The teaching of arithmetic

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Preface

This work has been prepared with a view to the needs of Teachers' Reading Circles and of those who are giving instruction or supervising tlie work in arithmetic in the elementary schools. The effort has been made to free it from the more difficult technicalities of mathematics as far as possible, and from that phraseology of the world of pedagogy that renders educational problems unnecessarily difficult. It is the author's desire to place before the large body of teachers of arithmetic, rather than the few who are interested in the technicalities of experiment, a brief summary of the development of the science and of the reasons for teaching it ; a statement of the subject matter, that may properly be selected for school purposes, and the arrangement of this matter in a course of study ; a consideration of a few of the technical features of arithmetic ; and a discussion of the work of the several school years. He feels it quite unnecessary, however, to repeat at length the simple explanations and devices that are found in any good textbook, believing that the teacher will usually find it advantageous in such matters to follow the book that she is using.

The problem of teaching arithmetic is not a particularly complex one. The world demands little of the school, hardly more than the four operations with integers, very simple fractions, and decimal fractions to two places ; a small number of tables of denominate numbers ; a few cases in mensuration ; and the ability to find a given per cent of a number. In America the schools are commonly given eight years in iv THE TEACHING OF ARITHMETIC

which to accomplish this work, and if they fail it is not for lack of time, but rather because they do not use this time wisely. If they introduce unnecessary features, such as obsolete tables or too elaborate explanations ; if they require business arithmetic that is beyond the understanding of the children ; and if they use hard methods of solving problems instead of simple ones, they cannot expect to bring the pupils to the standard that the world requires. This seems to be a part of the difficulty, but not all of it, and it is to assist teachers to find the difficulty, so far as it can be found at present, that this book is written.

If the author succeeds in making arithmetic more interesting to teacher and pupil, in making it touch more clearly the vital questions of daily life, in pointing out a line of work that is free from dangerous eccentricities, and in presenting a broad view of the subject as opposed to one in which small things obstruct the vision, the purposes that he has had in mind will be realized. It is with this hope that he submits this little work to his colleagues in the teaching profession.

DAVID EUGENE SMITH Teachers College Columbia University New York City

Contents

CHAPTER PAGE

I. The History of Arithmetic .... i

II. The Reasons for teaching Arithmetic ... ii

Chapter I The History Of Arithmetic

Of all the sciences, of all the subjects generally taught in the common schools, arithmetic is by far the oldest. Long before man had found for himself an alphabet, long before he first made rude ideographs upon wood or stone, he counted, he kept his tallies upon notched sticks, and he computed in some simple way by his fingers or by pebbles on the ground. He did not always count by tens, as in our decimal system ; indeed, this was a rather late device, and one suggested by his digits. At first he was quite content to count to two, and generations later to three, and then to four. Then he repeated his threes and had what we call a scale of three, and then, as time went on, he used a scale of four, and then a scale of five. In measurements he often used the scale of twelve, because he found that twelve is divisible by more factors than ten, and particularly by two and three and four ; but by the time he was ready to write his numbers the convenience of finger reckoning had become so generally recognized that ten was practically the universal radix. Nevertheless, there remain in our language and customs numerous relics of the duodecimal (scale of twelve) idea, such as the number of inches in a foot, of ounces in a troy pound, and of pence in a shilling, all influenced by the Roman inclination to make much use of twelve in practical computation.

The writing of numbers has undergone even more change than the number names. Not only was there usually a notation for each written language in ancient times, but some languages had several sets of numerals, as is seen in the three standard systems of Egypt, the two of Greece, and the somewhat varied forms in use in Rome. The Roman supremacy gave the numerals of these people great influence in Europe, and they were practically in universal use in the West until the close of the Middle Ages. The Romans themselves had no definite standard for their numerals. Whereas we write IV for four, IX for nine, and XL for forty, they usually wrote IIII, Villi, and XXXX. Occasionally they wrote IIXX for eighteen, and they would have written the year 1914 as MDCCCCXIIII rather than MCMXIV. Soon after books began to be printed a slight effort was made to use Roman numerals in fractions, as in the following facsimile from a German work of 1514:

VI 3^ip fcin © cd;i6 fl(^t<til/^«8 few f^etail &cr VIII Ad)t fltit g^n^ ma^m •

From Kobel's Arithmetic of 1514

THE HISTORY OF ARITHMETIC 3

The Romans, however, never wrote fractions hke this, and much that has been quite recently taught in the schools about these numerals is entirely modern.

It may sometimes be wondered how the world was able to handle numbers in, the days of Roman numerals. How, for example, did people add CLXVIand DXXIX, or multiply one of these numbers by another ? The answer is that they used small disks on a ruled board, or pebbles that were moved on wires or in grooves. The Latin name for "pebble " was calaibis, and from this we get our word "calculate." The following illustration shows the appearance of the medieval calculating board :

From the Title-page of Kobel's Arithmetic, 1514

Meantime there had arisen in the East, probably in India, although very likely subjected to influence from without, our present system of notation, and little by little this permeated the West. When it arose it was without a zero^ and hence without such place value as we use to-day ; but probably about the seventh century the zero appeared, and the conipleted system found its way northward into Persia and Arabia, and thence in due time it was transmitted to the West.

Some of the earliest Hindu forms are shown in the following table : ^

1 11 III nil

1 II III X IXIIX XX 7 ?

?J}JJ} Al^ll

I II + ,^

G A

-= ^ tp7 Pcco

^•3 J y'i'jf ^ o ^ CO ^

Table showing the Progress of Number Forms in India

After the zero was invented, the numerals appear in various manuscripts with forms like those on page 5.

All of this goes to show how long it took the world to settle upon the forms that we teach our children.

1 This and similar tables are from Smith and Karpinski's " The Hindu- Arabic Numerals," Ginn and Company, Boston, 191 1, to which reference is made- for complete descriptions.

THE HISTORY OF ARITHMETIC 5

1234567 890

Numerals used with Place Value

At first the subject of arithmetic was purely practical, the counting of arrows or sheep or men. For a long time this was all that number meant to the world, until the mystic age developed and philosophy began. Then numbers were differentiated, and odd and even were distinguished, and '" There 's luck in odd numbers " became a tenet of faith, while the even numbers became designated as earthly and feminine. The story is long and interesting,, that of the development of this mysticism, with its special prominence of three and

1 30uani a mi motto sikcnnimi : U.

First Page of the Treviso AriI-hmetic, 1478

seven. Furthermore, the movement led to a study of the properties of numbers, roots, and series. Number games became connected with the abacus, and so the practical and the mysterious were more or less blended even at times when they are generally regarded as widely separated.

THE HISTORY OF ARITHMETIC 7

The first printed arithmetic appeared at Treviso, a city situated north of Venice, in 1478. It may add a little to the interest of the subject to see the facsimile of the first page of this very rare book as shown on page 6. The first three lines read as follows : " Here beginneth a practica, very good and useful to anyone who wishes to use the mercantile art commonly called the art of the abacus." " Practica " and " art of the abacus " were then common names for business arithmetic.

The methods of multiplying have changed greatly in modern times. To illustrate this fact, two cases of multiplication, each showing the product of 456,789 by 987,654, taken from an Italian manuscript of the fifteenth century, are shown on page 8.

Long division has changed very greatly, and it will probably be of interest to the reader to see a facsimile of the first printed page in which our modern form occurs. It is from Calandri's arithmetic of 1491, printed at Florence, and is shown on page 9.

The growth of topics of arithmetic is also an interesting subject for investigation. We say that there are four fundamental operations, although once there was only one, and at another time the world recognized as many as nine. We operate chiefly with decimal fractions, as in working with dollars and cents, although these fractions are scarcely three hundred years old. We are impatient that a child stumbles over common fractions, and yet, so difficult did the world find the subject that for thousands of years only the unit fraction was used. We wonder how the long division form of greatest common divisor ever had place in arithmetic, and yet it was a practical necessity in business until about 1600 a.d. We feel that " partnership involving time " could never have been practical, and yet until a couple of centuries ago it was

From an Italian Manuscript of the Fifteenth Century

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o •?57r Uieitne 1 1 t4 uictine t^^s-

•paiti fo B g •paiti > e I

i\icni\c I (^o Uicnne o ff

From Calandri's Arithmetic, 1491 lO THE TEACHING OF ARITHMETIC

decidedly so. And thus it is with many topics of arithmetic, — they have changed from century to century, and even in our own time from year to year. It is well for a teacher to know a little of this history of the subject taught, although space does not allow for any serious consideration of the topic in this work. In the bibliography some reference will be found to sources easily available, and the teacher who wishes to see arithmetic in progress, as opposed to arithmetic stagnant and filled with the obsolete, should become acquainted with one or more of these works upon the subject. The history of arithmetic is the best single stimulus, to good method in teaching the subject.

Bibliography. Smith, The Teaching of Elementary Mathematics, New York, 1900; Smith, Rara Arithmetica, Boston, 1909; Smith and Karpinski, Hindu-Arabic Numerals, Boston, 191 1 ; Ball, A Primer of the History of Mathematics, London, 1895, and A Short Account of the History of Mathematics, London, 4th edition, 1908; Fink, History of Mathematics, translated by Beman and Smith, Chicago, igoo ; Cajori, History of Elementary Mathematics, New York, 1896, and History of Mathematics, New York, 1893 ; Jackson, The Educational Significance of Sixteenth Century Arithmetic, New York, 1906; Gow, A Short History of Greek Mathematics, Cambridge, 18S4; Conant, The Number Concept, New York, 1896; Brooks, Philosophy of Arithmetic, revised edition, Philadelphia, 1902. There are numerous works in German on the history of mathematics and of mathematical teaching, and a considerable number of works in French and Italian.

Chapter II The Reasons For Teaching Arithmetic

The ancients had less difficulty than we have in assigning a reason for teaching arithmetic, because they generally differentiated clearly between two phases of the subject. The Greeks, for example, called numerical calculation by the name logistic, and this subject was taught solely for practical purposes to those who were going into trade. A man might have been a very good philosopher or statesman or warrior without ever having learned to divide one long number by another. Such a piece of knowledge would probably have been looked upon as a bit of technical training, like our use of the slide rule or the arithmometer, two of the several modern machines by the use of which we can add, subtract, multiply, divide, raise to powers, and extract roots of numbers. On the other hand, the Greeks called their science of numbers arithmetic, a subject that had nothing whatever to do with addition, subtraction, multiplication, or division, and that excluded all applications to trade and industry. This subject was taught to the philosopher, and to the man of ""liberal education," as we still call him. It considered questions like the factorability of numbers, powers and roots, and series — topics having little if any practical application in the common walks of life. Therefore when a Greek was asked why he taught logistic, his answer was definite : It is to make a business man able to compute sufficiently well for his trade. If asked why he taught arithmetic, as the term was then used, his answer was still fairly clear, although open to debate : I teach it because it makes a man's mind more philosophic.

In the present day we have a somewhat more difficult task when we attempt to answer this question. Arithmetic with us includes the ancient logistic, and we teach the subject to all classes of people : to one who will become a day laborer, belonging to a class that never in the history of the world studied such a subject until very recently ; to the tradesman, who never uses or cares to use the chapter on prime numbers; to the statesman, who will probably have little opportunity to employ logarithms in any work that may come to him ; to the clergyman, to whom the metric system will soon be merely a name ; and to the housewife, the farmer, and all those who travel the multifarious walks of our complex human life. For us to tell why we teach the American arithmetic to all these people is by no means so easy as it was for the Greek to answer his simple question.

In general, however, we may say that as we have combined the ancient logistic (calculation) and arithmetic (theory) in one subject, so we have combined the Greek purposes, and teach this branch because it is useful in a business way to every one, and also because it gives a kind of training that other subjects do not give.

As to the first reason there can be no question. When the great mass of men were slaves the business phase was not so important ; but now that every man is to a great extent his own master, receiving money and spending it, some knowledge of calculation is necessary for every American citizen. To elaborate upon this point is superfluous. There is, however, one principle that should guide us in the consideration of this phase of the question : Whatever pretends to be practical in arithmetic should i-eally be so. We have no right to inject a mass of problems on antiquated investments, on obsolete forms of partnership, on forgotten methods of mercantile business, or on measures no longer in common use, and

THE REASONS FOR TEACHING ARITHMETIC 13

make the claim that these problems are practical. If we wish them for some other purpose, well and good ; but as practical problems they have no right to appear. To set up a false custom of the business world is as bad as to teach any other untruth ; it places arithmetic in particular, and education in general, in a false light before pupils and parents, and is unjustified by any reason that we can adduce. An obsolete business problem has just one reason for being, and this reason is that it has historical interest. We can secure the mental discipline as well by other means, and we have no right to handicap a child's mind with things that he will be forced to forget the minute he enters practical life.

Axiomatic as this statement may seem, however, it is by no means easy to limit the subject matter in arithmetic to the art of calculation and to problems that are representative only of the real business life of to-day. The custom of the school, the familiarity of the teacher with the traditional problem rather than with the demands of modern business, the genuine interest that children find in puzzle problems, the influence of the official curriculum — all militate against as rapid reform as might be desired. This condition of affairs is by no means limited to our own country; it is characteristic of the school everywhere and always. How it affects arithmetic in its details will be seen in the subsequent chapters.

There remains the side of mental discipline which I have elsewhere called, for want of another term and following various other writers, the culture side. What mental training does a child get from arithmetic that he does not get from biology, or Latin, or music .? This is a question so difficult to answer that no one has yet satisfied the world in his reply, and no one is likely to do so. There have been elaborate articles written to show that the proper study of arithmetic has an ethical value, though exactly what there is in the subject to make us treat our neighbor better it is a little difficult to say. Others have said that arithmetic, through its very rhythm, has an aesthetic value, as is doubtless true ; but that this is generally realized, or that it serves to make us more appreciative of the beautiful, is hardly to be argued with any seriousness. Still others have felt that by coming in contact with exact and provable truth an individual sets for himself a higher standard in all other lines of work, and this again is probably the case, although the measure of its influence has never been satisfactorily ascertained. And to these reasons may be added many more, such as the training of a deductive science, although elementary arithmetic is to a large extent inductive ; the training in concentration, although the untangling of a Latin construction requires quite as close attention ; the exaltation of mind that comes from the study of numbers that may increase or decrease indefinitely, and others of like nature. And out of it all, what shall we say ? That arithmetic can offer no m'ental discipline that other subjects do not give ? No one really feels this, in spite of the fact that the exact nature of this discipline is hard to formulate. Every one is conscious that he got something out of the study, aside from training in calculation and business applications, that has made him stronger, and the few really scientific investigations that have been made as to the effect of mathematical study bear out this intuitive feeling.

Nevertheless, if we ask a class of recently trained teachers if there is any value in the study of arithmetic except the utilitarian one, an animated and general response greets us in the familiar words, "The doctrine of formal discipline has been exploded." These words have been repeated so often that they have lost whatever of meaning they may once have had, and they usually signify but little more than a Latin phrase would to a child. If one asks, What is the mental

THE REASONS FOR TEACHING ARITHMETIC 15

discipline that has been exploded ? there comes no general reply, although a few may answer that " power acquired in one line cannot be transferred." And if we ask what this means, some of the more thoughtful of our hearers may say that the memorizing of rules in arithmetic does not help us to memorize the rules of grammar — if we do either of these things to-day.

Statements like these are interesting, but they mean little. The fact is, some one has set up a man of straw and, with the help of several thousand followers, is slaying him. No one claims any such mental discipline for arithmetic, nor does any one claim that rapidity in acquiring or reciting the multiplication table is going perceptibly to increase a child's rapidity in learning or reciting the names of the trunk lines that radiate from Chicago. No one believes that accuracy in multiplication i« a power that will be so transferred as to make a child perceptibly more accurate in map drawing. It would be absurd to assert that a child who comes to like arithmetic would thereby like his mother the better, or like to go to bed or to get up any more ardently. All such "transfers of power " are of course ridiculous, and no one makes or ever has made any such claims for mental discipline on the part of arithmetic. And if any one is tempted to believe that this form of mental discipline has been exploded, the only true reply is that no one ever asserted it.

It has, however, been asserted that arithmetic strengthens the reasoning powers, and there seems to be no doubt that this assertion has been made in times past with unwarranted emphasis. Every one knows of cases where a person reasons well in arithmetic and. not in other subjects. This is probably due to the fact that he has a greater interest in mathematics than in other branches of study, and hence gives less attention to the latter. At any rate, it is probable that the claim has been exaggerated, although the efforts of the psychologists to weigh the value of arithmetic, or of any other subject of study, have not progressed far enough to produce results that can be described as other than crude. It is impossible to weigh a child's mind before a certain subject has been studied, and again after it has been completed, and say that the subject has added or subtracted so much of mental power. We must, after all, fall back on our own experience, with such little help as experimental psychology can render.

Not to dismiss the mental-discipline side too summarily, however, and at the same time seeking to avoid the endless verbiage that usually characterizes the discussion, it is well to set forth more clearly some of the objects to be sought on the culture side of arithmetic. In the first place, we seek an absolute accuracy of operation that differs from the kind of accuracy we seek in science or linguistics or music. The fact that we have, in thousands of problems, sought a result so exact as to stand every test leads us to set a higher standard of accuracy in all lines than we could have set without it. This justifies the introduction of any part of theoretical arithmetic for which the pupil is mentally ready. It is one reason why cube root was formerly studied, when pupils were more mature than now, and in the same way it has justified progressions and a more elaborate treatment of primes than any business need would warrant. Here, then, is a reason for teaching arithmetic that is above and beyond the merely practical of the present moment.

A similar and related reason appears in the fact that mathematics in general, and arithmetic in particular, requires a helpful form of analysis that does not stand out so clearly in other studies. " I can prove this if I can prove that ; I can prove that if I can prove a third thing ; but I can prove that third thing ; hence I see my way to proving the first." This

THE REASONS FOR TEACHING ARITHMETIC IJ

is the analytic form that has come down to us from Plato. It appears more evidently in geometry, but is essentially the reasoning of arithmetic as well. "' I can find the cost of 2 1 yd. if I can find the cost of i yd. ; but I know the cost of 6\ yd., so I can find the cost of i yd. ; hence I can solve my problem," is the unworded line of the child's analysis. Such a training, unconsciously received and often unconsciously given, is valuable in every problem we meet, leading us to exclude the nonessential and hold with tenacity to a definite line of argument.

Furthermore, in mathematics in general, and in arithmetic in particular, we set down our results with a clearness and terseness of expression that is not found in other subjects. It is difficult to weigh the influence of a habit thus acquired. It may not show itself to-day or to-morrow, but when the need for it arises, possibly long after arithmetic as a study has been laid aside, it may again appear. The assertion that a statement is " mathematically exact " is not without meaning, and the habit acquired under some helpful, sympathetic, inspiring teacher, of setting forth the work in arithmetic neatly, clearly, and with no superfluous labor, is one of those mental acquisitions that may easily '" carry over " into the ordinary work of practical life.

These three phases of the culture side of arithmetic, the side of mental discipline, will then suffice for our present purpose, which is to show that such a side exists : ( i ) the contact with absolute truth ; (2) the acquisition of helpful forms of analytic reasoning; (3) the acquisition of certain habits that " carry over " into related fields of work. With respect to all three it may be said that our psychologists are more sympathetic to-day than they were a few years ago. That these exhaust the list of values of arithmetic from the culture standpoint must not, however, be inferred.

Bibliography. Smith, The Teaching of Elementary Mathematics, pp. 1-70; Smith, The Teaching of Geometry, Boston, 1911 ; Suzzallo, Teaching of Primary Arithmetic, Boston, 191 1 ; Young, Teaching of Mathematics, pp. 9, 41-52, 202-256, New Yorlc, 1907; Branford, A Study of Mathematical Education, Oxford, 1908; Stone, Arithmetical Abilities, Teachers College Series, 1908; Rietz and Shade, Correlation of Efficiency in Mathematics, etc.. Bulletin of University of Illinois, Vol. VI, No. 10 ; Hill, Educational Value of Mathematics, Educational Review, Vol. IX, p. 349 ; Hawkes, Educational Values in Mathematical Teaching, Mathematics Teacher, Vol. IV, p. 75 ; McCormack, Why do we study Matheniatics ? Journal of Proceedings of the National Educational Association, igio,,p. 505 ; Metzler, Formal Discipline, Mathematics Teacher, Vol. Ill, p. 9 ; see also Vol. II, p. 47, and Vol. I, p. 2 ; Slaught, Ideals in the Teaching of Mathematics, School Science and Mathematics, Vol. V, p. 703 ; Starch, Transfer of Training in Arithmetical Operations, Journal of Educational Psychology, Vol. II; Thorndike, Relation between Memory for Words and Memory for Numbers, American Journal of Psychology, Vol. XXI, p. 487 ; Winch, Accuracy in School QM^SAxexv, Journal of Educational Psychology, Vol. I, p. 557, and Vol. II. See also the Report on Mathematics in the Elementary Schools, and the Report of the American Commissioners of the International Commission on the Teaching of Mathematics, which is sent gratis, like the other American reports, on application to the Bureau of Education, Washington, D.C.

Chapter III What Arithmetic Should Include

If we look upon arithmetic only with a view to its utilitarian value, to its power to fit us for the computations that the average person needs to perform or know about in daily life, the range of subject matter is not great. Addition, particularly of money, but not involving very large or numerous amounts, is probably the most important topic. Very likely the making of change, one of the forms of subtraction, is next in frequency of use. Perhaps, for it is difficult to say with certainty, the simple fractions i and | are next in line of relative importance, including J of a sum of money, \ of a length or weight, and so on. Then may come easy multiplications, to find the cost of 5 pounds of sugar, or of i6 yards of cloth, given the price per pound or yard. A few of the most commonly used measures and their relations must then be known, as that 12 inches equal I foot and 16 ounces equal I pound. Given this equipment, the average run of humanity would be able to get along fairly well. But beyond this there lies a second field of work that every one may need, that a large minority will need, and that we should all at least know something about. This field includes all four fundamental operations with integers, with simple common fractions (say with denominators of one or two figures), with decimal fractions at least to hundredths, and with compound numbers of at least two denominations ; the common business cases of percentage, and their applications ; the common problems of business, all of which are applications of the operations above mentioned ; and a little knowledge of ratio and proportion, chiefly for understanding the meaning of these terms. From the standpoint of business needs this equipment would answer the purposes of nearly every one. Whatever of applied arithmetic lies beyond this is a part of the technical training of a very small minority.