mm ^amM IttioBtailg Slibrarg Jlttfaca, Ntai ^arfe %OU"lSHT WITH THE INCOME OF THE SAGE ENDOWMENT .FUND THE GIFT OF HENRY W. SAGE 1891 Cornell University Library QA 135.T49P9 The psychology of arithmetic, 3 1924 012 021 576 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012021576 Zbc ps^cboloas of tbe Hlcmentars Scbool Subjects Bt EDWARD L. THORNDIKE THE PSYCHOLOGY OF ARITHMETIC THE MACMILLAN COMPANY NEW YOSK • BOSTON ■ CHICAGO - DALLAS ATLANTA • SAN FRANCISCO MACMILLAN & CO.. Limited LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO THE PSYCHOLOGY OF ARITHMETIC BY EDWARD L. THORNDIKE TEACHERS COLLEGE, COLT7MBIA tTNIVEKSITY Weto gork THE MACMILLAN COMPANY 1922 All riglUs reeerved PBINTED IN THE UNITED STATES OP AMERICA COPTBIGHT, 1922, bt the macmillan company. Set up and electrotyped. Published January, 1922. N'ortDoot ^ttse J. S. Gushing Co. —Berwick & Smith Co. Norwood, Maas., U.S.A. PREFACE Within recent years there have been three lines of advance in psychology which are of notable significance for teach- ing. 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 satisfy- ingness of the result is the chief force that forms them, and that habit rules in the realm of thought as truly and as fully as in the realm of action. The second is the great increase in knowledge of the amount, rate, and conditions of improvement in those organized groups or hierarchies of habits which we call abiUties, such as ability to add or ability to read. Practice and improvement are no longer vague generaUties, but concern changes which are definable and measurable by standard tests and scales. The third is the better understanding of the so-called "higher processes" of analysis, abstraction, the formation of general notions, and reasoning. The older view of a mental chemistry whereby sensations were compounded into percepts, percepts were duplicated by images, percepts and images were amalgamated into abstractions and con- cepts, 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" VI PREFACE in a revised and clarified form has important applications in the teaching of all the school subjects. This book presents the applications of this newer dynamic psychology to the teaching of arithmetic. Its contents are substantially what have been included in a course of lectures on the psychology of the elementary school subjects given by the author for some years to students of elementary education at Teachers College. Many of these former students, now in supervisory charge of elementary schools, have urged that these lectures be made available to teachers in general. So they are now published in spite of the author's desire to clarify and reinforce certain matters by further researches. A word of explanation is necessary concerning the exercises and problems cited to illustrate various matters, espe- cially 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 equiva- lent problems such as represent accurately the spirit and intent of the originals. I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn and Company, D. C. Heath and Company, The Macnaillan Company, The Oxford University Press, Rand, McNally and Company, Dr. C. W. Stone, The Teachers College Bureau of Publications, and The World Book Company, in permitting various quotations. Edward L. Thdrndike. Teachers College Columbia University April 1, 1920 CONTENTS CHAPTER PAGB Introduction: The Psychology of the Elementary School Subjects xi I. The Nature op Arithmetical Abilities ... 1 Knowledge of the Meanings of Numbers Arithmetical Language Problem Solving Arithmetical Reasoning Summary The Sociology of Arithmetic II. The Measurement op Arithmetical Abilities . . 27 A Sample Measurement of an Arithmetical Ability AbiUty to Add Integers Measurements of Ability in Computation Measurements of Abihty in Applied Arithmetic: the Solution of Problems III. The Constitution op Arithmetical Abilities . . 51 The Elementary Functions of Arithmetical Learning Knowledge of the Meaning of a Fraction Learning the Processes of Computatibn IV. The Constitution or Arithmetical Abilities (con- tinued) 70 The Selection of the Bonds to Be Formed The Importance of Habit Formation Desirable Bonds Now Often Neglected Wasteful and Harmful Bonds Guiding Principles vm CONTENTS CHAPTER PAGE V. The Psychology of Drill in Arithmetic : the Strength OF Bonds 102 The Need of Stronger Elementary Bonds Early Mastery The Strength of Bonds for Temporary Service The Strength of Bonds with Technical Facts and Terms The Strength of Bonds Concerning the Reasons for Arithmetical Processes Propaedeutic Bonds VI. , The Psychology of Drill in Arithmetic : the Amount OF Practice and the Organization or Abilities 122 The Amount of Practice Under-learning and Over-learning The Organization of Abilities VII. The Sequence of Topics: the Order of Formation of Bonds 141 Conventional versus Effective Orders Decreasing Interference and Increasing Facilitation Interest General Principles VIII. The Distribution of Practice 156 The Problem Sample Distributions Possible Improvements IX. The Psychology of Thinking: Abstract Ideas and General Notions in Arithmetic . . . 169 Responses to Elements and Classes FaciUtating the Analysis of Elements Systematic and Opportunistic Stimuli to Analysis Adaptations to Elementary-school Pupils X. The Psychology of Thinking: Reasoning in Arith- metic 185 The Essentials of Arithmetical Reasoning Reasoning as the Cooperation of Organized Habits CONTENTS IX CHAPTER PAGE XI. Original Tendencies and Acquisitions bepobe School 195 The Utilization of Instinctive Interests The Order of Development of Original Tendencies Inventories of Arithmetical Knowledge and Skill The Perception of Number and Quantity The Early Awareness of Number XII. Interest in Arithmetic 209 Censuses of Pupils' Interests Relieving Eye Strain Significance for Related Activities Intrinsic Interest in Arithmetical Learning XIII. The Conditions of Learning 227 Ejcternal Conditions The Hygiene of the Eyes in Arithmetic The Use of Concrete Objects in Arithmetic Oral, Mental, and Written Arithmetic XrV. The Conditions of Learning: the Problem Attitude 266 Illustrative Cases General Principles Difficulty and Success as Stimuli False Inferences XV. Individual Differences 285 Nature and Amount Differences within One Class The Causes of Individual Differences The Interrelations of Individual Differences Bibliography of References 301 Index 311 GENERAL INTRODUCTION THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS The psychology of the elementary school subjects is concerned with the connections whereby a child is able to respond to the sight of printed words by thoughts of their meanings, to the thought of "six and eight" by thinking "fourteen," to certain sorts of stories, poems, songs, and pictures by appreciation thereof, to certain situations by acts of skill, to certain others by acts of courtesy and justice, and so on and on through the series of situations and re- sponses 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 pro- duction of changes in human nature represented by an almost countless Ust 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 Ufe outside of school confronts him with them. We are not at present able to define the work of the ele- mentary 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 "abiUty to read the vernacular," "abiUty to spell common words," "ability to add, sub- xi Xll GENERAL INTRODUCTION tract, multiply, and divide with integers," "knowledge of the history of the. United States," "honesty in examina- tions," and "appreciation of good music," defined by some general results obtained rather than by the elementary bonds which constitute them. The psychology of the school subjects begins where our common sense knowledge of these functions leaves off and tries to define the knowledge, interest, power, skill, or ideal in question more adequately, to measure improvement in it, to analyze it into its constituent bonds, to decide what bonds need to be formed and in what order as means to the most economical attainment of the desired improvement, to survey the original tendencies and the tendencies already acquired before entrance to school which help or hinder progress in the elementary school subjects, to examine the motives that are or may be used to make the desired connections satisfjdng, to examine any other special condi- tions of improvement, and to note any facts concerning individual differences that are of special importance to the conduct of elementary school work. Put in terms of problems, the task of the psychology of the elementary school subjects is, in each case : — (1) What is the function? For example, just what is "ability to read"? Just what does "the understanding of decimal notation" mean? Just what are "the moral effects to be sought from the teaching of literature " ? (2) How are degrees of ability or attainment, and degrees of progress or improvement in the function or a part of the function measured ? For example, how can we determine how well a pupil should write, or how hard words we expect him to spell, or what good taste we expect him to show? How can we define to ourselves what knowledge of the meaning of a fraction we shall try to secure in grade 4? GENERAL ESTTRODUCTION Xlll (3) What can be done toward reducing the function to terms of particular situation-response connections, whose formation can he more surely and easily controlled ? For example, how far does ability to spell involve the formation one by one of bonds between the thought of almost every word in the language and the thought of that word's letters in their correct order ; and how far does, say, the bond leading from the situation of the sound of ceive in receive and deceive to their correct spelling insure the correct spelling of that part of perceive? Does "ability to add" involve special bonds leading from "27 and 4" to "31," from "27 and 5" to "32," and "27 and 6" to "33 " ; or will the bonds leading from "7 and 4" to "11," "7 and 5" to "12" and "7 and 6" to "13" (each plus a simple inference) serve as well? What are the situations and responses that represent in actual behavior the quaUty that we call school patriotism? (4) In almost every case a certain desired change of knowl- edge or skill or power can he attained by any one of several sets of hands. Which of them is the hest? What are the advantages of each ? For example, learning to add may in- clude the bonds "0 and are 0," "0 and 1 are 1," "0 and 2 are 2," "1 and are 1," "2 and are 2," etc. ; or these may be all left unformed, the pupil being taught the habits of entering as the siun of a column that is composed of zeros and otherwise neglecting in addition. Are the rules of usage worth teaching as a means toward correct speech, or is the time better spent in detailed practice in correct speech itself? (5) A bond to be formed may be formed in any one of many degrees of strength. Which of these is, at any given stage of learning the subject, the most desirable, all things considered ? For example, shall the dates of all the early settlements of North America be learned so that the exact year will be XIV GENERAL INTRODUCTION 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 contain- ing them spoken at an ordinary rate of speaking? Or is slow speech permissible, and even imperative, on the part of the teacher, with gradual increase of rate? (6) In almost every case, any set of bonds may produce the desired change when presented in any one of several orders. Which is the best order? What are the advantages of each? Certain systems for teaching handwriting perfect the ele- mentary 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 coiu-se. Would it be better if they were formed early as a means of faciUtating knowl- edge of decimal fractions ? (7) What are the original tendencies and preschool acquisi- tions upon which the connection-forming of the elementary school may be based or which it has to counteract? For example, if a pupil knows the meaning of a heard word, he may read it understandingly from getting its sound, as by phonic reconstruction. What words does the average beginner so know? What are the individual differences in GENERAL INTRODUCTION XV this respect? What do the instincts of gregariousness, attention-getting, approval, and helpfulness jecommend concerning group-work versus individual-work, and concern- ing 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 con- fronted with the printed page, and what must we do with it in teaching reading? (8) What armament of satisfiers and annoyers, of positive and negative interests and motives, stands ready for use in the formation of tKe intrinsically uninteresting connections be- tween black marks and meanings, numerical exercises and their answers, words and their spelling, and the like? School practice has tried, more or less at random, incentives and deterrents from quasi-physical pain to the most sentimental fondling, from sheer cajolery to philosophical argument, from appeals to assumed savage and primitive traits to appeals to the interest in automobiles, flying-machines, and wireless telegraphy. Can not psychology give some rules for guidance, or at least limit experimentation to its more hopeful fields ? (9) The general conditions of efficient learning are de- scribed in manuals of educational psychology. How do these apply in the case of each task of the elementary school ? For example, the arrangement of school drills in addition and in short division in the form of practice experiments has been found very effective in producing interest in the work and in improvement at it. In what other arithmetical functions may we expect the same ? (10) Beside the general principles concerning the nature and causation of individual differences, there must obviously be, in existence or obtainable as a possible result of proper investi- XVI GENERAL INTRODUCTION gabion, 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 f What are the means of learning more of them ? Courtis finds that a child may be specially strong in addition, and yet be specially weak in subtraction in comparison with others of his age and grade. It even seems that such subtle and intricate tendencies are inherited. How far is such specialization the rule ? Is it, for example, the case that a child may have a special gift for spelling certain sorts of words, for drawing faces rather than flowers, for learning ancient history rather than modern? Such are our problems : this volume discusses them in the case of arithmetic. The student who wishes to relate the discussion to the general pedagogy of arithmetic may profitably read, in connection with this volume : The Teaching of Elementary Mathematics, by D. E. Smith ['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic, by the author ['21]. THE PSYCHOLOGY OF ARITHMETIC THE PSYCHOLOGY OF ARITHMETIC CHAPTER I THE NATURE OF ARITHMETICAL ABILITIES According to common sense, the task of the elementary school is to teach : — (1) the meanings of numbers, (2) the nature of our system of decimal notation, (3) the meanings of addition, subtraction, multiplication, and division, and (4) the nature and relations of certain common measures; to secure (5) the ability to add, subtract, multiply, and divide with integers, common and decimal fractions, and denominate numbers, (6) the abihty to apply the knowledge and power represented by (1) to (5) in solving problems, and (7) certain specific abihties to solve problems concerning percentage, interest, and other common occurrences in business life. This statement of the functions to be developed and im- proved is soimd 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 featiu-es of arithmetical training, and 1 2 PSYCHOLOGY OP ARITHMETIC put in forms of training that a wise educational plan would not tolerate. It is also the case that different leaders in arithmetical teaching, though they might all subscribe to the general statement of the previous paragraph, certainly do not in practice have identical notions of what arithmetic should be for the elementary school pupil. The ordinary view of the natvire of arithmetical learning is obscure or inadequate in four respects. It does not define what 'knowledge of the meanings of numbers' is; it does not take account of the very large amount of teaching of language which is done and should be done as a part of the teaching of arithmetic ; it does not distinguish between the ability to meet certain quantitative problems as Ufe offers them and the ability to meet the problems provided by textbooks and courses of study; it leaves 'the ability to apply arithmetical knowledge and power' as a rather mystical general faculty to be improved by some educational magic. The four necessary amendments may be discussed briefly. KNOWLEDGE OP THE MEANINGS OP NUMBERS Knowledge of the n^ftanings nf thf? H'aTpbftrs 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 ther series. meaning. ToL-knaw-the mpanin g nf 'six' in this, sense is -to know that it is one more than five and- one less 4han- seven — t hat it is between five and se ven in the pumber _ sfiries. 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, THE NATURE OP ARITHMETICAL ABILITIES 3 and the other customary objects of enumeratiiMi in the primary school. TVvig_wg_png.y pg.n t.hp rnl]^. r,f,'>,(M)meia,T)\-n^ To^n ow the m ea,ningj) f jsix in this spinsP! is tn b PiJjblp' tn na.me correctly-any^CQllection of six .separate;^ easily .distinguishable individual ohjeets. In the third place, knowledge of the numbers from one to ten may mean knowledge that two is twice whatever is called one, that t hree is three times whafr . ^ever is one, a nd so on. This is, of course, the ratio meaning To know the meaning of six in this sense is to know that if is one, a Une half a foot long is six, that if I I is one, | " | is about six, while if □ is one, | I is about six, and the like. In the fourth place, the meaning of a numberjnay be a smaller or larger fraction of its implications^ — its numerical, rgla^ tionS; facts about it. Tn know six in this rptisp. [sjhr>_Vnnw that it is more than five or four, less than seven or eight, twice thr ee, three times two , the^gum of five jjnd-oney-or-of four and ^two^ r of three and three, two less than eight — that with four it makes ten, that it is half of twelve, and the Uke. This we may call the 'nucleus of facts ' or relational meaning of a number. Ordinary school practice has conmionly 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 McLelkn and Dewey ['95] and Speer ['97], and the relational idea by Grube and his followers. This diversity of views concerning what the fvmction 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 4 PSYCHOLOGY OF ARITHMETIC samples of the sort of work at which children were kept employed for months by too ardent followers of Speer and Grube. THE^ERIES_IPEA OVEREMPHASIZED "This is essentially the counting period, and any words that can be arranged into a series furnish all that is necessary. Count- ing 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 multipUcation was developed out of counting, and not from addition as nearly all textbooks treat it. Miiltiplicatian ^is counting . When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations. I would emphasize again that during the counting period there is a somewhat spontaneous development of the number series- idea which Preyer has discussed in his Arithmogenesis ; that an immense momentum is given by a systematic series of names; and that these names are generally first learned and applied to objects later. A lady teacher told me that the Superintendent did not wish the teachers to allow the children to count on their fingers, but she failed to see why counting with horse-chestnuts was any better. Her children could hardly avoid using their fingers in counting other objects yet they followed the series to 100 without hesitation or reference to their fingers. This spon- taneous counting period, or naming and following the series, should precede its application to objects." [D. E. PhilKps, '97, p. 238.] THE RATIO IDEA OVEREMPHASIZED B FlQ. 1. THE NATURE OF ARITHMETICAL ABILITIES 5 " Ratios. — 1. Select solids having the relation, or ratio, of o, b, c, d, 0, e. 2. Name the soUds, a, b, c, d, o, e. The means of expressing must be as freely supplied as the means of discovery. The pupU is not expected to invent terms. 3. Tell all you can about the relation of these units. 4. Unite units and teU what the sum equals. 5. Make statements like this : o less e equals b. 6. c can be separated intb how many d's? into how many &'s? 7. c can be separated into how many 6's? What is the name of the largest unit that can be found in both c and d an exact number of times? 8. Each of the other units equals what part of c? 9. If 6 is 1, what is each of the other units? 10. If a is 1, what is each of the other units? 11. If 6 is 1, how many I's are there in each of the other units? 12. If d is 1 , how many 1 's and parts of 1 in each of the other units ? 13. 2 is the relation of what units ? 14. 3 is the relation of what units ? 16. I is the relation of what units ? 16. f is the relation of what units? 17. Which units have the relation f ? 18. Which unit is 3 times as large as § of 6? 19. c equals 6 times | of what unit ? 20. i of what unit equals | of c? 21. What equals | of c? d equals how many sixths of c? 22. o equals 5 times | of what unit ? 23. f of what unit equals | of o ? 24. f of d equals what unit? b equals how many thirds of d? 25. 2 is the ratio of d to ^ of what unit ? 3 is the ratio of d to | of what unit? 26. d equals f of what unit? f is the ratio of what units?" [Speer, '97, p. 9f.] THE RELATIONAL IDEA OVEREMPHASIZED An inspection of books of the eighties which followed the "Grube method" (for example, the New Elementary Arithmetic by E. E. White ['83]) wiU show undue emphasis on the relational ideas. There wiU be over a hundred and fifty successive tasks aU, or nearly all, on + 7 and — 7. There wiU be much written work of the sort shown below : PSYCHOLOGY OF ARITHMETIC Add: 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 2 which must have sorely tried the eyes of all concerned. Pupils are taught to " give the analysis and synthesis of each of the nine digits." Yet the author states that he does not carry the princi- ple of the Grube method " to the extreme of useless repetition and mechanism." It should be obvious that all four meanings have claims upon the attention of the elementary school. Four is the thing between three and five in the number series ; it is the name for a certain sized collection of discrete objects ; it is also the name for a continuous magnitude equal to four units — for four quarts of milk in a gallon pail as truly as for four separate quart-pails of milk ; it is also, if we know it well, the thing got by adding one to three or subtracting six from ten or taking two two's or half of eight. To know the meaning of a number means to know somewhat about it in all of these respects. The difficulty has been the narrow vision of the extremists. A child must not be left interminably counting; in fact the one-more-ness of the number series can almost be had as a by-product. A child must not be restricted to exercises with collections objectified THE NATURE OF ARITHMETICAL ABILITIES i i iJ © © © © © © © © © 4"i k. j^^ :!k.^ .*l^^ v , jl/.^. .\t/ .-x* "TS'T* /J^^ 'T'lv *T>/|>. TfvVS Vl^T^" ,'s!<.'>j-'.,\k .^A^i'^S:^ , ^^ 4^, .n I'n I /nI ^ lllllll lllllll 11 II II II II II II *-)(-^^* -X-)f^-X-^ -x-^-x-^-jt ^•x-^ ^■X--X- -Jf^-X- ^^^ ^-^^ H 12 U 18 Fig. 2. 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 imits — inches, feet, yards, glassf uls. 8 PSYCHOLOGY OF ARITHMETIC 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 measm-e 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 Uke, are simply idola^ trous ; these facts about eighteen, so far as they are needed, are better learned in the course of actual column- addition and -subtraction. ARITHMETICAL LANGUAGE The sgcond _improvemen t_to_be made in the orjdinary notion of what the fimction&ta be improved are in the case of arithmetic is to include among these fujictions the^ knowl- edge of certain words. The understanding of such words as both, aUj_in^ all, together, less^^ difference, sum, whole, part, equal, buy, sell, have left, measure, is contained in, and the ]^e^ is necessaiy in arithmetic as truly as is the under-, standing of nurnbers themselves. It must be provided for by the school; for pre-school and extra-school training does not furnish it, or furnishes it too late. It can be provided for much better in connection with the teaching of arithmetic than in connection with the teaching of English. It has not been provided for. An examination of the first fifty pages of eight recent textbooks for beginners in arith- metic reveals very sUght attention to this matter at the best and no attention at all in some cases. Three of the books do not even use the word sum, and one uses it only once in the fifty pages. In all the four himdred pages the word difference occurs only twenty times. When the words are used, no great ingenuity or care appears in THE NATURE OF ARITHMETICAL ABILITIES 9 the means of making sure that their meanings are under- stood. The chief reason why it has not been provided for is pre- cisely that the common notion of what the functions are that arithmetic is to develop has left out of account this function of intelligent response to quantitative terms, other than the names of the numbers and processes. Knowledge of language over a nauch 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 abiUty, 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 vmtil we have seen what the general function of problem-solving is. PROBLEM-SOLVING rChethiid respect in which the function, ' abiUty^in arith- metic/needs-clearer -definition,^ is this- .'probleiiksolying.' le aim of the elementary school isjfco provide for correct and econo noical respon se_to genuine 4a!Qblems, 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, com- puting areas, percentages, and discounts, estimating quanti- ties needed of certain materials to make certain household or shop products, and the Hke. 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 10 PSYCHOLOGY OF ARITHMETIC 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 de- scribed 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 Ust is there to be read) . To fit pupils to solve these real, personally imagined, or self-described problems, and ' described-by-another ' prob- lems, schools have reUed almost exclusively on training with problems of the last sort only. The following page taken almost at random from one of the best recent textbooks could be paralleled by thousands of others; and the oral problems put by teachers have, as a rule, no real situation supporting them. 1. At 70 cents per 100 pounds, what will be the amount of duty on an invoice of 3622 steel rails, each rail being 27 feet long and weighing 60 pounds to the yard? 2. A man had property valued at $6500. What will be his taxes at the rate of $10.80 per $1000? 3. Multiply seventy thousand fourteen hundred-thousandths by one hundred nine millionths, and divide the product by five hundred forty-five. 4. What number multiplied by 43f will produce 265f ? THE NATURE OP ARITHMETICAL ABILITIES 11 5. What decimal of a bushel is 3 quarts? 6. A man sells f of an acre of land for $93.75. What would be the value of his farm of 150f acres at the same rate ? 7. A coal dealer buys 375 tons coal at $4.25 per ton of 2240 pounds. He sells it at $4.50 per ton of 2000 pounds. What is his profit ? 8. Bought 60 yards of cloth at the rate of 2 yards for $5, and 80 yards more at the rate of 4 yards for $9. I immediately sold the whole of it at the rate of 5 yards for $12. How much did I gain? 9. A man piu-chased 40 bushels of apples at $1.50 per bushel. Twenty-five hundredths of them were damaged, and he sold them at 20 cents per peck. He sold the remainder at 50 cents per peck. How much did he gain or lose? 10. If oranges are 37^ cents per dozen, how many boxes, each containing 480, can be bought for $60? 11. A man can do a piece of work in 18f days. What part of it can he do in 6| days? 12. How old to-day is a boy that was born Oct. 29, 1896? [Walsh, '06, Part I, p. 165.] As a result, teachers and textbook writers have come to think of the functions of solving arithmetical problems as identical with the function of solving the described problems which they give in school in books, examination papers, and the like. If they do not think explicitly that this is so, they still act in training and in testing pupils as if it were so. It is not. Problems should be solved in school to the end that pupils may solve the problems which life offers. To know what change one should receive after a given real purchase, to keep one's accounts accurately, to adapt a recipe for six so as to make enough of the article for four persons, to estimate the amount of seed required for a plot of a given size from the statement of the amount required per acre, to make with surety the applications that the household, small stores, and ordinary trades require — such is the ability that the elementary school should develop. 12 PSYCHOLOGY OF ARITHMETIC Other things being equal, the school should set problems in arithmetic which Ufe then and later will set, should favor the situations which Ufe itself offers and the responses which life itself demands. Other things are not always equal. The same amount of time and effort will often be more productive toward the final end if directed during school to 'made-up' problems. The keeping of personal financial accounts as a school exercise is usually impracticable, partly because some of the children have no earnings or allowance — no accounts to keep, and partly because the task of supervising work when each child has a different problem is too great for the teacher. The use of real household and shop problems will be easy only when the school program includes the household arts and industrial education, and when these subjects them- selves 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 them- selves. All this is true, yet the general principle remains that, other things being equal, the school should favor real situa-. tions, should present issues as life will present them. Where other things make the use of verbally descr bed THE NATURE OF ARITHMETICAL ABILITIES 13 problems of the ordinary tjrpe 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 con- sider just what the situations of life really require and show clearly the application of that principle. For example, contrast these two problems applying cancel- lation : — A. A man sold 24 lambs at $18 apiece on each of six days, and bought 8 pounds of metal with the proceeds. How much did he pay per ounce for the metal? B. How taU must a rectangular tank 16" long by 8" wide be to hold as much as a rectangular tank 24" by 18" by 6"? The first problem not only presents a situation that would rarely or never occiu-, but also takes a way to find the answer that would not, in that situation, be taken since the price set by another would determine the amount. Much thought and ingenuity should in the future be expended in eUminating problems whose solution does not improve the real function to be improved by applied arith- metic, 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 limita- tions of school Hfe. The following illustrations will to some extent show con- cretely 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. 14 PSYCHOLOGY OF ARITHMETIC Samples of Desirable Applications of Arithmetic in Problems where the Situation is Actually Present to Sense in Whole or in Part Keeping the scores and deciding which side beat and by how much in appropriate classroom games, spelling matches, and the like. Computing costs, making and inspecting change, taking inventories, and the like with a real or play store. Mapping the school garden, dividing it into allotments, planning for the purchase of seeds, and the like. Measuring one's own achievement and progress in tests of word-knowledge, spelling, addition, subtraction, speed of writing, and the like. Measuring the rate of improvement per hour of practice or per week of school life, and the like. Estimating costs of food cooked in the school kitchen, articles made in the school shops, and the like. Computing the cost of telegrams, postage, expressage, for a real message or package, from the pubhshed tariffs. Computing costs from mail order catalogues and the like. Samples of Desirable Applications of Arithmetic where the Situation is Not Present to Sense The samples given here all concern the subtraction of fractions. Samples concerning any other arithmetical prin- ciple may be found in the appropriate pages of any text which contains problem-material selected with consideration of life's needs. 1. Dora is making jelly. The recipe calls for 24 cups cf sugar and she has only 21^. She has no time to go to the store so she has to borrow the sugar from a neighbor. How much must she get? THE NATURE OF ARITHMETICAL ABILITIES 15 Subtract 24 Think "^andi = l." Write §. 211 Think " 2 and 2 = 4." Write the 2. 2i 2. A box full of soap weighs 29| lb. The empty box weighs 3| lb. How much does the soap alone weigh ? 3. On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt. On July 15 there were just 11| lb. left. How much had he used in the two weeks ? 4. Grace promised to pick 30 qt. blueberries for her mother. So far she has picked 18| qt. How many more quarts must she pick? ( B This table of numbers tells Weight of Mary Adams what Nell's baby sister Mary When born 7f lb. weighed every two months from 2 months old llj lb. the time she was bom tiU she 4 months old 14| lb. was a year old. 6 months old 15f lb. 8 months old 17f lb. 10 months old 19| lb. 12 months old 21f lb. 1. How much did the Adams baby gain in the first two months ? . 2. Howmuch did the Adams baby gain in the second two months ? 3. In the third two months? 4. In the fourth two months? 5. From the time it was 8 months old till it was 10 months old? 6. In the last two months? 7. From the time it was born till it was 6 months old? C 1. Helen's exact average for December was 87|-. Kate's was 84§. How much higher was Helen's than Kate's? 87^ How do you think of i and ^? 8^ How do you think of If? How do you change the 4? 2. Find the exact average for each girl in the following list. Write the answers clearly so that you can see them easily. You will use them in solving problems 3, 4, 5, 6, 7, and 8. 16 PSYCHOLOGY OF ARITHMETIC Alice Dora Emma Grace Louise Mary Nell Rebecca Reading 91 87 83 81 79 77 76 73 Language 88 78 82 79 73 78 73 75 Arithmetic 89 85 79 75 84 87 89 80 Spelling 90 79 75 80 82 91 68 81 Geography 91 87 83 75 78 85 73 79 Writing 90 88 75 72 93 92 95 78 3. Which girl had the highest average ? 4. How much higher was her average than the next highest? 5. How much difference was there between the highest and the lowest girl? 6. Was Emma's average higher or lower than Louise's? How much? 7. How much difference was there between Alice's average and Dora's? 8. How much difference was there between Mary's average and Nell's? 9. Write five other problems about these averages, and solve each of them. Samples of Undesirable Applications of Arithmetic ^ Will has XXI marbles, XII jackstones, XXXVI pieces of string. How many things had he? George's kite rose CDXXXV feet and Tom's went LXIII feet higher. How high did Tom's kite rise? If from DCIV we take CCIV the result will be a number IV times as large as the mmiber of dollars Mr. Dane paid for his horse. How much did he pay for his horse? Hannah has f of a dollar, Susie ^, Nellie f , Norah ■^. How much money have they all together? A man saves 3|^ dollars a week. How much does he save in a year? A tree fell and was broken into 4 pieces, 13^ feet, lOf feet, 8| feet, and 7^ feet long. How tall was the tree ? 1 The following and later problems are taken from actual textbooks or courses of study or state examinations ; to avoid invidious comparisons, they are not exact quotations, but are equivalents in principle and form, as stated in the preface. THE NATURE OF ARITHMETICAL ABILITIES 17 Annie's father gave her 20 apples to divide among her friends. She gave each one 2f apples apiece. How many playmates had she ? John had 17f apples. He divided his whole apples into fifths. How many pieces had he in all? A landlady has 3f pies to be divided among her 8 boarders. How much will each boarder receive? There are twenty columns of spelling words in Mary's lesson and 16 words in each column. How many words are in her lesson? There are 9 nuts in a pint. How many pints in a pile of 5,888,673 nuts? The Adams school contains eight rooms; each room contains 48 pupils; if each pupil has eight cents, how much have they together? A pile of wood in the form of a cube contains 15| cords. What are the dimensions to the nearest inch? A man 6 ft. high weighs 175 lb. How tall is his wife who is of similar build, and weighs 125 lb. ? A stick of timber is in the shape of the frustum of a square pyramid, the lower base being 22 in. square and the upper 14 in. square. How many cubic feet in the log, if it is 22 ft. long? Mr. Ames, being asked his age, replied : "If you cube one half of my age and add 41,472 to the result, the sum wiU be one half the cube of my age. How old am I?" 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 hoiir to collect them; and I am confident that a thousand such problems describing situations that the pupil will never encoimter in real life, or putting questions that he will never be asked in real life, could easily be found in any ten text- books of the decade from 1900 to 1910. If there are 250 kernels of corn on one ear, how many are there on 24 ears of corn the same size? Maud is four times as old as her sister, who is 4 years old. What is the sum of their ages? 18 PSYCHOLOGY OF ARITHMETIC If the first century began with the year 1, with what year does it end? Every spider has 8 compound eyes. How many eyes have 21 spiders? jA nail 4 inches long is driven through a board so that it projects 1.695 inches on one side and 1.428 on the other. How thick is the board? Find the perimeter of an envelope 5 in. by 3 1 in. How many minutes in f of f of an hour? Mrs. Knox is f as old as Mr. Knox, who is 48 years old. Their son Edward is I- as old as his mother. How old is Edward? Suppose a pie to be exactly round and 101 miles in diameter. If it were cut into 6 equal pieces, how long would the curved edge of each piece be ? 8^% of a class of 36 boys were absent on a rainy day. 33|% of those present went out of the room to the school yard. How many were left in the room? Just after a ton of hay was weighed in market, a horse ate one pound of it. What was the ratio of what he ate to what was left? If a fan having 15 rays opens out so that the outer rays form a straight hne, how many degrees are there between any two adjacent rays? One half of the distance between St. Louis and New Orleans is 280 miles more than -^ of the distance ; what is the distance between these places? If the pressure of the atmosphere is 14.7 lb. per square inch what is the pressure on the top of a table IJ yd. long and f yd. wide? ^ of the total acreage of barley in 1900 was 100,000 acres ; what was the total acreage? What is the least number of bananas that a mother can exactly divide between her 2 sons, or among her 4 daughters, or among all her children? If AUce 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 AUce? 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 direc- tion, how many days wiU elapse before they will be together again? With only thirty or forty dollars a year to spend on a pupil's education, of which perhaps eight dollars are spent THE NATURE OF ARITHMETICAL ABILITIES 19 on improving his arithmetical abilities, the immediate guidance of his responses to real situations and personally initiated problems has to be supplemented largely by guidance of his responses to problems described in words, diagrams, pictures, and the like. Of these latter, words will be used most often. As a consequence the understanding of the words used in these descriptions becomes a part of the ability required in arithmetic. Such word knowledge is also required in so far as the problems to be solved in real life are at times described, as in advertisements, business letters, and the like. This is recognized by everybody in the case of words like remainder, 'profit, loss, gain, interest, cubic capacity, gross, net, and discount, but holds equally of let, suppose, balance, average, total, borrowed, retained, and many such semi- technical words, and may hold also of hundreds of other words unless the textbook and teacher are careful to use only words and sentence structiires 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 prob- lems to be solved simply for the sake of solving them, unnecessarily unreal problems, and clumsy descriptions, but it will remain to some extent as an important joint task for the 'arithmetic' and 'reading' of the elementary school. ARITHMETICAL REASONING The last respect in which the nature of arithmetical abilities requires definition concerns arithmeti cal rpagnnir^g An adequate treatment of the reasoning that may be expected of pupils in the elementary school and of the most 20 PSYCHOLOGY OF ARITHMETIC eflBcient ways to encourage and improve it cannot be given until we have studied the formation of habits. Forreasomng is essentially the orga nization and contro J of habits --Qf thoughi- Certain matters may, however, be decided here. The first concerns the use of computation and problems merely for discipline, — that is, the emphasis on training in reasoning regardless of whether the problem is otherwise worth reasoning about. It used to be thought that the mind was a set of faculties or abilities or powers which grew strong and competent by being exercised in a certain way, no matter on what they were exercised. Problems that could not occur in life, and were entirely devoid of any worthy interest, save the intellectual interest in solving them, were supposed to be nearly or quite as useful in training the mind to reason as the genuine problems of the home, shop, or trade. Anything that gave the mind a chance to reason would do ; and pupils labored to find when the minute hand and hour hand would be together, or how many sheep a shepherd had if half of what he had plus ten was one third of twice what he had ! We now know that the training depends largely on the particular data used, so that efficient discipline in reasoning requires that the pupil reason about matters of real impor- tance . There is no magic essence or faculty of reasoning that works in general and irrespective of the particular facts and re- lations reasoned about. Sqwe should try to find problems which not only stimulate the pupil to reason, but also direct Kis reasoning in uselulchanjifils.andj'ewaTdJi^^ olreal significanee. We should replace the purely discipiin^~ ary problems by problems that are also valuable as special training for important particular situations of life. Reason- ing sought for reasoning's sake alone is too wasteful an ex- penditure of time and is also Ukely to be inferior as reasoning. THE NATURE OF ARITHMETICAL ABILITIES 21 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 blimder, guide him rightly. Consider, for example, these four problems : 1. "A man bought ten dozen eggs for $2.50 and sold them for 30 cents a dozen. How many cents did he lose?" 2. "I went into Smith's store at 9 a.m. and remained until 10 A.M. 1 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 biU. How long was I in the store?" 3. "Whatmustyoudivide48by to get half of twice6?" ^ 4. " What must you add to 19 to get 30?" :0 - ? A- ^ ' _^ •■- — ■^' " H Xhejjpat,chl problem is now in disrepute, the _wise_teacher feeli ng by a sor t. Qf intnitirm t.ha,t.ix>-.wi11f-Hl1y^mqiiirft a. pupil to reasQn„tQ_a..rfisult..sharply:. contrary lai;hat to. which pre- vious habitsjead him is. risky. The foiir 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 modem books as a routine drill ! The first problem, on the contrary, all, save those who demand no higher criterion for a problem than that it make the pupil ' think,' would reject. It demands the reversal of fixed habits to no valid purpose; for in Hfe 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 ex- cuse the child's confidence in the training he has had. Prob- lems 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 22 PSYCHOLOGY OF ARITHMETIC saying, "I got up at 7 o'clock to multiply 9 by 2| and got 24| 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. Fabri- cated puzzle situations wherein a peculiar hidden element of the situation makes the good habits called up by the situation misleading are useful therefore rather as a relief and amusing variation in arithmetical work than as stimuli to thought. Problems like the third quoted above we might call puz- zling rather than 'catch' problems. They have value as drills in analysis of a situation into its elements that will amuse the gifted children, and as tests of certain abilities. They also require that of many confusing habits, the right one be chosen, rather than that ordinary habits be set aside by some hidden element in the situation. Not enough is known about their effect to enable us to decide whether or not the elementary school should include special facility with them as one of the arithmetical functions that it specially trains. The fourth 'catch' quoted above, which all would admit is a good problem, is good because it opposes a good habit for the sake of another good habit, forces the analysis of an element whose analysis Ufe 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 THE NATURE OF ARITHMETICAL ABILITIES 23 life the 'have 19, must get ... to have 30' situation is very frequent and important. On the whole, the ordinary problems which ordinary life proffers seem to be the sort that should be reasoned out, though the elementary school may include the less noxious forms of pure mental gymnastics for those pupils who like them. SUMMARY These discussions of the meanings of numbers, the lin- guistic demands of arithmetic, the distinction between scholastic and real applications of arithmetic, and the pos- sible restrictions of training in reasoning, — may serve as illustrations of the significance of the question, "What are the functions that the elementary school tries to improve in its teaching of arithmetic?" Other matters might well be considered in this connection, but the main outline of the work of the elementary school is now fairly clear. The arithmetical functions or abilities which it seeks to improve are, we may say : — (1) -Working knowledge of the meanings of numbers as Taames for certain sized collections, for certain relative magni- tudes, the magnitude of unity being known, and for certain centers or nuclei of relations to other numbers. (2) Working knowledge of the system of decimal notation. (3) Working knowledge of the meanings of addition, subtraction, multiplication, and division. (4) Working knowledge of the natiu-e and relations of certain common measures. (5) Working abiUty to add, subtract, multiply, and divide with integers, common and decimal fractions, and denomi- nate numbers, all being real positive numbers. (6) Working knowledge of words, symbols, diagrams, and 24 PSYCHOLOGY OF ARITHMETIC the like as reqiiired by life's simpler arithmetical demands or by economical preparation therefor. (7) The ability to apply all the above as required by Ufe's simpler arithmetical demands or by economical preparation therefor, including (7 a) certain specific abilities to solve problems concerning areas of rectangles, volumes of rec- tangular solids, percents, interest, and certain other com- mon occurrences in household, factory, and business life. THE SOCIOLOGY OF ARITHMETIC The phrase 'life's simpler arithmetical demands' is necessarily left vague. Just what use is being made of arithmetic in this country in 1920 by each person therein, we know only very roughly. What may be called a ' sociology' of arithmetic is very much needed to investigate this matter. For rare or difficult demands the ele- mentary school should not prepare; there are too many other desirable abilities that it should improve. A most interesting beginning at such an inventory of the actual uses of arithmetic has been made by Wilson ['19] and Mitchell.* Although their studies need to be much extended and checked by other methods of inquiry, two main facts seem fairly certain. First, the great majority of people in the great majority of their doings use only very elementary arithmetical processes. In 1737 cases of addition reported by Wilson, seven eighths were of five numbers or less. Over half of the multipliers reported were one-figure numbers. Over 95 per cent of the fractions operated with were included in this list : iilifltJfi. Three fourths of all the cases reported were simple one-step computations with integers or United States money. Second, they often use these very elementary processes, not because such are the quickest and most convenient, but because they have lost, or maybe never had, mastery of the more advanced processes which wotild 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 famihar instances of human avoidance of arithmetic. Wilson found very slight use of decimals ; and Mitchell found men computing with 1 The work of Mitchell has not been published, but the author has had the privilege of examining it.. THE NATURE OF ARITHMETICAL ABILITIES 25 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 "7f 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. Test Perform the operations indicated. Express all fractions in answers in lowest terms. Add: I+I+.25 4 yr. 6 mo. 1 jT. 2 mo. 6 yr. 9 mo. 3 yr. 6 mo. 4 yr. 5 mo. Subtract : 8.6-6.05007 |-f = Multiply : 29 ft. 6 in. 8 7X8X41 = 5i^~"2^— Divide : It seems probable that the school training in arithmetic of the past has not given enough attention to perfecting the more ele- mentary abilities. And we shaU later find further evidence of this. On the other hand, the fact that people in general do not at present use a process may not mean that they ought not to use it. Life's simpler arithmetical demands certainly do not include matters like the rules for finding cube root or true discount, which no sensible person uses. They should not include matters like computing the lateral surface or volume of pyramids and coneSj or knowing the customs of plasterers and paper hangers, which are used only by highly specialized trades. They should not in- clude matters like interest on call loans, usury, exact interest, and 26 PSYCHOLOGY OF ARITHMETIC the rediscounting of notes, which concern only brokers, bank clerks, and rich men. They should not include the technique of customs which are vanishing from efficient practice, such as simple interest on amount for times longer than a year, days of grace, or extremes and means in proportions. They should not include any elaborate practice with very large numbers, or decimals beyond thousandths, or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year. When we have an adequate sociology of arithmetic, stating accurately just who should use each arithmetical ability and how often, we shall be able to define the task of the elementary school in this respect. For the present, we may proceed by common sense, guided by two limiting rules. The first is, — " It is no more desirable for the elementary school to teach all the facts of arith- metic than to teach all the .words in the English language, or all the topography of the globe, or all the details of human physiology." The second is, — "It is not desirable to eliminate any element of arithmetical training until you have something better to put in its place." CHAPTER II THE MEASUREMENT OF ARITHMETICAL ABILITIES One of the best ways to clear up notions of what the functions are which schools should develop and improve is to get measures of them. If any given knowledge or skiU or power or ideal exists, it exists in some amoimt. A series of amounts of it, varying from less to more, defines the ability itself in a way that no general verbal description ^an do. Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to tell us what we mean by weight. By finding a series of words like only, smoke, another, pretty, answer, tailor, circus, telephone, saucy, and beginning, which are spelled correctly by known and decreasing percentages of children of the same age, or of the same school grade, we know better what we mean by 'spelling-difficulty.' Indeed, until we can measure the efficiency and improvement of a function, we are likely to be vague and loose in our ideas of what the fvmction is. A SAMPLE MEASUEEMENT OF AN ARITHMETICAL ABILITY : THE ABILITY TO ADD INTEGERS Consider first, as a sample, the measurement of ability to add integers. The following were the examples used in the measurements made by Stone ['08] : 27 28 PSYCHOLOGY OF ARITHMETIC 596 4695 428 872 2375 94 7948 4052 75 6786 6354 304 567 260 645 858 5041 984 9447 1543 897 7499 The scoring was as follows : Credit of 1 for each column added correctly. Stone combined measures of other abilities with this in a total score for amount done correctly in 12 minutes. Stone also scored the correctness of the additions in certain work in multiplication. Courtis uses a sheet of twenty-four tasks or 'examples/ each consisting of the addition of nine three-place numbers as shown below. Eight minutes is allowed. He scores the amount done by the number of examples, and also scores the number of examples done correctly, but does not sug- gest any combination of these two into a general-efficiency score. 927 379 756 837 924 110 854 965 344 The author long ago proposed that pupils be measured also with series like ato g shown below, in which the difficulty increases step by step. o. 32 2 3 2 2 1 2 2 3 12 4 5 5 1 42333222 THE MEASUREMENT OF ARETHMETICAL ABILITIES 29 b. 21 32 12 24 34 34 22 12 23 12 52 31 33 12 23 13 24 25 15 14 32 23 43 61 c. 22 3 4 35 32 83 22 3 3 31 3 2 33 11 3 21 38 45 52 52 2 4 33 64 d. 30 20 10 22 10 20 52 12 20 50 40 43 30 4 6 22 40 17 24 13 40 23 30 44 e. 4 5 20 12 12 20 10 20 30 3 40 4 11 20 20 10 30 20 4 1 23 7 2 20 2 40 23 40 11 10 30 20 20 10 11 20 22 30 25 /• 19 9 9 14 2 19 24 9 4 13 9 14 13 12 13 13 9 14 17 23 13 15 15 34 12 25 26 29 18 19 25 28 18 39 9- 13 13 9 14 12 9 9 13 12 9 14 24 23 19 19 29 9 9 13 21 28 26 26 14 8 8 29 23 29 16 15 19 17 19 19 22 Woody ['16] has constructed his well-known tests on this principle, though he uses only one example at each step of difficulty instead of eight or ten as suggested above. His test, so far as addition of integers goes, is : — gOO(MOiOI CO 1> ■* lO ^ (N tH COINI S?CDQ0iOCD C, lO CC 05 O 05 »0 5jl>il>.iCC30CDOOiOO(NI a,-^C5ooi:^ioa5(MTtiio .a tn £■ 05 Tt* (N »0 05 I C C<1 1— I T-H i-H I •a o o (A o •—I CO ' + CO "^ CO col ^t^ (Ml ^•co lOl c? o CO >o 1-H CO I ^St2i^ i-H (M " O IC CO CO tH (M 05 CO «• 00 lO (N -^ CO (N S? CO lO COI i (M (M i-h] s CO O iQ to § (N ■* 3| C CO lO 1-1,1 C3-2 XU4 CTjg ^''^ ^ '^> ^''^ ^^^ ' four figures in the quotient. 9. 10. 11. 12. 13. 22|253 22|2895 21|889r 22|290 32|l6;368 Check your results for 9, 10, 11, 12, and 13. 66 PSYCHOLOGY OF ARITHMETIC LONG DIVISION : INDUCTIVE EXPLANATION — Continued 1. The boys and girls of the Welfare Club plan to earn money to buy a victrola. There are 23 boys and girls. They can get a good second-hand victrola for 15.75. How much must each earn if they divide the cost equally? Here is the best way to find out : $.25 Think how many 23s there are in 57. 2 is right. 23\$5J5 ^" 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. THE PSYCHOLOGY OF THINKING 171 emphasize whatever appro'priate minor bonds frdnfi the element in question the learner already possesses. Thus, in teaching children to respond to the ' fiveness ' of various collections, we show five boys or five girls or five pencils, and say, " See how many boys are standing up. Is Jack the only boy that is standing here? Are there more than two boys standing? Name the boys while I point at them and count them. (Jack) is one, and (Fred) is one more, and (Henry) is one more. Jack and Fred make (two) boys. Jack and Fred and Henry make (three) boys." (And so on with the attentive counting.) The mental set or attitude is directed toward favoring the partial and predominant activity of ' how-many-ness ' as far as may be ; and the useful bonds that the ' fiveness,' the ' one and one and one and one and one-ness, ' already have, are emphasized as far as may be. The second of the means used to facihtate 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 sepa- rate each total response into an element bound to the A and an element bound to the V. C. Thus the chUd 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 con- comitants are also such as have preferred minor bonds of their own (the sight of a row of boys per se tends strongly to call up the ' Boys ' element of the response). The other elements of the responses (boys, girls, pencils, etc.) have each only a slight connec- tion with the ' fiveness ' element of the situations. These slight connections also in large part * counteract each other, leaving the field clear for whatever uninhibited bond the ' fiveness ' has. The third means used to facilitate analysis is having the learner respond to situations which', pair by pair, present the element in a certain context and present that same context with the opposite of the element in question, or with something at least very unlike the element. Thus, a child who is being taught to respond to ' one fifth ' is not only led to respond to ' one fifth of a cake,' ' one • They may, of course, also result in a fusion or an alternation of responses, but only rarely. 172 PSYCHOLOGY OP ARITHMETIC fifth of a pie,' ' one fifth of an apple,' ' one fifth of ten inches,' ' one fifth of an army of twenty soldiers,' and the Uke ; he is also led to respond to each of these in contrast with ' five cakes,' * five pies,' ' five apples,' ' five times ten laches,' ' five armies of twenty soldiers.' Similarly the ' place values ' of tenths, hundredths, and the rest are taught by contrast with the tens, hundreds, and thousands. These means utilize the laws of connection-forming to dis- engage a response element from gross total responses and attach it to some situation element. The forces of use, disuse, satis- faction, and discomfort are so maneuvered that an element which never exists by itself in nature can influence man almost as if it did so exist, bonds being formed with it that act almost or quite irrespective of the gross total situation in which it inheres. What happens can be most conveniently put in a general statement by using symbols. Denote by a+b, a+g, a+l, a+q, a+v, and a+B certain situations ahke in the element a and different in all else. Suppose that, by original nature or training, a child responds to these situations respectively by ri+r2, ri+r-,, ri+r^, ri-\-rv,, ri+ra, r\-\-r^. Suppose that man's neurones are capable of such action that Ti, Ti, r-i, ri2, r^i, and r^i can each be made singly. Case I. Varjdng Concomitants Suppose that a-\-h, •a-\-g, a+l, etc., occur once each. We have a+h responded to by ri+rj, a-\-g " ri+r-,, a+l " ri+ri2, a+q " ri +rv,, a+v " ri+r22, and a+B " n+r^, as shown in Scheme I Scheme I a h g I Q V B n 6 1 1 1 1 1 1 ri 1 rv 1 ri2 1 rn 1 Til 1 Til 1 THE PSYCHOLOGY OF THINKING 173 a is thus responded to by ri (that is, connected with ri) each time, or six in all, but only once each with b, g, I, q, v, and B. b, g, I, q, v, and B are connected once each with ri and once respectively with Vi, r?, ri2, etc. The bond from a to ri, has had six times as much exercise as the bond from a to r-i, or from a to rj, etc. In any new gross situation, a 0, a will be more predominant in determining response than it would otherwise have been ; and ri will be more likely to be made than r^, ri, r^, etc., the other previous associates in the response to a situation containing a. That is, the bond from the element a to the response ri has been notably strengthened. Case II. Contrasting Concomitants Now suppose that b and g are very dissimilar elements {e.g., white and black), that I and q are very dissimilar {e.g., long and short), and that v and B are also very dissimilar. To be very dissiimlar means to be responded to very differently, so that r^, the response to g, will be very unlike ra, the response to b. So r? may be thought of as rnot 2 or r_2. In the same way ri2 may be thought of as rnot 12 or r_i2, and r^j may be called rnot 22 or r_22. Then, if the situations ab, ag,al, aq,av, and a B are responded to, each once, we have : — a+b responded to by ri+r2, a+g a+l a+q a+v a+B " '•l+''not2, " ri+ri2, " ri+rnot 12, " ri+r22, and " ri+r„ot22, as shown in Scheme II. Scheme II a b g I (opp. of 6) (opp. of I) V B (opp. of v) ri 6 1 1 1 1 1 1 »'notl 1 ''not 2 ri2 1 1 ''not 12 '■22 1 1 rnot 22 1 Ti is connected to a by 6 repetitions. r2 and r„ot 2 are 1 . each CO nected to a by 1 repetition, but since they interfere, canceling each 174 PSYCHOLOGY OF ARITHMETIC other so to speak, the net result is for a to have zero tendency to call up 7-2 or j-not 2. ri2 and rnot 12 are each connected to a by 1 repetition, but they interfere with or cancel each other with the net result that a has zero tendency to call up ru or rnot 12. So with 7-22 and rnot 22- Here then the net result of the six connec- tions oiab,ag,al,aq,a v, and a B is to connect a with r, and with nothing else. Case III. Contrasting Concomitants and Contrasting Element Suppose now that the facts are as in Case II, but with the addition of six experiences where a certain element which is the opposite of, or very dissimilar to, a is connected with the response J"not 1, or r_i, which is opposite to, or very dissimilar to ri. Call this opposite of a, —a. That is, we have not only a+b responded to by ri+ri, a+g " " J-i+rnota, a+l " " ri+ri2, a+q " " ri+rnoti2, I a+v " " ri+r22, an a+B " " ri+rnot22, but also —a+b responded to by j-not i+Ti, -a+g -a+l —a+q —a+v -a+B It It ^not IT '"not 2) '"not 1+»'12, '"not IT^'not 12) ^not i+ra, and »'noti+''not22, as shown in Scheme III. Scheme III »'not2 a opp. b g I q V B of a (opp. of 5) (opp. of l) (opp. of v) 6 I 1 1 1 1 1 6 111111 1 1 2 1 1 2 THE PSYCHOLOGY OP THINKING 175 /•12 1 1 2 /•not 12 11 2 r22 1 1 2 rnot22 11. 2 In this series of twelve experiences a connects with ri six times and the opposite of a connects with rnot 1 six times, a connects equally often with three pairs of mutual destructives r2 and rnot 2, ri2 and rnot 12, r22 and rnot 22, and so has zero tendency to call them up. —a has also zero tendency to call up any of these responses except its opposite, r^^t. i- ^^ 9, 1, 1, i^> and B are made to connect equally often with ri and rnot i- So, of these elements, a is the only one left with a tendency to call up ri. Thus, by the mere action of frequency of connection, ri is con- nected with a ; the bonds from a to anything except ri are being counteracted, and the slight bonds from anything except a to ri are being counteracted. The element a becomes predominant in situations containing it; and its bond toward ri becomes relatively enormously strengthened and freed from competi- tion. These three processes occur in a similar, but more complicated, form if the situations a+b, a+gr, etc., are replaced by a+b+c+d +e+/, a+g+h+i+j-\-k, etc., and the responses ri+r2, rl+r^, ri+ri2, etc., are replaced by ri+r2+r3+r4+rB+r8, rl+r^+rs+rg ■^-rio+rii, etc. — provided the ri, r2, rg, Vi, etc., can be made singly. In so far as any one of the responses is necessarily co-active with any one of the others (so that, for example, rig always brings rze with it and vice versa), the exact relations of the numbers recorded in schemes hke schemes I, II, and III on pages 172 to 174 will change ; but, unless ri has such an inevitable co-actor, the general results of schemes I, II, and III wiU hold good. If ri does have such an inseparable co-actor, say r2, then, of course, a can never acquire bonds with ri alone, but everywhere that r^ or r^ appears in the preceding schemes the other element must appear also, ri rz would then have to be used as a unit in analysis. The ' a-\-b' ' a+g,' ' a+l,' . . . ' a+S ' situations may occur unequal numbers of times, altering the exact numerical relations of the connections formed and presented in schemes I, II, and III ; but the process in general remains the same. So much for the effect of use and disuse in attaching appropriate response elements to certain subtle elements of situations. There are three main series of effects of satisfaction and discomfort. 176 PSYCHOLOGY OF ARITHMETIC They serve, first, to emphasize, from the start, the desired bonds leading to the responses ri+r2, ri+r-,, 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 froni a tori; and to eliminate every tendency of a to connect with aught save ri, and of aught save a to connect with ri. Their third service is to strengthen the bonds produced of appropriate responses to a wherever it occurs, whether or not any formal comparisons and contrasts take place. The process of learning to respond to the difference of pitch in tones from whatever instrument, to the ' square-root-ness ' of whatever nimiber, 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, disuse7"satisf action, and discomfort. " WhgJt^^ppens in such cases is that the re- sponse, by being connectEa with many situations aUke 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 concomi- tants. Conversely any element is bound firmly . to any one response thatlslfiadeTO 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. Appro- priate 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 THE PSYCHOLOGY OF THINKING 177 of his training, very subtle elements of situations can so oper- ate. Learning by analysis does not often proceed in the care- fully organized way represented by the most ingenious marshaUng of comparing and contrasting activities. The associations with gross totals, whereby in the end an ele- ment is elevated to independent power to determine re- sponse, may come in a haphazard order over a long interval of time. Thus a gifted three-year-old boy will have the response element qf 'saying or thinking too,' 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 abstrac- tion the schools direct. Thus the kindergarten exercises- in analyzing out number, color, size, and shape commonly assume that 'one-ness' versus 'more-than-one-ness,' black and white, big and little, round and not round are, at least vaguely, active as elements responded to in some inde- pendence of their contexts. Moreover, the tests of actual trial and success in further undirected exercises usually cooperate to confirm and extend and refine what the system- atic 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 4X40 = 160, 4X400 = 1600, 800-80 = 720, 800-8 = 792, 800-' 800 = 0, 42X48 = 2016, 24X48 = 1152, and the like, are satisfying; while 4X40 = 16, 23X48 = 832, 800-8 = 0, and the Uke, are not. The__prQcess of analysis is the same in such casual, unsystematized formation of connections with elements as in the deUberately man- 178 PSYCHOLOGY OF ARITHMETIC iaged, piecemeal inspection, comparison, and contrast de- scribed above. SYSTEMATIC AND OPPOETUNISTIC STIMULI TO ANALYSIS The arrangement of a pupil's experiences so as to direct his attention to an element, vary its concomitants instruc- tively, stimulate comparison, and throw the element into re- lief by contrast may be by fixed, formal, systematic exer- cises. 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 de- signed to build up the power of correct response to the ele- ment, whereas the chief feature of the latter is that it uses especially such opportunities as occur by reason of the pupil's activities and interests. Each method has its advantages and disadvantages. The systematic method chooses experiences that are specially designed to stimulate the analysis; it provides these at a certain fixed time so that they may work together; it can then and there test the pupils to ascertain whether they really have the power to respond to the element or aspect or feature in question. Its disadvantages are, first, that many of the pupils will feel no need for and attach no interest or motive to these formal exercises ; second, that some of the pupils may memorize the answers as a verbal task instead of acquiring insight into the facts; third, that the abihty to respond to the element may remain restricted to the special cases devised for the systematic training, and not be available for the genuine uses of arithmetic. The opportunistic method is strong just where the sys- tematic is weak. Since it seizes upon opportunities created THE PSYCHOLOGY OF THINKING 179 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 Ufe, the power acquired is more Ukely to be applicable to life. Its disadvantage is that it is harder to manage. More thought and experimentation are required to find the best experiences; greater care is required to keep track of the development of an abstraction which is taught not in two days, but over two months; and one may forget to test, the pupils at the end. In so far as the textbook and teacher are able to overcome these disadvantages by ingenuity and care, the opportunistic method is better. ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS We may expect much improvement in the formation of abstract and general ideas in arithmetic from the applica- tion of three principles in addition to those already described. They are : (1) Provide enough actual experiences before asking the pupil to understand and use an abstract or general idea. (2). Develop such ideas gradually, not attempting to give complete and perfect ideas all at once. (3) Develop such ideas so far as possible from experiences which will be valuable to the pupil in and of themselves, quite apart from their merit as aids in developing the abstraction or general notion. Consider these three principles in order. Children, especially the less gifted intellectually, need more experiences as a basis for and as applications of an arithmetical abstraction or concept than are usually given them. For example, in paving the way for the principle, "Any number times equals 0," it is not safe to say, "John worked 8 days for minutes per day. How many minutes 180 PSYCHOLOGY OF ARITHMETIC did he work?" and "How much is times 4 cents?" It will be much better to spend ten or fifteen minutes as fol- lows:^ 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 yard? How many inches are there in 4 ft. ? In 2 ft. ? In ft. ? 7 pk. = qt. 5 pk. = . . . . qt. pk. = . . . . qt. A boy receives 60 cents an hour when he works. How much does he re- ceive when he works 3 hr.? 8 hr.? 6 hr.? hr.? A boy received 60 cents a day for days How muc*h did he receive? How much is times $600? How much is times $5000 ? How much is times a milUon dollars ? times any number equals .... 232 (At the blackboard.) time 232 equals what? 30 I write under the 0.^ 3 times 232 equals what ? 6960 Continue at the blackboard with 734 321 312 41 ^ _40 _30 60 etc." Pupils in the elementary school, except the most gifted, should not be expected to gain mastery over such concepts as common fraction, decimal fracUon, factor, and root quickly. They can learn a definition quickly and learn to use it in very easy cases, where even a vague and imperfect under- standing of it will guide response correctly. But complete 1 The more gifted children may be put to work using the principle after the first minute or two. 2 232 30 If desired this form may be used, with the appropriate difference 000 in the form of the questions and statements, 696 6900 THE PSYCHOLOGY OF THINKING 181 and exact understanding commonly requires them to take, not one intellectual step, but many; and mastery in use cormnonly comes only as a slow growth. For example, suppose that pupils are taught that .1, .2, .3, etc., mean Tu, TO, T^, etc., that .01, .02, .03, etc., mean tstt, tot, rh, etc., that .001, .002, .003, etc., mean rmv, xwir, rmu, 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, — j, .4, f , .07, .002, f, — 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 | and i, 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, — TOTT, -^2) "qcQ , $.25, — are common fractions, and which are decimals; or to write the decimal fractions which equal f and |. If now the teacher had given all at once the additional experiences needed to pro vide, the abiUty 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 re- lation to TTsS, totS, and;^'!T(nrffS, the mind will suffer from violent indigestion. A wise pedagogy in fact will break up the process of learn- 182 PSYCHOLOGY OF ARITHMETIC ing the meaning and use of decimal fractions into many- teaching units, for example, as follows : — (1) Such familiarity with fractions with large denomi- nators as is desirable for pupils to have, as by an exercise in reducing to lowest terms, ^, U, U, H , M, U, t^, t^, and the like. This is good as a review of cancellation, and as an extension of the idea of a fraction. (2) Objective work, showing ^ sq. ft., -^ sq. ft., rhs sq. ft., and xtnny sq. ft., and having these identified and the forms YTT sq. ft., tbtt sq. ft., and t^u^ sq. ft. learned. Finding how many feet = TV mile and tto mile. (3) Familiarity with xmys and nnnrs by reductions of" iinnr, Tinr, etc., to lowest terms and by writing the missing numerators in iinnr = TTnr=T5' and the like, and by finding ^, T^, and TTnnr of 3000, 6000, 9000, etc. (4) Writing ^ as .1 and tto as .01, ^, ^, ^, etc., as .11, .12, .13. United States money is used as the intro- duction. Application is made to miles. (5) Mixed numbers with a first decimal place. The cyclometer or speedometer. Adding numbers like 9.1, 14.7, 11.4, etc. (6) Place value in general from thousands to hundredths. (7) Review of (1) to (6). (8) Tenths and hundredths of a mile, subtraction when both numbers extend to hundredths, using a railroad table of distances. (9) Thousandths. The names 'decimal fractions or decimals,' and 'decimal mixed numbers or decimals.' Drill in reading any number to thousandths. The work will continue with gradual extension and refinement of the understanding of decimals by learning how to operate with them in various ways._ Such may seem a slow progress, but in fact it is not, and THE PSYCHOLOGY OF THINKING 183 many of these exercises whereby the pupil acquires his mastery of decimals are useful as organizations and applica- tions of other arithmetical facts. That, it will be remembered, was the third principle : — "Develop abstract and general ideas by experiences which will be intrinsically valuable." The reason is that, even with the best of teaching, some pupils will not, within any reasonable limits of time expended, acquire ideas that are fully complete, rigorous when they should be, flexible when they should be, and absolutely exact. Many children (and adults, for that matter) could not within any reasonable limits of time be so taught the nature of a fraction that they could decide unerringly in original exercises like : — 2 75 Is -^ a common fraction? 25 Is $.25 a decimal fraction? Is one xth of y a fraction? Can the same words mean both a common fraction and a decimal fraction? Express 1 as a common fraction. Express 1 as a decimal fraction. These same children can, however, be taught to operate correctly mth fractions in the ordinary uses thereof. And that is the chief value of arithmetic to them. They should not be deprived of it because they cannot master its subtler principles. So we seek to provide experiences that will teach all pupils something of value, while stimulating in those who have the ability the growth of abstract ideas and general principles. Finally, we should bear in mind that working with qual- ities 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 184 Psychology of arithmetic 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 ^ means; he then answers such questions as i of 10=? iof6=? i of 20 = ? ; by being told when he is right and when he is wrong, he gets from these experiences a better idea of ^; again he does his best with i=TTr) i=Ts-, etc., and as before refines and enlarges his concept of i. He adds i to |, etc., i to ■^, etc., ^ to ^, etc., and thereby gains still fvirther, and so on. What begins as a blind habit of manipulation started by imitation may thus grow into the power of correct response to the essential element. The pupil who has at the start no notion at aU of 'multiplying' may learn what multiply- ing is by his experience that '4 6 multiplying gives 24' ; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing something with numbers and differentiates right results, he will often reach in the end the abstractions which he is supposed to need in the beginning. It may even be the case with some of the abstractions required in arithmetic that elaborate provision for comprehension beforehand is not so efficient as the same amount of energy devoted partly to provision for analysis itself beforehand and partly to practice in response to the element in question without full comprehension. It certainly is not the best psychology and not the best educational theory to think that the pupil first masters a principle and then merely appUes it — first does some think- ing and then computes by mere routine. On the contrary, the appUcations 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 principle^ of arithmetic as a science. CHAPTER X THE PSYCHOLOGY OF THINKING: REASONING EST ARITHMETIC THE ESSENTIALS OF ARITHMETICAL REASONING We distinguish aimless reverie, as when a child dreams of a vacation trip, from purposive thinking, as when he tries to work out the answer to "How many weeks of vacation can a family have for $120 if the cost is $22 a week for board, $2.25 a week for laundry, and $1.76 a week for incidental expenses, and if the railroad fares for the round trip are $12?" We distinguish the process of response to famiUar 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 i 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 un- known tongue is learned. In proportion as thinking is pur- posive, 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 propor- tion as many bonds act together in an organized way to produce response, we call it reasoning. 185 186 PSYCHOLOGY OP ARITHMETIC 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 selec- tion of the elements that are deemed significant for the question at hand, the attachment of a certain amotoit of importance or weight to each of them, and their use in the right relations. Thought may fail because it has not suit- able facts, or does not select from them the right ones, or does not attach the right amount of weight to each, or does not put them together properly. In the world at large, many of our failm-es 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 probabiUties. 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 aU and that no two were aUke, he could not reason out the answer, no matter what his mastery of the theory of probabiUties. If a compe- tent thinker, he would first ask about the size and nature of tlie pack. In the actual practice of reasoning, that is, we have to survey our facts to see if we lack any that are necessary. If we do, the first task of reasoning is to acquire those facts. This is specially true of the reasoning about arithmetical facts in life. "Will 3i yards of this be enough for a dress ? " Reason directs you to learn how wide it is, what style of THE PSYCHOLOGY OF THINKING 187 dress you intend to make of it, how much material that style normally calls for, whether you are a careful or a wasteful cutter, and how big the person is for whom the dress is to be made. "How much cheaper as a diet is bread alone, than bread with butter added to the extent of 10% of the weight of the bread?" Reason directs you to learn the cost of bread, the cost of butter, the nutritive value of bread, and the nutritive value of butter. In the arithmetic of the school this feature of reasoning appears in cases where some fact about common measures must be brought to bear, or some table of prices or discounts must be consulted, or some business custom must be re- membered or looked up. Thus "How many badges, each 9 inches long, can be made from 2| yd. ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches. "At Jones' prices, which costs more, 3| lb. butter or 6^ lb. lard? How much more?" is a problem which directs, the thinker to ascertain Jones' prices. It may be noted that such problems are, other things being equal, somewhat better training in thinking than problems where all the data are given in the problem itself {e.g., "Which costs more, 3| lb. butter at 48)zS per lb. or 6|lb. lard at 27f5 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 appUed arith- metic as a problem where everything is given and he has only to manipulate the data. Life does not present its problems so. The process of selecting the right elements and attaching proper weight to them may be illustrated by the following problem : — "Which of these offers would you take, suppos- ing that you wish a D. C. K. upright piano, have $50 saved, 188 PSYCHOLOGY OF ARITHMETIC can save a little over $20 per month, and can borrow from your father at 6% interest ? " A Eeliable Piano. The Famous D. C. K. Upright. You pay $50 cash down and $21 a month for only a year and a half. >iVo interest to pay. We ask you to pay only for the piano and allow you plenty of time. B We offer the well-known D. C. K. Piano for $390. $50 cash and $20 a month thereafter. Regular interest at 6%. The interest soon is reduced to less than $1 a month. C The D. C. K. Piano. Special Offer, $375, cash. Compare our prices with those of any reliable firm. If you consider chiefly the "only," "No interest to pay," "only," and "plenty of time" in offer A, attaching much weight to them and little to the thought, "How much will $50 plus (18X$21) be?", you will probably decide wrongly. The situations of life are often complicated by many ele- ments 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, THE PSYCHOLOGY OF THINKING 189 and of ruling out all quantitative elements except those which should be considered. The first of the two simplifica- tions 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 re- quiring more selection. It should not, however, be a uni- versal custom; for in that case the pupils are tempted to think that in every problem they must use all the quantities given, as one must use all the pieces in a puzzle picture. It is obvious that the elements selected must not only be right but also be in the right relations to one another. For example, in the problems below, the 6 must be thought of in relation to a dozen and as being half of a dozen, and alsj as being 6 times 1. 1 must be mentally tied to "each." The 6 as half of a dozen must be related to the $1.00, $1.60, etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc. Buying in Quantity Doz. Each These are a grocer's prices for 1. Evaporated Milk $1.00 $.09 certain things by the dozen 2. Puffed Rice 1.60 .14 and for a single one. He sells 3. Puffed Wheat ... . 1.10 .10 a half dozen at half the price 4. Canned Soup 1.90 .17 of a dozen. Find out how 5. Sardines 1.80 .16 much you save by buying 6 6. Beans (No. 2 cans) 1.50 .13 aU at one time instead of buy- 7. Pork and Beans . . 1.70 .15 ing them one at a time. 8. Peas (No. 2 cans) 1.40 .12 9. Tomatoes (extra cans) 3.20 .28 10. Ripe olives (qt. cans) 7.20 .65 It is obvious also that in such arithmetical work as we 190 PSYCHOLOGY OF ARITHMETIC have been describing, the pupil, to be successful, must 'think things together.' Many bonds must cooperate to determine his final response. As a preface to reasoning about a problem we often have the discovery of the problem and the classification of just what it is, and as a postscript we have the critical inspection of the answer obtained to make sure that it is verified by experiment or is consistent with known facts. During the process of searching for, selecting, and weighting facts, there may be similar inspection and validation, item by item. EEASONING AS THE COOPERATION OF ORGANIZED HABITS The pedagogy of the past made two notable errors in practice based on two errors about the psychology of reason- ing. It considered reasoning as a somewhat magical power or essence which acted to counteract and overrule the ordi- nary laws of habit in man; and it separated too sharply the 'understanding of principles' by reasoning from the 'mechanical' work of computation, reading problems, re- membering facts and the like, done by 'mere' habit and memory. Reasoning or selective, inferential thinking is not at all opposed to, or independent of, the laws of habit, but really is their necessary result under the conditions imposed by man's nature and training. A closer examination of se- lective 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 as- sociative learning as described under the 'piecemeal' activity of situations ; and that attributing certain features of learning to mysterious faculties of abstraction or reason- ing gives no real help toward understanding or controlling them. THE PSYCHOLOGY OF THINKING 191 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 there- for, 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 Ukely to be one which rejects one after another response as their unfitness to satisfy a certain desideratum appears. What remains as the ap- parent course of thought includes only a few of the many bonds which did operate, but which, for the most part, were imsatisfying to the ruling attitude or adjustment. Successful responses to novel data, associations by similar- ity 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 re- sponses to novel data — as when he argues that the diagonal on a right -triangle of 796.278 mm. base and 137.294 mm. altitude will be 808.022 mm., or that Mary Jones, born this morning, will sometime die — are due to habits, notably the habits of response to certain elements or features, under the laws of piecemeal activity and assimilation. Nothing is less like the mysterious operations of a faculty of reasoning transcending the laws of connection-forming, than the behavior of men in response to novel situations. Let children who have hitherto confronted only such arith- metical tasks, in addition and subtraction with one- and two-place numbers and multiplication with one-place num- bers, as those exemplified in the first line below, be told to do the examples shown in the second line. 192 PSYCHOLOGY OF ARITHMETIC ^D Add Add Stobt. SCBT. MULTIPLT Mxn/nPLT MULTIPLT 8 37 35 8 37 8 9 6 5 24 68 23 19 5 24 5 -_ _3 32 MiTUnPLT 43 MtJi/nPLT 34 23 22 26 They will add the numbers, or subtract the lower from the upper number, or multiply 3X2 and 2X3, etc., getting 66, 86, and 624, or respond to the element of 'Multiply' at- tached to the two-place numbers by "I can't" or "I don't know what to do," or the Uke ; 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 '9 of the situation of the fact that ^ multiply' gave only 81, '10 and that 10_ multiply' gave only 100, or the like; and so may report an intelligent and justified "I can't," or reject the plan of 3X2 and 2X3, 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 'Mul- tiply,' 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. THE PSYCHOLOGY OP THINKING 193 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. abode leads to vwxyz because a has been bound to vwxyz by original natiu-e, exercise, or effect. Purposive behavior is the most important case of the influence of the attitude or set or adjustment of an organism in determining (1) which bonds shall act, and (2) which results shall satisfy. James early described the former fact, showing that the mechanism of habit can give the directed- ness or purposefulness in thought's products, provided that mechanism includes something paralleling the problem, the aim, or need, in question. The second fact, that the set or attitude of the man helps to determine which bonds shall satisfy, and which shall annoy, has commonly been somewhat obscured by vague assertions that the selection and retention is of what is "in point," or is "the right one," or is "appropriate," or the like. It is thus asserted, or at least hinted, that "the will," "the voluntary attention," "the consciousness of the problem," and other such entities are endowed with magic power to decide what is the "right" or "useful" bond and to kill off the others. The facts are that in purposive think- ing and action, as everywhere else, bonds are selected and retained by the satisfyingness, and are killed off by the dis- comfort, which they produce ; and that the potency of the man's set or attitude to make this satisfy and that annoy — to put certain conduction-units in readiness to act and others in unreadiness — is in every way as important as its potency to set certain conduction-units in actual operation. Reasoning is not a radically different sort of force operat- ing against habit but the organization and cooperation of many habits, thinking facts together. Reasoning is not 194 PSYCHOLOGY OF ARITHMETIC the negation of ordinary bonds, but the action of many of them, especially of bonds with subtle elements of the situa- tion. Some outside power does not enter to select and criticize ; the pupil's own total repertory of bonds relevant to the problem is what selects and rejects. An unsuitable idea is not killed off by some actus purus of intellect, but by the ideas which it itself calls up, in connection with the total set of mind of the pupil, and which show it to be inadequate. Almost nothing in arithmetic need be taught as a matter of mere unreasoning habit or memory, nor need anything, first taught as a principle, ever become a matter of mere habit or memory. 5X4 = 20 should not be learned as an isolated fact, nor remembered as we remember that Jones' telephone number is 648 J 2. Almost everything in arith- metic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come. The use of this organized hierarchy of habits to solve novel problems is reasoning. CHAPTER XI ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL THE UTILIZATION OF INSTINCTIVE INTERESTS The activities essential to acquiring ability in arithmetic can rely on little in man's instinctive equipment beyond the purely intellectual tendencies of curiosity and the satisfy- ingness of thought for thought's sake, and the general en- joyment of success rather than failure in an enterprise to which one sets oneseK. It is only by a certain amount of artifice that we can enlist other vehement inborn in- terests of childhood in the service of arithmetical knowl- edge 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 frac- tional parts, to cut strips, to fold paper, and the Uke will be useful. Various social instincts can be utiUzed in matches after the pattern of the spelhng match, contests between rows, certain number games, and the like. The scoring of both 195 196 PSYCHOLOGY OF ARITHMETIC the play and the work of the classroom is a useful field for control by the teacher of arithmetic. Hunt ['12] has noted the more important games which have some considerable amount of arithmetical training as a by-product and which are more or less suitable for class use. Flynn [' 12] has described games, most of them for home use, which give very definite arithmetical driU, though in many cases the drills are rather behind the needs of children old enough to understand and hke the game itself. It is possible to utihze 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 collec- tion 1'12, p. 69 f .] : — I. ADDITION " We must admit that there is nothing particularly interesting in a long column of numbers to be added. Let the teacher, how- ever, suggest that he can write the answer at sight, and the task will assume a totally different aspect. " A very simple niunber trick of this kind can be performed by making use of the principle of complementary addition. The arithmetical complement of a number with respect to a larger number is the difference between these two numbers. Most interesting results can be obtained by using complements with respect to 9. "The children may be called upon to suggest several numbers of two, three, or more digits. Below these write an equal number of addends and immediately announce the answer. The children, impressed by this apparently rapid addition, will set to work enthusiastically to test the results of this lightning calculation. "Example:— 3571 999 682 •A X3 793. 2997 642 1 317 ^B 206 J ORIGINAL TENDENCIES AND ACQUISITIONS 197 " Explanation : — The addends in group A are written down at random or suggested by the class. Those in group B are their cxjmplements. 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 mmiber of addends may be used and each addend may consist of any number of digits." Respect for arithmetic as a source of tricks and magic is very much less important than respect for its everyday services ; and computation to test such tricks is likely to be undertaken zealously only by the abler pupils. Conse- quently this source of interest should probably be used only sparingly, and perhaps the teacher should give such ex- hibitions 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-arith- metical entertainment, such as the demonstration of an adding machine, the story of primitive methods of counting, team races in computation, an exhibition of lightning calculation and intellectual sleight-of-hand by the teacher, or the voluntary study of arithmetical puzzles. The interest in achievement, in success, mentioned above is stronger in children than is often realized and makes advis- able the systematic use of the practice experiment as a method of teaching much of arithmetic. Children who thus compete with their own past records, keeping an exact score from week to week, make notable progress and enjoy hard work in making it. THE OEDBR OF DEVELOPMENT OF ORIGINAL TENDENCIES Negatively the difficulty of the work that pupils should be expected to do is conditioned by the gradual maturing 198 PSYCHOLOGY OF AEITHMETIC 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 6 and 6 as now, we may think it wise to provide somehow that certain minima of arithmetical ability be given them. There are, so far as is known, no special times and seasons at which the human animal by inner growth is specially ripe for one or another .section or aspect of arithmetic, except in so far as the general inner growth of intellectual powers makes the more abstruse and complex tasks smtable to later and later years. Indeed, very few of even the most enthusiastic devotees of the recapitulation theory or culture-epoch theory have attempted to apply either to the learning of arithmetic, and Branford is the only mathematician, so far as I know, who has advocated such application, even tempered by elaborate shif tings and reversals of the racial order. He says : — " Thus, for each age of the individual life — infancy, child- hood, 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 child- hood 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- ORIGINAL TENDENCIES AND ACQUISITIONS 199 chsean mathematics passing on through Greek and Hindu to medi- aeval European mathematics ; while the student is become suffi- ciently mature to begin the assimilation of modern and highly abstract European thought. The filling in of details must neces- sarily be left to the individual teacher, and also, within some such broadly marked limits, the precise order of the marshalling of the material for each age. For, though, on the whole, mathematical development has gone forward, yet there have been lapses from advances already made. Witness the practical world-loss of much valuable Hindu thought, and, for long centuries, the neglect of Greek thought : witness the world-loss of the invention by the Babylonians of the Zero, until re-invented by the Hindus, passed on by them to the Arabs, and by these to Europe. " Moreover, many blunders and false starts and false principles have marked the whole course of development. In a phrase, rivers have their backwaters. But it is precisely the teacher's function to avoid such racial mistakes, to take short cuts ulti- mately discovered, and to guide the young along the road ulti- mately fornid most accessible with such halts and retracings — returns up side-cuts — as the mental peculiarities of the pupils demand. " All this, the practical realization of the spirit of the principle, is to be wisely left to the mathematical teacher, familiar with the history of mathematical science and with the particular hmitations of his pupils and himself." ['08, p. 245.] The latitude of modification suggested by Branford re- duces the guidance to be derived from racial history to almost nil. Also it is apparent that the racial history in the case of arithmetical achievement is entirely a matter of acquisition and social transmission. Man's original nature is destitute of all arithmetical ideas. The human germs do not know even that one and one make two ! INVENTORIES OP ARITHMETICAL KNOWLEDGE AND SKILL A scientific plan for teaching arithmetic would begin with an exact inventory of the knowledge and skill which the pupils already possessed. Our ordinary notions of what a child knows at entrance to grade 1, or grade 2, or grade 3, 200 PSYCHOLOGY OF ARITHMETIC 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 text- books presupposing in their first fifty pages a knowledge of words which not half of the children can read even at the end of the 2 B grade. We do find just such evidence that ordinary ideas about the abilities of children at the beginning of systematic school training in arithmetic may be in gross error. For example, a reputable and in many ways admirable recent book has fourteen pages of exercises to teach the meaning of two and the fact that one and one make two ! As an example of the reverse error, consider putting all these words in the first twenty-five pages of a beginner's book : — absentees, attend- ance, blanks, continue, copy, during, examples, grouped, memo- rize, perfect, similar, splints, therefore, total ! Little, almost nothing, has been done toward providing an exact inventory compared with what needs to be done. We may note here (1) the facts relevant to arithmetic foimd by Stanley Hall, Hartmann, and others in their general in- vestigations of the knowledge possessed by children at en- trance to school, (2) the facts concerning the power of chil- dren to perceive differences in length, area, size of collection, and organization within a collection such as is shown in Fig. 24, and certain facts and theories about early awareness of number. In the Berlin inquiry of 1869, knowledge of the meaning of two, three, and four appeared in 74, 74, and 73 percent of the children upon entrance to school. Some of those re- corded as ignorant probably really knew, but failed to under- stand that they were expected to reply or were shy. Only ORIGINAL TENDENCIES AND ACQUISITIONS 201 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 Septem- ber, 1880, Stanley Hall found that 92 percent knew three, D 1 10 u 20 10 11 12 • • • • • • • • •■•A • • •• • • • • • • • •••• 8 9 10 Fig. 24.- -Objective presentation 83 percent knew four, and 71| 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 cotild repeat words spoken to them. In the Stanford form of the Binet tests counting four pennies is given as an abiUty of the typical four-year-old. Coimting 13 pennies correctly in at least one out of two 202 PSYCHOLOGY OF ARITHMETIC trials, and knowing three of the four coins, — penny, nickel, dime, and quarter, — are given as abilities of the typical six-year-old. THE PERCEPTION OP NUMBER AND QUANTITY We know that educated adults can tell how many lines or dots, etc., they see in a single glance (with an exposure too short for the eye to move) up to four or more, according to the clearness of the objects and their grouping. For example, Nanu ['04] reports that when a number of bright circles on a dark background are shown to educated adults for only .033 second, ten can be counted when arranged to form a parallelogram, but only five when arranged in a row. With certain groupings, of course, their 'perception' involves much inference, even conscious addition and multiplication. Similarly they can tell, up to twenty and beyond, the number of taps, notes, or other sounds in a series too rapid for single counting if the sounds are grouped in a conven- ient rhythm. These abiUties are, however, the product of a long and elaborate learning, including the learning of arithmetic itself. Elementary psychology and common experience teach us that the mere observation of groups or quantities, no matter how clear their number quality appears to the person who already knows the meanings of numbers, does not of itself create the knowledge of the meanings of numbers in one who does not. The experiments of Messenger ['03] and Burnett ['06] showed that there is no direct intuitive apprehension even of two as distinct from one. We have to learn to feel the two touches or see the two dots or lines as two. We do not know by exact measurements the growth in children of this ability to count or infer the number of ele- ments in a collection seen or series heard. Still less do we ORIGINAL TENDENCIES AND ACQUISITIONS 203 know what the growth would be without the influence of school training in counting, grouping, adding, and multiply- ing. Many textbooks and teachers seem to overestimate it greatly. Not all educated adults can, apart from measm-e- ment, 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. Children upon entering school have not been tested care- fully in respect to judgments of length and area, but we know from such studies as Gilbert's ['94] that the difference required in their case is probably over twice that required for children of 13 or 14. In judging weights, for example, a difference of 6 is perceived as easily by children 13 to 15 years of age as a difference of 15 by six-year-olds. A teacher who has adult powers of estimating length or area or weight and who also knows already which of the two 204 PSYCHOLOGY OF ARITHMETIC is longer or larger or heavier, may use two lines to illus- trate a difference which they really hide from the child. It is unlikely, for example, that the first of these Unes ^ 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 | of the length of the latter, rather than | of it or | of it or ^ of it or yi 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 arith- metically, the pupils' errors being all due to their inabil- ity to compare the lengths accurately. . _^ ' The quantities used should be such that their mere discrimination of- fers no difficulty even to a child of blunted sense powers. If | and 1 are to be compared, A and B are not allow- able. C, D, and E are much better. Teachers probably of- ten imderestimate or neglect the sensory difficulties of the tasks ■E I -^ P^^^'f" C^l ^^^y assign and of the material they use to illustrate absolute and relative magni- tudes. The result may be more pernicious when the pupils answer correctly than when they fail. For their correct B\ I ^1 I I I I I I I I I I I I I I I I I / ORIGINAL TENDENCIES AND ACQUISITIONS 205 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 aU the children. But their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be actually damaged by the exercise. THE EARLY AWARENESS OF NUMBER There has been some disagreement concerning the origin of awareness of number in the individual, in particular con- cerning the relative importance of the perception of how- many-ness and that of how-much-ness, of the perception of a defined aggregate and the perception of a defined ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and Decroly and Degand ['12].) The chief facts of significance for practice seem to be these : (1) Children with rare exceptions hear the names one, two, three, four, half, twice, two times, more, less, as many as, again, first, second, and third, long before they have analyzed out the quaUties and relations to which these words refer so as to feel them at all clearly. (2) Their knowledge of the quali- ties and relations is developed in the main in close associa^ tion 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 206 PSYCHOLOGY OF ARITHMETIC related to others in a series. For instance, if fairly gifted, a child comes, by the age of five, to see that a row of four cakes is an aggregate of four, seeing each cake as a part of the four and the four as the sum of its parts, to know that two of them are as many as the other two, that half of them would be two, and to think, when it is useful for him to do so, of four as a step beyond three on the way to five, or to think of hot as a step from warm on the way to very hot. The degree of development of these abilities depends upon the activity of the law of analysis in the individual and the character of his experiences. (4) He gets certain bad habits of response from the ambiguity of common usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears his parents or older children or others count pennies or rolls or eggs by saying one, two, three, four, and so on. He himself is perhaps misled into so counting. Thus the names properly belonging to a series of aggregations varying in amount come to be to him the names of the positions of the parts in a counted whole. This happens especially with numbers above 3 or 4, where the correct experience of the number as a name for the group has rarely been present. This attaching to the cardinal numbers above three or four the meanings of the ordinal numbers seems to affect many children on entrance to school. The numbering of pages in books, houses, streets, etc., and bad teaching of counting often prolong this error. (5) He also gets the habit, not necessarily bad, but often indirectly so, of using many names such as eight, nine, ten, eleven, fifteen, a hundred, a million, without any meaning. (6) The experiences of half, twice, three times as many, three times as long, etc., are rarer ; even if they were not, they would still be less easily productive of the analysis of the proper abstract element than are the experiences of ORIGINAL TENDENCIES AND ACQUISITIONS 207 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 foxirs, are very rare. Hence, the names, two, three, etc., mean to these children in the main, "one something and one something," "one something usually called one, and one something usually called one, and another something usually called one," and more rarely and imperfectly "two times anything," "three times anything," etc. With respect to Mr. Phillips' emphasis of the importance of the series-idea in children's minds, the matters of im- portance 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 magni- tudes varying by steps, is acquired later, as a result, not a cause, of awareness of numbers. With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers should be emphasized in schools, not because it is the main origin of the child's awareness of number, but because it is not, and because the ordinary prac- tical issues of child life do not adequately stimulate its action. It also seems both more economical and more scientific to introduce it through multiplication, division, and fractions rather than to insist that 4 and 5 shall from the start mean 4 or 5 times anything that is called 1, for instance, that 8 inches shall be called 4 two-inches, or 10 cents, 5 two-cents. 208 PSYCHOLOGY OF ARITHMETIC 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 suflSicient range of units for the first two school years. Teaching the meanings of | of 4, i of 6, I of 8, I of 10, I of 20, i of 6, ^ of 9, | of 30, J of 8, two 2s, five 2s, and the like, in early grades, each in con- nection with many different xmits of measure, provides a sufficient assurance that numbers will connect with relation- ships as well as with collections. CHAPTER XII INTEREST IN ARITHMETIC CENSUSES OF PUPILS' INTERESTS Arithmetic, although it makes little or no appeal to collecting, muscular manipulation, sensory curiosity, or the potent original interests in things and their mechanisms and people and their passions, is fairly well liked by children. The censuses of pupils' likes and disUkes 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 arith- metic 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, speUing, geography, and reUgion. Lobsien ['03], who asked one hundred children in each of the first five grades (Stufen) of the elementary schools of Kiel, "Which part of the school work (literally, 'which instruction period') do you like best?" found arithmetic led only by drawing and gymnastics in the case of the boys, and only by handwork in the case of the girls. 209 210 PSYCHOLOGY OF ARITHMETIC This is an exaggerated picture of the facts, since no count is made of those who especially dislike arithmetic. Arith- metic 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 gym- nastics for boys (28-1), handwork for girls (32-1 J), and draw- ing for both (16i-6). Writing (6|-4), arithmetic (14|-13), history (9-6^), reading (8|-8), and singing (6-7^) come next. Religion, nature study, physiology, geography, geom- etry, chemistry, language, and grammar are low. McKnight ['07] found with boys and girls in grades 7 and 8 of certain American cities that arithmetic was liked better than any of the school subjects except gymnastics and manual training. The vote as compared with history was : — Arithmetic 327 liked greatly, 96 disliked greatly. History 164 Uked greatly, 113 disliked greatly. In a later study Lobsien ['09] had 6248 pupils from 9 to 15 years old representing all grades of the elementary school report, so far as they could, the subject most disliked, the subject most liked, the subject next most liked, and the subject next in order. No child was forced to report all of these four judgments, or even any of them. Lobsien counts the likes and the dislikes for each subject. Gymnastics, handwork, and cooking are by far the most popular. History and drawing are next, followed by arithmetic and reading. Below these are geography, writing, singing, nature study, biblical history, catechism, and three minor subjects. Lewis ['13] secured records from English children in ele- mentary schools of the order of preference of all the studies INTEREST IN ARITHMETIC 211 listed below. He reports the results in the following table of percents : Top THinn of Middle Third of Lowest Third op Stddieb foh Studies fob Studies for Interest Interest Interest Drawing 78 20 2 Manual Subjects . . 66 26 8 History 64 24 12 Reading 53 38 9 Singing 32 48 20 Drill 20 55 25 Arithmetic .... 16 53 31 Science 23 37 40 Nature Study . . . 16 36 48 Dictation 4 57 39 Composition .... 18 28 54 Scripture 4 38 58 Recitation .... 9 23 68 Geography .... 4 24 72 Grammar 6 94 Brandell ['13] obtained data from 2137 Swedish children in Stockholm (327), Norrkoping (870), and Gothenburg (940). In general he found, as others have, that handwork, shop- work 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, speUing, and writing are, as in other studies; much less popular than arithmetic. 212 PSYCHOLOGY OP ARITHMETIC Other studies are by Lilius ['11] in Finland, Walsemann ['07], Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern ['13 and '14], in Germany. They confirm the general results stated. The reasons for the good showing that arithmetic makes are probably the strength of its appeal to the interest in definite achievement, success, doing what one attempts to do; and of its appeal, in grades 5 to 8, to the practical interest of getting on in the world, acquiring abilities that the world pays for. Of these, the former is in my opinion much the more potent interest. Arithmetic satisfies it es- pecially well, because, more than any other of the 'in- tellectual' studies of the elementary school, it permits the pupil to see his own progress and determine his own success or failure. The most important applications of the psychology of satisfiers and annoyers to arithmetic will therefore be in the direction of utilizing still more effectively this interest in achievement. Next in importance come the plans to attach to arithmetical learning the satisfyingness of bodily action, play, sociability, cheerfulness, and the hke, and of significance as a means of securing other desired ends than arithmetical abilities themselves. Next come plans to re- lieve arithmetical learning from certain discomforts such as the eyestrain of some computations and excessive copjdng of figures. These will be discussed here in the inverse order. RELIEVING EYESTRAIN At present arithmetical work is, hour for hour, probably more of a tax upon the eyes than reading. The task of copying numbers from a book to a sheet of paper is one of the very hardest tasks that the eyes of a pupil in the ele- INTEREST IN ARITHMETIC 213 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 computa- tion. 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 Uke 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 efiiciency in supervision. Arithmetical errors are not confused with errors of copying,^ 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. ' 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, wiU be making mistakes in carrying or copying." 7 2. 7^ 7%^ 7. 7 J eta 7.ib b* y J5 ^nh^^^ ^/.i 5^a ^ '>:3ii Jyf^ ^7- 9 G A /V 'TTX ^^3 /// !FiQ. 25 a. — Specimens taken at random from the computation work of eighth-grade pupils. This computation occurred in a genuine teat. In the original gray of the pencil marks the work is still harder to make out. 214 IL ^^/[iii If ^ / i^ ^ 7. // ^ rrn Fig. 25 b . 25 h. — Specimens taken at randoip 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. 215 216 PSYCHOLOGY OF ARITHMETIC Write the products : — D A. 3 4s= B. 5 7s= c. 9 2s= 5 2s= 8 3s= 4 4s= 7 2s= 4 2s= 2 7s= 1 6 = 4 5s= 6 4s= 1 3 = 4 7s= 5 5s= 3 7s= 5 9s= 3 6s= 4 ls= 7 5s= 3 2s= 6 8s = 7 ls= 3 9s= 9 8s= 6 3s= 5 ls= 4 3s= 4 9s= 8 6s= 2 4s= 3 5s= 8 4s= 2 2s= 9 6s= 8 5s= 8 7s= 2 5s= 7 9s= 5 8s= 5 4s= 6 2s= 7 6s= 8 2s= 7 4s= 7 3s= 8 9s = 9 3s= '• 4 20s = E. 9 60s= F. 40: x2=80 4 200s = 9 600s = 20 ><2= 6 30s = 5 30s= 30; ><2= 6 300s = 5 300s = 40; <2= 7x 50 = 8x 20 = 20; <3= 7x 500 = 8x200 = 30; <3= 3x 40 = 2x 70 = 300 X 3=900 3x 400 = 2x700 = 300 x2= INTEREST IN ARITHMETIC 217 Write the missing numbers : (r stands for remainder.) 25= . . 3s and . . r. 30= . . . 4s and . . r. 25= . .4s " . . . r. 30= . . . 5s " . . . r. 25= . .5s " . . . r. 30= . ..6s " . . . r. 25= . . . 6s " . . r. 30= . ..7s " .. . . r. 25= . ..7s " . . r. 30= . . .8s " . . . r. 25= . ..8s " . . . r. 30= . . .9s " . . . r. 25= . ..9s " . . . r. 26= . . . 3s and . . . r. 31= . . . 4s and . . . r 26= . ..4s " . . . r. 31= . ..5s " . . . r. 26= . . . 5s " . . . r. 31= . ..6s " . . . r 26= . . . 6s " . . . r. 31= . ..7s " . . . r 26= . . . 7s " . . . r. 31= . ..8s " . . . r 26= . . . 8s '•' . . . r. 31= . ..9s " . . . r 26 = .. . 9s '' . . . r. Write the whole numbers or mixed numbers which these fractions equal : — 5 4 7 4 8 4 n 4 4 3 5 3 6 3 7 5 Write the missing figures : — 8~4 4 2 10 9 5 4 2 7 3 11 3 8 8 2 8 9 9 16 8 4 8 13 8 8 5 6 6 "5 1 5~10 2 3 6 218 PSYCHOLOGY OF ARITHMETIC Write the missing numerators : — 1 = _ _ _ _ _ _ _ 2~12 8 10 4 16 6 14 1 = _ _ _ _ _ _ _ 3" 12 9 18 6 15 24 21 1 = _ _ _ _ _ _ _ 4 12 16 8 24 20 28 32 i=_ ______ 5~10 20 15 25 40 35 30 2^_ _ _ _ _ _ 3~12 18 21 6 15 24 9 3 ______ 4 8 16 12 20 24 32 28 Find the products. Cancel when you can : — 5 . 11 o 2 K I6^^= 12^'= 3^^ = 1.8= |xl5= 1x8 = mXEREST IN ARITHMETIC 219 SIGNIFICANCE FOR RELATED ACTIVITIES The use of bodily action, social games, and the like was dis- cussed in the section on original tendencies. ' ' Significance as a means of securing other desired ends than arithmetical learn- ing 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, house- keeping, shopwork, dressmaking, self-management, other school studies than arithmetic, and general school life and affairs. Significance as a means to future ends alone can also be more clearly and extensively attached to it than it now is. Whatever is done to supply greater strength of motive in studying arithmetic must be carefully devised so as not to get a strong but wrong motive, so as not to get abundant interest but in something other than arithmetic, and so as not to kill the goose that after all lays the golden eggs — the interest in intellectual activity and achievement itself. It is easy to secure an interest in laying out a baseball diamond, measuring ingredients for a cake, making a balloon of a certain capacity, or deciding the added cost of an extra trimming of ribbon for one's dress. The problem is to attach that interest to arithmetical learning. Nor should a teacher be satisfied with attaching the interest as a mere tail that steers the kite, so long as it stays on, or as a sugar- coating that deceives the pupil into swallowing the pill, or as an anodyne whose dose must be increased and increased if it is to retain its power. Until the interest permeates the arithmetical activity itself our task is only partly done, and perhaps is made harder for the next time. One important means of really interfusing the arith- metical 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 220 PSYCHOLOGY OF ARITHMETIC 'problem' attitude — and thus appreciate the technique which he actively hunts for to satisfy the need. In so far as arithmetical learning is organized to satisfy the practical demands of the pupil's life at the time, he should, so to speak, come part way to get its help. Even if we do not make the most skillful use possible of these interests derived from the quantitative problems of sports, housekeeping, shopwork, dressmaking, self-manage- ment, other school studies, and school hfe and affairs, the gain will still be considerable. To have them in mind will certainly preserve us from giving to children of grades 3 and 4 problems so devoid of relation to their interests as those shown below, all found (in 1910) in thirty successive pages of a book of excellent repute : — A chair has 4 legs. How many legs have 8 chairs? 5 chairs? A fly has 6 legs. How many legs have 3 flies ? 9 flies ? 7 flies ? (Eight more of the same sort.) In 1890 New York had 1,513,501 inhabitants, Milwaukee had 206,308, Boston had 447,720, San Francisco 297,990. How many had these cities together? (Five more of the same sort.) Milton was bom in 1608 and died in 1674. How many years did he hve ? (Several others of the same sort.) The population of a certain city was 35,629 in 1880 and 106,670 in 1890. Find the increase. (Several others of this sort.) A niunber of others about the words in various inaugural ad- dresses and the Psalms in the Bible. It also seems probable that with enough care other system- atic plans of textbooks can be much improved in this respect. From every point of view, for example, the early work in arith- metic should be adapted to some extent to the healthy child- ish interests in home affairs, the behavior of other children, and the activities of material things, animals, and plants. INTEREST IN ARITHMETIC 221 TABLE 9 Frequency op Appearance of Certain Words about Family Life, Play, and Action in Eight Elementary Textbooks in Arithmetic, pp. 1-50. A B c D E F 4 G H baby 2 brother . 2 6 1 1 1 family . 2 2 4 father . 1. 3 5 2 1 help . . home 2 4 4 2 2 7 1 mother . 4 2 9 5 5 1 7 sister . . 1 2 2 9 1 1 fork . . knife . . plate . . 4 2 2 1 spoon doU . . 10 1 10 6 10 9 game . . 1 3 5 5 jump . . marbles . 10 4 10 10 1 play . . 1 3 run . . 1 3 sing . . tag . . toy . . 1 car . . 2 4 2 3 1 cut . . 10 6 2 8 dig . . 2 flower . 1 4, 1 1 2 grow . . 1 plant . . 2 seed . . 3 1 string 1 10 1 1 wheel 5 10 222 PSYCHOLOGY OF ARITHMETIC 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 be- ginners' arithmetics was as shown in Table 9. The eight columns refer to the eight books (the first fifty pages of each). The numbers refer to the number of times the word in question appeared, the number 10 meaning 10 or more times in the fifty pages. Plurals, past tenses, and the like were counted. Help, fork, knife, spoon, jump, sing, and tag did not appear at all ! Toy and grow appeared each once in the 400 pages ! Play, run, dig, plant, and seed appeared once in a hundred or more pages. Baby did not appear as often as buggy. Family appeared no oftener than fence or Friday. Father appears about a third as often as farmer. Book A shows only 10 of these thirty words in the fifty pages ; book B only 4 ; book C only 12 ; and books D, E, F, G, and H only 13, 8, 14, 13, 10, respectively. The total number of appearances (counting the 10s as only 10 in each case) is 40 for A, 9 for B, 60 for C, 42 for D, 25 for E, 62 for F, 30 for G, and 37 for H. The five words — apple, egg, Mary, milk, and orange — are used oftener than all these thirty together. If it appeared that this apparent neglect of childish affairs and interests was deliberate to provide for a more systematic treatment of pure arithmetic, a better gradation of problems, and a better preparation for later genuine use than could be attained if the author of the textbook were tied to the child's apron strings, the neglect could be defended. It is not at all certain that children in grade 2 get much more enjoyment INTEREST IN ARITHMETIC 223 or ability from adding the costs of purchases for Christmas or Fourth of July, or multipljdng the number of cakes each child is to have at a party by the number of children who are to be there, than from adding gravestones or multiplying the number of hairs of bald-headed men. When, however, there is nothing gained by substituting remote facts for those of familiar concern to children, the safe policy is surely to favor the latter. In general, the neglect of childish data does not seem to be due to provision for some other end, but to the same inertia of tradition which has carried over the problems of laying walls and digging wells into city schools whose children never saw a stone wall or dug well. I shall not go into details concerning the arrangement of courses of study, textbooks, and lesson-plans to make de- sirable connections between arithmetical learning and sports, housework, shop work, and the rest. It may be worth while, however, to explain the term self -management, since this source of genuine problems of real concern to the pupils has been overlooked by most writers. By self -management is meant the pupil's use of his time, his abilities, his knowledge, and the like. By the time he reaches grade 5, and to some extent before then, a boy should keep some account of himself, of how long it takes him to do specified tasks, of how much he gets done in a specified time at a certain sort of work and with how many errors, of how much improvement he makes month by month, of which things he can do best, and the like. Such objective, matter-of-fact, quantitative study of one's be- havior 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 224 PSYCHOLOGY OF ARITHMETIC priggishness. On the contrary, this matter-of-fact study of what one is and does may well replace a certain amount of the exhortations and admonitions concerning what one ought to do and be. All this is still truer for a girl. The demands which such an accounting of one's own activities make of arithmetic have the special value of con- necting 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 shop- worker from 10 to 14 do not lead. Children have little money, but they have time in thousands of units ! They do laot get discounts or bonuses from commercial houses, but ihey can discount their quantity of examples done for the errors made, and credit themselves with bonuses of all sorts for extra achievements. INTRINSIC INTEREST IN ARITHMETICAL LEARNING There remains the most important increase of interest in arithmetical learning — an increase in the interest directly bound to achievement and success in arithmetic itself. "Arithmetic," says David Eugene Smith, "is a game and all boys and girls are players." It should not be a mere game for them and they should not merely play, but their unpractical interest in doing it because they can do it and can see how well they do do it is one of the school's most precious assets. Any healthy means to give this interest more and better stimulus should therefore be eagerly sought and cherished. Two such means have been suggested in other connections. The first is the extension of training in checking and verify- ing work so that the pupil may work to a standard of ap- ESTTEREST IN ARITHMETIC 225 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 trust- worthy idea of just how much better or faster he can do the same tasks than he could do a month or a year ago, and of just how much harder things he can do now than then. Another means of stimulating the essential interest in quantitative thinking itself is the arrangement of the work so that real arithmetical thinking is encouraged more than mere imitation and assiduity. This means the avoidance of long series of applied problems all of one type to be solved in the same way, the avoidance of miscellaneous series and review series which are almost verbatim repetitions of past problems, and in general the avoidance of excessive repetition of any one problem-situation. Stimulation to real arithmetical thinking is weak when a whole day's problem work requires no choice of methods, or when a review simply repeats without any step of organization or progress, or when a pupil meets a situation (say the 'buy x things, at y per thing, how much pay' situation) for the five- hundredth time. Another matter worthy of attention in this connection is the unwise tendency to omit or present in diluted form some of the topics that appeal most to real intellectual interests, just because they are hard. The best illustration, perhaps, is the problem of ratio or "How many times as large (long, heavy, expensive, etc.) as x is y ?" Mastery of the 'times as' relation is hard to acquire, but it is well worth acquiring, not only because of its strong intellectual appeal, but also because of its prime importance in the applications of arithmetic to science. In the older arithmetics it was con- fused by pedantries and verbal difficulties and penalized 226 PSYCHOLOGY OF ARITHMETIC by unreal problems about fractions of men doing parts of a job in strange and devious times. Freed from these, it should be reinstated, beginning as early as grade 5 with such simple exercises as those shown below and progressing to the problems of food values, nutritive ratios, gears, speeds, and the like in grade 8. John is 4 years old. Fred is 6 years old. Mary is 8 years old. Nell is 10 years old. Alice is 12 years old. Bert is 15 years old. Who is twice as old as John ? Who is half as old as Alice ? Who is three times as old as John ? Who is one and one half times as old as Nell ? Who is two thirds as old as Fred ? etc., etc., etc. Alice is ... . times as old as John. John is .... as old as Mary. Fred is ... . times as old as John. AMce is ... . times as old as Fred. Fred is .... as old as Mary, etc., etc., etc. Finally it should be remembered that all improvements in making arithmetic worth learning and helping the pupil to learn it will in the long run add to its interest. Pupils like to learn, to achieve, to gain mastery. Success is interest- ing. 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 at- tractions. CHAPTER XIII THE CONDITIONS OF LEARNING We shall consider in this chapter the influence of time of day, size of class, and amount of time devoted to arithmetic in the school program, the hygiene of the eyes in arith- metical 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. "^ EXTERNAL CONDITIONS Computation of one or another sort has been used by several investigators as a test of efficiency at different times in the day. When freed from the effects of practice on the one hand and lack of interest due to repetition on the other, the results uniformly show an increase in speed late in the school session with a falling off in accuracy that about balances it.^ There is no wisdom in putting arithmetic early in the session because of its difficulty. Lively and sociable exercises in mental arithmetic with oral answers in fact seem to be admirably fitted for use late in the session. Except for the general principles (1) of starting the dg^y with work that will set a good standard of cheerful, efficient pro- ' Facts concerning the conditions of learning in general will be found in the author's Educational Psychology, Vol. 2, Chapter 8, or in the Educational Psychology, Briefer Course, Chapter 15. 2 See Thorndike ['00], King ['07], and Heck ['13]. 227 228 PSYCHOLOGY OF ARITHMETIC d action and (2) of getting the least interesting features of the day's work done fairly early in the day, psychology permits practical exigencies to rule the program, so far as present knowledge extends. Adequate measurements of the effect of time of day on improvement have not been made, but there is no reason to believe that any one time between 9 A.M. and 4 p.m. is appreciably more favorable to arith- metical learning than to learning geography, history, spelUng, and the like. 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 ofiicers 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. ElUott ['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, supple- mented 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 con- comitants thereof. The effect of the amount of time devoted to arithmetic in the school program has been studied extensively by Rice ['02 and '03] and Stone ['08]. Dr. Rice ['02] measured the arithmetical abiUty of some THE CONDITIONS OF LEARNING 229 6000 children in 18 different schools in 7 different cities. The results of these measurements are summarized in Table 10. This table "gives two averages for each grade as well as for each school as a whole. Thus, the school at the top shows averages of 80.0 and 83.1, and the one at the bottom, 25.3 and 31.5. The first represents the percentage of answers which were absolutely correct; the second shows what per cent of the problems were correct in principle, i.e the average that would have been received if no mechanical errors had been made." The facts of Dr. Rice's table show that there is a positive relation between the general standing of a school system in the tests and the amount of time devoted to arithmetic by its program. The relation is not close, however, being that expressed by a correlation coefficient of .36|. Within any one school system there is no relation between the standing of a particular school and the amount of time de- voted 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 em- phasis on arithmetic in work set for the study periods at home. A still more elaborate investigation of this same topic was made by Stone ['08]. I quote somewhat fully from it, since it is an instructive sample of the sort of studies that will doubtless soon be made in the case of every elementary school subject. He foimd that school systems differed nota- bly 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 . 230 PSYCHOLOGY OF ARITHMETIC TABLE 10 Averages for Individxjal Schools in Arithmetic i 1 era Yeae 7th Year 8th Year School Avekage 1 1 1 1 3 1 +j 1 1^ III 1 79.3 80.3 81.1 82.3 91.7 93.9 80.0 83.1 3.7 53 I 1 80.4 81.5 64.2 67.2 80.9 82.8 76.6 80.3 4.6 60 I 2 80.9 83.4 43.5 50.9 72.7 79.1 69.3 75.1 7.7 25 I 3 72.2 74.0 63.5 66.2 74.6 76.6 67.8 72.2 6.1 46 I 4 69.9 72.2 54.6 57.8 66.5 69.1 64.3 70.3 8.5 45 II 1 71.2 75.3 33.6 35.7 36.8 40.0 60.2 64.8 7.1 60 III 2 43.7 45.0 63.9 56.7 51.1 53.1 64.6 58.9 7.4 60 IV 1 58.9 60.4 31.2 34.1 41.6 43.5 55.1 68.4 5.6 60 IV 2 69.8 63.1 — — 22.6 22.5 53.9 58.8 8.3 — IV 3 54.9 58.1 36.2 38.6 43.5 45.0 61.5 57.6 10.6 60 IV 4 42.3 45.1 16.1 19.2 48.7 48.7 42.8 48.2 11.2 — V 1 44.1 48.7 29.2 32.5 51.1 58.3 45.9 61.3 10.5 40 VI 1 68.3 71.3 33.5 36.6 26.9 30.7 39.0 42.9 9.0 33 VI 2 46.1 49.5 19.6 24.2 30.2 40.6 36.5 43.6 16.2 30 VI 3 34.5 36.4 30.5 35.1 23.3 24.1 36.0 42.5 15.2 48 VII 1 35.2 37.7 29.1 32.6 26.1 27.2 40.5 46.9 11.7 42 VII 2 35.2 38.7 16.0 16.4 19.6 21.2 36.6 40.6 10.1 76 VII 3 27.6 33.7 8.9 10.1 11.3 11.3 25.3 31.5 19.6 45 High achievement by a system in computation went with high achievement in solving the problems, the correlation being about .50 ; and the system that scored high in addition or subtraction or multiplication or division usually showed closely similar excellence in the other three, the correlations being about .90. Of the conditions under which arithmetical learning took place, the one most elaborately studied was the amount of time devoted to arithmetic. On the basis of replies by principals of schools to certain questions, he gave each of THE CONDITIONS OF LEARNING 231 TABLE 11 Scores Made by the Sixth-Grade Ptjpils op Each op Twenty-Six School Systems System ScoBB IN Tests with Score in Tests in Peoblems Computing 23 356 1841 24 429 3513 17 444 3042 4 464 3563 26 464 2167 22 468 2311 16 469 3707 20 491 2168 18 509 3758 15 532 2779 3 533 2845 8 538 2747 6 550 3173 1 552 2935 10 601 2749 2 615 2958 21 627 2951 13 636 3049 14 661 3561 9 691 3404 7 734 3782 12 736 3410 11 759 3261 26 791 3682 19 848 4099 .5 914 3569 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 computa- tion. With home study included there is little relation to 232 PSYCHOLOGY OF ARITHMETIC the achievement of the system in solving problems, but there is a clear effect on achievement in computation. The facts as given by Stone are : — TABLE 12 COREELATION OP TiME EXPENDITURES WITH ABILITIES r Reasoning and Without I Time Expenditure -.01 Home Study | Fundamentals and i Time Expenditure 09 r Reasoning and Including I Time Expenditure 13 Home Study ) Fimdamentals and I Time Expenditure 49 These correlations, it should be borne in mind, are for school systems, not for individual pupils. It might be that, though the system which devoted the most time to arithmetic did not show corresponding superiority in the product over the system devoting only half as much time, the pupils within the system did achieve in exact proportion to the time they gave to study. Neither correlation would permit inference concerning the effect of different amounts of time spent by the same pupil. Stone considered also the printed announcements of the courses of study in arithmetic in these twenty-six systems. Nineteen judges rated these announced courses of study for excellence according to the instructions quoted below : — CONCERNING THE RATING OF COURSES OP STUDY Judges please read before scoring I. Some Factors Determining Relative Excellence. (N. B. The following enumeration is meant to be suggestive rather than complete or exclusive. And each scorer is urged to rely primarily on his own judgment.) 1. Helpfulness to the teacher in. teaching the subject matter outlined. THE CONDITIONS OF LEARNING 233 2. Social value or concreteness of sources of problems. 3. The arrangement of subject matter. 4. The provision made for adequate drill. 5. A reasonable minimum requirement with suggestions for valuable additional work. 6. The relative values of any predominating so-called methods — such as Speer, Grube, etc. 7. The place of oral or so-called mental arithmetic. 8. The merit of textbook references. II. Cautions and Directions. (Judges please follow as implicitly as possible.) 1. Include references to textbooks as parts of the Course of Study. This necessitates judging the parts of the texts referred to. 2. As far as possible become equally familiar with all courses before scoring any. 3. When you are ready to begin to score, (1) arrange in serial order according to excellence, (2) starting with the middle one score it 50, then score above and below 50 according as courses are better or poorer, indicating rela- tive differences in excellence by relative differences in scores, i.e. in so far as you find that the courses differ by about equal steps, score those better than the middle one 51, 52, etc., and those poorer 49, 48, etc., but if you find that the courses differ by unequal steps show these inequalities by omitting numbers. 4. Write ratings on the slip of paper attached to each course. The systems whose courses of study were thus rated highest did not manifest any greater achievement in Stone's tests than the rest. The thirteen with the most approved an- nouncements of courses of study were in fact a little inferior in achievement to the other thirteen, and the correlation coefficients were sUghtly negative. Stone also compared eighteen systems where there was supervision of the work by superintendents or supervisors as well as by principals with four systems where the prin- cipals and teachers had no such help. The scores in his tests were very much lower in the four latter cities. 234 PSYCHOLOGY OF ARITHMETIC THE HYGIENE OF THE EYES IN AEITHMETIC We have already noted that the task of reading and copy- ing numbers is one of the hardest that the eyes have to per- form in the elementary school, and that it should be alleviated A B 623 635 555 54:5 554: 333 646 646 686 975 872 621 -196 -689 FiQ. 26. — Type too large. by arranging much of the work so that only answers need be written by the pupil. The figures to be read and copied THE CONDITIONS OF LEARNING 235 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. Size. — Type may be too large as well as too small, though the latter is the commoner error. If it is too large, as in Fig. 26, which is a duplicate of type actually used in a form of practice pad, the eye has to make too many fixations to take in a given content. All things considered, 12-point type in grades. 3 and 4, 11 -point in grades 5 and 6, and 10-point in grades 7 and 8 seem the most desirable sizes. These are shown in Fig. 27. Too small type occurs oftenest in fractions and in the dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30 are samples from actual school practice. Samples of the desirable size are shown in Figs. 31 and 32. The technique of modern typesetting makes it very difficult and expensive to make fractions of the horizontal type (l, f , f) large enough with- out making the whole-number figures with which they are mingled too large or giving an imcouth appearance to the total. Consequently fractions somewhat smaller than are desirable may have to be used occasionally in textbooks.^ There is no valid excuse, however, for the excessively small fractions which often are made in blackboard work. 0123456789 12 3 4 5 6 7 8 9 0123456789 Fig. 27. — 12-point, 11-point, and 10-point type. ' 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 236, PSYCHOLOGY OF ARITHMETIC This is a picture of Mary's garden. s How many feet is it around the garden ? Fio. 28. — Type of measurements too small. TI NTWENTY THIETY POETS FIFTY smr SEVENTY . . • 1 10 i-IO iO 10 .10 = 80 HO flO iO +10 10 .10 .10 •10 10 HO .10 .10 60 .10 .10 10 .10 »10 .10 = 70 .10 .10 .10 FiQ. 29. — Type too small. THE CONDITIONS OF LEARNING 237 Find the area of: A. B. D. ca|t> 2fyd. 7b -H— — 6i ~ft~ — 61 -11- RACK • fROWT FiQ. 30. — Numbeis too small and badly designed. 24 Feet P ^^ 8 Ft 16 Feet Fia. 31. — Figure 28 with suitable numbers. 238 PSYCHOLOGY OF ARITHMETIC Find the area of: A. 46in- B. 63 ■n-tt. C. D. c U ■ ■g-in. ■D 55 2fyd. ■«1|f< — 6^ ^\h* 6^— ^li> BACK FRONT 7^ ■ ■ FiQ. 32. — Figure 30 with suitable numbers. Style. — The ordinary type forms often have 3 and 8 so made as to require strain to distinguish them. 5 is some- times 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 repre- sented by its essential^ features without any distracting shading or knobs or turns . Figure 34 shows some of the types in common use. There are no dernonstrably great differ- ences amongst these. In fractions' there is a notable gain from using the slant form {^3, H) for exercises in addition 1 It will be still better if the 4 is replaced by an open-top 4. THE CONDITIONS OF LEARNING 239 A. 1. 2. 3. 4. 812 378 592 429 933 181 642 476 B. 8. 9. 10. 11. 765 365 546 238 495 195 327 87 C. 15. 16. 17. 18. 005 250 200 98 725 400 102 Fig. 33. — Block type; a very desirable type except that it is somewhat too heavy. 12 3 4 5 6 7 12 3 4 5 6 7 12 3 4 5 6 7 12 3 4 5 6 7 12 3 4 5 6 7 1234567 I 234567 1234567 1234567 Fig. 34. — Common styles of printed numbers. 240 PSYCHOLOGY OF ARITHMETIC and subtraction, and for almost all mixed numbers. This appears clearly to the eye in the comparison of Fig. 35 be- low, where the same fractions all in 10-point type are dis- played 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 sub- tracted 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. 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. 19^ 19f 6^ ^ 21 >^ H 9y2 ^ 155/8 15f Z% 3| 173/8 17| %% ^ Fig. 35. — Diagonal and horizontal fractions compared. The conditions of learning 24i $1.10 2.85 3.75 6.42 1.49 2.25 7.50 $25.36 $1.10 $2.85 $3.75 $6.42 $1.49 $2.25 $7.50 $25.36 B B iof6 = lof 27 = iof6 = iof 27 = 1 of 10 = iof8 = lof 12 = lof 15 = lof 18 = iof 18 = iofl2 = J of 16 = lof 10 = iof8 = iof 12 = ^of 15 = iof8 = iof 18 = iof 18 = iof 12 = iof 16 = iof 14 = iof8 = 1 of 14 = lof 40 = iof 18 = lof 40 = ■Jof 18 = iof 40 = • iof 36 = lof 40 = iof 18 = iof 36 = lof 32 = ■lof 18 = ^of 56 = iof 32 = iof 35 = iof 56 = ^of 35 = F F f of9 = |of 20 = f of9 = fof 20 = f of 16 = fof 20 = fof 20 = fof 15 = fof 16 = fof 20 = fof 20 = fof 15 = G G 1-1-1- 2T^2 ~ 3_(_3_ 414 ~ 2f+| = 1+1 = f+i = !+! = lHf = i+i = li+i = f+f= 21+1 = f+f = l+i = f+f = li+f = Fig. 36. — Good vertical spacing. Fig. 37. — Bad vertical spacing. 242 PSYCHOLOGY OF ARITHMETIC Find, without pencil, the loss or gain. Cost RAT'S OF Profit OE Loss Cost Rate of Pbofit OR LOSB Cost Rate of Profit OR Loss 1. $3000 20% 13. $3200 12i% 25. $900 25% 2. 7300 10% 14. 4000 62|% 26. 800 12i% 3. 4500 40% 15. 2700 66f% 27. 450 20% 4. 250 30% 16. 1600 15% 28. 600 30% 5. 3600 331% 17. 7200 25% 29. 1600 25% 6. 2400 37|% 18. 8500 50% 30. 950 10% 7. 4800 12*% 19. 4200 16f% 31. 2200 20% 8. 6000 8*% 20. 150 3% 32. 2500 8% 9. 1600 6i% 21. 7500 10% 33. 10,000 12*% 10. 1800 16f% 22. 3500 20% 34. 160 12|% 11. 2000 24% 23. 1800 25% 35. 1500 20% 12. 4500 66f% 24. 4200 16f% 36. 4000 37i% Rate op Rate of Rate of Cost Profit OR Loss Cost Profit OR Loss Cost Profit OR Loss 1. $3000 20% 13. $3200 12§% 25. $900 25% 2. 7300 10% 14. 4000 62§% 26. 800 12i% 3. 4500 40% 15. 2700 66f% 27. 450 20% 4. 250 30% 16. 1600 15% 28. 600 30% 5. 3600 33i% 17. 7200 25% 29. 1600 25% 6. 2400 37i% 18. 8500 50% 30. 950 10% 7. 4800 12i% 19. 4200 16f% 31. 2200 20% 8. 6000 84% 20. 150 3% 32. 2500 8% 9. 1600 6i% 21. 7500 10% 33. 10,000 12i% 10. 1800 16! % 22. 3500 20% 34. 160 12i% 11. 2000 2i% 23. 1800 25% 35. 1500 20% 12. 4500 66f% 24. 4200 16f% 36. 4000 37i% Fioa. 38 (above) and 39 (below). — Good and bad left- right spacing. THE CONDITIONS OP LEARNING 243 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 figiu-es 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 handwriting of words is often better than it needs to be for life's purposes, the Avriting of figures is usually much worse. The figures presented in books on pen- manship are also commonly bad, showing neglect or misunder- standing of the matter on the part of leaders in penmanship. Spacing. — Spacing up and down the column is rarely too wide, but very often too narrow. The specimens shown in Figs. 36 and 37 show good practice contrasted with the common fault. Spacing from right to left is generally fairly satisfactory in books, though there is a bad tendency to adopt some one routine throughout and so to miss chances to use reductions and increases of spacing so as to help the eye and the mind in special cases. Specimens of good and bad spacing are shown in Figs. 38 and 39. In the work of the pupils, the spacing from right to left is often too narrow. This crowd- ing of letters, together with unevenness of spacing, adds notably to the task of eye and mind. The composition or make-up of the page. — Other things being equal, that arrangement of the page is best which helps a child most to keep his place on a page and to find it after having looked away to work on the paper on which he computes, or for other good reasons. A good page and a bad page in this respect are shown in Figs. 40 and 41. 244 PSYCHOLOGY OF ARITHMETIC Suppose that you are a clerk selling butter and cheese at these prices. Find the cost of each purchase. Butter Standard Creamery 24^ per pound Oak Farm SOji " " Cedar Farm Special 34(4 " " XX Unsalted 45fi " " 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 16. 16. 17. 18. 19. 20. 21. 22. 23. f lb. Oak Farm butter. If lb. Oak Farm butter. I lb. Cottage cheese. ^ lb. Old EngUsh cheese. ^■^ lb. Old EngUsh cheese. \ lb. Swiss cheese, i lb. Swiss cheese. TS 7 lb. Oak Farm butter. 1,^ lb. Oak Farm butter. Ij^-lb. Swiss cheese. 2\ lb. Cedar Farm butter. ^ lb. Unsalted butter. ^ lb. Old English cheese. 4 lb. Standard creamery. 1,3^ lb. Swiss cheese. \^ lb. Swiss cheese. ^ lb. Oak Farm butter. 1| lb. Full cream cheese. 3| lb. Full cream cheese. 1^^ lb. Cottage cheese. 2f lb. Cedar Farm butter. 6 lb. Standard creamery. \^ lb. Old Enghsh cheese. 1^^ lb. Old English cheese. ' Cheese Cottage 16fS per pound Full Cream 22(i " " Old English 32i " " Sidss 38f5 " " 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 60. i'lb 3 Full cream cheese. f lb. Full cream cheese. 1^^ lb. Swiss cheese. 3yV lb. Old English cheese. 1| lb. Standard creamery. 2y\ lb. Oak Farm butter. If lb. Cedar Farm butter. 2^ lb. Old English cheese. Y*^ lb. Unsalted butter. 23 oz. Cottage cheese. 1 lb. 7 oz. Oak Farm butter. 2 lb. 3 oz. Old English. 1 lb. 5 oz. Swiss .cheese. 1 lb. 5 oz. Old English. 1 lb. 1 oz. Swiss cheese. 9 oz. Swiss cheese. I lb. 5 oz. Oak Farm. 5 oz. Unsalted butter. II oz. Swiss cheese. 20 oz. Cottage cheese. 1 lb. 6 oz. Oak Farm. 10 oz. Unsalted butter. 2 lb. 4 oz. Old English. Fig. 40. — A page well made up to suit the action of the eye. THE CONDITIONS OF LEARNING 245 Suppose that you are a clerk selling butter and cheese at these prices. Find the cost of each purchase. Butter : — Standard Creamery, 24j^ per pound ; Oak Farm, 30f5 per pound ; Cedar Farm Special, 34f5 per pound ; XX Unsalted, 45f5 per poimd. Cheese : — Cottage, 16^ per pound; Full Cream, 22^ per pound ; Old English, ^2j. per pound ; Swiss, SSjiS per pound. 1. I lb. Oak Farm butter. 2. If lb. Oak Farm butter. 3. | lb. Cottage cheese. 4. ^ lb. Old English cheese. 5. y% lb. Old English cheese. 6. | lb. Swiss cheese. 7. J lb. Swiss cheese. 8. -^ lb. Swiss cheese. 9. -^ lb. Oak Farm butter. 10. 1,^ lb. Oak Farm butter. 11. 1^^ lb. Swiss cheese. 12. 2\ lb. Cedar Farm butter. 13. ^^ lb. Unsalted butter. 14. -f^ lb. Old Enghsh cheese. 15. 4 lb. Standard creamery. 16. 1^ lb. Swiss cheesei 17. fl lb. Swiss cheese. 18. ^^ lb. Oak Farm butter. 19. 1| lb. Full cream cheese. 20. 3| lb. Full cream cheese. 21. 1-j^ lb. Cottage cheese. 22. 2| lb. Cedar Farm butter. 23. 6 lb. Stand- ard creamery. 24. \^ lb. Old Enghsh cheese. 25. 1^;^ lb. Old Enghsh cheese. 26. 2\ lb. Cedar Farm butter. 27. | lb. Full cream cheese. 28. l^ lb. Old Enghsh. 29. f lb. Full cream cheese. 30. 1^ lb. Swiss cheese. 31. 3^^^ lb. Old English cheese. 32. 1\ lb. Standard creamery. 33. 2^^ lb. Oak Farm butter. 34. If lb. Cedar Farm butter. 35. 2^ lb. Old English cheese. 36. ^ lb. Unsalted butter. 37. 23 oz. Cottage cheese. 38. 1 lb. 7 oz. Oak Farm butter. 39. 2 lb. 3 oz. Old English. 40. 1 lb. 5 oz. Swiss cheese. 41. 1 lb. 5 oz. Old English. 42. 1 lb. 1 oz. Swiss cheese. 43. 9 oz. Swiss cheese. 44." 1 lb. 5 oz. Oak Farm. 45. 5 oz. Unsalted butter. 46. 11 oz. Swiss cheese. 47. 20 oz. Cottage cheese. 48. 1 lb. 6 oz. Oak Farm. 49. 10 oz. Unsalted butter. 50. 2 lb. 4 oz. Old English. Fig. 41. — The same matter as in Fig. 40, much less well made up. 246 PSYCHOLOGY OF ARITHMETIC Objective presentations. — Pictures, diagrams, maps, and other presentations should not tax the eye unduly, (a) by requiring too fine distinctions, or (6) by inconvenient arrangement of the data, preventing easy counting, measuring, comparison, or whatever the task is, or (c) by putting too many facts in one picture so that the eye and mind, when trying to make out any one, are confused by the others. Illustrations of bad practices in these respects are shown in Figs. 42 to 52. A few specimens of work well arranged for the eye are shown in Figs. 53 to 56. Good rules to remember are : — Other things being equal, make distinctions by the clearest method, fit material to the tendency of the eye to see an 'eyeful' at a time (roughly 1| inch by ^ inch in a book; 1| ft. by I ft. on the blackboard), and let one picture teach only one fact or relation, or such facts and relations as do not interfere in perception. The general conditions of seating, illumination, pa;per, and the like are even more important when the eyes are used with numbers than when they are used with words. •^XJ ' ' /''- mM S^i ^^ '•^s ^rJSSK etp0ST:^;-iME> TpoSr^ls: jLL ^^^^M ^'^^^^ fSB^^ ^^^^^%i. m .'^^^S ^^^ ^^^^'' ^^^^/W ^ ^^fe^i^^ ^^^1 I^Sfll^p^ -ff^^^^ ^^piij3&t^^^^^a^^^^^ ^^HmI B^^BBBwss. ^^B ^^^^^^'~ £& f^^r-^^^^^HB P^^^^^H ^p HH T^i. ^r^^SS^S^^SSS^*!9fFMf. ^m ^^ v^ %!^^^':- lfc'^»sa.-**nE **pArf^!s3fe^^^'^^^^ ^s^w ^^fsT Pio. 42. — Try to count the rungs on the ladder, or the shocks in the wagon. THE CONDITIONS OF LEARNING 247 FlQ. 43. — How many oars do you see ? How many birds ? How many fish 7 Fig. 44. — Count the birds in each of the three flocks of birds. 248 PSYCHOLOGY OF ARITHMETIC .|ilil4l.|.|klkli Fig. 45. — Note the lack of clear division of the hundreds. Consider the difficulty of counting one of these columns of dots. THE CONDITIONS OF LEARNING 249 • • • ^ M M M M M M ^ M M U * M ^1 M M ^ M M ■ Fig. 46. — What do you suppose these pictures are intended to show? ^^ -^ / TWENTY -S/X >, I I i , .-V . . ! C" TiVEA/Tr "^ • ! i > A... __ -v..-.. ! ....A ■ } r/nsT TEN r ^ccoaid ten t s/x \ . Mf~ rw£:NTr-s/x "3j Fig. 47. — Would a beginner know that after THIRTEEN he was to switch around and begin at the other end ? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2 ? Does this picture illustrate or obfuscate ? 250 PSYCHOLOGY OF ARITHMETIC I- MINUEND. WTuT / 1 1 1 IT'ElNl I \ \ \ I 1 I r eInI i 1 1 n.-MINUEND (Rearranged), m SUBTRAHEND, /,„ { I c^ E IN 1 II ill III 1 DIFFERENCE ( 1 1 1 IT E N,J „„ I ■ i i i Fig. 48. — How long did it take you to find out what these pictures mean ? ****** ?K*** ********** ***** ***** ^^^^^ ***** **** **** ***» **** **** ******* ^^^^m^ ^^^^^^ *** *** *** ^5K* *** *** *** Fig. 49. — Count the figures in the first row, using your eyes alone ; have some one make lines of 10, 11, 12, 13, and more repetitions of this figure spaced closely as here. Count 20 or 30 such lines, using the eye unaided by fingers, pencil, etc. THE CONDITIONS OF LEARNING 251 These oblongs show what numbers? FlO.i^SO. — Can you answer the question without measuring ? Could a child of seven or eight ? — T" ■^ ~~ ~| 1 y-' 1 ^T^ — — -i- — — 4 ... — T -J. — 1 — +- ■ — i- ^-h :z — — IT ^ n: — "^ ~" b: — _.,-., 1 , — " ■i- : i ^ ^ 1 r ! 1 I — ]._ . — ... .. .. •i- .. ..J ... ._- ..-i. i 1 1 , ' — r- TT" t t" ! i +- 1 ^ J i i i 1 1 J 1 — FlO. 51. — What are these drawings intended to show? Why do they show the facts only obscurely and dubiously ? 2. FlO. 62. What are these drawings intended to show ? What simple change would make them show the facts much more clearly ? 252 PSYCHOLOGY OF ARITHMETIC • •• • • • • • • • • '• • !• • • • i* ** • • !••• • • ««» , • • • • •• • • • e * i • •• • •• • • • • • • • • • • •• • • •• • • • ••• M •J • • • e • • • • •II • • • • • • • • • '• III* • • • • 1* *' • 1* * • • •• • •• • ' - • • • • • • • • • • • -.';.- ••• • •• • • • Fio. 53. — Arranged inconvenient "eye-fulU." BE Tell which bar has — 1. About 5 percent of its length black. 2. About 10 percent of its length black. 3. About 25 percent of its length black. 4. About 75 percent of its length black. 5. About 90 percent of its length black. , 6. About 95 percent of its length black. Fig. 54. — Clear, simple, and easy of bpmparison. THE CONDITIONS OF LEARNING 253 Halves Thirds Fourths Sixths FiQ. 55. — Clear, simple, and well spaced. Eighths 1-i 2-8 1-A 2 12 Fia. 56. - -Well arranged, though a little wider spacing between the squares would make it even better. THE USE OF CONCRETE OBJECTS IN ARITHMETIC We mean by concrete objects actual things, events, and relations presented to sense, in contrast to words and num- bers 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 six- teen ounce weights are such concrete events. A pint beside a quart, an inch beside a foot, an apple shown cut in halves display such concrete relations to a pupil who is attentive to the issue. Concrete presentations are obviously useful in arithmetic to teach meanings under the general law that a word or number or sign or symbol acquires meaning by being con- nected with actual things, events, qualities, and relations. 254 PSYCHOLOGY OP ARITHMETIC We have also noted their usefubiess 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| yd. of ribbon?" by using lOX^, draws a line 3| yd. long and divides it into 5-inch lengths. Concrete experiences are useful whenever the meaning of a number, like 9 or | or .004, or of an operation, like multiply- ing 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 nexi facts, relations, and pro- cedures are to be taught. How much concrete material shall be presented will de- pend 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 f , f , f , I) h f ! I) ^nd 4 than in the case of a dull child or one who only knows f and f . 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, spUnts, beans, and the like. If 1|-|-| = 2 is taught early in grade 3, there will be more gain from the use of 1| inches and ^ inch on the foot rule than if the same relations were taught in connection with the general addition of Uke fractions late in grade 4. Sometimes the under- standing can be had either by connecting the idea with the reality directly, or by connecting the two indirectly via some other idea. The amount of concrete material to be THE CONDITIONS OP LEARNING 255 used will depend on its relative advantage per unit of time spent. Thus it might be more economical to connect -y^, ^, and ii with real meanings indirectly by caUing up the resemblance to the f , |, |, f , |, f , |, f, and | already studied, than by showing -^ of an apple, ts oi & yard, xi of a foot, and the like. In general the economical course is to test the under- standing of the matter from time to time, using more con- crete 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 con- crete experiences to get a meaning which they could have got themselves by pure thought. We should also remember that the new idea, say of the meaning of decimal fractions, will be improved and clarified by using it (see page 183 f.), so that the attainment of a 'perfect conception of decimal frac- tions before doing anything with them is unnecessary and probably very wasteful. A few illustrations may make these principles more in- structive. (a) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000, need more concrete aids than are commonly given. Guessing contests about the value in dollars of the school building and other buildings, the area of the schoolroom floor and other surfaces in square inches, the number of minutes in a week, and year, and the like, together with proper computations and measurements, are very useful to reenf orce the concrete presentations and supply genuine prob- lems in multiplication and subtraction with large numbers. (b) Numbers very much smaller than one, such as ^, -it, .04, and .002, also need some concrete aids. A diagram Uke that of Fig. 57 is useful. 256 PSYCHOLOGY OF ARITHMETIC (c) Majority and plurality should be understood by every citizen. They can be understood without concrete aid, but ' 1 A B c • ■ D Fig. 57. — Concrete aid to understanding fractions with large denominators. A= TrSnr sq. ft. ; B= -dn sq. ft. ; C= «^ sq. ft. ; D= A sq. ft. an actual vote is well worth while for the gain in vividness and surety. (d) Insurance against loss by fire can be taught by ex- planation and analogy alone, but it will be economical to have some actual insuring and payment of premiums and a genuine loss which is reimbursed. THE CONDITIONS OF LEARNING 257 (e) Four play banks in the corners of the room, receiving deposits, cashing checks, and later discounting notes will give good educational value for the time spent. (/) Trade discount, on the contrary, hardly requires more concrete illustration than is found in the very problems to which it is applied. {g) The process of finding the number of square units in a rectangle by multiplying with the appropriate numbers representing length and width is probably rather hindered than helped by the ordinary objective presentation as an introduction. The usual form of objective introduction is as follows : — Fia. 58. How long is this rectangle? How large is each square? How many square inches are there in the top row ? How many rows are 258 PSYCHOLOGY OF ARITHMETIC there? How many square inches are there in the whole rec- tangle ? Since there are three rows each containing 4 square inches, we have 3X4 square inches =12 square inches. Draw a rectangle 7 inches long and 2 inches wide. If you divide it into inch squares how many rows will there be? How many inch squares will there be in each row? How many square inches are there in the rectangle? _L Fig. 59. It is better actually to hide the individual square units as in Fig. 59. There are four reasons : (1) The concrete rows and columns rather distract attention from the essential thing to be learned. This is not that "x rows one square wide, y squares in a row will make xy squares in all," but that "by using proper units and the proper operation the area of any rectangle can be found from its length and width." (2) Children have little difficulty in learning to THE CONDITIONS OF LEARNING 259 multiply rather than add, subtract, or divide when computing area. (3) The habit so formed holds good for areas like If by 4|, with fractional dimensions, in which any effort to count up the areas of rows is very troublesome and con- fusing. (4) The notion that a square inch is an area 1 ' by 1 ' rather than ^ by 2' or f in. by 3 in. or 1| in. by f in. is likely to be formed too emphatically if much time is spent upon the sort of concrete presentation shown above. It is then better to use concrete counting of rows of small areas as a means of verification after the procedure is learned, than as a means of deriving it. There has been, especially in Germany, much argument concerning what sort of number-pictures (that is, arrange- ment of dots, hues, or the like, as shown in Fig. 60) is best for use in connection with the number names in the early years of the teaching of arithmetic. Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and others have measured the accuracy of children in estimating the number of dots. in arrangements of one or more of these different types. ^ Many writers interpret a difference in favor of estimating, say, the square arrange- ments 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 1 For an account in English of their main findings see Howell ['14], pp. 149-251. O O O o o o o o o b s o .a o o o .a o o o o o O O DO O o o o O O O O o o O o o o o a o o 1 > O M a 260 2 THE CONDITIONS OF LEARNING 261 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 ex- posures. 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. Such drills are probably useful if undertaken with zeal, and if kept as supplementary to more realistic objective work with play money, children marching, material to be distributed, garden-plot lengths to be measured, and the like, and if so administered that the pupils soon get the generalized abstract meaning of the numbers freed from dependence on an inner picture of any sort. This freedom is so important that it may make the use of many types of number-pictures advisable rather than the use of the one which in and of itself is best. As Meumann says: "Perceptual reckoning can be over- done. It had its chief significance for the surety and clear- ness 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 de- velop from these elementary ones, it necessarily works as a retarding force and holds back the natural development of arithmetic. This moves on to work with abstract number and with mechanical association and reproduction." ['07, Vol. 2, p. 357.] Such drills are commonly overdone by those who make 262 PSYCHOLOGY OF ARITHMETIC use of them, being given too often, and continued after their instructiveness has waned, and used instead of more signif- icant, interesting, and varied work in counting and estimating and measuring real things. Consequently, there is now rather a prejudice against them in oiu* better schools. They should probably be reinstated but to a moderate and ju- dicious use. ORAL, ■ MENTAL, AND WRITTEN ARITHMETIC There has been much dispute over the relative merits of oral and written work in arithmetic — a question which is much confused by the different meanings of 'oral' and 'written.' Oral has meant (1) work where the situations are presented orally and the pupil's final responses are given orally, or (2) work where the situations are presented orally and the pupils' final responses are written or partly written and partly oral, or (3) work where the situations are presented in writing or print and the final responses are oral. Written has meant (1) work where the situations are presented in writing or print and the final responses are made in writing, or (2) work where also many of the intermediate responses are written, or (3) work where the situations are presented orally but the final responses and a large percentage of the intermediate computational responses are written. There are other meanings than these. It is better to drop these very ambiguous terms and ask clearly what are the merits and demerits, in the case of any specified arithmetical work, of auditory and of visual pres- entation of the situations, and of sajdng and of writing each specified step in the response. The disputes over mental versiis written arithmetic are also confused by ambiguities in the use of 'mental.' Mental has been used to mean "done without pencil and paper" THE CONDITIONS OF LEARNING 263 and also "done with few overt responses, either written or spoken, between the setting of the task and the announce- ment of the answer." In neither case is the word mental specially appropriate as a description of the total fact. As before, we should ask clearly, "What are the merits and demerits of making certain specified intermediate responses in inner speech or imaged sounds or visual images or imageless thought — that is, without actual writing or overt speech?" It may be said at the outset that oral, written, and inner presentations of initial situations, oral, written, and inner announcements of final responses, and oral, written, and inner management of intermediate processes have varying degrees of merit according to the particular arithmetical exercise, pupil, and context. Devotion to oralness or mental- ness as such is simply fanatical. Various combinations, such as the written presentation of the situation with inner management of the intermediate responses and oral an- nouncement of the final response have their special merits for particular cases. These merits the reader can evaluate for himself for any given sort of work for a given class by considering : (1) The amount of practice received by the class per hour spent; (2) the ease of correction of the work ; (3) the ease of under- standing the tasks ; (4) the prevention of cheating ; (5) the cheerfulness and sociabiUty of the work; (6) the freedom from eyestrain, and other less important desiderata. It should be noted that the stock schemes A, B, C, and D below are only a few of the many that are possible and that schemes E, F, G, and H have special merits. 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 hfe itself 264 PSYCHOLOGY OF ARITHMETIC Presentation of Initial Situation Management of Intebuediate Processes A. Printed or written Written B. " " Inner C. Oral (by teacher) Written D.. " " Inner Announcement of Final Response Written Oral by one pupil, inner by the rest Written E. As in A or C Oral by one pupil, inner by the rest A mixture, the pupil As in A or B or H writing what he needs F. The real situation As in E itself, in part at least G. Both read by the As in E pupil and put orally by the teacher H. As in A or C or G As in E As in A or B or H As in A or B or H ' Written by all pu- pils, announced or- ally by one pupil 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 pro- duction, reenforcement of training already given, and prepara- tion for training to be given. There is nothing intellectually criminal about using a pencil as well as inner thought ; on the other hand there is no magical value in writing out for the teacher's inspection figures that the pupil does not need in order to attain, preserve, verify, or correct his result, THE CONDITIONS OF LEARNING 265 The common practice of having the final responses of all easy tasks given orally has no sure justification. On the contrary, the great advantage of having all pupils really do the work should be secured in the easy work more than anywhere else. If the time cost of copying the figures is eUminated 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 eye- strain 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, sup- posing them to give inner answers to every exercise done by the class as a whole ; it makes sure that the dull pupils who would rarely get an inner answer at the rate demanded by the oral exercise, do as much as they are able to do. Two arguments often made for the oral statement of problems by the teacher are that problems so put are better xmderstood, especially in the grades up through the fifth, and that the problems are more hkely to be genuine and related to the Ufe the pupils know. When these state- ments are true, the first is a stiU better argument for having the pupils read the problems aided by the teacher's oral state- ment of them. For the difficulty is largely that the pupils cannot read well enough; and it is better to help them to sm-mount the difficulty rather than simply evade it. The second is not an argument for oralness versus writtenness, but for good problems versus bad ; the teacher who makes up such good problems should, in fact, take special care to write them down for later use, which may be by voice or by the blackboard or by printed sheet, as is best. CHAPTER XrV THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE Dewey, and others following him, have emphasized the desirability of having pupils do their work as active seekers, conscious of problems whose solution satisfies some real need of their own natures. Other things being equal, it is unwise, they argue, for pupils to be led along blindfold as it were by the teacher and textbook, not knowing where they are going or why they are going there. They ought rather to have some living purpose, and be zealous for its attain- ment. 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 fol- lowers, and that a rather detailed treatment of it should be given here. ILLUSTRATIVE CASES Consider first some cases where time spent in making pupils understand the end to be attained before attacking the task by which it is attained, or care about attaining the end (well or iU understood) is well spent, 266 CONDITIONS OF LEARNING: PROBLEM ATTITUDE 267 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 coimting or measuring, as in 'hidden' addition and measurement by inference. For example, the teacher has three pencils counted and put under a book; has two more counted and put under the book; and asks, "How many pencils are there under the book?" Answers, when obtained, are verified or refuted by actual counting and measuring. The time here is well spent because the children can do the necessary thinking if the tasks are well chosen ; because they are thereby prevented from beginning their study of addition by the bad habit of pseudo-adding by looking at the two groups of objects and counting their number instead of real adding, that is, thinking of the two numbers and infer- ring their sum ; and further, because facing the problem of adding as a real problem is in the end more economical for learning arithmetic and for intellectual training in general than being enticed into adding by objective or other processes which conceal the difficulty while helping the pupil to master it. The manipulation of short multiplication may be intro- duced 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 multi- plication, 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 num- bers is beyond all save the most gifted and probably beyond them.) Correct solution by multiplication by some child 268 PSYCHOLOGY OP ARITHMETIC who happens to have learned it elsewhere should be ac- cepted. Let the main proof of the trustworthiness of the manipulation be by measurement and by addition. Proof by the stock arguments from the place-values of numbers may also be used. If no child hits on the manipulation in question, the problem of finding the length without adding may be set. If they still fail, the problem may be made easier by being put as "4 times 22 gives the answer. Write down what you think 4 times 22 will be." Other reductions of the difficulty of the problem may be made, or the teacher may give the answer without very great harm being done. The important requirement is that the pupils should be aware of the problem and treat the manipulation as a solu- tion 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 situa- tion used here can probably be devised. The time spent in this way is well spent (1) because all but the very duU 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) be- cause there is no way of beginning training in short multi- plication that is much better. In the same fashion multiplication by two-place numbers may be introduced by confronting pupils with the problem of the number of sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square inches in 35 square feet, or the number of days in 32 years, or whatever similar problem can be brought up so as to be felt as a problem. Suppose that it is the 35 square feet. Solutions by (5X144) -I- (30X144), however arranged, or by (10X144)-}- (10X144)-!- (10X144)-!- (5X144), or by 3500-1- (35X40) 4- CONDITIONS OF LEARNING: PROBLEM ATTITUDE 269 (35X4), or by 7 X (5X144), 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 objec- tive work, or by a mixture of short multiplication and ad- dition, 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 53X9 or 84X7 may be verified by the reverse short multiplica- tion. The deductive proof of the correctness of the manipu- lation may be given in whole or in part in connection with exercises like 10X2= 30X14 = 10X3= 3X44 = 10X4= 30X44 = 10X14= 3X144 = 10X44= 20X144 = 10X144= ■ 40X144 = 20X2= 30X144 = 20X3= 5X144 = 30X3= 35 = 30+ 30X4= 30X144 added to 5X144 = 3X14 = Certain wrong answers may be shown to be wrong in many ways; e.g., 432,720 is too big, for 35 times a thousand square inches is only 35,000; 1152 is too small, for 35 times a hundred square inches would be 3500, or more than 1152. The time spent in realizing the problem here is fairly well spent because (1) any successful original manipulation in 270 PSYCHOLOGY OF ARITHMETIC this case represents an excellent exercise of thought, be- cause (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 reaUze the problem in a very few minutes. It may, however, be still better to give the pupils the right method just as soon as the problem is realized, without having them spend more time in trying to solve it. Thus : — 1 square foot has 144 square inches. How many square inches are there in 35 square feet (marked out in chalk on the floor as a piece 10 ft.X3 ft. plus a piece 5 ft.Xl ft.) ? 1 yard = 36 inches. How many inches long is this wall (found by measure to be 13 yards) ? Here is a quick way to find the answers : — 144 720 432 5040 sq. inches in 35 sq. ft. 36 13 108 36_ 468 inches in 13 yd. Consider now the following introduction to dividing by a decimal : — Dividing by a Decimal 1, How many minutes will it 16.9 take a motorcycle to go .7B\12.675 12.675 miles at the rate 75 of .75 mi. per minute ? ^17 450 675 675 CONDITIONS OF LEARNING: PROBLEM ATTITUDE 271 2. Check by multiplying 16.9 by .75. 3. How do you know that the quotient cannot be as Uttle as 1.69 ? 4. How do you know that the quotient cannot be as large as 169 ? 5. Find the quotient for 3.75-5-1.5. 6. Check your result by multiplying the quotient by the divisor. 7. How do you know that the quotient cannot be .25 or 25 ? 8. Look at this problem. .25|7.5 How do you know that 3.0 is wrong for the quotient? How do you know that 300 is wrong for the quotient? State which quotient is right for each of these : — ■02 1 or .2 1 or 2.1 or 21 or 210 .021 or .21 or 21 or 210 9. 1.813.78 10. 1.8|37:8 .03 or .3 or 3 or 30 or 300 1.813.78 .03 or .3 or 3 or 30 or 300 1.25|37.5 .05 or .5 or 5 or 50 or 500 11. 1.25137.5 12. 12.5137.5 .05 or .5 or 5 or 50 or 500 13. 1.25|6:25 14. 12.5|6:25 15. Is this rule true ? If it is true, learn it. In a correct result, the number of decimal places in the divisor and quotient together equals the number of decimal places in the dividend. These and similar exercises excite the problem attitude in children who have a general interest in getting right answers. Such a series carefully arranged is a desirable introduction to a statement of the rule for placing the decimal point in division with decimals. For it attracts attention to the general principle (divisor X quotient should equal dividend), which is more important than the rule for convenient loca- tion 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 re- member this method by inspection long after he has for- gotten the jBxed rule. It is well for the pupil to be introduced to many arith- 272 PSYCHOLOGY OF ARITHMETIC 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 : — GRADE 3 To be Done at Home Look at a watch. Has it any hands besides the hour hand and the minute hand? Find out all that you can about how a watch tells seconds, how long a second is, and how many seconds make a minute. GRADE 5 Measuring Rainfall Rainfall per Week 1. In which weeks was the rainfall 1 (cu. in. per sq. in. of area) or more ? June 1-7 1.056 2. Which week of August had the largest rainfall for that month? 3. Which was the driest week of the sunmier? (Driest means with the least rainfall.) July 6-12 .782 4. Which week was the next to the driest? 5. In which weeks was the rainfall between .800 and 1.000? Aug. 3- 9 .512 6. Look down the table and estimate whether the average rainfall for one week was about .5, or about .6, or about .7, or about .8, or about .9. 1-7 1.056 8-14 1.103 15-21 1.040 22-28 .960 29-July5 .915 6-12 .782 13-19 .790 20-26 .670 27-Aug. 2 .503 3-9 .512 10-16 .240 17-23 .215 24-30 .811 CONDITIONS OF LEARNING: PROBLEM ATTITUDE 273 Dairy Records Record of Star Elsie Jan. Feb. Mar. Apr. May June Poimds of Milk 1742 1690 1574 1226 1202 1251 Butter-Fat per Pound of Milk .0461 .0485 .0504 .0490 .0466 .0481 Read this record of the milk given by the cow Star Elsie. The first column tells the number of pounds of milk given by Star Elsie each month. The second column tells what fraction of a pound of butter-fat each pound of milk contained. 1. Read the first line, saying, " In January this cow gave 1742 pounds of milk. There were 461 ten thousandths of a pound of butter-fat per pound of milk." Read the other hnes in the same way. 2. How many pounds of butter-fat did the cow produce in Jan. ? 3. In Feb.? 4. In Mar.? 5. In Apr.? 6. In May? 7. In June? GRADE 5 OR LATER Using Recipes to Make Larger or Smaller Quantities I. State how much you would use of each material in the following recipes : (a) To make double the quantity. (6) To make half the quantity, (c) To make 1| times the quantity. You may use pencil and paper when you cannot find the right amount mentally. 1. Peanut Pentjche 1 tablespoon butter 2 cups brown sugar J cup milk or cream f cup chopped peanuts I teaspoon salt 2. Molasses Candy i cup butter 2 cups sugar 1 cup molasses IJ cups boiling water 3. Raisin Opera Caramels 2 cups light brown sugar 5 cup thin cream 5 cup raisins Walnut Molasses Squares 2 tablespoons butter 1 cup molasses i cup sugar § cup walnut meats 274 PSYCHOLOGY OF ARITHMETIC 6. Reception Rolls 6. Graham Raised Loaf 1 cup scalded milk 2 cups milk 1| tablespoons sugar 6 tablespoons molasses 1 teaspoon salt 1| teaspoons salt i cup lard | yeast cake 1 yeast cake J cup lukewarm water i cup lukewarm water 2 cups sifted Graham flour White of 1 egg j cup Graham bran 31 cups flour J cup flour (to knead) II. How much would you use of each material in the following recipes: (a) To make f as large a quantity? (6) To make 1| times as much ? (c) To make 2§ times as much ? 1. English Dttmplings 2. White Mountain Angel Cake I pound beef suet li cups egg whites IJ cups flour Ij cups sugar 3 teaspoons baking' powder 1 teaspoon cream of tartar 1 teaspoon salt 1 cup bread flour J teaspoon pepper i teaspoon salt 1 teaspoon minced parsley 1 teaspoon vanilla i cup cold water 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. Play- ing 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 situa- tions suggesting problems which may bring a living purpose into arithmetical learning in grade 2. These are all avail- able under ordinary conditions of class instruction. A sample of such problems for a higher grade (6) is shown below. Estimating Areas The children in the geography class had a contest in estimating the areas of different surfaces. Each child wrote his estimates CONDITIONS OF LEARNING: PROBLEM ATTITUDE 275 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. Write your estimates for A, B, C, D, and E. Then study the next 6 pages and learn how to find the exact areas. (The next 6 pages comprise training in the mensuration of parallelograms and triangles.) In some cases the affairs of individual pupils include prob- lems 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, keep- ing the accoimts of a Girls' Canning Club, or the like. It requires much time and very great skill to direct the work of thirty or more pupils each busy with a special type of his own, so as to make the work instructive for each, but in some cases the expense of time and skill is justified. GENERAL PRINCIPLES In general what should be meant when one says that it is desirable to have pupils in the problem-attitude when they are studying arithmetic is substantially as follows : — First. — Information that comes as an answer to questions 276 PSYCHOLOGY OF ARITHMETIC is better attended to, understood, and remembered than information that just comes. Second. — Similarly, movements that come as a step toward achieving an end that the pupil has in view are better connected with their appropriate situations, and such connections are longer retained, than is the case with move- ments that just happen. Third. — The more the pupil is set toward getting the question answered or getting the end achieved, the greater is the satisfyingness attached to the bonds of knowledge or skill which mean progress thereto. Fourth. — It is bad policy to rely exclusively on the purely intellectualistic problems of "How can I do this?" "How can I get the right answer ? " " What is the reason for this ? ' ' "Is there a better way to do that ? " and the like. It is bad poUcy 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 intellectuaUstic 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 fiu-- thered by arithmetic then and there. It is good policy to spend time in establishing certain mental sets — stimulat- ing, or even creating, certain needs — setting up problems themselves — when the time so spent brings a sufficient improvement in the quality and quantity of the pupils' interest in arithmetical learning. Fifth. — It would be still worse policy to rely exclusively CONDITIONS OF LEARNING: PROBLEM ATTITUDE 277 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 intellectuaUstic needs and problems should be considered along with all others, and given whatever weight their educational value deserves. DIFFICULTY AND SUCCESS AS STIMULI There are certain misconceptions of the doctrine of the problem-attitude. The most noteworthy is that difficulty — temporary failure — an inadequacy of already existing bonds — is the essential and necessary stimulus to thinking and learning. Dewey himself does not, as I understand him, mean this, but he has been interpreted as meaning it by some of his followers.^ Difficulty — temporary failure, inadequacy of existing bonds — on the contrary does nothing whatsoever in and of itself ; and what is done by the annoying lack of success which sometimes accompanies difficulty sometimes hinders thinking and learning. Mere difficulty, mere failure, mere inadequacy of existing bonds, does nothing. It is hard for me to add three eight- place numbers at a glance ; I have failed to find as effective illustrations for pages 276 to 277 as I wished ; my existing sensori-motor connections are inadequate to playing a golf course in 65. But these events and conditions have done nothing to stimulate me in respect to the behavior in ques- tion. In the first of the three there is no annoying lack and no dynamic influfence at all ; in the second there was to some 1 In his How We Think. 278 PSYCHOLOGY OF ARITHMETIC degree an annoying lack — a slight irritation at not getting just what I wanted, — and this might have impelled me to further thinking (though it did not, and getting one tiptop illustration would as a rule stimulate me to hunt for others more than failing to get such). In the third case the lack of the 65 does not annoy me or have any noteworthy dynamic effect. The lack of 90 instead of 95-100 is annoying and is at times a stimulus to further learning, though not nearly so strong a stimulus as the attainment of the 90 would be ! At other times this annoying lack is distinctly inhibitory — a stimulus to ceasing to learn. In the intellectual life the inhibitory effect seems far the commoner of the two. Not getting answers seems as a rule to make us stop trying to get them. The annoying lack of success with a theoretical problem most often makes us desert it for problems to whose solution the existing bonds promise to be more adequate. ■ The real issue in all this concerns the relative strength, in the pupil's intellectual life, of the "negative reaction" of behavior in general. An animal whose life processes are interfered with so that an annoying state of affairs is set up, changes his behavior, making one after another responses as his instincts and learned tendencies prescribe, until the annoying state of affairs is terminated, or the animal dies, or suffers the annoyance as less than the alternatives which his responses have produced. When the annoying state of affairs is characterized by the failure of things as they are to minister to a craving — as in cases of hunger, loneliness, sex-pursuit, and the like, — we have stimulus to action by an annoying lack or need, with relief from action by the satisfaction of the need. Such is in some measure true of man's intellectual life. In recalling a forgotten name, in solving certain puzzles, or in simplifying an algebraic complex, there is an annoying CONDITIONS OF LEARNING: PROBLEM ATTITUDE 279 lack of the name, solution, or factor, a trial of one after another response, until the annoyance is relieved by success or made less potent by fatigue or distraction. Even here the difficulty does not do anything — but only the annoying interference with our intellectual peace by the problem. Further, although for the particular problem, the annoying lack stimulates, and the successful attainment stops thinldng, the later and more important general effect on thinking is the reverse. Successful attainment stops our thinking on that problem but makes us more predisposed later to thinking in general. Overt negative reaction, however, plays a relatively small part in man's intellectual life. Filling intellectual voids or relieving intellectual strains in this way is much less frequent than being stimulated positively by things seen, words read, and past connections acting under modified circumstances. The notion of thinking as conoing 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 horn's of smooth straightforward use and adaptation of existing connections, and they might as truly be called hindrances to thought — barriers which past successes help the thinker to surmount. Problems themselves come more often as cherished issues which new facts reveal, and whose contemplation the thinker enjoys, than as strains or lacks or 'problems which I need to solve.' It is just as true that the thinker gets many of his problems as results from, or bonuses along with, his information, as that he gets much of his information as results of his efforts to solve problems. As between difficulty and success, success is in the long run more productive of thinking. Necessity is not the mother of invention. Knowledge of previous inventions is 280 PSYCHOLOGY OF ARITHMETIC the mother; original ability is the father. The solutions of previous problems are more potent in producing both new problems and their solutions than is the mere awareness of problems and desire to have them solved. In the case of arithmetic, learning to cancel instead of getting the product of the dividends and the product of the divisors and dividing the former by the latter, is a clear case of very valuable learning, with ease emphasized rather than difficulty, with the adequacy of existing bonds (when slightly redirected) as the prime feature of the process rather than their inadequacy, and with no sense of failure or lack or conflict. It would be absurd to spend time in arousing in the pupil, before beginning cancellation, a sense of a difficulty — viz., that the full multiplying and dividing takes longer than one would like. A pupil in grade 4 or 5 might well contemplate that difficulty for years to no ad- vantage. 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 multipljdng is not inadequate ; it is admirable in every respect. The issue of its inadequacy does not truly appear until the new method is found. It is the best way until the better way is mastered. In the same way it is unwise to spend time in making pupils aware of the annoying lacks to be suppUed 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 CONDITIONS OF LEARNING: PROBLEM ATTITUDE 281 square erected on the linear wait as base. The annoying lack will be unproductive, while the learning takes place readily as a modification of existing habits, and is suf- ficiently appreciated as soon as it does take place. The multipUcation tables come when instead of merely counting by 7s from up saying "7, 14, 21," etc., the pupil counts by 7s from up saying "Two sevens make 14, three sevens make 21, four sevens make 28," etc. The division tables come as easy selections from the known multiplications ; the casting out of nines comes as an easy device. The computation of the area of a rectangle is best facilitated, not by awareness of the lack of a process for doing it, but by awareness of the success of the process as verified objectively. In all these cases, too, the pupil would be misled if we aroused first a sense of the inadequacy of counting, adding, and objective division, an awareness of the difficulties which the multiplication and division tables and nines device and area theorem relieve. The displaced processes are admirable and no imnecessary fault should be found with them, and they are not inadequate until the shorter ways have been learned. FALSE INFEEENCES One false inference about the problem-attitude is that the p upil should always unde rstand_thej;i m or issue b efore. be- ginning tojorm thebpnds which give the method or profifiss thatjproyide.§. the MutiQiL. On the contrary, he will of ten- get the process more easily and value it more highly if he is taught what it is for gradually while he is learning it. The system of decimal notation, for example, may better be taken first as a mere fact, just as we teach a child to talk without trying first to have him understand the value of ver- bal intercourse, or to keep clean without trying first to have him understand the bacteriological consequences of filth. - 282 PSYCHOLOGY OF ARITHMETIC A second infereiice — that the pupil should always b e taught to care about an issue and crave a proc ess for man ag- ing it bef ore beginning to learn the proceis^^is equalTy lalse. 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 '". 16661 X= divide by 6," ".333iX = divide by 3," "mul- jtiply by .875 = divide the number by 8 and subtract the .quotient from the number." A third unwisg te ndency is to degrade the mere ^ivin p ; o f information — to belittle the value of facts acquired in an y other way Than in the course of deliberate effort by the pupi l to relieve a problem or conflict or difficulty. As a protest against merely verbal knowledge, and merely memoriter knowledge, and neglect of the active, questioning search for knowledge, this tendency to belittle mere facts has been healthy, but as a general doctrine it is itself equally one- sided. Mere facts not got by the pupil's thinking are often of enormous value. They may stimulate to active thinking just as truly as that may stimulate to the reception of facts. In arithmetic, for example, the names of the numbers, the use of the fractional form to signify that the upper number is divided by the lower number, the early use of the decimal point in U. S. money to distinguish dollars from cents, and the meanings of "each," "whole," "part," "together," "in all," "sum," "difference," "product," "quotient," and the like are self -justifying facts. A fourth false inference is that whatever tpanhing jmglcea l^^JP^Eili§£§-5'.SH5?ii5S ^^^ 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 CONDITIONS OF LEARNING: PROBLEM ATTITUDE 283 appropriate for that pupil for that question. Merely to think may be of little value. To rely much on formal dis- cipline is just as pernicious here as elsewhere. The tendency to emphasize the methods of learning arithmetic at the ex- pense 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 discipUnary rather than content value has led in the study of languages and grammar, or in the old puzzle prob- lems of arithmetic. The last false inference that I shall discuss here is the inference that most of the proble ms b y which arit hmf.ticaU learnin g is stimulated had better be external to arithmetic jtssli — problems about Noah's Ark or Easter Flowers or the Merry Go Round or A Trip down the Rhine. Outside interests should be kept in mind, as has been abimdantly 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, frac- tions, and decimals, and to work out quantitative relations. In some measure, learning arithmetic is like learning to typewrite. The learner of the latter has little desire to present attractive-looking excuses for being late, or to save expense for paper. He has no desire to hoard copies of such and such literary gems. He may gain zeal from the fact that a school party is to be given and invitations are to be sent out, but the problem "To typewrite better" is after all his main problem. Learning arithmetic is in some measure a game whose moves are motivated by the general 284 PSYCHOLOGY OF ARITHMETIC 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 con- cerning getting rid of the ball, or having the man at first- base possess it, or putting out an. opponent against whom he has a grudge, but because that skill is required by the game as a whole, so the pupil, in some measure, learns the technique of arithmetic, not because of particular concrete problems whose solutions it furnishes, but because that technique is required by the game of arithmetic — a game that has intrinsic worth and many general recommendations. CHAPTER XV INDIVIDUAL DIFFERENCES The general facts concerning individual variations in abilities — that the variations are large, that they are con- tinuous, and that for children of the same age they usually cluster around one typical or modal ability, becoming less and less frequent as we pass to very high or very low degrees of the ability — are all well illustrated by arithmetical abilities. NATURE AND AMOUNT The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In these diagrams each space along the base- line 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 fovu-, and so on, in a test with problems (stated in words). Figure 61 shows that these adults varied from no prob- lems 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 aU reached the same school grade and so had rather 285 286 PSYCHOLOGY OF ARITHMETIC similar educational opportunities in arithmetic, the varia- tion is still very great. It requires a range from 16 to over 30 examples right to include even nine tenths of them. It should, however, be noted that if each individual had been scored by the average of his work on eight or ten differ- FiQ. 61. — The scores of 1000 soldiers in the National Army born in English- speaking countries, in Test 2 of the Army Alpha. The score is the number of correct answers obtained in five minutes. Probably 10 to 15 percent of these men were unable to read or able to read only very easy sentences at a very slow rate. Data furnished by the Division of Psychology in the office of the Surgeon General. ent days instead of by his work in just one test, the varia- bility would have been somewhat less than appears in Figs. 61, 62, and 63. It is also the case that if each individual had been scored. ^=i){ 20 40 60 80 100 120 140 160 180 200 220 240 Fig. 62. — Tlje scores of 100 1 1-year-old pupils in a test of computation. Estimated from the data givenby Burt ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the number of correct figures. 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 INDIVIDUAL DIFFERENCES 287 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 85 87 29 81 83 35 37 Fig. 63. — The scores of pupils in grade 6 in city schools in the Woody Division Test A. The score is the number of correct answers obtained in 20 minutes. From Woody ['16, p. 61]. matter what grade they had reached, the effect would have been to increase the variabiUty. 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 be- cause promotion is on a more general basis than arithmetical ability so that some very able arithmeticians are deUberately 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 inac- curacy in classifying and promoting pupils. In a composite score made up of the sum of the scores in "Woody tests, — Add. A, Subt. A, Mult. A, and Div. A, and two tests in problem-solving (ten and six graded problems, 288 PSYCHOLOGY OF ARITHMETIC with maximum attainable credits of 30 and 18), Kruse ['18] fomid 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. Fio. 64. 70 80 90 100 110 120 130 UO 160 160 170 Fio. 66. Figs. 64, 65, and 66. — The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a composite of tests in computation and problem- solving. The time was about 120 minutes. The maximum score attainable was 196. INDIVIDUAL DIFFERENCES 289 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 thafi the average pupil in grade 6. TABLE 13 Relative Frequencies op Scores in an Extensive Team of Arithmetical Tests. ' In Pebcents SCOBB Gbade 6 Grade 7 Grade 8 70 to 79 1.3 .9 .4 80 " 89 5.5 2.3 .4 90 " 99 10.6 4.3 2.9 100 " 109 19.4 5.2 4.4 110 " 119 19.8 18.5 5.8 120 " 129 23.5 16.2 16.8 130 " 139 12.6 17.5 16.8 140 " 149 4.6 13.9 22.9 • 150 " 159 1.7 13.6 17.1 160 " 169 1.2 4.8 9.4 170 " 179 2.5 3.3 DIFFERENCES WITHIN ONE CLASS The variation within a single class for which a single teacher has to provide is great. Even when teaching is departmental and promotion is by subjects, and when also the school is a large one and classification within a grade is by abiUty — 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 hmiting conditions for the teaching of arithmetic. Many methods appropriate 1 Compiled from data on p. 89 of Kruse ['18]. 290 PSYCHOLOGY OF ARITHMETIC 3 4 6 6 7and8 9 10 and 11 12 and 13 14 16 and 16 17 18 and 18 3 4 6 6 7and8 9 10 and 11 12 and 13 14 16 and 16 17 18 and 19 Fig. 67. to the top quarter of the class will be almost useless for the bottom quarter, and vice versa. Figures 67 and 68 show the scores of ten classes taken at INDIVIDUAL DIFFURBNCES 291 =1 Pupil ^^^^^^ 1^ -^.my//M/M. 3 4 5 6 7and8 9 I0andlll2andl3 14 ISandlB 17 18andl9 3 4 5 6 7 and S 9 10 and lllSand 13 14 15 and 16 17 18 and 19 Fig. 68. Figs. 67 and 68. — The scores of ten 6 B classes in a 12-minute test in computation with integers (the Courtis Test 7). The score is the number of units done. Certain long tasks are counted as two units. random from ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of computation done in 12 minutes. Ob- serve the very wide variation present in the case of every 292 PSYCHOLOGY OF ARITHMETIC 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 gener- ous 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. 18 and 19 Fig. 69. — A conservative estimate of the amount of variation to be expected within a single class of 32 pupils in grade 6, in the number of units done in Courtis Test 7 when all chance variations are eliminated. The variations within a class in respect to what processes are understood so as to be done with only occasional errors may be illustrated further as follows : — A teacher in grade 4 at or near the middle of the year in a city doing the cus- tomary work in arithmetic will probably find some pupil in her class who cannot do column addition even without /8- 9 78\ carrying, or the easiest written subtraction \5 3 or 37/, who does not know his multiplication tables or how to derive them, or understand the meanings of 4- — X and -r-, or have any useful ideas whatever about division. There will probably be some child in the class who can do such work as that shown below, and with very few errors. INDIVIDUAL DIFFERENCES 293 Add Subtract l+f+l+l 2h 31 14.3 6T^8 10.00 4 yd. 1ft. 6 in. 3.49 2 yd. 2 ft. 3 in. ' HXS 16 145 95 ±8 206 Divide 2)13.50 25)9750 The invention of means of teaching thirty so different children at once with the maximum help and minimum hindrance from their different capacities and acquisitions is one of the great opportunities for applied science. Com-tis, emphasizing the social demand for a certain moderate arithmetical attainment in the case of nearly all elementary school children of, say, grade 6, has lu-ged that definite special means be taken to bring the deficient children up to certain standards, without causing undesirable 'over- learning' by the more gifted children. Certain experi- mental 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 stand- ards for all or nearly all pupils can be arranged. THE CAUSES OF INDIVIDUAL DIPPEEENCES The differences found among children of the same grade in the same city are due in large measure to inborn differ- ences in their original natures. If, by a miracle, the children studied by Courtis, or by Woody, or by Kruse had all re- 294 PSYCHOLOGY OF ARITHMETIC ceived exactly the same nurture from birth to date, they would still have varied greatly in arithmetical ability, per- haps almost as much as they now do vary. The evidence for this is the general evidence that varia- tion in original nature is responsible for much of the eventual variation foimd in intellectual and moral traits, plus certain special evidence in the case of arithmetical abihties them- selves. Thomdike found ['05] that in tests with addition and multiplication twins were very much more alike than siblings ^ two or three years apart in age, though the re- semblance 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. If the differences found, say among children in grade 6 in addition, were due to differences in the quantity and quality of training in addition which they have had, then by giving each of them 200 minutes of additional identical training the differences should be reduced. For the 200 minutes of identical training is a step toward equalizing training. It has been found in many investigations of the matter that when we make training in arithmetic more nearly equal for any group the variation within the group is not reduced. On the contrary, equalizing training seems rather to in- crease 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 ex- ample, compare the gains of different individuals due to ' Siblings is used for children of the same parents. INDIVIDUAL DIFFERENCES 295 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] .^ TABLE 14 The Effect op Equal Amounts of Peactice upon Individual Differ- ence IN THE Multiplication of Three-Place Numbers Amount Percentaqe of COBKECT FiGTIEES - Initial Score Gain Initial Score Gain Initially highest five individuals next five " next six " next six " next six " 85 56 46 38 29 61 •51 22 8 24 70 68 74 58 56 18 10 8 12 14 THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES Achievement in arithmetic depends upon a number of different abilities. For example, acciiracy in copying num- bers depends upon eyesight, ability to perceive visual de- tails, 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 prob- lems 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. ' Similar results have been obtained in the case of arithmetical and other abihties by Thomdike ['08, '10, '15, '16], Whitley ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and Thomdike ['13], Hahn and Thomdike ['14], and on a very large scale by Race in a study as yet unpublished. 296 PSYCHOLOGY OF ARITHMETIC Since the abilities which together constitute arithmetical ability are thus specialized, the individual who is the best of a thousand of his age or grade in respect to, say, adding integers, may occupy different stations, perhaps, from 1st to 600th, in multiplying with integers, placing the decimal point in division with decimals, solving novel problems, copying figures, etc., etc. Such specialization is in part due to his having had, relatively to the others in the thousand, more or better training in certain of these abilities than in others, and to various circumstances of life which have caused him to have, relatively to the others in the thousand, greater interest in certain of these achievements than in others. The specialization is not wholly due thereto, how- ever. Certain inborn characteristics of an individual pre- dispose him to different degrees of superiority or inferiority to other men in different features of arithmetic. We measure the extent to which ability of one sort goes with or fails to go with ability of some other sort by the coefficient of correlation between the two. If every in- dividual keeps the same rank in the second abihty — 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 meaning that the best individual in ability A is no more likely to be in first place in abifity B than to be in any other rank. The meanings of coefficients of correlation of .90, .70, .50, and arie shown by Tables 15, 16, 17 and 18.^ 1 Unless he has a thorough understanding of the underlying theory, the student should be very cautious in making inferences from coefficients of correlation. INDIVIDUAL DIFFERENCES 297 TABLE 15 Distribution op Arrays in Successive Tenths of the Group When r = .90 lOlH 9th 8th 7th 6th 5th 4th 3d 2d 1st 1st tenth . .1 .4 1.8 6.6 22.4 68.7 2d tenth .1 .4 1.4 4.7 11.5 23.5 36.0 22.4 3d tenth . .1 .5 2.1 5.8 12.8 21.1 27.4 23.5 6.6 4th tenth . .4 2.1 6.4 12.8 20.1 23.8 21.2 11.5 1.8 6th tenth . .1 1.4 5.8 12.8 19.3 22.6 20.1 12.8 4.7 .4 6th tenth . .4 4.7 12.8 20.1 22.6 19.3 12.8 ,, 5.8 1.4 .1 7th tenth . 1.8 11.5 21.2 23.8 20.1 12.8 6.4 2.1 .4 8th tenth . 6.6 23.5 27.4 211 12.8 5.8 2.1 .5 .1 9th tenth . 22.4 36.0 23.5 11.5 4.7 1.4 .4 .1 10th tenth . 68.7 22.4 6.6 1.8 .4 .1 TABLE 16 Distribution op Arrays in Successive Tenths op the Group When r = .70 10th 9th 8th 7th 6th 5th 4th 3d 2d 1st 1st tenth .2 .7 1.5 2.8 4.8 8.0 13.0 22.3 46.7 2d tenth .2 1.2 2.6 4.5 7.0 9.8 13.4 17 3 21.7 22.3 3d tenth .7 2.6 5.0 7.3 10.0 12.5 14.9 16.7 17.3 13.0 4th tenth . 1.5 4.5 7.3 9.8 12.0 13.7 14.8 14.9 13.4 8.0 5th tenth . 2.8 7.0 10.0 12.0 13.4 14.0 13.7 12.5 9.8 4.8 6th tenth . 4.8 9.8 12.5 13.7 14.0 13.4 12.0 10.0 7.0 2.8 7th tenth . 8.0 13.4 14.9 14.8 13.7 12.0 9.8 7.3 4.5 1.5 8th tenth . 13.0 17.3 16.7 14.9 12.5 10.0 7.3 5.0 2.6 .7 9th tenth . 22.3 21.7 17.3 13.4 9.8 7.0 4.5 2.6 1.2 .2 10th tenth . 46.7 22.3 13.0 8.0 4.8 2.8 1.5 .7 .2 298 PSYCHOLOGY OP ARITHMETIC TABLE 17 DiSTEIBUTION OF ARRAYS OF SUCCESSIVE TENTHS OF THE GbOUP When r = .50 10th 9th 8th 7th 6th 5th 4th 3d 2d IST 1st tenth .8 2.0 3.2 4.6 6.2 8.1 10.5 13.9 18.0 31.8 2d tenth . 2.0 4.1 5.7 7.3 8.8 10.5 12.2 14.1 16.4 18.9 3d tenth . 3.2 5.7 7.4 8.9 1.00 11.2 12.3 13.3 14.1 13.9 4th tenth . 4.6 7.3 8.8 9.9 10.8 11.6 12.0 12.3 12.2 10.5 5th tenth . 6.2 8.8 10.0 10.8 11.3 11.5 11.6 11.2 10.5 8.1 6th tenth . 8.1 10.5 11.2 11.6 11.5 11.3 10.8 10.0 8.8 6.2 7th tenth . 10.5 12.2 12.3 12.0 11.6 10.8 9.9 8.8 7.5 4.6 8th tenth 13.9 14 1 13.3 12.3 11.2 10.0 8.8 7.4 5.7 3.2 9th tenth . 18.9 16.4 14.1 12.2 10.5 8.8 7.3 5.7 4.1 2.0 10th tenth . 31.8 18.9 13.9 10.5 8.1 6.2 4.6 3.2 2.0 .8 TABLE 18 Distribution op Arrays, in Successive Tenths op the Group When r = .0 10th 9th 8th 7th 6th 5th 4th 3d 2d 1st 1st tenth 10 10 10 10 10 10 10 10 10 10 2d tenth . 10 10 10 10 10 10 10 10 10 10 3d tenth . 10 ■10 10 10 10 10 10 10 10 10 4th tenth 10 10 10 10 10 10 10 10 10 10 5th tenth . 10 10 10 10 10 ,10 10 10 10 10 6th tenth . 10 10 10 10 10 10 10 10 10 10 ' 7th tenth . 10 10 10 10 10 10 10 10 10 10 8th tenth . 10 10 10 10 10 10 10 10 10 10 9th tenth . 10 10 10 10 10 10 10 10 10 10 10th tenth . 10 10 10 10 10 10 10 10 10 10 The significance of any coefiicient of correlation depends upon the group of individuals for which it is determined. A correlation of .40 between computation and problem-solving INDIVIDUAL DIFFERENCES 299 in eighth-grade pupils of 14 years would mean a much closer real relation than a correlation of .40 in all 14-year-olds, and a very, very much closer relation than a correlation of .40 for all children 8 to 15. Unless the individuals concerned are very elaborately tested on several days, the correlations obtained are "at- tenuated" toward by the "accidental" errors in the original measurements. This effect was not known until 1904 ; consequently the correlations in the earlier studies of arithmetic are all too low. In general, the correlation between ability in any one important feature of computation and abiUty in any other important feature of computation is high. If we make enough tests to measure each individual exactly in : — (A) Subtraction with integers and decimals, (B) Multiplication with integers and decimals, {€) Division with integers and decimals, (jD) Multiplication and division with common fractions, and (E) Computing with percents, we shall probably find the intercorrelations for a thousand 14-year-olds to be near .90. Addition of integers {F) will, however, correlate less closely with any of the above, being apparently dependent on simpler and more isolated abilities. The correlation between problem-solving (G) and computa- tion will be very much less, probably not over .60. It should be noted that even when the correlation is as high as .90, there will be some individuals very high in one ability and very low in the other. Such disparities are to some extent, as Courtis ['13, pp. 67-75] and Cobb ['17] have argued, due to inborn characteristics of the individual in question which predispose him to very special sorts of 300 PSYCHOLOGY OF ARITHMETIC strength and weakness. They are often due, however, to defects m his learning whereby he has acquired more ability than he needs in one line of work or has failed to acquire some needed ability which was well within his capacity. In general, all correlations between an individual's diver- gence 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 counter- balances the effects that robbing Peter to pay Paul may have. Speed and accuracy are thus positively correlated. The individuals who do the most work in ten minutes will be above the average in a test of accuracy. The common notion that speed is opposed to accuracy is correct when it means that the same person will tend to make more errors if he works at too rapid a rate ; but it is entirely wrong when it means that the kind of person who works more rapidly than the average person is likely to be less accurate than the average person. Interest in arithmetic and ability at arithmetic are probably correlated positively in the sense that the pupil who has more interest than other pupils of his age tends in the long run to have more abiUty than they. They are certainly correlated in the sense that the pupil who 'likes' arithmetic better than geography or history tends to have relatively more ability in arithmetic, or, in other words, that the pupil who is more gifted at arithmetic than at drawing or English tends also to like it better than he likes these. These correlations are high. It is correct then to think of mathematical ability as, in a sense, a unitary ability of which any one individual may have much or little, most individuals possessing a moderate INDIVIDUAL DIFFERENCES 301 amount of it^ This is consistent, however, with the oc- casional appearance of individuals possessed of very great talents for this or that particular feature of mathematical ability and equally notable deficiencies in other fea- tures. 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 in- dications thereof. BIBLIOGRAPHY OF REFERENCES MADE IN THE TEXT Ames, A. F., and McLellan, J. F '00 Public School Arithmetic. Ballou, F. W '16 Determining the Achievements of Pupils in the Addition of Frac- tions. School Dqcument No. 3, 1916, Boston Public Schools. Brandell, G '13 Skolbarns intressen. Translated ['15] by W. Stern as, Das In- teresse der SchuUdnder an den Unterrichtsfachern. Brandford, B '08 A Study of Mathematical Educa- tion. Brown, J. C '11, '12 An Investigation on the Value of Drill Work in the Fundamental Operations in Arithmetic. Jour- nal of Educational Psychology, vol. 2, pp. 81-88, vol. 3, pp. 485- 492 and 561-570. Brown, J. C, and Coffman, L. D '14 How to Teach Arithmetic. Burgerstein, L '91 Die Arbeitscurve einer Schulstunde. Zeitschrift ftir Schulgesundheits- pflege, vol. 4, pp. 543-562 and 607-627. 302 ^PSYCHOLOGY OF ARITHMETIC Burnett, C. J '06 Burt, C '17 Chapman, J. C '14 Chapman, J. C '17 Cobb, M. V '17 Coffman, L. D., and Brown, J.C '14 Coffman, L. D., and Jessup, W. A '16 Courtis, S. A. . . . '09, '10, '11 Courtis, S. A '11, '12 Courtis, S. A. Courtis, S. A. '13 '14 The Estimation of Number. Har- vard Psychological Studies, vol. 2, pp. 349-404. The Distribution and Relations of Educational Abilities. Report of the London Coimty Council, No. 1868. Individual Differences in Ability and Improvement and Their Cor- relations. Teachers College Contributions to Education, No. 63. The Scientific Measurement of Classroom Products. (With G. P. Rush.) A Preliminary Study of the Inherit- ance of Arithmetical Abilities. Jour, of Educational Psychology, vol. 8, pp. 1-20. Jan., 1917. How to Teach Arithmetic. The Supervision of Arithmetic. Measurement of Growth and Effi- ciency in Arithmetic. Elemen- tary School Teacher, vol. 10, pp. 58-74 and 177-199, vol. 11, pp. 171-185, 360-370, and 528- 539. Report on Educational Aspects of the Public School System of the City of New York. Part II, Subdivision 1, Section D. Re- port on the Courtis Tests in Arithmetic. Courtis Standard Tests. Second Annual Accounting. Manual of Instructions for Giving and Scoring the Courtis Standard INDIVIDUAL DIFFERENCES 303 Decroly, M., and Degand, J. '12 Degand, J. See Decroly. De Voss, J. C See Monroe, De Voss, and Kelly. Dewey, J '10 Dewey, J., and McLellan, J. A. '95 Donovan, M. E., and Thorndike, E. L '13 ElUott, C. H '14 Flynn, F. J '12 Freeman, F. N '10 Friedrich, J '97 Gilbert, J. A '94 Tests in the Three R's. Depart- ment of Comparative Research. 82 Eliot St., Detroit, Mich., 1914. L'evolution des notions de quan- tity continues et discontinues chez I'enfant. Archives de psy- chologic, vol. 12, pp. 81-121. How We Think. Psychology of Nimiber and Its Applications to Methods of Teaching Arithmetic. Improvement in a Practice Experi- ment under School Conditions. American Journal oi Psychology, vol. 24, pp. 426-428. Variation in the Achievements of Pupils. Teachers College, Colum- bia University, Contributions to Education, No. 72. Mathematical Games — Adapta- tions from Games Old and New. Teachers College Record, vol. 13, pp. 399-412. Untersuchungen iiber den Auf- merksamkeitsumfang und die Zahlauffassung. Padagogische- Psychologische Arbeiten, I, 88- 168. Untersuchungen iiber die Einflussa der Arbeitsdauer und die Arbeits- pausen auf die geistige Leist- ungsfahigkeit der Schulkinder. Zeitschrift fiir Psyehologie, vol. 13, pp. 1-53. Researches on the Mental and 304 PSYCHOLOGY OF ARITHMETIC Greenleaf, B 73 Hahn, H. H., and Thorndike, E. L '14 HaU,G.S '83 Hartmann, B '90 Heck, W. H '13 Heck, W. H '13 Hoffmann, P '11 Hoke, K. J., and Wilson, G. M. '20 Holmes, M. E '95 Howell, H.B '14 Hunt, C. W '12 Physical Development of School Children. Studies from the Yale Psychological Laboratory, vol. 2, pp. 40-100. Practical Arithmetic. Some Results of Practice in Addi- tion under School Conditions. Journal of Educational Psy- chology, vol. 6, No. 2, pp. 65-84. The Contents of Children's Minds on Entering School. Princeton Review, vol. II, pp. 249-272, May, 1883. Reprinted in As- pects of Child Life and Educa- tion, 1907. Die Analyze des Kindlichen Ge- danken-Kreises als die Naturge- masse des Ersten Schulunter- richts, 1890. A Study of Mental Fatigue. A Second Study in Mental Fatigue in the Daily School Program. Psychological Clinic, vol. 7, pp. 29-34. Das Interesse der Schtiler an den Unterrichtsfachern. Zeitschrift flir padagogische Psychologic, XII, 458-470. How to Measure. The Fatigue of a School Hour. Pedagogical Seminary, vol. 3, pp. 213-234. A Foundation Study in the Peda- gogy of Arithmetic. Play and Recreation in Arithmetic. Teachers College Record, vol. 13, pp. 388-398. INDIVIDUAL DIFFERENCES 305 Jessup, W. A., and Coffman, L. D '16 Kelly, F. J. See Monroe, Be Voss and Kelly. King, A. C '07 Kirbir, T. J '13 Klapper, P '16 Kruse, P.J '18 Laser, H '94 Lay, W. A '98 Lay, W.A '07 I^wis, E. '13 Lobsien, M '03 Lobsien, M '09 McCall, W. A '21 McDougle, E. C '14 The Supervision of Arithmetic. The Daily Program ia Elementary Schools. MSS. Practice in the Case of School Children. Teachers College Con- tributions to Education, No. 58. The Teaching of Arithmetic. The Overlapping of Attaioments in Certain Sixth, Seventh, and Eighth Grades. Teachers College, Colxmibia University, Contributions to Education, No. 92. Ueber geistige Ermudung beim Schulunterricht. Zeitschrift fiir Schulgesundheitspflege, vol. 7, pp. 2-22. Fiihrer durch den ersten Rechen- unterricht. Fiihrer durch den Rechenunterricht der Unterstufe. Popular and Unpopular School- Subjects. The Journal of Ex- perimental Pedagogy, vol. 2, pp. 89-98. Kinderideale. Zeitschrift fiir pa- dagogische Psychologie, V, 323- 344 and 457-494. Beliebtheit und Unbeliebtheit der Unterrichtsfacher. Padagog- isches Magazin, Heft 361. How to Measure in Education. A Contribution to the Pedagogy of Arithmetic. Pedagogical Semi- nary, vol. 21, pp. 161-218. 306 PSYCHOLOGY OF ARITHMETIC McKnight, J. A. . '07 McLellan, J. A., and Dewey, J. '95 McLellan, J. A., and Ames, A. F '00 Messenger, J. F '03 Meumann, E '07 MitcheU, H. E '20 Monroe, W. S., De Voss, J. C, andKeUy, F. J '17 Nanu, H. A. '04 National Intelligence Tests . '20 Fhillipte, D. E '97 Pommer, 0. ...... '14 Rice, J. M '02 Rice, J. M. ...... '03 Rush, G. P '17 Differentiation of the Curriculum in the Upper Grammar Grades. MSS. in the library of Teachers College, Columbia University. Psychology of Number and Its AppUcations to Methods of Teaching. PubUc School Arithmetic. The Perception of Number. Psy- chological Review, Monograph Supplement No. 22. Vorlesungen zur Einfiihrung in die experimentelle Padagogik. Unpubhshed studies of the uses of arithmetic in factories, shops, farms, and the hke. Educational Tests and Measure- ments. Zur Psychologie der Zahl Auffas- sung. Scale A, Form 1, Edition 1. Number and Its Application Psy- chologically Considered. Pedago- gical Seminary, vol. 5, pp. 221- 281. Die Erforschimg der Beliebtheit der Unterrichtsfacher. Ihre psy- chologischen Grundlagen und ihre padagog. Bedeutung. VII. Jahresber. des k. k. Ssaatsgymn. im XVIII Bez. v. Wien. Test in Arithmetic. Forum, vol. 34, pp. 281-297. Causes of Success and Failure in Arithmetic. Forum, vol. 34, pp. 437-452. The Scientific Measurement of INDIVIDUAL DIFFERENCES 307 Seekel, E '14 Selkin, F. B '12 Smith, D. E '01 Smith, D. E '11 Speer, W. W '97 Starch, D '11 Starch, D '16 Stem, W '05 Stem, C, and Stem, W. '13 Stem, W '14 Classroom Products. (With J. C. Chapman.) Ueber dieBeziehung zwischen der Beliebtheit und der Schwierig- keit der Schulfacher. Ergeb- nisse einer Erhebung. Zeit- schrif t fiir Angewandte Psycholo- gie 9. S. 268-277. Number Games Bordering on Arith- metic and Algebra. Teachers College Record, vol. 13, pp. 452-495. The Teaching of Elementary Math- ematics. The Teaching of Arithmetic. Arithmetic : Elementary for Pupils. Transfer of Training in Arithmeti- cal Operations. Journal of Edu- cational Psychology, vol. 2, pp. 306-310. Educational Measurements. Ueber Beliebtheit und Unbelieb- theit der Schulfacher. Zeit- schrift fur padagogische Psychol- ogic, VII, 267-296. Beliebtheit und Schwierigkeit der Schulfacher. (Freie Schulge- meinde Wickersdorf.) Auf Grund der von Herrn Luserke beschafften Materialien bear- beitet. In : " Die Ausstellung zur vergleichenden Jungendkunde der Geschlechter in Breslau.'' Arbeit 7 des Bundes fiir Schul- reform. S. 24-26. Zur vergleichenden Jugendkunde der Geschlechter. Vortrag. III. Deutsch. Kongr. f. Jugendkunde 308 PSYCHOLOGY OP ARITHMETIC Stone, C. W '08 Suzzallo, H '11 Thorndike, E. L '00 Thorndike, E. L. . . . . '08 Thorndike, E. L '10 Thorndike, E. L., and Donovan, M. E '13 Thorndike, E. L., and Donovan, M. E., and Hahn, H. H. . '14 Thorndike, E.L '15 Thorndike, E. L. '16 Walsh, J. H '06 WeUs, F. L '12 usw. Arbeiten 8 des Bundes fur Schuheform. S. 17-38. Arithmetical Abilities and Some Factors Determining Them. Teachers College Contributions to Education, No. 19. The Teaching of Primary Arith- metic. Mental Fatigue. Psychological Review, vol. 7, pp. 466-482 and 547-579. The Effect of Practice in the Case of a Purely Intellectual Function. American Journal of Psychology, vol. 10, pp. 374-384. Practice in the case of Addition. American Journal of Psychology, vol. 21, pp. 483-486. Improvement in a Practice Experi- ment under School Conditions. American Journal of Psychology, vol. 24, pp. 426-428. Some Results of Practice in Addi- tion under School Conditions. Journal of Educational Psychol-, ogy, vol. 5, No. 2, pp. 65-84. , The Relation between Initial Abil- ity and Improvement in a Sub- stitution Test. School and So- ciety, vol. 12, p. 429. Notes on Practice, Improvability, and the Ciu've of Work. Ameri- can Journal of Psychology, vol. 27, pp. 550-565. Grammar School Arithmetic, The Relation of Practice to Indi- vidual Differences. American INDIVIDUAL DIFFERENCES 309 White, E.E. ...... '83 Whitley, M. T '11 Wiederkehr, G '07 Wilson, G.M '19 Wilson, G. M., Woody, G. . . and Hoke, K. J. '20 '16 Journal of Psychology, vol. 23, pp. 75-88. A New Elementary Arithmetic. An Empirical Study of Certain Tests for Individual Differences. Archives of Psychology, No. 19. Statistiche Untersuchungen iiber die Art und den Grad des In- teresses bei Kindern der Volks- schule. Neue Bahnen, vol. 19, pp. 241-251, 289-299. A Survey of the Social and Business Usage of Arithmetic. Teachers College Contributions to Educa- tion, No. 100. How to Measure. Measurements of Some Achieve- ments in Arithmetic. , Teachers College Contributions to Educar tion, No. 80. INDEX Abilities, arithmetical, nature of, 1 ff. ; measurement of, 27 ff. ; constitution of, 51 ff. ; organization of, 137 ff. Abstract numbers, 85 ff. Abstraction, 169 ff. Accuracy, in relation to speed, 31 ; in fundamental operations, 102 ff. Addition, measurement of, 27 ff., 34; constitution of, 52 f . ; habit in rela- tion to, 71 f. ; in the higher decades, 75 f. ; accuracy in, 108 f . ; amount of practice in, 122 ff. ; interest in, 196 f. Aims of the teaching of arithmetic, 23 f. Ames, A. F., 89 Analysis, learning by, 169 ff . ; systematic and opportunistic stimuli to, 178 £. ; gradual progress in, 180 ff. Area, 257 f., 275 Arithmetic, sociology of, 24 ff. Arithmetical abilities. See Abilities. Arithmetical language, 8 f., 19, 89 ff., 94 ff. Arithmetical learning, before school, 199 ff. ; conditions of, 227 ff. ; in relation to time of day, 227 ff. ; in relation to time devoted to arithmetic, 228 ff. Arithmetical reasoning. See Reasoning. Arithmetical terms, 8, 19 Averages, 40 f . ; 135 f . Ballou, F. W., 34, 38 Banking, 256 f . BiNET, A., 201 Bonds, selection of, 70 ff. ; strength of, 102 ff. ; for temporary service. 111 ff. ; order of formation of, 141 ff. See also Habits. Bkandell, G., 211 Bhandpord, B., 198 f . Brown, J. C, xvi, 103 BURGERSTEIN, L., 103 Burnett, C. J., 202 Burt, C, 286 Cardinal and ordinal numbers contused, 206 Catch problems, 21 ff. Chapman, J. C, 49 Class, size of, in relation to arithmetical learning, 228; variation withiji a, 289 ff. Cobb, M. V., 299 COFFMAN, L. D., xvi Collection meaning of numbers, 3 ff. Computation, measurements of, 33 ff. ; explanations of the processes in, 60 ff. ; accuracy in, 102 ff. See also Addition, Subtraction, Multiplica- tion, Division, Fractions, Decimal numbers, Percents. Concomitants, law of varying, 172 ff. ; law of contrasting, 173 ff. Concrete numbers, 85 ff. Concrete objects, use of, 253 ff. Conditions of arithmetical learning, 227 ff. Constitution of arithmetical abilities, 51 ff. Copying of numbers, eyestrain due to, 212 f. Correlations of arithmetical abilities, 295 ff. Courses of study, 232 f . Courtis, S. A., 28 ff., 43 ff., 49, 103. 291, 293, 299 Crutches, 112 f. Culture-epoch theory, 198 f. Dairy records, 273 Decimal numbers, uses of, 24 f . ; meas- • urement of ability with, 36 ff. ; learning, 181 ff. ; division by, 270 f . De Cholt, M., 205 Deductive reasoning, 60 ff., 185 ff. Degand, J., 205 Denominate numbers, 141 f ., 147 f . Described problems, 10 ff. Development of knowledge of number; 205 ff. 311 312 INDEX Db Vosa, J. C, 49 Dewey, J., 3, 83, 150, 205, 207, 208, 219, 266, 277 Differences in arithmetical ability, 285 ff. ; within a class, 289 ff. DiflElculty as a stimulus, 277 ff. Drill, 102 ff. Discipline, mental, 20 Distribution of practice, 156 S. Division, measurement of, 35 f ., 37 ; constitution of, 57 ff. ; deductive explanations of, 63, 64 t. ; inductive explanations of, 63 f., 65 f . ; habit in relation to, 72 ; with remainders, 76 ; with fractions, 78 ff . ; amount of practice in, 122 ff. ; distribution of practice in, 167 ; use of the prob- lem attitude in teaching, 270 f . Donovan, M. E., 295 Elements, responses to, 169 ff. Eleven, multiples of, 85 Elliott, C. H., 228 Equation form, importance of, 77 f . Explanations of the processes of com- putation, 60 ff. ; memory of, 115 f . ; time for giving, 154 ff. Eyestrain in arithmetical work, 212 ff. Facilitation, 143 ff. Figures, printing of, 235 ff. ; writing of, 214 f., 241 Fltnn, F. J., 196 Fractions, uses of, 24 f . ; measurement of ability with, 36 ff. ; knowledge of the meaning of, 54 ff. Freeman, F. N., 259, 261 Fribdeich, J., 103 Generalization, 169 ff. Gilbert, J. A., 203 Graded tests, 28 ff., 36 ff. Greatest common divisor, 88 f . Habits, importance of, in arithmetical learning, 70 ff. ; now neglected, 75 ff.; harmful or wasteful, 83 ff.; 91 ff. ; propsedeutic, 117 ff. ; organization of, 137 ff. ; arrangement of, 141 ff. Hahn, H. H., 295 Hall, G. S., 200 f. Hahtmann, B., 200 f. Heck, W. H., 227 Heredity in arithmetical abilities, 293 ff. Highest common factor, 88 f . Hoke, K. J., 49 HoLMBS, M. E., 103 Howell, H. B., 259 Hunt, C. W., 196 Hygiene of arithmetic, 212 ff., 234 ff. Individual differences, 285 ff. Inductive reasoning, 60 ff., 169 ff. Insurance, 256 Interest as a principle determining the order of topics, 150 ff. Interests, instinctive 195 ff . ; censuses of, 209 ff.; neglect of childish, 220 ff. ; in self-management, 223 f . ; in- trinsic, 224 ff. Interference, 143 ff. Inventories of arithmetical knowledge and skiU, 199 ff. Jessup, W. a., xvi Kbllt, F. J., 49 King, A. C, 103, 227 Kirbt, T. J., 76 f., 104, 295 Klapper, p., xvi Ketjse, p. J., 289, 293 Ladder tests, 28 ff., 36 ff. Language in arithmetic, 8 f., 19, 89 ff., 94 ff. Laser, H.. 103 Lay, W. a., 259, 261 Learning, nature of arithmetical, 1 ff. Least common multiple, 88 f . Lewis, E. O., 210 f . LoBSiBN, M., 209 f . McCall, W. a., 49 McDouGLB, E. C, 85 ff. McKnight, J, A.. 210 McLellan, J. A., 3, 83, 89, 205, 207 Manipulation of numbers, 60 ff. Meaning, of numbers, 2 ff., 171; of a fraction, 54 ff. ; of decimals, 181 f . Measurement of arithmetical abilities, 27 ff. Mental arithmetic, 262 ff. Messenger, J. F., 202 Metric system, 147 Meumann, E., 261 MrtcHELL, H. E., 24 INDEX 313 MoNHOE, W. S., 49 Multiplication, measurement of, 35, 36; constitution of, 51; deductive explanations of, 61.; inductive ex- planations of, 61 f . ; with fractions, 78 ff. ; by eleven, 85 ; amount of practice in, 122 ff. ; order of learning the elementary facts of, 144 f . ; dis- tribution of practice in, 158 ff. ; use of the problem attitude in teaching, 267 ff. Nanu, H. a., 202 National Intelligence Tests, 49 f . Negative reaction in intellectual life, 278 f. Number pictures, 259 ff. Numbers, meaning of, 2 ; as measures of continuous quantities, 75 ; abstract and concrete, 85 S. ; denominate, 141 f ., 147 f . ; use of large, 145 f . ; perception of, 202 ff. ; early aware- ness of, 205 ff. ; confusion of cardinal and ordinal, 206. See also Decimal numbers and Fractions. Objective aids, used for verification, 154 ; in general, 243 ff. Oral arithmetic, 262 ff. Order of topics, 141 ff. Ordinal numbers, confused with cardi- nal, 206 Original tendencies and arithmetic, 195 ff . Overleaming, 134 ff. Percents, 80 f . Perception of number, 202 ff. Phillips, D. E., 3, 4, 205, 207 Pictures, hygiene of, 246 ff. ; number, 259 ff. POMMER, O., 212 Practice, amount of, 122 ff. ; distribu- tion of, 156 ff. Precision in fimdamental operations, 102 ff. Problem attitude, 266 ff. Problems, 9 ff. ; "catch,'' 21 ff. ; meas- urement of ability with, 42 ff. ; whose answer must be known in order to frame them, 93 f . ; verbal form of. 111 f . ; interest in, 220 ff. ; as introductions to arithmetical learn- ing, 266 ff. . Propaedeutic bonds, 117 ff. Purposive thinking, 193 ff. Quantity, number and, 85 ff. ; per- ception of, 202 ff. Race, H., 295 Rainfall, 272 Ratio, 225 f. ; meaning of numbers, 3 ff. Reaction, negative, 278 f . Reality, in problems, 9 ff. Reasoning, arithmetical, nature of, 19 ff. ; measurement of ability in, 42 ff. ; derivation of tables by, 58 f. ; about the rationale of computations, 60 ff. ; habit in relation to, 73 f., 190 ff. ; problems which provoke false, 100 f . ; the essentials of arithmetical, 185 ff. ; selection in, 187 ff. ; as the cooperation of organized habits, 190 ff. Recapitulation theory, 198 f . Recipes, 273 f . Rectangle, area of, 257 f. Rice, J. M., 228 ff. Rush, G. P., 49 Seekel, E., 212 Selkin, F. B., 196 f. Sequence of topics, 141 ff. Series meaning of numbers, 2 ff. Size of class in relation to arithmetical learning, 228 Smith, D. E., xvi, 224 Social instincts, use of, 195 f . Sociology of arithmetic, 24 ff. Speed in relation to accuracy, 31, 108 Speer, W. W., 3, 5, 83 Spiral order, 141, 145 Starch, D., 49, 295 Stern, W., 210, 212 Stone, C. W., 27 ff., 42 ff., 228 ff. Subtraction, measurement of, 34 f. ; constitution of, 57 f . ; amount of practice in, 122 ff. Supervision, 233 f. Sdzzallo, H., xvi Temporary bonds, 111 fl. Terms, 113 f. Tests of arithmetical abilities, 27 ff. Thohndike, E. L., 34, 38 ff., 227, 294 Time, devoted to arithmetic, 228 ff. ; of day, in relation to arithmetical learning, 227 f . 314 INDEX Type, hygiene of, 235 B. Underlearning, 134 ff. United States money, 148 ff. Units of measure, arbitrary, 5, 83 t. Variation, among individuals, 285 ff. Variety, in teaching, 153 Verification, 81 f . ; aided by greater strength of the fundamental bonds, 107 ff. Walsh, J. H., 11 Wells, F. L., 295 White, E. E., 5 Whitley, M. T., 295 WlBD^RKEHK, G.. 212 Wilson, G. M., 24, 49 Woody, C, 29 ff., 52, 287, 293 Words. See Language and Terms. Written arithmetic, 262 ff. Zero in multiplication, 179 f . liiili Ijlliiliii I