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THE PSYCHOLOGY OF
ARITHMETIC

BY
EDWARD L. THORNDIKE

TEACHERS COLLEGE, COLUMBIA
UNIVERSITY

New York
THE MACMILLAN COMPANY
1929
All rights reserved

Copyright, 1922,
By THE MACMILLAN COMPANY.

Set up and electrotyped. Published January, 1922.
Reprinted October, 1924; May, 1926; August, 1927; October, 1929.

· PRINTED IN THE UNITED STATES OF AMERICA ·

PREFACE

Within recent years there have been three lines of advance in psychology which are of notable significance for teaching. The first is the new point of view concerning the general process of learning. We now understand that learning is essentially the formation of connections or bonds between situations and responses, that the satisfyingness of the result is the chief force that forms them, and that habit rules in the realm of thought as truly and as fully as in the realm of action.

The second is the great increase in knowledge of the amount, rate, and conditions of improvement in those organized groups or hierarchies of habits which we call abilities, such as ability to add or ability to read. Practice and improvement are no longer vague generalities, but concern changes which are definable and measurable by standard tests and scales.

The third is the better understanding of the so-called "higher processes" of analysis, abstraction, the formation of general notions, and reasoning. The older view of a mental chemistry whereby sensations were compounded into percepts, percepts were duplicated by images, percepts and images were amalgamated into abstractions and concepts, and these were manipulated by reasoning, has given way to the understanding of the laws of response to elements or aspects of situations and to many situations or elements thereof in combination. James' view of reasoning as "selection of essentials" and "thinking things together" in a revised and clarified form has important applications in the teaching of all the school subjects.

This book presents the applications of this newer dynamic psychology to the teaching of arithmetic. Its contents are substantially what have been included in a course of lectures on the psychology of the elementary school subjects given by the author for some years to students of elementary education at Teachers College. Many of these former students, now in supervisory charge of elementary schools, have urged that these lectures be made available to teachers in general. So they are now published in spite of the author's desire to clarify and reinforce certain matters by further researches.

A word of explanation is necessary concerning the exercises and problems cited to illustrate various matters, especially erroneous pedagogy. These are all genuine, having their source in actual textbooks, courses of study, state examinations, and the like. To avoid any possibility of invidious comparisons they are not quotations, but equivalent problems such as represent accurately the spirit and intent of the originals.

I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn and Company, D. C. Heath and Company, The Macmillan Company, The Oxford University Press, Rand, McNally and Company, Dr. C. W. Stone, The Teachers College Bureau of Publications, and The World Book Company, in permitting various quotations.

Edward L. Thorndike.

Teachers College
Columbia University
April 1, 1920

CONTENTS

GENERAL INTRODUCTION

THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS

The psychology of the elementary school subjects is concerned with the connections whereby a child is able to respond to the sight of printed words by thoughts of their meanings, to the thought of "six and eight" by thinking "fourteen," to certain sorts of stories, poems, songs, and pictures by appreciation thereof, to certain situations by acts of skill, to certain others by acts of courtesy and justice, and so on and on through the series of situations and responses which are provided by the systematic training of the school subjects and the less systematic training of school life during their study. The aims of elementary education, when fully defined, will be found to be the production of changes in human nature represented by an almost countless list of connections or bonds whereby the pupil thinks or feels or acts in certain ways in response to the situations the school has organized and is influenced to think and feel and act similarly to similar situations when life outside of school confronts him with them.

We are not at present able to define the work of the elementary school in detail as the formation of such and such bonds between certain detached situations and certain specified responses. As elsewhere in human learning, we are at present forced to think somewhat vaguely in terms of mental functions, like "ability to read the vernacular," "ability to spell common words," "ability to add, subtract, multiply, and divide with integers," "knowledge of the history of the United States," "honesty in examinations," and "appreciation of good music," defined by some general results obtained rather than by the elementary bonds which constitute them.

The psychology of the school subjects begins where our common sense knowledge of these functions leaves off and tries to define the knowledge, interest, power, skill, or ideal in question more adequately, to measure improvement in it, to analyze it into its constituent bonds, to decide what bonds need to be formed and in what order as means to the most economical attainment of the desired improvement, to survey the original tendencies and the tendencies already acquired before entrance to school which help or hinder progress in the elementary school subjects, to examine the motives that are or may be used to make the desired connections satisfying, to examine any other special conditions of improvement, and to note any facts concerning individual differences that are of special importance to the conduct of elementary school work.

Put in terms of problems, the task of the psychology of the elementary school subjects is, in each case:—

(1) What is the function? For example, just what is "ability to read"? Just what does "the understanding of decimal notation" mean? Just what are "the moral effects to be sought from the teaching of literature"?

(2) How are degrees of ability or attainment, and degrees of progress or improvement in the function or a part of the function measured? For example, how can we determine how well a pupil should write, or how hard words we expect him to spell, or what good taste we expect him to show? How can we define to ourselves what knowledge of the meaning of a fraction we shall try to secure in grade 4?

(3) What can be done toward reducing the function to terms of particular situation-response connections, whose formation can be more surely and easily controlled? For example, how far does ability to spell involve the formation one by one of bonds between the thought of almost every word in the language and the thought of that word's letters in their correct order; and how far does, say, the bond leading from the situation of the sound of ceive in receive and deceive to their correct spelling insure the correct spelling of that part of perceive? Does "ability to add" involve special bonds leading from "27 and 4" to "31," from "27 and 5" to "32," and "27 and 6" to "33"; or will the bonds leading from "7 and 4" to "11," "7 and 5" to "12" and "7 and 6" to "13" (each plus a simple inference) serve as well? What are the situations and responses that represent in actual behavior the quality that we call school patriotism?

(4) In almost every case a certain desired change of knowledge or skill or power can be attained by any one of several sets of bonds. Which of them is the best? What are the advantages of each? For example, learning to add may include the bonds "0 and 0 are 0," "0 and 1 are 1," "0 and 2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these may be all left unformed, the pupil being taught the habits of entering 0 as the sum of a column that is composed of zeros and otherwise neglecting 0 in addition. Are the rules of usage worth teaching as a means toward correct speech, or is the time better spent in detailed practice in correct speech itself?

(5) A bond to be formed may be formed in any one of many degrees of strength. Which of these is, at any given stage of learning the subject, the most desirable, all things considered? For example, shall the dates of all the early settlements of North America be learned so that the exact year will be remembered for ten years, or so that the exact date will be remembered for ten minutes and the date with an error plus or minus of ten years will be remembered for a year or two? Shall the tables of inches, feet, and yards, and pints, quarts, and gallons be learned at their first appearance so as to be remembered for a year, or shall they be learned only well enough to be usable in the work of that week, which in turn fixes them to last for a month or so? Should a pupil in the first year of study of French have such perfect connections between the sounds of French words and their meanings that he can understand simple sentences containing them spoken at an ordinary rate of speaking? Or is slow speech permissible, and even imperative, on the part of the teacher, with gradual increase of rate?

(6) In almost every case, any set of bonds may produce the desired change when presented in any one of several orders. Which is the best order? What are the advantages of each? Certain systems for teaching handwriting perfect the elementary movements one at a time and then teach their combination in words and sentences. Others begin and continue with the complex movement-series that actual words require. What do the latter lose and gain? The bonds constituting knowledge of the metric system are now formed late in the pupil's course. Would it be better if they were formed early as a means of facilitating knowledge of decimal fractions?

(7) What are the original tendencies and pre-school acquisitions upon which the connection-forming of the elementary school may be based or which it has to counteract? For example, if a pupil knows the meaning of a heard word, he may read it understandingly from getting its sound, as by phonic reconstruction. What words does the average beginner so know? What are the individual differences in this respect? What do the instincts of gregariousness, attention-getting, approval, and helpfulness recommend concerning group-work versus individual-work, and concerning the size of a group that is most desirable? The original tendency of the eyes is certainly not to move along a line from left to right of a page, then back in one sweep and along the next line. What is their original tendency when confronted with the printed page, and what must we do with it in teaching reading?

(8) What armament of satisfiers and annoyers, of positive and negative interests and motives, stands ready for use in the formation of the intrinsically uninteresting connections between black marks and meanings, numerical exercises and their answers, words and their spelling, and the like? School practice has tried, more or less at random, incentives and deterrents from quasi-physical pain to the most sentimental fondling, from sheer cajolery to philosophical argument, from appeals to assumed savage and primitive traits to appeals to the interest in automobiles, flying-machines, and wireless telegraphy. Can not psychology give some rules for guidance, or at least limit experimentation to its more hopeful fields?

(9) The general conditions of efficient learning are described in manuals of educational psychology. How do these apply in the case of each task of the elementary school? For example, the arrangement of school drills in addition and in short division in the form of practice experiments has been found very effective in producing interest in the work and in improvement at it. In what other arithmetical functions may we expect the same?

(10) Beside the general principles concerning the nature and causation of individual differences, there must obviously be, in existence or obtainable as a possible result of proper investigation, a great fund of knowledge of special differences relevant to the learning of reading, spelling, geography, arithmetic, and the like. What are the facts as far as known? What are the means of learning more of them? Courtis finds that a child may be specially strong in addition and yet be specially weak in subtraction in comparison with others of his age and grade. It even seems that such subtle and intricate tendencies are inherited. How far is such specialization the rule? Is it, for example, the case that a child may have a special gift for spelling certain sorts of words, for drawing faces rather than flowers, for learning ancient history rather than modern?

Such are our problems: this volume discusses them in the case of arithmetic. The student who wishes to relate the discussion to the general pedagogy of arithmetic may profitably read, in connection with this volume: The Teaching of Elementary Mathematics, by D. E. Smith ['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic, by the author ['21].

THE PSYCHOLOGY OF ARITHMETIC

THE PSYCHOLOGY OF
ARITHMETIC

CHAPTER I

THE NATURE OF ARITHMETICAL ABILITIES

According to common sense, the task of the elementary school is to teach:—(1) the meanings of numbers, (2) the nature of our system of decimal notation, (3) the meanings of addition, subtraction, multiplication, and division, and (4) the nature and relations of certain common measures; to secure (5) the ability to add, subtract, multiply, and divide with integers, common and decimal fractions, and denominate numbers, (6) the ability to apply the knowledge and power represented by (1) to (5) in solving problems, and (7) certain specific abilities to solve problems concerning percentage, interest, and other common occurrences in business life.

This statement of the functions to be developed and improved is sound and useful so far as it goes, but it does not go far enough to make the task entirely clear. If teachers had nothing but the statement above as a guide to what changes they were to make in their pupils, they would often leave out important features of arithmetical training, and put in forms of training that a wise educational plan would not tolerate. It is also the case that different leaders in arithmetical teaching, though they might all subscribe to the general statement of the previous paragraph, certainly do not in practice have identical notions of what arithmetic should be for the elementary school pupil.

The ordinary view of the nature of arithmetical learning is obscure or inadequate in four respects. It does not define what 'knowledge of the meanings of numbers' is; it does not take account of the very large amount of teaching of language which is done and should be done as a part of the teaching of arithmetic; it does not distinguish between the ability to meet certain quantitative problems as life offers them and the ability to meet the problems provided by textbooks and courses of study; it leaves 'the ability to apply arithmetical knowledge and power' as a rather mystical general faculty to be improved by some educational magic. The four necessary amendments may be discussed briefly.

KNOWLEDGE OF THE MEANINGS OF NUMBERS

Knowledge of the meanings of the numbers from one to ten may mean knowledge that 'one' means a single thing of the sort named, that two means one more than one, that three means one more than two, and so on. This we may call the series meaning. To know the meaning of 'six' in this sense is to know that it is one more than five and one less than seven—that it is between five and seven in the number series. Or we may mean by knowledge of the meanings of numbers, knowledge that two fits a collection of two units, that three fits a collection of three units, and so on, each number being a name for a certain sized collection of discrete things, such as apples, pennies, boys, balls, fingers, and the other customary objects of enumeration in the primary school. This we may call the collection meaning. To know the meaning of six in this sense is to be able to name correctly any collection of six separate, easily distinguishable individual objects. In the third place, knowledge of the numbers from one to ten may mean knowledge that two is twice whatever is called one, that three is three times whatever is one, and so on. This is, of course, the ratio meaning. To know the meaning of six in this sense is to know that if _________ is one, a line half a foot long is six, that if [ __ ] is one, [ ____________ ] is about six, while if [ _ ] is one, [ ______ ] is about six, and the like. In the fourth place, the meaning of a number may be a smaller or larger fraction of its implications—its numerical relations, facts about it. To know six in this sense is to know that it is more than five or four, less than seven or eight, twice three, three times two, the sum of five and one, or of four and two, or of three and three, two less than eight—that with four it makes ten, that it is half of twelve, and the like. This we may call the 'nucleus of facts' or relational meaning of a number.

Ordinary school practice has commonly accepted the second meaning as that which it is the task of the school to teach beginners, but each of the other meanings has been alleged to be the essential one—the series idea by Phillips ['97], the ratio idea by McLellan and Dewey ['95] and Speer ['97], and the relational idea by Grube and his followers.

This diversity of views concerning what the function is that is to be improved in the case of learning the meanings of the numbers one to ten is not a trifling matter of definition, but produces very great differences in school practice. Consider, for example, the predominant value assigned to counting by Phillips in the passage quoted below, and the samples of the sort of work at which children were kept employed for months by too ardent followers of Speer and Grube.

THE SERIES IDEA OVEREMPHASIZED

"This is essentially the counting period, and any words that can be arranged into a series furnish all that is necessary. Counting is fundamental, and counting that is spontaneous, free from sensible observation, and from the strain of reason. A study of these original methods shows that multiplication was developed out of counting, and not from addition as nearly all textbooks treat it. Multiplication is counting. When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations.

I would emphasize again that during the counting period there is a somewhat spontaneous development of the number series-idea which Preyer has discussed in his Arithmogenesis; that an immense momentum is given by a systematic series of names; and that these names are generally first learned and applied to objects later. A lady teacher told me that the Superintendent did not wish the teachers to allow the children to count on their fingers, but she failed to see why counting with horse-chestnuts was any better. Her children could hardly avoid using their fingers in counting other objects yet they followed the series to 100 without hesitation or reference to their fingers. This spontaneous counting period, or naming and following the series, should precede its application to objects." [D.E. Phillips, '97, p. 238.]

THE RATIO IDEA OVEREMPHASIZED

Fig. 1.

"Ratios.—1. Select solids having the relation, or ratio, of a, b, c, d, o, e.

2. Name the solids, a, b, c, d, o, e.

The means of expressing must be as freely supplied as the means of discovery. The pupil is not expected to invent terms.

3. Tell all you can about the relation of these units.

4. Unite units and tell what the sum equals.

5. Make statements like this: o less e equals b.

6. c can be separated into how many d's? into how many b's?

7. c can be separated into how many b's? What is the name of the largest unit that can be found in both c and d an exact number of times?

8. Each of the other units equals what part of c?

9. If b is 1, what is each of the other units?

10. If a is 1, what is each of the other units?

11. If b is 1, how many 1's are there in each of the other units?

12. If d is 1, how many 1's and parts of 1 in each of the other units?

13. 2 is the relation of what units?

14. 3 is the relation of what units?

15. ¹⁄₂ is the relation of what units?

16. ²⁄₃ is the relation of what units?

17. Which units have the relation ³⁄₂?

18. Which unit is 3 times as large as ¹⁄₂ of b?

19. c equals 6 times ¹⁄₃ of what unit?

20. ¹⁄₃ of what unit equals ¹⁄₆ of c?

21. What equals ¹⁄₂ of c? d equals how many sixths of c?

22. o equals 5 times ¹⁄₃ of what unit?

23. ¹⁄₃ of what unit equals ¹⁄₅ of o?

24. ²⁄₃ of d equals what unit? b equals how many thirds of d?

25. 2 is the ratio of d to ¹⁄₃ of what unit? 3 is the ratio of d to ¹⁄₂ of what unit?

26. d equals ³⁄₄ of what unit? ³⁄₄ is the ratio of what units?" [Speer, '97, p. 9f.]

THE RELATIONAL IDEA OVEREMPHASIZED

An inspection of books of the eighties which followed the "Grube method" (for example, the New Elementary Arithmetic by E.E. White ['83]) will show undue emphasis on the relational ideas. There will be over a hundred and fifty successive tasks all, or nearly all, on + 7 and − 7. There will be much written work of the sort shown below:

Add:

which must have sorely tried the eyes of all concerned. Pupils are taught to "give the analysis and synthesis of each of the nine digits." Yet the author states that he does not carry the principle of the Grube method "to the extreme of useless repetition and mechanism."

It should be obvious that all four meanings have claims upon the attention of the elementary school. Four is the thing between three and five in the number series; it is the name for a certain sized collection of discrete objects; it is also the name for a continuous magnitude equal to four units—for four quarts of milk in a gallon pail as truly as for four separate quart-pails of milk; it is also, if we know it well, the thing got by adding one to three or subtracting six from ten or taking two two's or half of eight. To know the meaning of a number means to know somewhat about it in all of these respects. The difficulty has been the narrow vision of the extremists. A child must not be left interminably counting; in fact the one-more-ness of the number series can almost be had as a by-product. A child must not be restricted to exercises with collections objectified as in Fig. 2 or stated in words as so many apples, oranges, hats, pens, etc., when work with measurement of continuous quantities with varying units—inches, feet, yards, glassfuls, pints, quarts, seconds, minutes, hours, and the like—is so easy and so significant. On the other hand, the elaboration of artificial problems with fictitious units of measure just to have relative magnitudes as in the exercises on page 5 is a wasteful sacrifice. Similarly, special drills emphasizing the fact that eighteen is eleven and seven, twelve and six, three less than twenty-one, and the like, are simply idolatrous; these facts about eighteen, so far as they are needed, are better learned in the course of actual column-addition and -subtraction.

Fig. 2.

ARITHMETICAL LANGUAGE

The second improvement to be made in the ordinary notion of what the functions to be improved are in the case of arithmetic is to include among these functions the knowledge of certain words. The understanding of such words as both, all, in all, together, less, difference, sum, whole, part, equal, buy, sell, have left, measure, is contained in, and the like, is necessary in arithmetic as truly as is the understanding of numbers themselves. It must be provided for by the school; for pre-school and extra-school training does not furnish it, or furnishes it too late. It can be provided for much better in connection with the teaching of arithmetic than in connection with the teaching of English.

It has not been provided for. An examination of the first fifty pages of eight recent textbooks for beginners in arithmetic reveals very slight attention to this matter at the best and no attention at all in some cases. Three of the books do not even use the word sum, and one uses it only once in the fifty pages. In all the four hundred pages the word difference occurs only twenty times. When the words are used, no great ingenuity or care appears in the means of making sure that their meanings are understood.

The chief reason why it has not been provided for is precisely that the common notion of what the functions are that arithmetic is to develop has left out of account this function of intelligent response to quantitative terms, other than the names of the numbers and processes.

Knowledge of language over a much wider range is a necessary element in arithmetical ability in so far as the latter includes ability to solve verbally stated problems. As arithmetic is now taught, it does include that ability, and a large part of the time of wise teaching is given to improving the function 'knowing what a problem states and what it asks for.' Since, however, this understanding of verbally stated problems may not be an absolutely necessary element of arithmetic, it is best to defer its consideration until we have seen what the general function of problem-solving is.

PROBLEM-SOLVING

The third respect in which the function, 'ability in arithmetic,' needs clearer definition, is this 'problem-solving.' The aim of the elementary school is to provide for correct and economical response to genuine problems, such as knowing the total due for certain real quantities at certain real prices, knowing the correct change to give or get, keeping household accounts, calculating wages due, computing areas, percentages, and discounts, estimating quantities needed of certain materials to make certain household or shop products, and the like. Life brings these problems usually either with a real situation (as when one buys and counts the cost and his change), or with a situation that one imagines or describes to himself (as when one figures out how much money he must save per week to be able to buy a forty-dollar bicycle before a certain date). Sometimes, however, the problem is described in words to the person who must solve it by another person (as when a life insurance agent says, 'You pay only 25 cents a week from now till—and you get $250 then'; or when an employer says, 'Your wages would be 9 dollars a week, with luncheon furnished and bonuses of such and such amounts'). Sometimes also the problem is described in printed or written words to the person who must solve it (as in an advertisement or in the letter of a customer asking for an estimate on this or that). The problem may be in part real, in part imagined or described to oneself, and in part described to one orally or in printed or written words (as when the proposed articles for purchase lie before one, the amount of money one has in the bank is imagined, the shopkeeper offers 10 percent discount, and the printed price list is there to be read).

To fit pupils to solve these real, personally imagined, or self-described problems, and 'described-by-another' problems, schools have relied almost exclusively on training with problems of the last sort only. The following page taken almost at random from one of the best recent textbooks could be paralleled by thousands of others; and the oral problems put by teachers have, as a rule, no real situation supporting them.

1. At 70 cents per 100 pounds, what will be the amount of duty on an invoice of 3622 steel rails, each rail being 27 feet long and weighing 60 pounds to the yard?

2. A man had property valued at $6500. What will be his taxes at the rate of $10.80 per $1000?

3. Multiply seventy thousand fourteen hundred-thousandths by one hundred nine millionths, and divide the product by five hundred forty-five.

4. What number multiplied by 43¾ will produce 265⁵⁄₈?

5. What decimal of a bushel is 3 quarts?

6. A man sells ⁵⁄₈ of an acre of land for $93.75. What would be the value of his farm of 150¾ acres at the same rate?

7. A coal dealer buys 375 tons coal at $4.25 per ton of 2240 pounds. He sells it at $4.50 per ton of 2000 pounds. What is his profit?

8. Bought 60 yards of cloth at the rate of 2 yards for $5, and 80 yards more at the rate of 4 yards for $9. I immediately sold the whole of it at the rate of 5 yards for $12. How much did I gain?

9. A man purchased 40 bushels of apples at $1.50 per bushel. Twenty-five hundredths of them were damaged, and he sold them at 20 cents per peck. He sold the remainder at 50 cents per peck. How much did he gain or lose?

10. If oranges are 37½ cents per dozen, how many boxes, each containing 480, can be bought for $60?

11. A man can do a piece of work in 18¾ days. What part of it can he do in 6²⁄₃ days?

12. How old to-day is a boy that was born Oct. 29, 1896? [Walsh, '06, Part I, p. 165.]

As a result, teachers and textbook writers have come to think of the functions of solving arithmetical problems as identical with the function of solving the described problems which they give in school in books, examination papers, and the like. If they do not think explicitly that this is so, they still act in training and in testing pupils as if it were so.

It is not. Problems should be solved in school to the end that pupils may solve the problems which life offers. To know what change one should receive after a given real purchase, to keep one's accounts accurately, to adapt a recipe for six so as to make enough of the article for four persons, to estimate the amount of seed required for a plot of a given size from the statement of the amount required per acre, to make with surety the applications that the household, small stores, and ordinary trades require—such is the ability that the elementary school should develop. Other things being equal, the school should set problems in arithmetic which life then and later will set, should favor the situations which life itself offers and the responses which life itself demands.

Other things are not always equal. The same amount of time and effort will often be more productive toward the final end if directed during school to 'made-up' problems. The keeping of personal financial accounts as a school exercise is usually impracticable, partly because some of the children have no earnings or allowance—no accounts to keep, and partly because the task of supervising work when each child has a different problem is too great for the teacher. The use of real household and shop problems will be easy only when the school program includes the household arts and industrial education, and when these subjects themselves are taught so as to improve the functions used by real life. Very often the most efficient course is to make sure that arithmetical procedures are applied to the real and personally initiated problems which they fit, by having a certain number of such problems arise and be solved; then to make sure that the similarity between these real problems and certain described problems of the textbook or teacher's giving is appreciated; and then to give the needed drill work with described problems. In many cases the school practice is fairly well justified in assuming that solving described problems will prepare the pupil to solve the corresponding real problems actually much better than the same amount of time spent on the real problems themselves.

All this is true, yet the general principle remains that, other things being equal, the school should favor real situations, should present issues as life will present them.

Where other things make the use of verbally described problems of the ordinary type desirable, these should be chosen so as to give a maximum of preparation for the real applications of arithmetic in life. We should not, for example, carelessly use any problem that comes to mind in applying a certain principle, but should stop to consider just what the situations of life really require and show clearly the application of that principle. For example, contrast these two problems applying cancellation:—

A. A man sold 24 lambs at $18 apiece on each of six days, and bought 8 pounds of metal with the proceeds. How much did he pay per ounce for the metal?

B. How tall must a rectangular tank 16" long by 8" wide be to hold as much as a rectangular tank 24" by 18" by 6"?

The first problem not only presents a situation that would rarely or never occur, but also takes a way to find the answer that would not, in that situation, be taken since the price set by another would determine the amount.

Much thought and ingenuity should in the future be expended in eliminating problems whose solution does not improve the real function to be improved by applied arithmetic, or improves it at too great cost, and in devising problems which prepare directly for life's demands and still can fit into a curriculum that can be administered by one teacher in charge of thirty or forty pupils, under the limitations of school life.

The following illustrations will to some extent show concretely what the ability to apply the knowledge and power represented by abstract or pure arithmetic—the so-called fundamentals—in solving problems should mean and what it should not mean.

Samples of Desirable Applications of Arithmetic in Problems where the Situation is Actually Present to Sense in Whole or in Part

Keeping the scores and deciding which side beat and by how much in appropriate classroom games, spelling matches, and the like.

Computing costs, making and inspecting change, taking inventories, and the like with a real or play store.

Mapping the school garden, dividing it into allotments, planning for the purchase of seeds, and the like.

Measuring one's own achievement and progress in tests of word-knowledge, spelling, addition, subtraction, speed of writing, and the like. Measuring the rate of improvement per hour of practice or per week of school life, and the like.

Estimating costs of food cooked in the school kitchen, articles made in the school shops, and the like.

Computing the cost of telegrams, postage, expressage, for a real message or package, from the published tariffs.

Computing costs from mail order catalogues and the like.

Samples of Desirable Applications of Arithmetic where the Situation is Not Present to Sense

The samples given here all concern the subtraction of fractions. Samples concerning any other arithmetical principle may be found in the appropriate pages of any text which contains problem-material selected with consideration of life's needs.

A

1. Dora is making jelly. The recipe calls for 24 cups of sugar and she has only 21½. She has no time to go to the store so she has to borrow the sugar from a neighbor. How much must she get?

Subtract

2. A box full of soap weighs 29½ lb. The empty box weighs 3½ 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 11½ 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 18½ qt. How many more quarts must she pick?

B

This table of numbers tells what Nell's baby sister Mary weighed every two months from the time she was born till she was a year old.

1. How much did the Adams baby gain in the first two months?

2. How much did the Adams baby gain in the second two months?

3. In the third two months?

4. In the fourth two months?

5. From the time it was 8 months old till it was 10 months old?

6. In the last two months?

7. From the time it was born till it was 6 months old?

C

1. Helen's exact average for December was 87¹⁄₃. Kate's was 84¹⁄₂. How much higher was Helen's than Kate's?

87¹⁄₃
84¹⁄₂
———

How do you think of ¹⁄₂ and ¹⁄₃?

How do you think of 1²⁄₆?

How do you change the 4?

2. Find the exact average for each girl in the following list. Write the answers clearly so that you can see them easily. You will use them in solving problems 3, 4, 5, 6, 7, and 8.

3. Which girl had the highest average?

4. How much higher was her average than the next highest?

5. How much difference was there between the highest and the lowest girl?

6. Was Emma's average higher or lower than Louise's? How much?

7. How much difference was there between Alice's average and Dora's?

8. How much difference was there between Mary's average and Nell's?

9. Write five other problems about these averages, and solve each of them.

Samples of Undesirable Applications of Arithmetic[1]

Will has XXI marbles, XII jackstones, XXXVI pieces of string. How many things had he?

George's kite rose CDXXXV feet and Tom's went LXIII feet higher. How high did Tom's kite rise?

If from DCIV we take CCIV the result will be a number IV times as large as the number of dollars Mr. Dane paid for his horse. How much did he pay for his horse?

Hannah has ⁵⁄₈ of a dollar, Susie ⁷⁄₂₅, Nellie ³⁄₄, Norah ¹³⁄₁₆. How much money have they all together?

A man saves 3¹⁷⁄₈₀ dollars a week. How much does he save in a year?

A tree fell and was broken into 4 pieces, 13¹⁄₆ feet, 10³⁄₇ feet, 8¹⁄₂ feet, and 7¹⁶⁄₂₁ feet long. How tall was the tree?

Annie's father gave her 20 apples to divide among her friends. She gave each one 2²⁄₉ apples apiece. How many playmates had she?

John had 17²⁄₅ apples. He divided his whole apples into fifths. How many pieces had he in all?

A landlady has 3³⁄₇ pies to be divided among her 8 boarders. How much will each boarder receive?

There are twenty columns of spelling words in Mary's lesson and 16 words in each column. How many words are in her lesson?

There are 9 nuts in a pint. How many pints in a pile of 5,888,673 nuts?

The Adams school contains eight rooms; each room contains 48 pupils; if each pupil has eight cents, how much have they together?

A pile of wood in the form of a cube contains 15½ cords. What are the dimensions to the nearest inch?

A man 6 ft. high weighs 175 lb. How tall is his wife who is of similar build, and weighs 125 lb.?

A stick of timber is in the shape of the frustum of a square pyramid, the lower base being 22 in. square and the upper 14 in. square. How many cubic feet in the log, if it is 22 ft. long?

Mr. Ames, being asked his age, replied: "If you cube one half of my age and add 41,472 to the result, the sum will be one half the cube of my age. How old am I?"

These samples, just given, of the kind of problem-solving that should not be emphasized in school training refer in some cases to books of forty years back, but the following represent the results of a collection made in 1910 from books then in excellent repute. It required only about an hour to collect them; and I am confident that a thousand such problems describing situations that the pupil will never encounter in real life, or putting questions that he will never be asked in real life, could easily be found in any ten textbooks of the decade from 1900 to 1910.

If there are 250 kernels of corn on one ear, how many are there on 24 ears of corn the same size?

Maud is four times as old as her sister, who is 4 years old. What is the sum of their ages?

If the first century began with the year 1, with what year does it end?

Every spider has 8 compound eyes. How many eyes have 21 spiders?

A nail 4 inches long is driven through a board so that it projects 1.695 inches on one side and 1.428 on the other. How thick is the board?

Find the perimeter of an envelope 5 in. by 3¼ in.

How many minutes in ⁵⁄₉ of ⁹⁄₄ of an hour?

Mrs. Knox is ³⁄₄ as old as Mr. Knox, who is 48 years old. Their son Edward is ⁴⁄₉ as old as his mother. How old is Edward?

Suppose a pie to be exactly round and 10½ miles in diameter. If it were cut into 6 equal pieces, how long would the curved edge of each piece be?

8¹⁄₃% of a class of 36 boys were absent on a rainy day. 33¹⁄₃% of those present went out of the room to the school yard. How many were left in the room?

Just after a ton of hay was weighed in market, a horse ate one pound of it. What was the ratio of what he ate to what was left?

If a fan having 15 rays opens out so that the outer rays form a straight line, how many degrees are there between any two adjacent rays?

One half of the distance between St. Louis and New Orleans is 280 miles more than ¹⁄₁₀ of the distance; what is the distance between these places?

If the pressure of the atmosphere is 14.7 lb. per square inch what is the pressure on the top of a table 1¼ yd. long and ²⁄₃ yd. wide?

¹³⁄₂₈ of the total acreage of barley in 1900 was 100,000 acres; what was the total acreage?

What is the least number of bananas that a mother can exactly divide between her 2 sons, or among her 4 daughters, or among all her children?

If Alice were two years older than four times her actual age she would be as old as her aunt, who is 38 years old. How old is Alice?

Three men walk around a circular island, the circumference of which is 360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a day. If they start together and walk in the same direction, how many days will elapse before they will be together again?

With only thirty or forty dollars a year to spend on a pupil's education, of which perhaps eight dollars are spent on improving his arithmetical abilities, the immediate guidance of his responses to real situations and personally initiated problems has to be supplemented largely by guidance of his responses to problems described in words, diagrams, pictures, and the like. Of these latter, words will be used most often. As a consequence the understanding of the words used in these descriptions becomes a part of the ability required in arithmetic. Such word knowledge is also required in so far as the problems to be solved in real life are at times described, as in advertisements, business letters, and the like.

This is recognized by everybody in the case of words like remainder, profit, loss, gain, interest, cubic capacity, gross, net, and discount, but holds equally of let, suppose, balance, average, total, borrowed, retained, and many such semi-technical words, and may hold also of hundreds of other words unless the textbook and teacher are careful to use only words and sentence structures which daily life and the class work in English have made well known to the pupils. To apply arithmetic to a problem a pupil must understand what the problem is; problem-solving depends on problem-reading. In actual school practice training in problem-reading will be less and less necessary as we get rid of problems to be solved simply for the sake of solving them, unnecessarily unreal problems, and clumsy descriptions, but it will remain to some extent as an important joint task for the 'arithmetic' and 'reading' of the elementary school.

ARITHMETICAL REASONING

The last respect in which the nature of arithmetical abilities requires definition concerns arithmetical reasoning. An adequate treatment of the reasoning that may be expected of pupils in the elementary school and of the most efficient ways to encourage and improve it cannot be given until we have studied the formation of habits. For reasoning is essentially the organization and control of habits of thought. Certain matters may, however, be decided here. The first concerns the use of computation and problems merely for discipline,—that is, the emphasis on training in reasoning regardless of whether the problem is otherwise worth reasoning about. It used to be thought that the mind was a set of faculties or abilities or powers which grew strong and competent by being exercised in a certain way, no matter on what they were exercised. Problems that could not occur in life, and were entirely devoid of any worthy interest, save the intellectual interest in solving them, were supposed to be nearly or quite as useful in training the mind to reason as the genuine problems of the home, shop, or trade. Anything that gave the mind a chance to reason would do; and pupils labored to find when the minute hand and hour hand would be together, or how many sheep a shepherd had if half of what he had plus ten was one third of twice what he had!

We now know that the training depends largely on the particular data used, so that efficient discipline in reasoning requires that the pupil reason about matters of real importance. There is no magic essence or faculty of reasoning that works in general and irrespective of the particular facts and relations reasoned about. So we should try to find problems which not only stimulate the pupil to reason, but also direct his reasoning in useful channels and reward it by results that are of real significance. We should replace the purely disciplinary problems by problems that are also valuable as special training for important particular situations of life. Reasoning sought for reasoning's sake alone is too wasteful an expenditure of time and is also likely to be inferior as reasoning.

The second matter concerns the relative merits of 'catch' problems, where the pupil has to go against some customary habit of thinking, and what we may call 'routine' problems, where the regular ways of thinking that have served him in the past will, except for some blunder, guide him rightly.

Consider, for example, these four problems:

1. "A man bought ten dozen eggs for $2.50 and sold them for 30 cents a dozen. How many cents did he lose?"

2. "I went into Smith's store at 9 A.M. and remained until 10 A.M. I bought six yards of gingham at 40 cents a yard and three yards of muslin at 20 cents a yard and gave a $5.00 bill. How long was I in the store?"

3. "What must you divide 48 by to get half of twice 6?"

4. "What must you add to 19 to get 30?"

The 'catch' problem is now in disrepute, the wise teacher feeling by a sort of intuition that to willfully require a pupil to reason to a result sharply contrary to that to which previous habits lead him is risky. The four illustrations just given show, however, that mere 'catchiness' or 'contra-previous-habit-ness' in a problem is not enough to condemn it. The fourth problem is a catch problem, but so useful a one that it has been adopted in many modern books as a routine drill! The first problem, on the contrary, all, save those who demand no higher criterion for a problem than that it make the pupil 'think,' would reject. It demands the reversal of fixed habits to no valid purpose; for in life the question in such case would never (or almost never) be 'How many cents did he lose?' but 'What was the result?' or simply 'What of it?' This problem weakens without excuse the child's confidence in the training he has had. Problems like (2) are given by teachers of excellent reputation, but probably do more harm than good. If a pupil should interrupt his teacher during the recitation in arithmetic by saying, "I got up at 7 o'clock to multiply 9 by 2¾ and got 24¾ for my answer; was that the right time to get up?" the teacher would not thank fortune for the stimulus to thought but would think the child a fool. Such catch questions may be fairly useful as an object lesson on the value of search for the essential element in a situation if a great variety of them are given one after another with routine problems intermixed and with warning of the general nature of the exercise at the beginning. Even so, it should be remembered that reasoning should be chiefly a force organizing habits, not opposing them; and also that there are enough bad habits to be opposed to give all necessary training. Fabricated puzzle situations wherein a peculiar hidden element of the situation makes the good habits called up by the situation misleading are useful therefore rather as a relief and amusing variation in arithmetical work than as stimuli to thought.

Problems like the third quoted above we might call puzzling rather than 'catch' problems. They have value as drills in analysis of a situation into its elements that will amuse the gifted children, and as tests of certain abilities. They also require that of many confusing habits, the right one be chosen, rather than that ordinary habits be set aside by some hidden element in the situation. Not enough is known about their effect to enable us to decide whether or not the elementary school should include special facility with them as one of the arithmetical functions that it specially trains.

The fourth 'catch' quoted above, which all would admit is a good problem, is good because it opposes a good habit for the sake of another good habit, forces the analysis of an element whose analysis life very much requires, and does it with no obvious waste. It is not safe to leave a child with the one habit of responding to 'add, 19, 30' by 49, for in life the 'have 19, must get .... to have 30' situation is very frequent and important.

On the whole, the ordinary problems which ordinary life proffers seem to be the sort that should be reasoned out, though the elementary school may include the less noxious forms of pure mental gymnastics for those pupils who like them.

SUMMARY

These discussions of the meanings of numbers, the linguistic demands of arithmetic, the distinction between scholastic and real applications of arithmetic, and the possible restrictions of training in reasoning,—may serve as illustrations of the significance of the question, "What are the functions that the elementary school tries to improve in its teaching of arithmetic?" Other matters might well be considered in this connection, but the main outline of the work of the elementary school is now fairly clear. The arithmetical functions or abilities which it seeks to improve are, we may say:—

(1) Working knowledge of the meanings of numbers as names for certain sized collections, for certain relative magnitudes, the magnitude of unity being known, and for certain centers or nuclei of relations to other numbers.

(2) Working knowledge of the system of decimal notation.

(3) Working knowledge of the meanings of addition, subtraction, multiplication, and division.

(4) Working knowledge of the nature and relations of certain common measures.

(5) Working ability to add, subtract, multiply, and divide with integers, common and decimal fractions, and denominate numbers, all being real positive numbers.

(6) Working knowledge of words, symbols, diagrams, and the like as required by life's simpler arithmetical demands or by economical preparation therefor.

(7) The ability to apply all the above as required by life's simpler arithmetical demands or by economical preparation therefor, including (7 a) certain specific abilities to solve problems concerning areas of rectangles, volumes of rectangular solids, percents, interest, and certain other common occurrences in household, factory, and business life.

THE SOCIOLOGY OF ARITHMETIC

The phrase 'life's simpler arithmetical demands' is necessarily left vague. Just what use is being made of arithmetic in this country in 1920 by each person therein, we know only very roughly. What may be called a 'sociology' of arithmetic is very much needed to investigate this matter. For rare or difficult demands the elementary school should not prepare; there are too many other desirable abilities that it should improve.

A most interesting beginning at such an inventory of the actual uses of arithmetic has been made by Wilson ['19] and Mitchell.[2] Although their studies need to be much extended and checked by other methods of inquiry, two main facts seem fairly certain.

First, the great majority of people in the great majority of their doings use only very elementary arithmetical processes. In 1737 cases of addition reported by Wilson, seven eighths were of five numbers or less. Over half of the multipliers reported were one-figure numbers. Over 95 per cent of the fractions operated with were included in this list: ¹⁄₂ ¹⁄₄ ³⁄₄ ¹⁄₃ ²⁄₃ ¹⁄₈ ³⁄₈ ¹⁄₅ ²⁄₅ ⁴⁄₅. Three fourths of all the cases reported were simple one-step computations with integers or United States money.

Second, they often use these very elementary processes, not because such are the quickest and most convenient, but because they have lost, or maybe never had, mastery of the more advanced processes which would do the work better. The 5 and 10 cent stores, the counter with "Anything on this counter for 25¢," and the arrangements for payments on the installment plan are familiar instances of human avoidance of arithmetic. Wilson found very slight use of decimals; and Mitchell found men computing with 49ths as common fractions when the use of decimals would have been more efficient. If given 120 seconds to do a test like that shown below, leading lawyers, physicians, manufacturers, and business men and their wives will, according to my experience, get only about half the work right. Many women, finding on their meat bill "7³⁄₈ lb. roast beef $2.36," will spend time and money to telephone the butcher asking how much roast beef was per pound, because they have no sure power in dividing by a mixed number.

Test

Perform the operations indicated. Express all fractions in answers in lowest terms.

Add:

Subtract:

Multiply:

Divide:

4½ ÷ 7 =

It seems probable that the school training in arithmetic of the past has not given enough attention to perfecting the more elementary abilities. And we shall later find further evidence of this. On the other hand, the fact that people in general do not at present use a process may not mean that they ought not to use it.

Life's simpler arithmetical demands certainly do not include matters like the rules for finding cube root or true discount, which no sensible person uses. They should not include matters like computing the lateral surface or volume of pyramids and cones, or knowing the customs of plasterers and paper hangers, which are used only by highly specialized trades. They should not include matters like interest on call loans, usury, exact interest, and the rediscounting of notes, which concern only brokers, bank clerks, and rich men. They should not include the technique of customs which are vanishing from efficient practice, such as simple interest on amount for times longer than a year, days of grace, or extremes and means in proportions. They should not include any elaborate practice with very large numbers, or decimals beyond thousandths, or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year.

When we have an adequate sociology of arithmetic, stating accurately just who should use each arithmetical ability and how often, we shall be able to define the task of the elementary school in this respect. For the present, we may proceed by common sense, guided by two limiting rules. The first is,—"It is no more desirable for the elementary school to teach all the facts of arithmetic than to teach all the words in the English language, or all the topography of the globe, or all the details of human physiology." The second is,—"It is not desirable to eliminate any element of arithmetical training until you have something better to put in its place."

CHAPTER II

THE MEASUREMENT OF ARITHMETICAL ABILITIES

One of the best ways to clear up notions of what the functions are which schools should develop and improve is to get measures of them. If any given knowledge or skill or power or ideal exists, it exists in some amount. A series of amounts of it, varying from less to more, defines the ability itself in a way that no general verbal description can do. Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to tell us what we mean by weight. By finding a series of words like only, smoke, another, pretty, answer, tailor, circus, telephone, saucy, and beginning, which are spelled correctly by known and decreasing percentages of children of the same age, or of the same school grade, we know better what we mean by 'spelling-difficulty.' Indeed, until we can measure the efficiency and improvement of a function, we are likely to be vague and loose in our ideas of what the function is.

A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: THE ABILITY TO ADD INTEGERS

Consider first, as a sample, the measurement of ability to add integers.

The following were the examples used in the measurements made by Stone ['08]:

The scoring was as follows: Credit of 1 for each column added correctly. Stone combined measures of other abilities with this in a total score for amount done correctly in 12 minutes. Stone also scored the correctness of the additions in certain work in multiplication.

Courtis uses a sheet of twenty-four tasks or 'examples,' each consisting of the addition of nine three-place numbers as shown below. Eight minutes is allowed. He scores the amount done by the number of examples, and also scores the number of examples done correctly, but does not suggest any combination of these two into a general-efficiency score.

927
379
756
837
924
110
854
965
344
———

The author long ago proposed that pupils be measured also with series like a to g shown below, in which the difficulty increases step by step.

Woody ['16] has constructed his well-known tests on this principle, though he uses only one example at each step of difficulty instead of eight or ten as suggested above. His test, so far as addition of integers goes, is:—

SERIES A. ADDITION SCALE (in part)

By Clifford Woody

In his original report, Woody gives no scheme for scoring an individual, wisely assuming that, with so few samples at each degree of difficulty, a pupil's score would be too unreliable for individual diagnosis. The test is reliable for a class; and for a class Woody used the degree of difficulty such that a stated fraction of the class can do the work correctly, if twenty minutes is allowed for the thirty-eight examples of the entire test.

The measurement of even so simple a matter as the efficiency of a pupil's responses to these tests in adding integers is really rather complex. There is first of all the problem of combining speed and accuracy into some single estimate. Stone gives no credit for a column unless it is correctly added. Courtis evades the difficulty by reporting both number done and number correct. The author's scheme, which gives specified weights to speed and accuracy at each step of the series, involves a rather intricate computation.

This difficulty of equating speed and accuracy in adding means precisely that we have inadequate notions of what the ability is that the elementary school should improve. Until, for example, we have decided whether, for a given group of pupils, fifteen Courtis attempts with ten right, is or is not a better achievement than ten Courtis attempts with nine right, we have not decided just what the business of the teacher of addition is, in the case of that group of pupils.

There is also the difficulty of comparing results when short and long columns are used. Correctness with a short column, say of five figures, testifies to knowledge of the process and to the power to do four successive single additions without error. Correctness with a long column, say of ten digits, testifies to knowledge of the process and to the power to do nine successive single additions without error. Now if a pupil's precision was such that on the average he made one mistake in eight single additions, he would get about half of his five-digit columns right and almost none of his ten-digit columns right. (He would do this, that is, if he added in the customary way. If he were taught to check results by repeated addition, by adding in half-columns and the like, his percentages of accurate answers might be greatly increased in both cases and be made approximately equal.) Length of column in a test of addition under ordinary conditions thus automatically overweights precision in the single additions as compared with knowledge of the process, and ability at carrying.

Further, in the case of a column of whatever size, the result as ordinarily scored does not distinguish between one, two, three, or more (up to the limit) errors in the single additions. Yet, obviously, a pupil who, adding with ten-digit columns, has half of his answer-figures wrong, probably often makes two or more errors within a column, whereas a pupil who has only one column-answer in ten wrong, probably almost never makes more than one error within a column. A short-column test is then advisable as a means of interpreting the results of a long-column test.

Finally, the choice of a short-column or of a long-column test is indicative of the measurer's notion of the kind of efficiency the world properly demands of the school. Twenty years ago the author would have been readier to accept a long-column test than he now is. In the world at large, long-column addition is being more and more done by machine, though it persists still in great frequency in the bookkeeping of weekly and monthly accounts in local groceries, butcher shops, and the like.

The search for a measure of ability to add thus puts the problem of speed versus precision, and of short-column versus long-column additions clearly before us. The latter problem has hardly been realized at all by the ordinary definitions of ability to add.

It may be said further that the measurement of ability to add gives the scientific student a shock by the lack of precision found everywhere in schools. Of what value is it to a graduate of the elementary school to be able to add with examples like those of the Courtis test, getting only eight out of ten right? Nobody would pay a computer for that ability. The pupil could not keep his own accounts with it. The supposed disciplinary value of habits of precision runs the risk of turning negative in such a case. It appears, at least to the author, imperative that checking should be taught and required until a pupil can add single columns of ten digits with not over one wrong answer in twenty columns. Speed is useful, especially indirectly as an indication of control of the separate higher-decade additions, but the social demand for addition below a certain standard of precision is nil, and its disciplinary value is nil or negative. This will be made a matter of further study later.

MEASUREMENTS OF ABILITIES IN COMPUTATION

Measurements of these abilities may be of two sorts—(1) of the speed and accuracy shown in doing one same sort of task, as illustrated by the Courtis test for addition shown on page 28; and (2) of how hard a task can be done perfectly (or with some specified precision) within a certain assigned time or less, as illustrated by the author's rough test for addition shown on pages 28 and 29, and by the Woody tests, when extended to include alternative forms.

The Courtis tests, originated as an improvement on the Stone tests and since elaborated by the persistent devotion of their author, are a standard instrument of the first sort for measuring the so-called 'fundamental' arithmetical abilities with integers. They are shown on this and the following page.

Tests of the second sort are the Woody tests, which include operations with integers, common and decimal fractions, and denominate numbers, the Ballou test for common fractions ['16], and the "Ladder" exercises of the Thorndike arithmetics. Some of these are shown on pages 36 to 41.

Courtis Test

Arithmetic. Test No. 1. Addition

Series B