadding="2" cellspacing="0" summary="Table 15"> +<thead class="bb bt"> +<tr><th> </th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr> +</thead> +<tbody> +<tr><td align='left'>1st tenth</td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'></td><td align='center'> .1</td><td align='center'> .4</td><td align='center'> 1.8</td><td align='center'> 6.6</td><td align='center'>22.4</td><td align='center'>68.7</td></tr> +<tr><td align='left'>2d tenth</td><td align='center'></td><td align='center'></td><td align='center'> .1</td><td align='center'> .4</td><td align='center'> 1.4</td><td align='center'> 4.7</td><td align='center'>11.5</td><td align='center'>23.5</td><td align='center'>36.0</td><td align='center'>22.4</td></tr> +<tr><td align='left'>3d tenth</td><td align='center'></td><td align='center'> .1</td><td align='center'> .5</td><td align='center'> 2.1</td><td align='center'> 5.8</td><td align='center'>12.8</td><td align='center'>21.1</td><td align='center'>27.4</td><td align='center'>23.5</td><td align='center'> 6.6</td></tr> +<tr><td align='left'>4th tenth</td><td align='center'></td><td align='center'> .4</td><td align='center'> 2.1</td><td align='center'> 6.4</td><td align='center'>12.8</td><td align='center'>20.1</td><td align='center'>23.8</td><td align='center'>21.2</td><td align='center'>11.5</td><td align='center'> 1.8</td></tr> +<tr><td align='left'>5th tenth</td><td align='center'> .1</td><td align='center'> 1.4</td><td align='center'> 5.8</td><td align='center'>12.8</td><td align='center'>19.3</td><td align='center'>22.6</td><td align='center'>20.1</td><td align='center'>12.8</td><td align='center'> 4.7</td><td align='center'> .4</td></tr> +<tr><td align='left'>6th tenth</td><td align='center'> .4</td><td align='center'> 4.7</td><td align='center'>12.8</td><td align='center'>20.1</td><td align='center'>22.6</td><td align='center'>19.3</td><td align='center'>12.8</td><td align='center'> 5.8</td><td align='center'> 1.4</td><td align='center'> .1</td></tr> +<tr><td align='left'>7th tenth</td><td align='center'> 1.8</td><td align='center'>11.5</td><td align='center'>21.2</td><td align='center'>23.8</td><td align='center'>20.1</td><td align='center'>12.8</td><td align='center'> 6.4</td><td align='center'> 2.1</td><td align='center'> .4</td><td> </td></tr> +<tr><td align='left'>8th tenth</td><td align='center'> 6.6</td><td align='center'>23.5</td><td align='center'>27.4</td><td align='center'>21.1</td><td align='center'>12.8</td><td align='center'> 5.8</td><td align='center'> 2.1</td><td align='center'> .5</td><td align='center'> .1</td><td> </td></tr> +<tr><td align='left'>9th tenth</td><td align='center'>22.4</td><td align='center'>36.0</td><td align='center'>23.5</td><td align='center'>11.5</td><td align='center'> 4.7</td><td align='center'> 1.4</td><td align='center'> .4</td><td align='center'> .1</td><td align='center'></td><td> </td></tr> +<tr><td align='left'>10th tenth</td><td align='center'>68.7</td><td align='center'>22.4</td><td align='center'> 6.6</td><td align='center'> 1.8</td><td align='center'> .4</td><td align='center'> .1</td><td align='center'></td><td align='center'></td><td align='center'> </td><td> </td></tr> +</tbody> +</table></div> + +<p> </p> + +<p class="tabcap">TABLE 16</p> + +<p class="center"><span class="smcap">Distribution of Arrays in Successive Tenths of the Group +When</span> <i>r</i> = .70</p> + +<div class='center'> +<table border="1" frame="hsides" rules="cols" cellpadding="2" cellspacing="0" summary="Table 16"> +<thead class="bb bt"> +<tr><th> </th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr> +</thead> +<tbody> +<tr><td align='left'>1st tenth</td><td align='center'></td><td align='center'> .2</td><td align='center'> .7</td><td align='center'> 1.5</td><td align='center'> 2.8</td><td align='center'> 4.8</td><td align='center'> 8.0</td><td align='center'>13.0</td><td align='center'>22.3</td><td align='center'>46.7</td></tr> +<tr><td align='left'>2d tenth</td><td align='center'> .2</td><td align='center'> 1.2</td><td align='center'> 2.6</td><td align='center'> 4.5</td><td align='center'> 7.0</td><td align='center'> 9.8</td><td align='center'>13.4</td><td align='center'>17.3</td><td align='center'>21.7</td><td align='center'>22.3</td></tr> +<tr><td align='left'>3d tenth</td><td align='center'> .7</td><td align='center'> 2.6</td><td align='center'> 5.0</td><td align='center'> 7.3</td><td align='center'>10.0</td><td align='center'>12.5</td><td align='center'>14.9</td><td align='center'>16.7</td><td align='center'>17.3</td><td align='center'>13.0</td></tr> +<tr><td align='left'>4th tenth</td><td align='center'> 1.5</td><td align='center'> 4.5</td><td align='center'> 7.3</td><td align='center'> 9.8</td><td align='center'>12.0</td><td align='center'>13.7</td><td align='center'>14.8</td><td align='center'>14.9</td><td align='center'>13.4</td><td align='center'> 8.0</td></tr> +<tr><td align='left'>5th tenth</td><td align='center'> 2.8</td><td align='center'> 7.0</td><td align='center'>10.0</td><td align='center'>12.0</td><td align='center'>13.4</td><td align='center'>14.0</td><td align='center'>13.7</td><td align='center'>12.5</td><td align='center'> 9.8</td><td align='center'> 4.8</td></tr> +<tr><td align='left'>6th tenth</td><td align='center'> 4.8</td><td align='center'> 9.8</td><td align='center'>12.5</td><td align='center'>13.7</td><td align='center'>14.0</td><td align='center'>13.4</td><td align='center'>12.0</td><td align='center'>10.0</td><td align='center'> 7.0</td><td align='center'> 2.8</td></tr> +<tr><td align='left'>7th tenth</td><td align='center'> 8.0</td><td align='center'>13.4</td><td align='center'>14.9</td><td align='center'>14.8</td><td align='center'>13.7</td><td align='center'>12.0</td><td align='center'> 9.8</td><td align='center'> 7.3</td><td align='center'> 4.5</td><td align='center'> 1.5</td></tr> +<tr><td align='left'>8th tenth</td><td align='center'>13.0</td><td align='center'>17.3</td><td align='center'>16.7</td><td align='center'>14.9</td><td align='center'>12.5</td><td align='center'>10.0</td><td align='center'> 7.3</td><td align='center'> 5.0</td><td align='center'> 2.6</td><td align='center'> .7</td></tr> +<tr><td align='left'>9th tenth</td><td align='center'>22.3</td><td align='center'>21.7</td><td align='center'>17.3</td><td align='center'>13.4</td><td align='center'> 9.8</td><td align='center'> 7.0</td><td align='center'> 4.5</td><td align='center'> 2.6</td><td align='center'> 1.2</td><td align='center'> .2</td></tr> +<tr><td align='left'>10th tenth</td><td align='center'>46.7</td><td align='center'>22.3</td><td align='center'>13.0</td><td align='center'> 8.0</td><td align='center'> 4.8</td><td align='center'> 2.8</td><td align='center'> 1.5</td><td align='center'> .7</td><td align='center'> .2</td><td> </td></tr> +</tbody> +</table></div> + + +<p><span class='pagenum'><a name="Page_298" id="Page_298">[Pg 298]</a></span></p> + +<p> </p> + +<p class="tabcap">TABLE 17</p> + +<p class="center"><span class="smcap">Distribution of Arrays of Successive Tenths of the Group +When</span> <i>r</i> = .50</p> + + +<div class='center'> +<table border="1" frame="hsides" rules="cols" cellpadding="2" cellspacing="0" summary="Table 17"> +<thead class="bb bt"> +<tr><th> </th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr> +</thead> +<tbody> +<tr><td align='left'>1st tenth</td><td align='center'> .8</td><td align='center'> 2.0</td><td align='center'> 3.2</td><td align='center'> 4.6</td><td align='center'> 6.2</td><td align='center'> 8.1</td><td align='center'>10.5</td><td align='center'>13.9</td><td align='center'>18.0</td><td align='center'>31.8</td></tr> +<tr><td align='left'>2d tenth</td><td align='center'> 2.0</td><td align='center'> 4.1</td><td align='center'> 5.7</td><td align='center'> 7.3</td><td align='center'> 8.8</td><td align='center'>10.5</td><td align='center'>12.2</td><td align='center'>14.1</td><td align='center'>16.4</td><td align='center'>18.9</td></tr> +<tr><td align='left'>3d tenth</td><td align='center'> 3.2</td><td align='center'> 5.7</td><td align='center'> 7.4</td><td align='center'> 8.9</td><td align='center'>10.0</td><td align='center'>11.2</td><td align='center'>12.3</td><td align='center'>13.3</td><td align='center'>14.1</td><td align='center'>13.9</td></tr> +<tr><td align='left'>4th tenth</td><td align='center'> 4.6</td><td align='center'> 7.3</td><td align='center'> 8.8</td><td align='center'> 9.9</td><td align='center'>10.8</td><td align='center'>11.6</td><td align='center'>12.0</td><td align='center'>12.3</td><td align='center'>12.2</td><td align='center'>10.5</td></tr> +<tr><td align='left'>5th tenth</td><td align='center'> 6.2</td><td align='center'> 8.8</td><td align='center'>10.0</td><td align='center'>10.8</td><td align='center'>11.3</td><td align='center'>11.5</td><td align='center'>11.6</td><td align='center'>11.2</td><td align='center'>10.5</td><td align='center'> 8.1</td></tr> +<tr><td align='left'>6th tenth</td><td align='center'> 8.1</td><td align='center'>10.5</td><td align='center'>11.2</td><td align='center'>11.6</td><td align='center'>11.5</td><td align='center'>11.3</td><td align='center'>10.8</td><td align='center'>10.0</td><td align='center'> 8.8</td><td align='center'> 6.2</td></tr> +<tr><td align='left'>7th tenth</td><td align='center'>10.5</td><td align='center'>12.2</td><td align='center'>12.3</td><td align='center'>12.0</td><td align='center'>11.6</td><td align='center'>10.8</td><td align='center'> 9.9</td><td align='center'> 8.8</td><td align='center'> 7.5</td><td align='center'> 4.6</td></tr> +<tr><td align='left'>8th tenth</td><td align='center'>13.9</td><td align='center'>14.1</td><td align='center'>13.3</td><td align='center'>12.3</td><td align='center'>11.2</td><td align='center'>10.0</td><td align='center'> 8.8</td><td align='center'> 7.4</td><td align='center'> 5.7</td><td align='center'> 3.2</td></tr> +<tr><td align='left'>9th tenth</td><td align='center'>18.9</td><td align='center'>16.4</td><td align='center'>14.1</td><td align='center'>12.2</td><td align='center'>10.5</td><td align='center'> 8.8</td><td align='center'> 7.3</td><td align='center'> 5.7</td><td align='center'> 4.1</td><td align='center'> 2.0</td></tr> +<tr><td align='left'>10th tenth</td><td align='center'>31.8</td><td align='center'>18.9</td><td align='center'>13.9</td><td align='center'>10.5</td><td align='center'> 8.1</td><td align='center'> 6.2</td><td align='center'> 4.6</td><td align='center'> 3.2</td><td align='center'> 2.0</td><td align='center'> .8</td></tr> +</tbody> +</table></div> + +<p> </p> + +<p class="tabcap">TABLE 18</p> + +<p class="center"><span class="smcap">Distribution of Arrays, in Successive Tenths of the Group +When</span> <i>r</i> = .0</p> + +<div class='center'> +<table border="1" frame="hsides" rules="cols" cellpadding="2" cellspacing="0" summary="Table 18"> +<thead class="bb bt"> +<tr><th> </th><th>10<small>TH</small></th><th>9<small>TH</small></th><th>8<small>TH</small></th><th>7<small>TH</small></th><th>6<small>TH</small></th><th>5<small>TH</small></th><th>4<small>TH</small></th><th>3<small>D</small></th><th>2<small>D</small></th><th>1<small>ST</small></th></tr> +</thead> +<tbody> +<tr><td align='left'>1st tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>2d tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>3d tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>4th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>5th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>6th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>7th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>8th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>9th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +<tr><td align='left'>10th tenth</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td><td align='center'>10</td></tr> +</tbody> +</table></div> + +<p> </p> +<p>The significance of any coefficient of correlation depends +upon the group of individuals for which it is determined. A +correlation of .40 between computation and problem-solving +<span class='pagenum'><a name="Page_299" id="Page_299">[Pg 299]</a></span> +in eighth-grade pupils of 14 years would mean a much closer +real relation than a correlation of .40 in all 14-year-olds, +and a very, very much closer relation than a correlation of +.40 for all children 8 to 15.</p> + +<p>Unless the individuals concerned are very elaborately +tested on several days, the correlations obtained are "attenuated" +toward 0 by the "accidental" errors in the +original measurements. This effect was not known until +1904; consequently the correlations in the earlier studies of +arithmetic are all too low.</p> + +<p>In general, the correlation between ability in any one +important feature of computation and ability in any other +important feature of computation is high. If we make +enough tests to measure each individual exactly in:—</p> + +<p>(<i>A</i>) Subtraction with integers and decimals,</p> + +<p>(<i>B</i>) Multiplication with integers and decimals,</p> + +<p>(<i>C</i>) Division with integers and decimals,</p> + +<p>(<i>D</i>) Multiplication and division with common fractions, +and</p> + +<p>(<i>E</i>) Computing with percents,</p> + +<p class="noidt">we shall probably find the intercorrelations for a thousand +14-year-olds to be near .90. Addition of integers (<i>F</i>) +will, however, correlate less closely with any of the +above, being apparently dependent on simpler and more +isolated abilities.</p> + +<p>The correlation between problem-solving (<i>G</i>) and computation +will be very much less, probably not over .60.</p> + +<p>It should be noted that even when the correlation is as +high as .90, there will be some individuals very high in one +ability and very low in the other. Such disparities are to +some extent, as Courtis ['13, pp. 67-75] and Cobb ['17] have +argued, due to inborn characteristics of the individual in +question which predispose him to very special sorts of +<span class='pagenum'><a name="Page_300" id="Page_300">[Pg 300]</a></span> +strength and weakness. They are often due, however, to +defects in his learning whereby he has acquired more ability +than he needs in one line of work or has failed to acquire +some needed ability which was well within his capacity.</p> + +<p>In general, all correlations between an individual's divergence +from the common type or average of his age for one +arithmetical function, and his divergences from the average +for any other arithmetical function, are positive. The +correlation due to original capacity more than counterbalances +the effects that robbing Peter to pay Paul may +have.</p> + +<p>Speed and accuracy are thus positively correlated. The +individuals who do the most work in ten minutes will be +above the average in a test of accuracy. The common notion +that speed is opposed to accuracy is correct when it means +that the same person will tend to make more errors if he +works at too rapid a rate; but it is entirely wrong when it +means that the kind of person who works more rapidly +than the average person is likely to be less accurate than the +average person.</p> + +<p>Interest in arithmetic and ability at arithmetic are +probably correlated positively in the sense that the pupil +who has more interest than other pupils of his age tends in +the long run to have more ability than they. They are +certainly correlated in the sense that the pupil who 'likes' +arithmetic better than geography or history tends to have +relatively more ability in arithmetic, or, in other words, +that the pupil who is more gifted at arithmetic than at +drawing or English tends also to like it better than he likes +these. These correlations are high.</p> + +<p>It is correct then to think of mathematical ability as, in +a sense, a unitary ability of which any one individual may +have much or little, most individuals possessing a moderate +<span class='pagenum'><a name="Page_301" id="Page_301">[Pg 301]</a></span> +amount of it. This is consistent, however, with the occasional +appearance of individuals possessed of very great +talents for this or that particular feature of mathematical +ability and equally notable deficiencies in other features.</p> + +<p>Finally it may be noted that ability in arithmetic, though +occasionally found in men otherwise very stupid, is usually +associated with superior intelligence in dealing with ideas +and symbols of all sorts, and is one of the best early indications +thereof.</p> +<p> </p> + +<div class="footnotes"> +<h3>FOOTNOTES</h3> +<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The following and later problems are taken from actual textbooks or +courses of study or state examinations; to avoid invidious comparisons, they +are not exact quotations, but are equivalents in principle and form, as stated +in the preface.</p></div> + +<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> The work of Mitchell has not been published, but the author has had +the privilege of examining it.</p></div> + +<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> The form of Test 6 quoted here is that given by Courtis ['11-'12, p. 20]. +This differs a little from the other series of Test 6, shown on pages 43 and 44.</p></div> + +<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> Eight or ten times <i>in all</i>, not eight or ten times for each fact of the +tables.</p></div> + +<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> The facts concerning the present inaccuracy of school work in arithmetic +will be found on pages 102 to 105.</p></div> + +<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> McLellan and Ames, <i>Public School Arithmetic</i> [1900].</p></div> + +<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> These concern allowances for two errors occurring in the same example +and for the same wrong answer being obtained in both original work and +check work.</p></div> + +<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> The very early learning +of 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 3 × 3, and perhaps +a few more multiplications is not considered here. It is advisable. +The treatment of 0 × 0, 0 × 1, 1 × 0, etc., is not considered here. It is probably +best to defer the '× 0' bonds until after all the others are formed and +are being used in short multiplication, and to form them in close connection +with their use in short multiplication. The '0 ×' bonds may well be deferred +until they are needed in 'long' multiplication, 0 × 0 coming last of all.</p></div> + +<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> See page 76.</p></div> + +<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> At the end of a volume or part, the count may be from as few as 5 or as +many as 12 pages.</p></div> + +<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> Certain paragraphs in this and the following chapter are taken from the +author's <i>Educational Psychology</i>, with slight modifications.</p></div> + +<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> It should be noted that just as concretes give rise to abstractions, so these +in turn give rise to still more abstract abstractions. Thus fourness, fiveness, +twentyness, and the like give rise to 'integral-number-ness.' Similarly just +as individuals are grouped into general classes, so classes are grouped into still +more general classes. Half, quarter, sixth, and tenth are general notions, but +'one ...th' is more general; and 'fraction' is still more general.</p></div> + +<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> They may, of course, also result in a fusion or an alternation of responses, +but only rarely.</p></div> + +<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> The more gifted children may be put to work using the principle after +the first minute or two.</p></div> + +<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> +If desired this form may be used, with the appropriate difference in the form of the questions and statements.</p> + +<table border="0" cellpadding="4" cellspacing="0" summary=""> +<tr><td align='right'> 232<br /> +<span class="u"> 30</span><br /> + 000<br /> +<span class="u"> 696 </span><br /> +6960</td></tr> +</table> +</div> + +<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> Courtis finds in the case of addition that "of all the individuals making +mistakes at any given time in a class, at least one third, and usually two +thirds, will be making mistakes in carrying or copying."</p></div> + +<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> Facts concerning the conditions of learning in general will be found in +the author's <i>Educational Psychology</i>, Vol. 2, Chapter 8, or in the <i>Educational +Psychology, Briefer Course</i>, Chapter 15.</p></div> + +<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> See Thorndike ['00], King ['07], and Heck ['13].</p></div> + +<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> A special type could be constructed that would use a large type body, +say 14 point, with integers in 10 or 12 point and fractions much larger than +now.</p></div> + +<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> It will be still better if the 4 is replaced by an open-top 4.</p></div> + +<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> For an account in English of their main findings see Howell ['14], pp. +149-251.</p></div> + +<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> In his <i>How We Think</i>.</p></div> + +<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> Compiled from data on p. 89 of Kruse ['18].</p></div> + +<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> Siblings is used for children of the same parents.</p></div> + +<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> Similar results have been obtained in the case of arithmetical and other +abilities by Thorndike ['08, '10, '15, '16], Whitley ['11], Starch ['11], Wells +['12], Kirby ['13], Donovan and Thorndike ['13], Hahn and Thorndike ['14], +and on a very large scale by Race in a study as yet unpublished.</p></div> + +<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> Unless he has a thorough understanding of the underlying theory, the +student should be very cautious in making inferences from coefficients of +correlation.</p></div> + + +</div> + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_302" id="Page_302">[Pg 302]</a></span></p> +<h2><a name="BIBLIOGRAPHY" id="BIBLIOGRAPHY"></a>BIBLIOGRAPHY OF REFERENCES MADE IN +THE TEXT</h2> + + +<div class='center'> +<table border="0" cellpadding="4" cellspacing="20" summary=""> +<colgroup><col width="25%" /><col width="10%" /><col width="65%" /></colgroup> +<tr><td align='left'>Ames, A. F., and McLellan, J. F.</td><td align='right'>'00</td><td align='left'>Public School Arithmetic.</td></tr> +<tr><td align='left'>Ballou, F. W.</td><td align='right'>'16</td><td align='left'>Determining the Achievements of Pupils in the Addition of Fractions. School Document No. 3, 1916, Boston Public Schools.</td></tr> +<tr><td align='left'>Brandell, G.</td><td align='right'>'13</td><td align='left'>Skolbarns intressen. Translated ['15] by W. Stern as, Das Interesse der Schulkinder an den Unterrichtsfächern.</td></tr> +<tr><td align='left'>Brandford, B.</td><td align='right'>'08</td><td align='left'>A Study of Mathematical Education.</td></tr> +<tr><td align='left'>Brown, J. C.</td><td align='right'>'11, '12</td><td align='left'>An Investigation on the Value of Drill Work in the Fundamental Operations in Arithmetic. Journal of Educational Psychology, vol. 2, pp. 81-88, vol. 3, pp. 485-492 and 561-570.</td></tr> +<tr><td align='left'>Brown, J. C. and Coffman, L. D.</td><td align='right'>'14</td><td align='left'>How to Teach Arithmetic.</td></tr> +<tr><td align='left'>Burgerstein, L.</td><td align='right'>'91</td><td align='left'>Die Arbeitscurve einer Schulstunde. Zeitschrift für Schulgesundheitspflege, vol. 4, pp. 543-562 and 607-627.<span class='pagenum'><a name="Page_303" id="Page_303">[Pg 303]</a></span></td></tr> +<tr><td align='left'>Burnett, C. J.</td><td align='right'>'06</td><td align='left'>The Estimation of Number. Harvard Psychological Studies, vol. 2, pp. 349-404.</td></tr> +<tr><td align='left'>Burt, C.</td><td align='right'>'17</td><td align='left'>The Distribution and Relations of Educational Abilities. Report of The London County Council, No. 1868.</td></tr> +<tr><td align='left'>Chapman, J. C.</td><td align='right'>'14</td><td align='left'>Individual Differences in Ability and Improvement and Their Correlations. Teachers College Contributions to Education, No. 63.</td></tr> +<tr><td align='left'>Chapman, J. C.</td><td align='right'>'17</td><td align='left'>The Scientific Measurement of Classroom Products. (With G. P. Rush.)</td></tr> +<tr><td align='left'>Cobb, M. V.</td><td align='right'>'17</td><td align='left'>A Preliminary Study of the Inheritance of Arithmetical Abilities. Jour. of Educational Psychology, vol. 8, pp. 1-20. Jan., 1917.</td></tr> +<tr><td align='left'>Coffman, L. D., and Brown, J. C.</td><td align='right'>'14</td><td align='left'>How to Teach Arithmetic.</td></tr> +<tr><td align='left'>Coffman, L. D., and Jessup, W. A.</td><td align='right'>'16</td><td align='left'>The Supervision of Arithmetic.</td></tr> +<tr><td align='left'>Courtis, S. A.</td><td align='right'>'09, '10, '11</td><td align='left'>Measurement of Growth and Efficiency in Arithmetic. Elementary School Teacher, vol. 10, pp. 58-74 and 177-199, vol. 11, pp. 171-185, 360-370, and 528-539.</td></tr> +<tr><td align='left'>Courtis, S. A.</td><td align='right'>'11, '12</td><td align='left'>Report on Educational Aspects of the Public School System of the City of New York. Part II, Subdivision 1, Section D. Report on the Courtis Tests in Arithmetic.</td></tr> +<tr><td align='left'>Courtis, S. A.</td><td align='right'>'13</td><td align='left'>Courtis Standard Tests. Second Annual Accounting.</td></tr> +<tr><td align='left'>Courtis, S. A.</td><td align='right'>'14</td><td align='left'>Manual of Instructions for Giving and Scoring the Courtis Standard <span class='pagenum'><a name="Page_304" id="Page_304">[Pg 304]</a></span>Tests in the Three R's. Department of Comparative Research, 82 Elliot St., Detroit, Mich., 1914.</td></tr> +<tr><td align='left'>Decroly, M., and Degand, J.</td><td align='right'>'12</td><td align='left'>L'evolution des notions de quantités continues et discontinues chez l'enfant. Archives de psychologie, vol. 12, pp. 81-121.</td></tr> +<tr><td align='left'>Degand, J. <i>See</i> Decroly.</td></tr> +<tr><td align='left'>De Voss, J. C. <i>See</i> Monroe, De Voss, and Kelly.</td></tr> +<tr><td align='left'>Dewey, J.</td><td align='right'>'10</td><td align='left'>How We Think.</td></tr> +<tr><td align='left'>Dewey, J., and McLellan, J. A.</td><td align='right'>'95</td><td align='left'>Psychology of Number and Its Applications to Methods of Teaching Arithmetic.</td></tr> +<tr><td align='left'>Donovan, M. E., and Thorndike, E. L.</td><td align='right'>'13</td><td align='left'>Improvement in a Practice Experiment under School Conditions. American Journal of Psychology, vol. 24, pp. 426-428.</td></tr> +<tr><td align='left'>Elliott, C. H.</td><td align='right'>'14</td><td align='left'>Variation in the Achievements of Pupils. Teachers College, Columbia University, Contributions to Education, No. 72.</td></tr> +<tr><td align='left'>Flynn, F. J.</td><td align='right'>'12</td><td align='left'>Mathematical Games—Adaptations from Games Old and New. Teachers College Record, vol. 13, pp. 399-412.</td></tr> +<tr><td align='left'>Freeman, F. N.</td><td align='right'>'10</td><td align='left'>Untersuchungen über den Aufmerksamkeitsumfang und die Zahlauffassung. Pädagogische-Psychologische Arbeiten, I, 88-168.</td></tr> +<tr><td align='left'>Friedrich, J.</td><td align='right'>'97</td><td align='left'>Untersuchungen über die Einflüsse der Arbeitsdauer und die Arbeitspausen auf die geistige Leistungsfähigkeit der Schulkinder. Zeitschrift für Psychologie, vol. 13, pp. 1-53.</td></tr> +<tr><td align='left'>Gilbert, J. A.</td><td align='right'>'94</td><td align='left'>Researches on the Mental and<span class='pagenum'><a name="Page_305" id="Page_305">[Pg 305]</a></span> Physical Development of School Children. Studies from the Yale Psychological Laboratory, vol. 2, pp. 40-100.</td></tr> +<tr><td align='left'>Greenleaf, B.</td><td align='right'>'73</td><td align='left'>Practical Arithmetic.</td></tr> +<tr><td align='left'>Hahn, H. H., and Thorndike, E. L.</td><td align='right'>'14</td><td align='left'>Some Results of Practice in Addition under School Conditions. Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.</td></tr> +<tr><td align='left'>Hall, G. S.</td><td align='right'>'83</td><td align='left'>The Contents of Children's Minds on Entering School. Princeton Review, vol. II, pp. 249-272, May, 1883. Reprinted in Aspects of Child Life and Education, 1907.</td></tr> +<tr><td align='left'>Hartmann, B.</td><td align='right'>'90</td><td align='left'>Die Analyze des Kindlichen Gedanken-Kreises als die Naturgemässedes Ersten Schulunterrichts, 1890.</td></tr> +<tr><td align='left'>Heck, W. H.</td><td align='right'>'13</td><td align='left'>A Study of Mental Fatigue.</td></tr> +<tr><td align='left'>Heck, W. H.</td><td align='right'>'13</td><td align='left'>A Second Study in Mental Fatigue in the Daily School Program. Psychological Clinic, vol. 7, pp. 29-34.</td></tr> +<tr><td align='left'>Hoffmann, P.</td><td align='right'>'11</td><td align='left'>Das Interesse der Schüler an den Unterrichtsfächern. Zeitschrift für pädagogische Psychologie, XII, 458-470.</td></tr> +<tr><td align='left'>Hoke, K. J., and Wilson, G. M.</td><td align='right'>'20</td><td align='left'>How to Measure.</td></tr> +<tr><td align='left'>Holmes, M. E.</td><td align='right'>'95</td><td align='left'>The Fatigue of a School Hour. Pedagogical Seminary, vol. 3, pp. 213-234.</td></tr> +<tr><td align='left'>Howell, H. B.</td><td align='right'>'14</td><td align='left'>A Foundation Study in the Pedagogy of Arithmetic.</td></tr> +<tr><td align='left'>Hunt, C. W.</td><td align='right'>'12</td><td align='left'>Play and Recreation in Arithmetic. Teachers College Record, vol. 13, pp. 388-398.<span class='pagenum'><a name="Page_306" id="Page_306">[Pg 306]</a></span></td></tr> +<tr><td align='left'>Jessup, W. A., and Coffman, L. D.</td><td align='right'>'16</td><td align='left'>The Supervision of Arithmetic.</td></tr> +<tr><td align='left'>Kelly, F. J. <i>See</i> Monroe, De Voss and Kelly.</td></tr> +<tr><td align='left'>King, A. C.</td><td align='right'>'07</td><td align='left'>The Daily Program in Elementary Schools. MSS.</td></tr> +<tr><td align='left'>Kirby, T. J.</td><td align='right'>'13</td><td align='left'>Practice in the Case of School Children. Teachers College Contributions to Education, No. 58.</td></tr> +<tr><td align='left'>Klapper, P.</td><td align='right'>'16</td><td align='left'>The Teaching of Arithmetic.</td></tr> +<tr><td align='left'>Kruse, P. J.</td><td align='right'>'18</td><td align='left'>The Overlapping of Attainments in Certain Sixth, Seventh, and Eighth Grades. Teachers College, Columbia University, Contributions to Education, No. 92.</td></tr> +<tr><td align='left'>Laser, H.</td><td align='right'>'94</td><td align='left'>Ueber geistige Ermüdung beim Schulunterricht. Zeitschrift für Schulgesundheitspflege, vol. 7, pp. 2-22.</td></tr> +<tr><td align='left'>Lay, W. A.</td><td align='right'>'98</td><td align='left'>Führer durch den ersten Rechenunterricht.</td></tr> +<tr><td align='left'>Lay, W. A.</td><td align='right'>'07</td><td align='left'>Führer durch den Rechenunterricht der Unterstufe.</td></tr> +<tr><td align='left'>Lewis, E. O.</td><td align='right'>'13</td><td align='left'>Popular and Unpopular School-Subjects. The Journal of Experimental Pedagogy, vol. 2, pp. 89-98.</td></tr> +<tr><td align='left'>Lobsien, M.</td><td align='right'>'03</td><td align='left'>Kinderideale. Zeitschrift für pädagogische Psychologie, V, 323-344 and 457-494.</td></tr> +<tr><td align='left'>Lobsien, M.</td><td align='right'>'09</td><td align='left'>Beliebtheit und Unbeliebtheit der Unterrichtsfächer. Pädagogisches Magazin, Heft 361.</td></tr> +<tr><td align='left'>McCall, W. A.</td><td align='right'>'21</td><td align='left'>How to Measure in Education.</td></tr> +<tr><td align='left'>McDougle, E. C.</td><td align='right'>'14</td><td align='left'>A Contribution to the Pedagogy of Arithmetic. Pedagogical Seminary, vol. 21, pp. 161-218.<span class='pagenum'><a name="Page_307" id="Page_307">[Pg 307]</a></span></td></tr> +<tr><td align='left'>McKnight, J.A.</td><td align='right'>'07</td><td align='left'>Differentiation of the Curriculum in the Upper Grammar Grades. MSS. in the library of Teachers College, Columbia University.</td></tr> +<tr><td align='left'>McLellan, J.A., and Dewey, J.</td><td align='right'>'95</td><td align='left'>Psychology of Number and Its Applications to Methods of Teaching.</td></tr> +<tr><td align='left'>McLellan, J.A., and Ames, A.F.</td><td align='right'>'00</td><td align='left'>Public School Arithmetic.</td></tr> +<tr><td align='left'>Messenger, J.F.</td><td align='right'>'03</td><td align='left'>The Perception of Number. Psychological Review, Monograph Supplement No. 22.</td></tr> +<tr><td align='left'>Meumann, E.</td><td align='right'>'07</td><td align='left'>Vorlesungen zur Einführung in die experimentelle Pädagogik.</td></tr> +<tr><td align='left'>Mitchell, H.E.</td><td align='right'>'20</td><td align='left'>Unpublished studies of the uses of arithmetic in factories, shops, farms, and the like.</td></tr> +<tr><td align='left'>Monroe, W.S., De Voss, J.C., and Kelly, F.J.</td><td align='right'>'17</td><td align='left'>Educational Tests and Measurements.</td></tr> +<tr><td align='left'>Nanu, H.A.</td><td align='right'>'04</td><td align='left'>Zur Psychologie der Zahl Auffassung.</td></tr> +<tr><td align='left'>National Intelligence Tests</td><td align='right'>'20</td><td align='left'>Scale A, Form 1, Edition 1.</td></tr> +<tr><td align='left'>Phillips, D.E.</td><td align='right'>'97</td><td align='left'>Number and Its Application Psychologically Considered. Pedagogical Seminary, vol. 5, pp. 221-281.</td></tr> +<tr><td align='left'>Pommer, O.</td><td align='right'>'14</td><td align='left'>Die Erforschung der Beliebtheit der Unterrichtsfächer. Ihre psychologischen Grundlagen und ihre pädagog. Bedeutung. VII. Jahresber. des k.k. Ssaatsgymn. im XVIII Bez. v. Wien.</td></tr> +<tr><td align='left'>Rice, J.M.</td><td align='right'>'02</td><td align='left'>Test in Arithmetic. Forum, vol. 34, pp. 281-297.</td></tr> +<tr><td align='left'>Rice, J.M.</td><td align='right'>'03</td><td align='left'>Causes of Success and Failure in Arithmetic. Forum, vol. 34, pp. 437-452.</td></tr> +<tr><td align='left'>Rush, G.P.</td><td align='right'>'17</td><td align='left'>The Scientific Measurement of Classroom Products. (With J. C. Chapman.)<span class='pagenum'><a name="Page_308" id="Page_308">[Pg 308]</a></span></td></tr> +<tr><td align='left'>Seekel, E.</td><td align='right'>'14</td><td align='left'>Ueber die Beziehung zwischen der Beliebtheit und der Schwierigkeit der Schulfächer. Ergebnisse einer Erhebung. Zeitschrift für Angewandte Psychologie 9. S. 268-277.</td></tr> +<tr><td align='left'>Selkin, F. B.</td><td align='right'>'12</td><td align='left'>Number Games Bordering on Arithmetic and Algebra. Teachers College Record, vol. 13, pp. 452-493.</td></tr> +<tr><td align='left'>Smith, D. E.</td><td align='right'>'01</td><td align='left'>The Teaching of Elementary Mathematics.</td></tr> +<tr><td align='left'>Smith, D. E.</td><td align='right'>'11</td><td align='left'>The Teaching of Arithmetic.</td></tr> +<tr><td align='left'>Speer, W. W.</td><td align='right'>'97</td><td align='left'>Arithmetic: Elementary for Pupils.</td></tr> +<tr><td align='left'>Starch, D.</td><td align='right'>'11</td><td align='left'>Transfer of Training in Arithmetical Operations. Journal of Educational Psychology, vol. 2, pp. 306-310.</td></tr> +<tr><td align='left'>Starch, D.</td><td align='right'>'16</td><td align='left'>Educational Measurements.</td></tr> +<tr><td align='left'>Stern, W.</td><td align='right'>'05</td><td align='left'>Ueber Beliebtheit und Unbeliebtheit der Schulfächer. Zeitschrift für pädagogische Psychologie, VII, 267-296.</td></tr> +<tr><td align='left'>Stern, C., and Stern, W.</td><td align='right'>'13</td><td align='left'>Beliebtheit und Schwierigkeit der Schulfächer. (Freie Schulgemeinde Wickersdorf.) Auf Grund der von Herrn Luserke beschafften Materialien bearbeitet. In: "Die Ausstellung zur vergleichenden Jungendkunde der Geschlechter in Breslau." Arbeit 7 des Bundes für Schulreform. S. 24-26.</td></tr> +<tr><td align='left'>Stern, W.</td><td align='right'>'14</td><td align='left'>Zur vergleichenden Jugendkunde der Geschlechter. Vortrag. III. Deutsch. Kongr. f. Jugendkunde <span class='pagenum'><a name="Page_309" id="Page_309">[Pg 309]</a></span>usw. Arbeiten 8 des Bundes für Schulreform. S. 17-38.</td></tr> +<tr><td align='left'>Stone, C.W.</td><td align='right'>'08</td><td align='left'>Arithmetical Abilities and Some Factors Determining Them. Teachers College Contributions to Education, No. 19.</td></tr> +<tr><td align='left'>Suzzallo, H.</td><td align='right'>'11</td><td align='left'>The Teaching of Primary Arithmetic.</td></tr> +<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'00</td><td align='left'>Mental Fatigue. Psychological Review, vol. 7, pp. 466-482 and 547-579.</td></tr> +<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'08</td><td align='left'>The Effect of Practice in the Case of a Purely Intellectual Function. American Journal of Psychology, vol. 10, pp. 374-384.</td></tr> +<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'10</td><td align='left'>Practice in the case of Addition. American Journal of Psychology, vol. 21, pp. 483-486.</td></tr> +<tr><td align='left'>Thorndike, E.L., and Donovan, M.E.</td><td align='right'>'13</td><td align='left'>Improvement in a Practice Experiment under School Conditions. American Journal of Psychology, vol. 24, pp. 426-428.</td></tr> +<tr><td align='left'>Thorndike, E.L., and Donovan, M.E., and Hahn, H.H.</td><td align='right'>'14</td><td align='left'>Some Results of Practice in Addition under School Conditions. Journal of Educational Psychology, vol. 5, No. 2, pp. 65-84.</td></tr> +<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'15</td><td align='left'>The Relation between Initial Ability and Improvement in a Substitution Test. School and Society, vol. 12, p. 429.</td></tr> +<tr><td align='left'>Thorndike, E.L.</td><td align='right'>'16</td><td align='left'>Notes on Practice, Improvability and the Curve of Work. American Journal of Psychology, vol 27, pp. 550-565.</td></tr> +<tr><td align='left'>Walsh, J.H.</td><td align='right'>'06</td><td align='left'>Grammar School Arithmetic.</td></tr> +<tr><td align='left'>Wells, F.L.</td><td align='right'>'12</td><td align='left'>The Relation of Practice to Individual Differences. <span class='pagenum'><a name="Page_310" id="Page_310">[Pg 310]</a></span>American Journal of Psychology, vol. 23, pp. 75-88.</td></tr> +<tr><td align='left'>White, E. E.</td><td align='right'>'83</td><td align='left'>A New Elementary Arithmetic.</td></tr> +<tr><td align='left'>Whitley, M. T.</td><td align='right'>'11</td><td align='left'>An Empirical Study of Certain Tests for Individual Differences. Archives of Psychology, No. 19.</td></tr> +<tr><td align='left'>Wiederkehr, G.</td><td align='right'>'07</td><td align='left'>Statistiche Untersuchungen über die Art und den Grad des Interesses bei Kindern der Volksschule. Neue Bahnen, vol. 19, pp. 241-251, 289-299.</td></tr> +<tr><td align='left'>Wilson, G. M.</td><td align='right'>'19</td><td align='left'>A Survey of the Social and Business Usage of Arithmetic. Teachers College Contributions to Education, No. 100.</td></tr> +<tr><td align='left'>Wilson, G. M., and Hoke, K. J.</td><td align='right'>'20</td><td align='left'>How to Measure.</td></tr> +<tr><td align='left'>Woody, C.</td><td align='right'>'10</td><td align='left'>Measurements of Some Achievements in Arithmetic. Teachers College Contributions to Education, No. 80.</td></tr> +</table></div> + + +<hr style="width: 100%;" /> +<p><span class='pagenum'><a name="Page_311" id="Page_311">[Pg 311]</a></span></p> +<h2><a name="INDEX" id="INDEX"></a>INDEX</h2> + + +<div class="pblockquot"><p class="noidt"> +Abilities, arithmetical, nature of, <a href="#Page_1">1 ff.</a>;<br /> + measurement of, <a href="#Page_27">27 ff.</a>;<br /> + constitution of, <a href="#Page_51">51 ff.</a>;<br /> + organization of, <a href="#Page_137">137 ff.</a><br /> +<br /> +Abstract numbers, <a href="#Page_85">85 ff.</a><br /> +<br /> +Abstraction, <a href="#Page_169">169 ff.</a><br /> +<br /> +Accuracy, in relation to speed, <a href="#Page_31">31</a>;<br /> + in fundamental operations, <a href="#Page_102">102 ff.</a><br /> +<br /> +Addition, measurement of, <a href="#Page_27">27 ff.</a>, <a href="#Page_34">34</a>;<br /> + constitution of, <a href="#Page_52">52 f.</a>;<br /> + habit in relation to, <a href="#Page_71">71 f.</a>;<br /> + in the higher decades, <a href="#Page_75">75 f.</a>;<br /> + accuracy in, <a href="#Page_108">108 f.</a>;<br /> + amount of practice in, <a href="#Page_122">122 ff.</a>;<br /> + interest in <a href="#Page_196">196 f.</a><br /> +<br /> +Aims of the teaching of arithmetic, <a href="#Page_23">23 f.</a><br /> +<br /> +<span class="smcap">Ames, A. F.</span>, <a href="#Page_89">89</a><br /> +<br /> +Analysis, learning by, <a href="#Page_169">169 ff.</a>;<br /> + systematic and opportunistic stimuli to, <a href="#Page_178">178 f.</a>;<br /> + gradual progress in, <a href="#Page_180">180 ff.</a><br /> +<br /> +Area, <a href="#Page_257">257 f.</a>, <a href="#Page_275">275</a><br /> +<br /> +Arithmetic, sociology of, <a href="#Page_24">24 ff.</a><br /> +<br /> +Arithmetical abilities. <i>See</i> Abilities.<br /> +<br /> +Arithmetical language, <a href="#Page_8">8 f.</a>, <a href="#Page_19">19</a>, <a href="#Page_89">89 ff.</a>, <a href="#Page_94">94 ff.</a><br /> +<br /> +Arithmetical learning, before school, <a href="#Page_199">199 ff.</a>;<br /> + conditions of, <a href="#Page_227">227 ff.</a>;<br /> + in relation to time of day, <a href="#Page_227">227 ff.</a>;<br /> + in relation to time devoted to arithmetic, <a href="#Page_228">228 ff.</a><br /> +<br /> +Arithmetical reasoning. <i>See</i> Reasoning.<br /> +<br /> +Arithmetical terms, <a href="#Page_8">8</a>, <a href="#Page_19">19</a><br /> +<br /> +Averages, <a href="#Page_40">40 f.</a>; <a href="#Page_135">135 f.</a><br /> +<br /> +<br /> +<span class="smcap">Ballou, F. W.</span>, <a href="#Page_34">34</a>, <a href="#Page_38">38</a><br /> +<br /> +Banking, <a href="#Page_256">256 f.</a><br /> +<br /> +<span class="smcap">Binet, A.</span>, <a href="#Page_201">201</a><br /> +<br /> +Bonds, selection of, <a href="#Page_70">70 ff.</a>;<br /> + strength of, <a href="#Page_102">102 ff.</a>;<br /> + for temporary service, <a href="#Page_111">111 ff.</a>;<br /> + order of formation of, <a href="#Page_141">141 ff.</a><br /> + <i>See also</i> Habits.<br /> +<br /> +<span class="smcap">Brandell, G.</span>, <a href="#Page_211">211</a><br /> +<br /> +<span class="smcap">Brandford, B.</span>, <a href="#Page_198">198 f.</a><br /> +<br /> +<span class="smcap">Brown, J. C.</span>, <a href="#Page_xvi">xvi</a>, <a href="#Page_103">103</a><br /> +<br /> +<span class="smcap">Burgerstein, L.</span>, <a href="#Page_103">103</a><br /> +<br /> +<span class="smcap">Burnett, C. J.</span>, <a href="#Page_202">202</a><br /> +<br /> +<span class="smcap">Burt, C.</span>, <a href="#Page_286">286</a><br /> +<br /> +<br /> +Cardinal and ordinal numbers confused, <a href="#Page_206">206</a><br /> +<br /> +Catch problems, <a href="#Page_21">21 ff.</a><br /> +<br /> +<span class="smcap">Chapman, J. C.</span>, <a href="#Page_49">49</a><br /> +<br /> +Class, size of, in relation to arithmetical learning, <a href="#Page_228">228</a>;<br /> + variation within a, <a href="#Page_289">289 ff.</a><br /> +<br /> +<span class="smcap">Cobb, M. V.</span>, <a href="#Page_299">299</a><br /> +<br /> +<span class="smcap">Coffman, L. D.</span>, <a href="#Page_xvi">xvi</a><br /> +<br /> +Collection meaning of numbers, <a href="#Page_3">3 ff.</a><br /> +<br /> +Computation, measurements of, <a href="#Page_33">33 ff.</a>;<br /> + explanations of the processes in, <a href="#Page_60">60 ff.</a>;<br /> + accuracy in, <a href="#Page_102">102 ff.</a><br /> + <i>See also</i> Addition, Subtraction, Multiplication, Division, Fractions, Decimal numbers, Percents.<br /> +<br /> +Concomitants, law of varying, <a href="#Page_172">172 ff.</a>;<br /> + law of contrasting, <a href="#Page_173">173 ff.</a><br /> +<br /> +Concrete numbers, <a href="#Page_85">85 ff.</a><br /> +<br /> +Concrete objects, use of, <a href="#Page_253">253 ff.</a><br /> +<br /> +Conditions of arithmetical learning, <a href="#Page_227">227 ff.</a><br /> +<br /> +Constitution of arithmetical abilities, <a href="#Page_51">51 ff.</a><br /> +<br /> +Copying of numbers, eyestrain due to, <a href="#Page_212">212 f.</a><br /> +<br /> +Correlations of arithmetical abilities, <a href="#Page_295">295 ff.</a><br /> +<br /> +Courses of study, <a href="#Page_232">232 f.</a><br /> +<br /> +<span class="smcap">Courtis, S. A.</span>, <a href="#Page_28">28 ff.</a>, <a href="#Page_43">43 ff.</a>, <a href="#Page_49">49</a>, <a href="#Page_103">103</a>, <a href="#Page_291">291</a>, <a href="#Page_293">293</a>, <a href="#Page_299">299</a><br /> +<br /> +Crutches, <a href="#Page_112">112 f.</a><br /> +<br /> +Culture-epoch theory, <a href="#Page_198">198 f.</a><br /> +<br /> +<br /> +Dairy records, <a href="#Page_273">273</a><br /> +<br /> +Decimal numbers, uses of, <a href="#Page_24">24 f.</a>;<br /> + measurement of ability with, <a href="#Page_36">36 ff.</a>;<br /> + learning, <a href="#Page_181">181 ff.</a>;<br /> + division by, <a href="#Page_270">270 f.</a><br /> +<br /> +<span class="smcap">De Croly, M.</span>, <a href="#Page_205">205</a><br /> +<br /> +Deductive reasoning, <a href="#Page_60">60 ff.</a>, <a href="#Page_185">185 ff.</a><br /> +<br /> +<span class="smcap">Degand, J.</span>, <a href="#Page_205">205</a><br /> +<br /> +Denominate numbers, <a href="#Page_141">141 f.</a>, <a href="#Page_147">147 f.</a><br /> +<br /> +Described problems, <a href="#Page_10">10 ff.</a><br /> +<br /> +Development of knowledge of number, <a href="#Page_205">205 ff.</a><br /> +<span class='pagenum'><a name="Page_312" id="Page_312">[Pg 312]</a></span><br /> +<span class="smcap">De Voss, J. C.</span>, <a href="#Page_49">49</a><br /> +<br /> +<span class="smcap">Dewey, J.</span>, <a href="#Page_3">3</a>, <a href="#Page_83">83</a>, <a href="#Page_150">150</a>, <a href="#Page_205">205</a>, <a href="#Page_207">207</a>, <a href="#Page_208">208</a>, <a href="#Page_219">219</a>, <a href="#Page_266">266</a>, <a href="#Page_277">277</a><br /> +<br /> +Differences in arithmetical ability, <a href="#Page_285">285 ff.</a>;<br /> + within a class, <a href="#Page_289">289 ff.</a><br /> +<br /> +Difficulty as a stimulus, <a href="#Page_277">277 ff.</a><br /> +<br /> +Drill, <a href="#Page_102">102 ff.</a><br /> +<br /> +Discipline, mental, <a href="#Page_20">20</a><br /> +<br /> +Distribution of practice, <a href="#Page_156">156 ff.</a><br /> +<br /> +Division, measurement of, <a href="#Page_35">35 f.</a>, <a href="#Page_37">37</a>;<br /> + constitution of, <a href="#Page_57">57 ff.</a>;<br /> + deductive explanations of, <a href="#Page_63">63</a>, <a href="#Page_64">64 f.</a>;<br /> + inductive explanations of, <a href="#Page_63">63 f.</a>, <a href="#Page_65">65 f.</a>;<br /> + habit in relation to, <a href="#Page_72">72</a>;<br /> + with remainders, <a href="#Page_76">76</a>;<br /> + with fractions, <a href="#Page_78">78 ff.</a>;<br /> + amount of practice in, <a href="#Page_122">122 ff.</a>;<br /> + distribution of practice in, <a href="#Page_167">167</a>;<br /> + use of the problem attitude in teaching, <a href="#Page_270">270 f.</a><br /> +<br /> +<span class="smcap">Donovan, M. E.</span>, <a href="#Page_295">295</a><br /> +<br /> +<br /> +Elements, responses to, <a href="#Page_169">169 ff.</a><br /> +<br /> +Eleven, multiples of, <a href="#Page_85">85</a><br /> +<br /> +<span class="smcap">Elliott, C. H.</span>, <a href="#Page_228">228</a><br /> +<br /> +Equation form, importance of, <a href="#Page_77">77 f.</a><br /> +<br /> +Explanations of the processes of computation, <a href="#Page_60">60 ff.</a>;<br /> + memory of, <a href="#Page_115">115 f.</a>;<br /> + time for giving, <a href="#Page_154">154 ff.</a><br /> +<br /> +Eyestrain in arithmetical work, <a href="#Page_212">212 ff.</a><br /> +<br /> +<br /> +Facilitation, <a href="#Page_143">143 ff.</a><br /> +<br /> +Figures, printing of, <a href="#Page_235">235 ff.</a>;<br /> + writing of, <a href="#Page_214">214 f.</a>, <a href="#Page_241">241</a><br /> +<br /> +<span class="smcap">Flynn, F. J.</span>, <a href="#Page_196">196</a><br /> +<br /> +Fractions, uses of, <a href="#Page_24">24 f.</a>;<br /> + measurement of ability with, <a href="#Page_36">36 ff.</a>;<br /> + knowledge of the meaning of, <a href="#Page_54">54 ff.</a><br /> +<br /> +<span class="smcap">Freeman, F. N.</span>, <a href="#Page_259">259</a>, <a href="#Page_261">261</a><br /> +<br /> +<span class="smcap">Friedrich, J.</span>, <a href="#Page_103">103</a><br /> +<br /> +<br /> +Generalization, <a href="#Page_169">169 ff.</a><br /> +<br /> +<span class="smcap">Gilbert, J. A.</span>, <a href="#Page_203">203</a><br /> +<br /> +Graded tests, <a href="#Page_28">28 ff.</a>, <a href="#Page_36">36 ff.</a><br /> +<br /> +Greatest common divisor, <a href="#Page_88">88 f.</a><br /> +<br /> +<br /> +Habits, importance of, in arithmetical learning, <a href="#Page_70">70 ff.</a>;<br /> + now neglected, <a href="#Page_75">75 ff.</a>;<br /> + harmful or wasteful, <a href="#Page_83">83 ff.</a>; <a href="#Page_91">91 ff.</a>;<br /> + propædeutic, <a href="#Page_117">117 ff.</a>;<br /> + organization of, <a href="#Page_137">137 ff.</a>;<br /> + arrangement of, <a href="#Page_141">141 ff.</a><br /> +<br /> +<span class="smcap">Hahn, H. H.</span>, <a href="#Page_295">295</a><br /> +<br /> +<span class="smcap">Hall, G. S.</span>, <a href="#Page_200">200 f.</a><br /> +<br /> +<span class="smcap">Hartmann, B.</span>, <a href="#Page_200">200 f.</a><br /> +<br /> +<span class="smcap">Heck, W. H.</span>, <a href="#Page_227">227</a><br /> +<br /> +Heredity in arithmetical abilities, <a href="#Page_293">293 ff.</a><br /> +<br /> +Highest common factor, <a href="#Page_88">88 f.</a><br /> +<br /> +<span class="smcap">Hoke, K. J.</span>, <a href="#Page_49">49</a><br /> +<br /> +<span class="smcap">Holmes, M. E.</span>, <a href="#Page_103">103</a><br /> +<br /> +<span class="smcap">Howell, H. B.</span>, <a href="#Page_259">259</a><br /> +<br /> +<span class="smcap">Hunt, C. W.</span>, <a href="#Page_196">196</a><br /> +<br /> +Hygiene of arithmetic, <a href="#Page_212">212 ff.</a>, <a href="#Page_234">234 ff.</a><br /> +<br /> +<br /> +Individual differences, <a href="#Page_285">285 ff.</a><br /> +<br /> +Inductive reasoning, <a href="#Page_60">60 ff.</a>, <a href="#Page_169">169 ff.</a><br /> +<br /> +Insurance, <a href="#Page_256">256</a><br /> +<br /> +Interest as a principle determining the order of topics, <a href="#Page_150">150 ff.</a><br /> +<br /> +Interests, instinctive <a href="#Page_195">195 ff.</a>;<br /> + censuses of, <a href="#Page_209">209 ff.</a>;<br /> + neglect of childish, <a href="#Page_226">226 ff.</a>;<br /> + in self-management, <a href="#Page_223">223 f.</a>;<br /> + intrinsic, <a href="#Page_224">224 ff.</a><br /> +<br /> +Interference, <a href="#Page_143">143 ff.</a><br /> +<br /> +Inventories of arithmetical knowledge and skill, <a href="#Page_199">199 ff.</a><br /> +<br /> +<br /> +<span class="smcap">Jessup, W. A.</span>, <a href="#Page_xvi">xvi</a><br /> +<br /> +<br /> +<span class="smcap">Kelly, F. J.</span>, <a href="#Page_49">49</a><br /> +<br /> +<span class="smcap">King, A. C.</span>, <a href="#Page_103">103</a>, <a href="#Page_227">227</a><br /> +<br /> +<span class="smcap">Kirby, T. J.</span>, <a href="#Page_76">76 f.</a>, <a href="#Page_104">104</a>, <a href="#Page_295">295</a><br /> +<br /> +<span class="smcap">Klapper, P.</span>, <a href="#Page_xvi">xvi</a><br /> +<br /> +<span class="smcap">Kruse, P. J.</span>, <a href="#Page_289">289</a>, <a href="#Page_293">293</a><br /> +<br /> +<br /> +Ladder tests, <a href="#Page_28">28 ff.</a>, <a href="#Page_36">36 ff.</a><br /> +<br /> +Language in arithmetic, <a href="#Page_8">8 f.</a>, <a href="#Page_19">19</a>, <a href="#Page_89">89 ff.</a>, <a href="#Page_94">94 ff.</a><br /> +<br /> +<span class="smcap">Laser, H.</span>, <a href="#Page_103">103</a><br /> +<br /> +<span class="smcap">Lay, W. A.</span>, <a href="#Page_259">259</a>, <a href="#Page_261">261</a><br /> +<br /> +Learning, nature of arithmetical, <a href="#Page_1">1 ff.</a><br /> +<br /> +Least common multiple, <a href="#Page_88">88 f.</a><br /> +<br /> +<span class="smcap">Lewis, E. O.</span>, <a href="#Page_210">210 f.</a><br /> +<br /> +<span class="smcap">Lobsien, M.</span>, <a href="#Page_209">209 f.</a><br /> +<br /> +<br /> +<span class="smcap">McCall, W. A.</span>, <a href="#Page_49">49</a><br /> +<br /> +<span class="smcap">McDougle, E. C.</span>, <a href="#Page_85">85 ff.</a><br /> +<br /> +<span class="smcap">McKnight, J. A.</span>, <a href="#Page_210">210</a><br /> +<br /> +<span class="smcap">McLellan, J. A.</span>, <a href="#Page_3">3</a>, <a href="#Page_83">83</a>, <a href="#Page_89">89</a>, <a href="#Page_205">205</a>, <a href="#Page_207">207</a><br /> +<br /> +Manipulation of numbers, <a href="#Page_60">60 ff.</a><br /> +<br /> +Meaning, of numbers, <a href="#Page_2">2 ff.</a>, <a href="#Page_171">171</a>;<br /> + of a fraction, <a href="#Page_54">54 ff.</a>;<br /> + of decimals, <a href="#Page_181">181 f.</a><br /> +<br /> +Measurement of arithmetical abilities, <a href="#Page_27">27 ff.</a><br /> +<br /> +Mental arithmetic, <a href="#Page_262">262 ff.</a><br /> +<br /> +<span class="smcap">Messenger, J. F.</span>, <a href="#Page_202">202</a><br /> +<br /> +Metric system, <a href="#Page_147">147</a><br /> +<br /> +<span class="smcap">Meumann, E.</span>, <a href="#Page_261">261</a><br /> +<br /> +<span class="smcap">Mitchell, H. E.</span>, <a href="#Page_24">24</a><br /> +<span class='pagenum'><a name="Page_313" id="Page_313">[Pg 313]</a></span><br /> +<span class="smcap">Monroe, W. S.</span>, <a href="#Page_49">49</a><br /> +<br /> +Multiplication, measurement of, <a href="#Page_35">35</a>, <a href="#Page_36">36</a>;<br /> + constitution of, <a href="#Page_51">51</a>;<br /> + deductive explanations of, <a href="#Page_61">61</a>;<br /> + inductive explanations of, <a href="#Page_61">61 f.</a>;<br /> + with fractions, <a href="#Page_78">78 ff.</a>;<br /> + by eleven, <a href="#Page_85">85</a>;<br /> + amount of practice in, <a href="#Page_122">122 ff.</a>;<br /> + order of learning the elementary facts of, <a href="#Page_144">144 f.</a>;<br /> + distribution of practice in, <a href="#Page_158">158 ff.</a>;<br /> + use of the problem attitude in teaching, <a href="#Page_267">267 ff.</a><br /> +<br /> +<br /> +<span class="smcap">Nanu, H. A.</span>, <a href="#Page_202">202</a><br /> +<br /> +National Intelligence Tests, <a href="#Page_49">49 f.</a><br /> +<br /> +Negative reaction in intellectual life, <a href="#Page_278">278 f.</a><br /> +<br /> +Number pictures, <a href="#Page_259">259 ff.</a><br /> +<br /> +Numbers, meaning of, <a href="#Page_2">2</a>;<br /> + as measures of continuous quantities, <a href="#Page_75">75</a>;<br /> + abstract and concrete, <a href="#Page_85">85 ff.</a>;<br /> + denominate, <a href="#Page_141">141 f.</a>, <a href="#Page_147">147 f.</a>;<br /> + use of large, <a href="#Page_145">145 f.</a>;<br /> + perception of, <a href="#Page_205">205 ff.</a>;<br /> + early awareness of, <a href="#Page_205">205 ff.</a>;<br /> + confusion of cardinal and ordinal, <a href="#Page_206">206</a>.<br /> +<i>See also</i> Decimal numbers <i>and</i> Fractions.<br /> +<br /> +<br /> +Objective aids, used for verification, <a href="#Page_154">154</a>;<br /> + in general, <a href="#Page_243">243 ff.</a><br /> +<br /> +Oral arithmetic, <a href="#Page_262">262 ff.</a><br /> +<br /> +Order of topics, <a href="#Page_141">141 ff.</a><br /> +<br /> +Ordinal numbers, confused with cardinal, <a href="#Page_206">206</a><br /> +<br /> +Original tendencies and arithmetic, <a href="#Page_195">195 ff.</a><br /> +<br /> +Overlearning, <a href="#Page_134">134 ff.</a><br /> +<br /> +<br /> +Percents, <a href="#Page_80">80 f.</a><br /> +<br /> +Perception of number, <a href="#Page_202">202 ff.</a><br /> +<br /> +<span class="smcap">Phillips, D. E.</span>, <a href="#Page_3">3</a>, <a href="#Page_4">4</a>, <a href="#Page_205">205</a>, <a href="#Page_207">207</a><br /> +<br /> +Pictures, hygiene of, <a href="#Page_246">246 ff.</a>;<br /> + number, <a href="#Page_259">259 ff.</a><br /> +<br /> +<span class="smcap">Pommer, O.</span>, <a href="#Page_212">212</a><br /> +<br /> +Practice, amount of, <a href="#Page_122">122 ff.</a>;<br /> + distribution of, <a href="#Page_156">156 ff.</a><br /> +<br /> +Precision in fundamental operations, <a href="#Page_102">102 ff.</a><br /> +<br /> +Problem attitude, <a href="#Page_266">266 ff.</a><br /> +<br /> +Problems, <a href="#Page_9">9 ff.</a>;<br /> + "catch," <a href="#Page_21">21 ff.</a>;<br /> + measurement of ability with, <a href="#Page_42">42 ff.</a>;<br /> + whose answer must be known in order to frame them, <a href="#Page_93">93 f.</a>;<br /> + verbal form of, <a href="#Page_111">111 f.</a>;<br /> + interest in, <a href="#Page_220">220 ff.</a>;<br /> + as introductions to arithmetical learning, <a href="#Page_266">266 ff.</a><br /> +<br /> +Propædeutic bonds, <a href="#Page_117">117 ff.</a><br /> +<br /> +Purposive thinking, <a href="#Page_193">193 ff.</a><br /> +<br /> +<br /> +Quantity, number and, <a href="#Page_85">85 ff.</a>;<br /> + perception of, <a href="#Page_202">202 ff.</a><br /> +<br /> +<br /> +<span class="smcap">Race, H.</span>, <a href="#Page_295">295</a><br /> +<br /> +Rainfall, <a href="#Page_272">272</a><br /> +<br /> +Ratio, <a href="#Page_225">225 f.</a>;<br /> + meaning of numbers, <a href="#Page_3">3 ff.</a><br /> +<br /> +Reaction, negative, <a href="#Page_278">278 f.</a><br /> +<br /> +Reality, in problems, <a href="#Page_9">9 ff.</a><br /> +<br /> +Reasoning, arithmetical, nature of, <a href="#Page_19">19 ff.</a>;<br /> + measurement of ability in, <a href="#Page_42">42 ff.</a>;<br /> + derivation of tables by, <a href="#Page_58">58 f.</a>;<br /> + about the rationale of computations, <a href="#Page_60">60 ff.</a>;<br /> + habit in relation to, <a href="#Page_73">73 f.</a>, <a href="#Page_190">190 ff.</a>;<br /> + problems which provoke false, <a href="#Page_100">100 f.</a>;<br /> + the essentials of arithmetical, <a href="#Page_185">185 ff.</a>;<br /> + selection in, <a href="#Page_187">187 ff.</a>;<br /> + as the coöperation of organized habits, <a href="#Page_190">190 ff.</a><br /> +<br /> +Recapitulation theory, <a href="#Page_198">198 f.</a><br /> +<br /> +Recipes, <a href="#Page_273">273 f.</a><br /> +<br /> +Rectangle, area of, <a href="#Page_257">257 f.</a><br /> +<br /> +<span class="smcap">Rice, J. M.</span>, <a href="#Page_228">228 ff.</a><br /> +<br /> +<span class="smcap">Rush, G. P.</span>, <a href="#Page_49">49</a><br /> +<br /> +<br /> +<span class="smcap">Seekel, E.</span>, <a href="#Page_212">212</a><br /> +<br /> +<span class="smcap">Selkin, F. B.</span>, <a href="#Page_196">196 f.</a><br /> +<br /> +Sequence of topics, <a href="#Page_141">141 ff.</a><br /> +<br /> +Series meaning of numbers, <a href="#Page_2">2 ff.</a><br /> +<br /> +Size of class in relation to arithmetical learning, <a href="#Page_228">228</a><br /> +<br /> +<span class="smcap">Smith, D. E.</span>, <a href="#Page_xvi">xvi</a>, <a href="#Page_224">224</a><br /> +<br /> +Social instincts, use of, <a href="#Page_195">195 f.</a><br /> +<br /> +Sociology of arithmetic, <a href="#Page_24">24 ff.</a><br /> +<br /> +Speed in relation to accuracy, <a href="#Page_31">31</a>, <a href="#Page_108">108</a><br /> +<br /> +<span class="smcap">Speer, W. W.</span>, <a href="#Page_3">3</a>, <a href="#Page_5">5</a>, <a href="#Page_83">83</a><br /> +<br /> +Spiral order, <a href="#Page_141">141</a>, <a href="#Page_145">145</a><br /> +<br /> +<span class="smcap">Starch, D.</span>, <a href="#Page_49">49</a>, <a href="#Page_295">295</a><br /> +<br /> +<span class="smcap">Stern, W.</span>, <a href="#Page_210">210</a>, <a href="#Page_212">212</a><br /> +<br /> +<span class="smcap">Stone, C. W.</span>, <a href="#Page_27">27 ff.</a>, <a href="#Page_42">42 ff.</a>, <a href="#Page_228">228 ff.</a><br /> +<br /> +Subtraction, measurement of, <a href="#Page_34">34 f.</a>;<br /> + constitution of, <a href="#Page_57">57 f.</a>;<br /> + amount of practice in, <a href="#Page_122">122 ff.</a><br /> +<br /> +Supervision, <a href="#Page_233">233 f.</a><br /> +<br /> +<span class="smcap">Suzzallo, H.</span>, <a href="#Page_xvi">xvi</a><br /> +<br /> +<br /> +Temporary bonds, <a href="#Page_111">111 ff.</a><br /> +<br /> +Terms, <a href="#Page_113">113 f.</a><br /> +<br /> +Tests of arithmetical abilities, <a href="#Page_27">27 ff.</a><br /> +<br /> +<span class="smcap">Thorndike, E. L.</span>, <a href="#Page_34">34</a>, <a href="#Page_38">38 ff.</a>, <a href="#Page_227">227</a>, <a href="#Page_294">294</a><br /> +<br /> +Time, devoted to arithmetic, <a href="#Page_228">228 ff.</a>;<br /> + of day, in relation to arithmetical learning, <a href="#Page_227">227 f.</a><br /> +<span class='pagenum'><a name="Page_314" id="Page_314">[Pg 314]</a></span><br /> +Type, hygiene of, <a href="#Page_235">235 ff.</a><br /> +<br /> +<br /> +Underlearning, <a href="#Page_134">134 ff.</a><br /> +<br /> +United States money, <a href="#Page_148">148 ff.</a><br /> +<br /> +Units of measure, arbitrary, <a href="#Page_5">5</a>, <a href="#Page_83">83 f.</a><br /> +<br /> +<br /> +Variation, among individuals, <a href="#Page_285">285 ff.</a><br /> +<br /> +Variety, in teaching, <a href="#Page_153">153</a><br /> +<br /> +Verification, <a href="#Page_81">81 f.</a>;<br /> + aided by greater strength of the fundamental bonds, <a href="#Page_107">107 ff.</a><br /> +<br /> +<br /> +<span class="smcap">Walsh, J. H.</span>, <a href="#Page_11">11</a><br /> +<br /> +<span class="smcap">Wells, F. L.</span>, <a href="#Page_295">295</a><br /> +<br /> +<span class="smcap">White, E. E.</span>, <a href="#Page_5">5</a><br /> +<br /> +<span class="smcap">Whitley, M. T.</span>, <a href="#Page_295">295</a><br /> +<br /> +<span class="smcap">Wiederkehr, G.</span>, <a href="#Page_212">212</a><br /> +<br /> +<span class="smcap">Wilson, G. M.</span>, <a href="#Page_24">24</a>, <a href="#Page_49">49</a><br /> +<br /> +<span class="smcap">Woody, C.</span>, <a href="#Page_29">29 ff.</a>, <a href="#Page_52">52</a>, <a href="#Page_287">287</a>, <a href="#Page_293">293</a><br /> +<br /> +Words. <i>See</i> Language <i>and</i> Terms.<br /> +<br /> +Written arithmetic, <a href="#Page_262">262 ff.</a><br /> +<br /> +<br /> +Zero in multiplication, <a href="#Page_179">179 f.</a><br /> +</p></div> + +<p> </p> +<div class="notebox"> +<p class="noidt"><b>TRANSCRIBER'S NOTE: </b> +Obvious errors in spelling and punctuation have been silently closed. +Images have been moved from the middle of a paragraph to the closest paragraph break. +Footnotes have been consolidated at the end of the HTML ebook.</p> +</div> + + + + + + + + + +<pre> + + + + + +End of the Project Gutenberg EBook of The Psychology of Arithmetic, by +Edward L. 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