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 The teaching of 
 
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 OCT 6 1931 
 
 FEB 1? 1932
 
 By DAVID EUGENE SMITH, LL.D. 
 
 Professor of Mathematics in Teachers College, Columbia University 
 
 -2. 2. & d 
 
 Reprinted, with revisions and additions, from the Teachers College 
 Record, Vol. X, No. i, January, 1909 
 
 FOURTH EDITION 
 
 PUBLISHED BY 
 
 , doliunbta 
 NEW YORK CITY 
 1911 
 
 922
 
 Copyright, 1909, by Teachers College, Columbia University 
 
 PKESS OF 
 
 THE BRANDOW FEINTING COMPANY 
 ALBANT, N. Y.
 
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 crp- 
 
 PREFACE 
 
 The Teachers College Record for March, 1903, contained an 
 article on Mathematics in the Elementary School by the author's 
 colleague Professor McMurry and himself. This number, how- 
 ever, has long since been out of print, and as a result of this fact 
 it was thought best in the autumn of 1908 to prepare a new 
 number of the Record on The Teaching of Arithmetic. This 
 was done by the author, and the article appeared in January, 1909. 
 Although it was thought that the unusually large edition was 
 sufficient for all demands for some years to come, it was ex- 
 hausted within a few weeks, and it became necessary to print the 
 article in book form. In spite of the fact that the work was 
 originally written in a popular style, to the end that it might be 
 read by those who have no more interest in mathematics than 
 in the various other subjects of the curriculum, it has been 
 thought best to make but a few changes in arranging for its 
 publication in the present form. Intended as it is for those who 
 are teaching or supervising the work in arithmetic in the elemen- 
 tary schools, it would hardly serve its purpose if it departed 
 widely from the practical and entered the domain of pure theory. 
 As between influencing the few or the many on a topic of such 
 general interest it has been thought better to adopt the latter 
 course, and to prepare a book that might have place in educational 
 reading circles generally and serve as a basis for the work in the 
 class-room for the training of teachers.
 
 TABLE OF CONTENTS 
 
 CHAPTER I. ^ The History of the Subject 5 
 
 CHAPTER II. The Reasons for Teaching Arithmetic 8 
 
 CHAPTER III^ What Arithmetic should include 12 
 
 CHAPTER IV. The Nature of the Problems 15 
 
 CHAPTER V. The Arrangement of Material 19 
 
 CHAPTER VI. Method 22 
 
 CHAPTER VII. Mental or Oral Arithmetic 26 
 
 CHAPTER VIII. Written Arithmetic 30 
 
 CHAPTER IX. Children's Analyses 35 
 
 CHAPTER X. Interest and Effort 39 
 
 CHAPTER XI. Improvements in the Technique of Arithmetic 42 
 
 CHAPTER XII. Certain Great Principles of Teaching Arithmetic. . 52 
 
 CHAPTER XIII. General Subjects for Experiment 55 
 
 CHAPTER XIV. Details for Experiment 65 
 
 CHAPTER XV. The Work of the First School Year 76 
 
 CHAPTER XVI. The Work of the Second School Year 85 
 
 CHAPTER XVII. The Work of the Third School Year 91 
 
 CHAPTER XVIII. The Work of the Fourth School Year 98 
 
 CHAPTER XIX. The Work of the Fifth School Year 102 
 
 CHAPTER XX. The Work of the Sixth School Year 107 
 
 CHAPTER XXL The Work of the Seventh School Year m 
 
 CHAPTER XXII. The Work of the Eighth School Year 115
 
 THE TEACHING OF ARITHMETIC 
 
 CHAPTER I 
 
 S/t>. 7- 1 
 
 THE HISTORY OF THE SUBJECT 
 2. 2.2. 8<D 
 
 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. At one time he seems to have 
 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 became ready to write his numbers 
 the convenience of finger reckoning had become so generally 
 recognized that ten became practically the universal radix. 
 Nevertheless there remain in our language and customs numerous 
 relics of the duodecimal 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 more change than 
 even the number names. Not only was there a notation for each 
 language in ancient times, as to-day in the Orient, but some lan- 
 guages 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
 
 6 The Teaching of Arithmetic 
 
 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. 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 with- 
 out such place value as we use to-day; but probably about the 
 seventh century the zero appeared, and the completed system 
 found its way northward into Persia and Arabia, and thence in 
 due time it was transmitted to the West. 
 
 At first the subject was purely practical, a counting of arrows 
 or of sheep or of men. For a long time this was all that number 
 meant to the world, until the mystic age developed and phil- 
 osophy began. Then numbers were differentiated, and odd and 
 even were distinguished, and " There's luck in odd numbers " 
 became a tenet of faith, and 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 seven, particularly so as the movement led to a 
 study of the properties of numbers, to roots, and to series. It 
 is connected with number games, and these in turn led -to the 
 abacus, and so the practical and the mysterious are more or less 
 blended even at times when they are generally regarded as widely 
 separated. 
 
 The growth of topics of arithmetic is also an interesting sub- 
 ject 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 busi- 
 ness 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 intensely so. And thus it is with many 
 topics of arithmetic, they have changed from century to cen-
 
 The History of the Subject 7 
 
 tury, 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 arith- 
 metic is the best single stimulus to good method in teaching the 
 subject. 
 
 BIBLIOGRAPHY: Smith, The Teaching of Elementary Mathe- 
 matics, New York, 1900; Smith, Rara Arithmetica, Boston, 
 1909; Smith and Karpinski, Hindu-Arabic Numerals, Boston, 
 1911; Ball, A Primer of the History of Mathematics, Lon- 
 don, 1895, and A Short Account of the History of Mathematics, 
 London, 4th edition, 1908 ; Fink, History of Mathematics, trans- 
 lated by Beman and Smith, Chicago, 1900; 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, Cam- 
 bridge, 1884; Conant, The Number Concept, New York, 1896; 
 Brooks, Philosophy of Arithmetic, revise! edition, Philadelphia, 
 1902. There are numerous works in German on the history of 
 mathematics and of mathematical teaching, and a considerable 
 number 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 differenti- 
 ated 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 knowl- 
 edge would probably have been looked upon as a bit of technical 
 training, like our use of the slide rule or the arithmometer. On 
 the other hand the Greeks called their science of numbers arith- 
 metic, 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 num- 
 bers, 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 
 for his trade. If he was asked why he taught arithmetic, as the 
 term was then used, his answer was still fairly a unit : 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 in- 
 cludes 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 
 
 3
 
 The Reasons for Teaching Arithmetic 9 
 
 in any work that may come to him; to the clergyman, to whom 
 the metric system will soon be merely a name ; to the housewife, 
 the farmer, and to all those who travel the multifarious walks of 
 our complex human life. For us to tell why we teach the Ameri- 
 can 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 that we 
 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 sub- 
 jects 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 
 really be so. We have no right to inject a mass of problems on 
 antiquated investments, on obsolete forms of partnership, on for- 
 gotten methods of mercantile business, or on measures that are 
 no longer common, and 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 obso- 
 lete business problem has just one reason for being, and this 
 reason is that it ha historical interest. We can secure the mental 
 discipline as well by other means, and we have no right to handi- 
 cap a child's mind with things that he will be forced to forget 
 the minute he enters practical life. 
 
 There remains the side of mental discipline, which I have else- 
 where 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
 
 io The Teaching of Arithmetic 
 
 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 
 liable 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 com- 
 ing 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 accomplished. 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 induc- 
 tive; 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 has no mental disci- 
 pline 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 calculation and business applications, 
 that has made him stronger, and the few really scientific investi- 
 gations that have been made, as to the effect of mathematical 
 study, bear out this intuitive feeling. And this being so, we 
 might be justified even if we did not attempt to define just what 
 this is, any more than we should attempt too seriously to define 
 time, or love, or God, or eternity. 
 
 Not to dismiss the mental discipline side too summarily, how- 
 ever, 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 thou- 
 sands of problems, sought a result so exact as to stand every
 
 The Reasons for Teaching Arithmetic n 
 
 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 there, 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 mathe- 
 matics 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 is the analytic form that has 
 come down to us from Plato. It more evidently appears in 
 geometry, but is essentially the reasoning of arithmetic as well. 
 " I can find the cost of 2 l / 2 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 non-essential -and hold with tenacity 
 to a definite line of argument. 
 
 These two 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. 
 
 BIBLIOGRAPHY: Smith, Teaching of Elementary Mathematics, 
 pp. 1-70; Smith, Teaching of Geometry, Boston, 1911; Suz- 
 zallo, Teaching of^- Primary Arithmetic, Boston, 1911; Young, 
 The Teaching of Mathematics, New York, 1907, pp. 9, 41-52, 
 202-256 ; Bran ford, A Study of Mathematical Education, Oxford, 
 1908 ; Stone, Arithmetical Abilities, Teachers College Series, 
 1908; Rietz and Shade, Correlation of Efficiency in Mathe- 
 matics, etc., University of Illinois Bulletin, vol. VI, no. 10; Hill, 
 Educational Value of Mathematics, Educational Review, vol. IX, 
 p. 349 ; Schubert, Mathematical Essays and Recreations, Chicago, 
 1899, p. 27.
 
 CHAPTER III 
 WHAT ARITHMETIC SHOULD INCLUDE 
 
 If we taught arithmetic only for its utilitarian value, to fit a 
 person for the computations that the average man needs to per- 
 form or know about in daily life, the range of subject matter 
 would not be great. Addition, particularly of money, but not 
 involving very large or numerous amounts, is probably the most 
 important topic. Perhaps, for it is difficult to say with certainty, 
 the simple fractions ^ and *4 af e next in line of relative impor- 
 tance, including y 2 of a sum of money, l /4 of a length or weight, 
 and so on. Very likely the making of change, one of the forms 
 of subtraction, is next in frequency of use. Then may come easy 
 multiplications, to find the cost of 5 Ib. of sugar, or of 16 yd. 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 must 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 stand- 
 point 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. 
 Apothecary's measures form part of the technical training of the 
 drug clerk and the physician ; the average citizen has long since 
 
 12
 
 What Arithmetic Should Include 13 
 
 forgotten them, and happily so. Compound proportion is never 
 used practically, and any mathematician if called upon to solve 
 its problems would employ another and a better method. Duo- 
 decimals, while interesting historically and philosophically, from 
 the practical standpoint are used by so few as to place them 
 also in the technical training of the very small minority. Subjects 
 like discount and interest are, of course, included under the com- 
 mon applications of percentage. Similarly with stocks and bonds, 
 for although such securities are purchased by relatively few 
 people, their nature and uses should be understood by all, par- 
 ticularly as we seem to have entered upon an era of extensive 
 cooperation upon a stock basis. The general nature of applica- 
 tions, however, will be discussed later. 
 
 If we taught arithmetic only from the standpoint of mental 
 discipline we might use all the material here mentioned, and any 
 other topics that allowed for securing accurate results by clear 
 reasoning processes. Obsolete measures, obsolete methods, pro- 
 gressions, cube root and even higher roots, compound proportion, 
 all such topics might have place if we were seeking only the 
 discipline of arithmetic. When, however, we consider that we 
 are seeking to unite these two considerations, and are attempting 
 to make the subject both practical and disciplinary, then we are 
 met by the necessity for mutual concessions. The practical side 
 must concede to the disciplinary by having its processes clearly 
 understood^ and by developing the reason at every step ; the 
 disciplinary side must concede to the practical by selecting its 
 topics in such way as to give no false notions of business, and 
 as to encourage the pupils to an interest in the quantitative side 
 of the world about them. On the one side we must not teach 
 business arithmetic by mere arbitrary rules that are not under- 
 stood, since this would be to eliminate the disciplinary nature; 
 on the other side we must not introduce a style of time draft that 
 is now obsolete in America, or artificial examples in compound 
 proportion, because these inculcate wrong ideas of the business 
 world about us, nor extensive work in equation of payments 
 because this is part of the technical training of such a very 
 small minority that we can use our time to better advantage by 
 dwelling upon other topics. 
 
 A word should also be said as to the tendency of some teachers
 
 14 The Teaching of Arithmetic 
 
 to feel that worthy results are attained when children have been 
 drilled to unnecessary facility in one line or another. It is not diffi- 
 cult to train children to add two columns of figures at a time, 
 or to multiply by cross multiplication, or to detect six or eight 
 cubes at a glance or to display various other forms of arithmetical 
 ability analogous to the acrobatical feats sometimes allowed in 
 a gymnasium. Such activities have some value as games, and 
 they attract attention on the part of visitors, but it is a question 
 whether the pupil derives any real benefit from most of this kind 
 of work. It is because of this feeling that such features are not 
 found in our textbooks, the space being devoted to those things 
 that the business man finds useful in everyday life. 
 
 Thus it happens that the modern American arithmetic is a 
 fair compromise between the practical without theory and the 
 theoretical without practice, the two distinct phases of the old 
 Greek number work. To keep this balance true is one of the 
 missions of teachers to-day. The tendency is to obtain the mental 
 discipline of arithmetic from problems that are practical, and that 
 this tendency is a healthy one there seems to be no room for 
 reasonable doubt. 
 
 BIBLIOGRAPHY: Smith, Teaching of Elementary Mathematics, 
 p. 19 ; Young, pp. 23-242 ; Spencer, The Teaching of Elementary 
 Mathematics in the English Public Elementary Schools, London, 
 1911.
 
 CHAPTER IV 
 THE NATURE OF THE PROBLEMS 
 
 In no way has arithmetic changed as much of late years as in 
 the nature of the problems and the arrangement of the material. 
 The former has come about from two causes, (i) the needs of 
 society, and (2) the study of child psychology. The latter, the 
 arrangement of the material, has been determined almost entirely 
 by psychological considerations. In this chapter it is proposed to 
 speak of the former, the nature of the problems. 
 
 It should definitely be stated, however, that this emphasis laid 
 upon applied problems should not be construed to mean that the 
 abstract number work is not quite so important as the concrete. 
 To be sure we have some advocates of only the concrete, what the 
 Germans call the " clothed problem," to the entire exclusion of 
 the abstract drill work of the Pestalozzi school. Such extremists 
 are, however, not numerous, and they have but a small following. 
 Every one who looks into the subject is aware that on the score 
 of interest a child prefers to work with abstract numbers, while 
 as to the final results upon his education we seem to neglect 
 altogether too much the ability to get exact results quickly and 
 with a certainty as to their exactness. This phase of the work 
 will be mentioned in a later chapter, and for the time being we 
 may well consider the applied problem. 
 
 Within the last few years the question of the practical uses of 
 arithmetic has been a vital one in educational circles, especially 
 in Germany and America, resulting in considerable literature 
 upon the subject. These needs, while generally similar in various 
 countries, differ more or less in details. Thus a country whose 
 business was chiefly farming would need to emphasize agricultural 
 problems ; one that derived its wealth from its metals or its coal 
 would emphasize mining; a manufacturing nation would find 
 certain lines of problems of the factory peculiarly suited to its 
 needs, while one that derived its wealth chiefly from shipping 
 
 15
 
 16 The Teaching of Arithmetic 
 
 would require those relating to foreign commerce. The mathe- 
 matical foundation would be the same in all cases, but the material 
 content of the problem would vary. Now in America we are 
 unusually cosmopolitan in our needs in this respect, ranking high 
 in all these particulars save only (at present) in ocean traffic. 
 We are therefore very fortunate in having at our disposal unlim- 
 ited problem material that relates to our wide range of national 
 resources and industries. The advantage of using this material 
 instead of the obsolete inherited problems that came down to us 
 from Italy, through England, ought to be so evident as to require 
 no argument. There will always be some who cry out against 
 what they call encyclopedic information in an arithmetic, but 
 surely if a problem is to contain any facts at all it is better that 
 these facts be American and of the twentieth century than Italian 
 and of the fifteenth. Can there be any doubt that an American 
 boy or girl will get more breadth of view, more interest, and 
 possibly more directly useful information from a problem about 
 the mixing of plant foods for a southern farm, than one about 
 the mixing of teas that are never mixed in the way the text-book 
 says? If a pupil is to study about goods being transported, is 
 it not better for him to take a practical case relating to our rail- 
 roads than the old-time one of pedlars carrying their packs? It 
 is well, however, to avoid the unfortunate tendency manifested 
 by some recent writers to introduce problem material that no 
 child is ready to understand and with which teachers themselves 
 should not be expected to be familiar. Technical information 
 of trades, scientific nomenclature that belongs to the college or to 
 the later years in the high school, problems of the civil engineer 
 or the food chemist, these have no more place in the arithmetic 
 class than has the apothecary's table or the subject of equation 
 of payments. Common information, the subjects of interest to 
 the general public, and those matters that are topics of conversa- 
 tion in the usual walks of life are the bases upon which we may 
 reasonably build our problems. Undertaken in this spirit we need 
 not fear if critics accuse us of making our schools encyclopedic. 
 Every usable school arithmetic has always been an encyclopedia ; 
 what we have to determine is whether it shall now be an encyclo- 
 pedia of vital, modern facts, or one of obsolete, dull, useless 
 information. The needs of society demand the former ; vis inertia
 
 17 
 
 holds to the latter. The earnest teacher, awake to the needs of the 
 business community in which a school is located, can hardly fail 
 to introduce genuine problems with local color to enliven the 
 work in arithmetic. No text-book can fit the needs of every 
 locality, and original problems are easily found by the children 
 themselves if the opportunity is given. The awakening of inter- 
 est in such work, however, will not come from the style of text- 
 book of a generation ago; it will only come in connection with 
 the study of a book that is itself filled with this spirit. Fortu- 
 nately most of our modern writers are working earnestly to 
 meet the needs of to-day, and our American arithmetics may 
 well lay claim to being among the most progressive that are 
 appearing in this generation. 
 
 But what as to the effect of the study of child psychology? 
 Here too there has been made very great progress in recent years. 
 Although a problem may represent all that business needs suggest, 
 it still may not be suited to a particular school year. In other 
 words, we have to consider from grade to grade the interests and 
 powers of the child. We would not think of giving to a child 
 in the first grade the problem, If A has 2 shares of railroad stock 
 and B has 3 shares, how many have they together? For while 
 the child can add 2 and 3, he has no knowledge of stocks, nor 
 any interest in them. Change the subject to marbles or apples 
 or tops, and it is suited to his mind, but not otherwise. Thus 
 it has come about that teachers are trying to decide what are the 
 larger interests, actual or potential, of the children in the various 
 school years, to the end that the problems of arithmetic may be 
 the better apperceived. A beginning has been made, but the 
 future will see the work extended. We know that pride in our 
 national resources renders interesting a style of problem in the 
 fifth school year that would be of no value in the second, even 
 with smaller numbers. On the other hand we are equally aware 
 that certain problems involving children's games that are part of 
 the genuine applied mathematics of the third year would have no 
 interest whatever in the eighth. And so in general, teachers are 
 seriously attempting at the present time to coordinate the inter- 
 ests of children, the needs of society, and the mathematical pow- 
 ers in each of the grades from the first year through the ele- 
 mentary school.
 
 i8 The Teaching of Arithmetic 
 
 BIBLIOGRAPHY: Smith, Teaching of Elementary Mathematics, 
 p. 21 ; Smith, Teaching of Geometry, Boston, 1911; Young, 
 pp. 97, 210. Consult also the author's text-books on arithmetic 
 and works like Saxelby, Practical Mathematics, London, 1905 ; 
 Consterdine and Andrew, Practical Arithmetic, London, 1907; 
 Consterdine and Barnes, Practical Mathematics, London, 1908; 
 Castle, Workshop Mathematics, London, 1907; Castle, Manual 
 of Practical Mathematics, London, 1906; Cracknell, Practical- 
 Mathematics, 4th edition, London, 1906. For statistics for prob- 
 lems The World Almanac, New York, is an inexpensive and 
 valuable source.
 
 CHAPTER V 
 THE ARRANGEMENT OF MATERIAL 
 
 As already stated, the two most noteworthy changes in arith- 
 metic in recent years have related to the nature of the problems 
 and the arrangement of material. The latter has been the result 
 of a more or less serious study of child psychology, namely of 
 the powers of the individual in the various school years. 
 
 Formerly, say a century or more ago, it was the custom to 
 study arithmetic from a single book, after the boy (for the girl 
 seldom understood mathematics of any kind) could read and 
 write. The child was mature enough to understand the subject 
 after a fashion, and he " went through " the book. But as arith- 
 metic began to work its way down to the earliest grades it was 
 found impracticable to follow this plan, the subject being too 
 difficult for young minds. The ordinary text-book was therefore 
 preceded by a primer on arithmetic, and thus a two-book series 
 was formed. From this beginning numerous experiments have 
 proceeded, seeking to carry the improvement still farther. We 
 have had three-book series, eight-book series, lesson leaflets, two- 
 book courses arranged by grades, and so on. We have had 
 spirals of various degrees of turning, efforts to resume the 
 topical arithmetic, books arranged to follow certain narrow lines 
 of manual training, and so on, all serious efforts for better- 
 ment, but many of them too hastily considered to have any 
 material influence. 
 
 In the midst of it all, what have the practical teachers of the 
 country done? In every city, in several states, and by numerous 
 associations, courses of study have been arranged setting forth 
 the material that experience has shown can be used to advantage 
 in the several school years, and selections have been made from 
 the current arithmetics to supply what was needed. In other 
 words the practical teachers of America arranged their own 
 books to a great extent, basing their selection of material upon 
 
 19
 
 2O The Teaching of Arithmetic 
 
 an empirical psychology, and giving to the child what his inter- 
 ests and capabilities suggested. 
 
 It is out of such a movement, spontaneous but thoroughly 
 sound, that the later American text-books have arisen, and the 
 care and earnestness given to their preparation by various authors 
 and publishers should have the commendation of all. 
 
 These books are and probably will continue to be of two dis- 
 tinct types, each with strong merits of its own, and each capable 
 of producing the best work. First there is the topical arithmetic, 
 that is, a book arranged by topics, a subject like percentage being 
 studied once for all, the pupil staying with it until it is thoroughly 
 mastered. Such a book has two great merits; it tends to keep 
 the child upon each subject long enough to give him a feeling 
 of mastery that he would not have if he studied some of the 
 scrappy books constructed on the extreme spiral system, and it 
 allows a teacher who wishes to adopt a moderate spiral to do this 
 in a manner that will meet the local conditions. By this latter 
 is meant that some classes seem to need more work in a subject 
 like simple interest than other classes do, when it is first taken 
 up. A teacher may, therefore, select as much or as little as is 
 necessary when a topic is taken up for the first or the second 
 time, and this is perhaps more easily done from a topical than 
 from another form of book. 
 
 The second general type of text-book is one that attempts to 
 fit the course of study in a large number of schools. In general 
 these books agree in a number of particulars: (i) Certain sub- 
 jects, such as the most commonly used operations, should appear 
 several times in the school course; (2) others, such as the busi- 
 ness application of simple interest, may appear perhaps two or 
 three times, with gradually increasing difficulty; (3) still others, 
 like board measure, may be sufficiently treated once for all ; (4) 
 the closing year of arithmetic should be devoted to a study of 
 such higher problems of business as can be understood by the 
 children. These are some of the articles of agreement, and they 
 go to show the common-sense principles on which our modern 
 courses of study and text-books in arithmetic are based. 
 
 As to which of these two types is the better it is impossible 
 to give a general decision. It depends largely upon the school 
 and the teacher. With the one book the teacher arranges the
 
 The Arrangement of Material 21 
 
 matter to suit the pupils' needs ; in the other the matter is already 
 arranged to suit the average pupil. Neither will fit each indi- 
 vidual case, and the great thing is not so much the arrangement 
 of the matter as it is the modern spirit displayed in the omission 
 of the obsolete and the introduction of new, vital, interesting, 
 intelligible problems of to-day. 
 
 It should also be stated in this connection that a similar move- 
 ment is taking place with reference to algebra and geometry, and 
 that it is destined to reach still further up the grade and into 
 college mathematics. Algebra is old, but algebra text-books are 
 relatively modern. They were at first based upon the arithmetics 
 that preceded them, and as far as possible they followed their 
 arrangement of matter. The ordinary school algebra is, there- 
 fore, merely an old-style arithmetic with letters used instead of 
 numerals, and with a considerable number of its problems taken 
 from the arithmetical collections of earlier days. The arrange- 
 ment of matter is confessedly not scientific, and we are even now 
 seeing the sequence of topics challenged, and a serious effort to 
 improve the nature of the problems. The next move of impor- 
 tance in the teaching of elementary mathematics will be the re- 
 arrangement of the material and the enriching of the applications 
 of algebra and, probably to a less degree, of geometry. 
 
 BIBLIOGRAPHY: Consult the latest text-books, comparing the 
 two types and noting the distinctive advantages of each. Consult 
 also those extreme forms in which the spiral arrangement is 
 carried too far. This, and in general the other questions con- 
 sidered in this monograph, will be found discussed in Professor 
 Suzzallo's noteworthy work, Teaching of Primary Arithmetic, 
 Boston, 1911.
 
 CHAPTER VI 
 METHOD 
 
 Of all the terms used in educational circles " Method " is per- 
 haps the most loosely defined. Efforts have been made to limit 
 its meaning, to divide its responsibilities with such terms as 
 " Mode " and " Manner," but it still stands and is likely to stand 
 as- a convenient name for all sorts of ideas and theories and 
 devices. Nevertheless it has been most often used in arithmetic 
 to speak of the general plan of some individual for introducing 
 the subject, as when we speak of the Pestalozzi or Tillich or 
 Grube Methods, although it is also applied to such arrangements 
 of material as is indicated by the expressions Topical Method 
 and Spiral Method, and to such an emphasis of some particular 
 feature as has given name to the Ratio Method. It is not the 
 intention to attempt any definition of the term that shall include 
 all of these ramifications, but to take it as it stands, to characterize 
 briefly some of these " Methods," and then to speak briefly of the 
 subject as a whole. 
 
 Pestalozzi's method was really a creation of his followers. 
 What this great master did for arithmetic was to introduce it 
 much earlier into the school course, to use objects more system- 
 atically to make the number relations clear, to abandon arbitrary 
 rules, to drill incessantly on abstract oral work, and to emphasize 
 the unit by considering a number like 6 as " 6 times I." For the 
 time in which he lived (about 1800) all this was a healthy protest 
 against the stagnant education that he found. To-day it is only 
 an incidental lesson to the teacher, although Pestalozzi's spirit 
 and several of his ideas may well command the admiration and 
 respect of all who study the results of his great work. 
 
 The " Method " of Tillich, who followed Pestalozzi by a few 
 years consisted largely of making a systematic use of sticks cut 
 in various lengths, say from i inch to 10 inches. It is evident 
 that such a collection allowed for emphasizing the notion of tens, 
 
 22
 
 Method 23 
 
 for treating fractions as ratios, and for visualizing in a very good 
 way the simpler number relations. On the other hand it is also 
 evident that the use of only a single kind of material is based 
 upon a much narrower idea than that of Pestalozzi, who pur- 
 posely made use of as wide a range of material as he could. 
 
 The Grube Method, that created such a stir in America a 
 generation ago, was not very original with Grube (1842). Es- 
 sentially it was an adaptation of the concentric circle plan that 
 had already been used, a kind of spiral arrangement of matter to 
 meet the growing powers of the child. It contained some absurd- 
 ities, such as the exhausting of the work on one number before 
 proceeding to the next, as of studying 25 in all its relations before 
 learning 26 ; but on the whole it made somewhat for progress by 
 assisting to develop a sane form of the spiral idea. 
 
 It is not worth while to speak of other individual " Methods," 
 since they have little of value to the practical teacher, and the 
 student of the history of education can easily have access to them. 
 Enough has been said to show that one of the easiest things in 
 the teaching of arithmetic is the creation of " Method " and one 
 of the most useless. We may start off upon the idea that all 
 number is measure, and hence that arithmetic must consist of 
 measuring everything in sight, and we have a " Measuring 
 Method." It will be a narrow idea, we shall neglect much that is 
 important, but if we put energy back of it we shall attract attention 
 and will very likely turn out better computers than a poor teacher 
 will who is wise enough to have no " Method," in this narrow 
 sense of the term. Again, we may say that every number is a 
 fraction, the numerator being an integral multiple of the denomi- 
 nator in the case of whole numbers. From this assumption we 
 may proceed to teach arithmetic only as the science of fractions. 
 It will be hard work, but, given enough energy and patience and 
 skill, the children will survive it and will learn more of arithmetic 
 than may be the case with listless teaching on a better plan. We 
 might also start with the idea that every lesson should be a unit, 
 and that in it should come every process of arithmetic, so far as 
 this is possible, and we could stir up a good deal of interest in our 
 " Unit Method." Or, again, we could begin with the idea that 
 all action demands reaction, and that every lesson containing 
 addition should also contain subtraction ; that 6 + 4 = 10 should
 
 24 The Teaching of Arithmetic 
 
 be followed by 10 6 = 4 and 10 4 6 ; and that 2 X 5 = 10 
 should be followed by 10 -r- 2 5 and 10 -r- 5 = 2. By sufficient 
 ingenuity a very taking scheme could be evolved, and the " Inverse 
 Method " would begin to make a brief stir in the world. This in 
 fact has been the genesis, rise, and decline of Methods ; given a 
 strong but narrow-minded personality, with some little idea such 
 as those above mentioned ; this idea is exploited as a panacea ; it 
 creates some little stir in circles more or less local ; it is tried in a 
 greater or less number of schools ; the author and his pupils die, 
 and in due time the Method is remembered, if at all, only by some 
 inscription in those pedagogical graveyards known as histories of 
 education. 
 
 The object in writing thus is manifest. For the teacher with 
 but little experience there is a valuable lesson, namely, that there 
 is no " Method " that will lead to easy victory in the teaching of 
 arithmetic. There are a few great principles that may well be 
 taken to heart, but any single narrow plan and any single line of 
 material must be looked upon . with suspicion. Certain of the 
 general principles of Pestalozzi are eternal, but the Reckoning- 
 chest of Tillich is practically forgotten. 
 
 And here it is proper to say a word as to what schools of 
 observation and practice should stand for in these matters of 
 method and purpose. It would be a very easy thing to concentrate 
 on some single point, some device of teaching, some particular line 
 of problems, and to carry the work to an extreme that would 
 attract attention and produce results that would be remarked 
 upon. This is the temptation of those who direct such schools. 
 But is it a wise policy? These schools are established to train 
 teachers to well-balanced leadership, not to be extremists when 
 this means the neglect of essential features of education. The 
 graduates should know the best that there is in every theory of 
 education, but they should also avoid the worst. The prime 
 desideratum in arithmetic is the ability to work accurately, with 
 reasonable rapidity, and with interest, and to know how to apply 
 number to the ordinary affairs of life. To secure accuracy alone, 
 to secure speed alone, to have arithmetic a play spell without 
 accuracy and speed, or to know how to apply number to life in a 
 slovenly way, these are extremes that should be avoided at any 
 cost, including the tempting cost of sensationalism. It is the
 
 Method 25 
 
 mission of the training school or college to make the earnest, 
 well-balanced teacher, first of all. With this duty goes the 
 laudable one of reasonable experiment, of trying out sugges- 
 tions from whatever source; but normal schools and teachers 
 colleges must at all hazards guard against having it appear that 
 an experiment is an accomplished result, or of sacrificing our 
 children in unnecessary quasi-clinical work that is doomed to 
 failure. In this connection one of the resolutions adopted by 
 the National Education Association in 1908 may be read with 
 profit as voicing the sentiment of that s^ner element in education 
 that is, after all, the strength of our profession: 
 
 " We recommend the subordination of highly diversified and 
 overburdened courses of study in the grades to a thorough drill 
 in essential subjects; and the sacrifice of quantity to an improve- 
 ment in the quality of instruction. The complaints of business 
 men that pupils from the schools are inaccurate in results an& 
 careless of details is a criticism that should be removed. The 
 principles of sound and accurate training are as fixed as natural 
 laws and should be insistently followed. Ill-considered experi- 
 ments and indiscriminate methodising should be abandoned, and 
 attention devoted to the persevering and continuous drill necessary 
 for accurate and efficient training; and we hold that no course of 
 study in any public school should be so advanced or so rigid as 
 to prevent instruction to any student, who may need it, in the 
 essential and practical parts of the common English branches." 
 
 BIBLIOGRAPHY: The author's Teaching of Elementary Mathe- 
 matics, pp. 71-97, a rather extensive discussion; Seeley, Grube's 
 Method, New York, 1888; Soldan, Grube's Method, Chicago, 
 1878; C. A. McMurry, Special Method in Arithmetic, New 
 York, 1905 ; McLellan and Dewey, The Psychology of Number, 
 New York, 1895; Young, pp. 53-150; Hornbrook, Laboratory 
 Method of Teaching Mathematics, New York, 1895 ; Perry, The 
 Teaching of Mathematics, London, 1902 ; Suzzallo, loc. cit.; 
 Ballard, The Teaching of Mathematics in London Public Ele- 
 mentary Schools, London, 1911.
 
 CHAPTER VII 
 MENTAL OR ORAL ARITHMETIC 
 
 The objection to the expression " Mental Arithmetic " is fully 
 a generation old. It is argued that written arithmetic is quite as 
 mental as any other kind, and that the opposite to written is oral. 
 As to this there can be no argument, but the word " mental " has 
 so long been used to apply to that phase of arithmetic that is not 
 dependent upon written help that, like a person's proper name, it 
 need not be held strictly to account for what it literally signifies. 
 The expression " Mental Arithmetic " is therefore employed as 
 well as " Oral Arithmetic " in this article simply because it is 
 historical and of well-understood significance. 
 
 What, now, are the relative claims of written and mental arith- 
 metic? Historically, the mental long preceded the written, but 
 only in very simple problems, chiefly involving counting and easy 
 addition. As soon as the writing of numbers was introduced, 
 written arithmetic or else the arithmetic of some form of the 
 abacus became practically universal. In Japan to-day a native 
 shopkeeper will multiply 2 by 6 upon the soroban (abacus), and 
 such mechanical aids were not only not discarded in western 
 Europe until the sixteenth century, but they are still universal in 
 Russia. About the beginning of the last century, however, mental 
 arithmetic underwent a great revival, largely through the influence 
 of Pestalozzi in Europe and Warren Colburn in this country, in 
 each case as a protest against the intellectual sluggishness, lack 
 of reasoning, and slowness of operation of the old written arith- 
 metic. For a long time the mental form was emphasized, in 
 America doubtless unduly so, and was naturally followed by such 
 a reaction that it lost practically all of its standing. The question 
 for teachers to-day is this, what are the fair claims of these 
 two phases of the subject upon the time and energy of pupil and 
 teacher ? 
 
 There are two points of view in the matter, the practical and 
 
 26
 
 Mental or Oral Arithmetic 27 
 
 the educational or psychological, and fortunately they seem to 
 lead to the same conclusion. Practically a person of fair intelli- 
 gence should not need a pencil and paper to find the cost of 6 
 articles at 2 cents each, or of 5^4 yards at 16 cents a yard. The 
 ordinary purchase of household supplies requires a practical ability 
 in the mental arithmetic of daily life, and this ability comes to the 
 mind only through repeated exercise. As will be seen later, it is 
 a fair inference from statistical investigations that a person may 
 be rapid and accurate in written work but slow and uncertain in 
 oral solutions. Therefore, it will not do, from the practical stand- 
 point, to drill children only in written arithmetic if we expect 
 them to be reasonably ready in purely mental work. On psycho- 
 logical grounds, too, the neglect of mental arithmetic is unwise. 
 It is a familiar law that the memory is stronger on a fact that is 
 known in several ways (a convenient phrase, if not scientific) 
 than on a fact that is known in only one way. A man who knows 
 a foreign word only through the eye may forget it rather easily, 
 but if his tongue has been taught to pronounce it, even though he 
 be deaf, he can the more readily recall it. If in addition to this 
 his ear has often heard it he is the more strongly fortified, and if 
 he has also often written it, by pen or by typewriter, there is 
 this further chain that holds it to the memory. In other words, 
 the greater number of stimuli that we can bring to bear, the 
 more certain the reaction. Now arithmetic furnishes merely a 
 special case of this general law. If a child could simply see 
 9 X 8 = 72 often enough he would come to be able to write it 
 in due time, even if he did not know the meaning. If in addition 
 to this he knows the meaning of these symbols and recalls having 
 taken 9 bundles of 8 sticks each and finding that he. had 72 sticks, 
 then the impression on the brain is the more lasting. If, further- 
 more, he has been trained to say " nine times eight are seventy- 
 two " repeatedly, the impression is still stronger, and if he has 
 repeatedly heard this expression (and here is one of the advan- 
 tages of class recitation) he has then a still further mental grip 
 upon the fact. In other words, mental arithmetic in the form of 
 rapid oral work, with both individual and class recitation, is a 
 valuable aid, psychologically, to the retention of number facts. 
 
 There is, however, a danger to be recognized. It is asserted 
 that a child tires more quickly of abstract work than of genuine
 
 28 The Teaching of Arithmetic 
 
 concrete problems; problems, that is, that are not manifestly 
 " made up " but that represent some of his actual quantitative 
 experiences. Whether he really tires more quickly of the abstract 
 than of the concrete is by no means certain ; for he seems to have 
 more interest in the former than in the latter, probably from the 
 added difficulty that the concrete problem presents in requiring 
 him to know what operations he must perform. At any rate he 
 tires of both, as he does of any other intellectual exercise. It 
 therefore follows that if five minutes of mental work produce a 
 certain efficiency, thirty minutes will by no means produce six 
 times that efficiency. If, now, this mental work is valuable, how 
 much time and energy should be allotted to it? Possibly we 
 shall have a statistical reply to this question sometime, although 
 it will be a sorry day for good teaching if we should ever accept 
 such a reply as final, any more than we should accept the crude 
 statistics of the health department as determining the prescription 
 our physician gives us for indigestion. The statistics may help 
 us, but they can never control us. But in absence of even their 
 assistance, what shall we give as an empirical answer? It seems 
 to be the experience of teachers generally that a little mental work, 
 rapid, spirited, perhaps with some healthy, generous rivalry to 
 add spice to the exercise, should form part of every recitation 
 throughout the course in arithmetic. There will often be excep- 
 tions, but in general it is a pretty good rule to devote from three 
 to five minutes daily, and sometimes much more time, to this kind 
 of work. In this way a child never gets out of practice, save 
 during the summer holidays, and the practical and psychological 
 benefits can hardly be estimated. 
 
 What should be the nature of this mental work? On the 
 applied side there is no better test for the teacher's ability to 
 adapt herself to her environment, educationally, than this, for the 
 answer varies with the school year, the locality, the related sub- 
 jects in the course, and with many other factors. In general, 
 however, it may be said that mental arithmetic offers the best 
 means for correlating the subject with the pupil's other work, 
 both within and without the school. To limit it to this field, 
 however, would be an evident mistake, the work with abstract 
 number demanding the major part of the time assigned to this 
 feature. To acquire perfect mechanical reaction to a given
 
 Mental or Oral Arithmetic 29 
 
 stimulus much exercise is required, and for a child to think 72 
 when stimulated by the ideas 9X8 and 8X9 demands repeated 
 practice, not merely in relatively few applications but in a multi- 
 tude of questions involving abstract numbers. Nor is this practice 
 any more irksome than is the solution of applied problems, as any 
 teacher knows. It was almost exclusively by this abstract work 
 that Pestalozzi developed calculators of such ability with con- 
 crete problems as astonished those who visited his school, al- 
 though, if we may place confidence in the results of Dr. Stone's 
 recent investigations, ability in either of these lines does not 
 necessarily imply ability in the other. 
 
 In conclusion, we have two lines of work in mental arithmetic ; 
 (i) the concrete, in which the teacher has an excellent oppor- 
 tunity for correlation, for local color, and for stimulating the 
 interest in the uses of arithmetic; (2) the abstract, in which the 
 text-book may be trusted to furnish a considerable part of the 
 material. Each must be cultivated, and ability in one does not 
 necessarily mean a corresponding standard of ability in the other, 
 although a failure in the abstract line must lead to a failure in 
 the concrete. One leads to the acquisition of number-facts, the 
 other to the ability to rationally use these facts in applied 
 problems. 
 
 As a practical question for the teacher, how is the material 
 for this oral work to be found ? The answer is evident ; it must 
 be found exactly as we find material in geography, in history, 
 and in written arithmetic, from a text-book. No teacher can 
 make up on the spur of the moment all of the oral examples 
 necessary, and arrange them properly, and cover all of the impor- 
 tant phases of drill work. Either, then, a book must be used that 
 supplies both the oral and the written work, or else two books 
 must be used, one for the oral and one for the written. In either 
 case there should be a good supply of oral problems, to be supple- 
 mented by the teacher with such local problems and such correla- 
 tion with other work as may be advisable. 
 
 BIBLIOGRAPHY: The author's Teaching of Elementary Mathe- 
 matics, p. 117; Handbook to Arithmetics, Boston, 1904, p. 6; 
 Young, p. 230; C. W. Stone, Arithmetical Abilities; Wentworth- 
 Smith, Oral Arithmetic, Boston, 1909.
 
 CHAPTER VIII 
 WRITTEN ARITHMETIC 
 
 What has been said of mental arithmetic naturally leads to 
 some question as to the nature of the written work. What shall 
 this be? If the difference in longitude between two ships (since- 
 standard time by one system or another is now coming to be 
 universal on land) is 33 45', how shall a pupil find the difference 
 in time? Here are a few possibilities: 
 
 (I) (2) 
 
 2 hr. 15 min 2 hr. 15 min. 
 
 15)33 45' I5)33 45' 
 
 30 30 
 
 3 45' = 225' 3 45' = 225' 
 
 15 15 
 
 75 75 
 
 75 75 
 
 o (3) (4) 
 
 33 45' 33 45' 
 
 60 
 
 1980 
 45 
 
 30 
 
 15)2025(135 min. = 2 hr. 15 min. 
 
 52 
 
 45 
 
 75 
 
 (5) 
 
 33 45' = 33M 
 33#XY,. hr. = 2# hr. 
 
 30
 
 Written Arithmetic 31 
 
 Numerous other forms could be suggested, but these will suf- 
 fice for our purposes. Which of these should be preferred? In 
 general, should we recommend the form that gives us the result 
 most quickly, or some other that may show clearer reasoning? 
 In Nos. i and 4 the forms indicate that we divide degrees by an 
 abstract number and get hours instead of degrees; in No. 2 we 
 seem to divide degrees by degrees and get hours instead of an 
 abstract number ; in No. 3 we seem to multiply degrees and get 
 an abstract number, and to divide one abstract number by an- 
 other and get concrete time in the quotient; in No. 5 we omit 
 part of work of reduction but otherwise the solution is a truth- 
 ful one, with none of the errors of reasoning of the rest. .This 
 problem has been selected as the first for consideration because 
 it opens at once such a wide range of possibilities of form, but 
 essentially the same question repeatedly occurs from the very 
 first grade through the pupil's school life. If i yard of cloth 
 costs I5c., what wiH 6 yards cost? is a simpler question to 
 consider. Here we have these possibilities, among others: 
 
 (i) (2) (3) 
 
 IS 15 i5c. 
 
 666 
 
 90 9oc. pec. 
 
 (4) 6X15 = 90 (5) 6Xi5 = 9oc. 
 
 (6) 6Xi5c. = goc. (7) 15 X 6 = cpc. 
 
 Out of all these, which shall a child use in writing a solution? 
 
 In each of these problems the fundamental question is the 
 same: Shall written work be considered from the standpoint of 
 the answer only, as a business man would be inclined to do, or 
 from the standpoint of the logic of the school, the often non- 
 practical school? 
 
 The answer to such a question ought not to be dogmatic to 
 the extent of saying that any one form is always the best, al- 
 though it may say that those forms that are untrue in statement 
 are always bad. That is to say, sometimes it is better to write 
 
 15)33 45 15 
 
 2 15)225 6 
 
 15 
 2hr. 15 min. 90 
 
 At other times the step form, with the denomination accurately
 
 32 The Teaching of Arithmetic 
 
 set forth, is better. We need to distinguish between two lines of 
 work, equally important; the one relates to accuracy and speed 
 in operation, the getting of an answer as a business man would, 
 with no circumlocution and no superfluous symbols or opera- 
 tions ; this is the mechanical part of the problem and there must 
 be abundant exercise on this side. Then there is the "equally im- 
 portant side of the reasoning, explaining why the mechanical 
 work is performed as it is, why we multiply instead of divide, and 
 how we know that the result is hours instead of degrees, or cents 
 instead of yards of cloth. Here the step form of analysis may 
 be depended upon to show the pupil's line of reasoning. These 
 two lines of written work are, therefore, legitimate. What, then, 
 is illegitimate in written work, and what are the dangers to be 
 guarded against in that which we do adopt? As to the first, it 
 may be laid down as axiomatic that a form that states or seems 
 to state a falsehood is illegitimate. That is, 30 -r- 15 = 2 hr. 
 is a false statement; it is not even excusable on the score of 
 brevity, since 30 X a / 15 nr - = 2 nr - is as brief, is true, and is as 
 easily explained as any form. So 6 X 15 = goc. is a false state- 
 ment and should not be tolerated, although 6 X 15 = 90, or 
 6 X I5c. = Qoc., is legitimate. And as to the dangers against 
 which to guard, the following advice may be given : ( I ) To 
 require that every applied problem should be solved in steps is 
 to encourage arithmetical dawdling; the pupils should continu- 
 ally be exercised in rapid solution, the correct answer speedily 
 ascertained, as a business man would get it, being the aim. A 
 pupil who lets his mind continually dwell upon dollar signs and 
 well written steps cannot help but drop away from strict atten- 
 tion to rapidity and accuracy of calculation. (2) To split hairs 
 on questions of such forms as 9 X I5c. or 150 X 9 is to get away 
 from the essential point ; we must recognize the fact that there is 
 good authority for both, although the former, writing the sym- 
 bols in the order they are read, is coming into rather general use 
 in America. The great question is to see, in these analyses, that 
 the thought is clear, and that a pupil is not thinking in a hazy 
 way of "15 cents times 9." (3) To require no analyses of the 
 applied problems is an extreme that is about as bad as to require 
 them for all, and perhaps worse. It is quite sure to result in 
 looseness of reasoning that makes correct results mere matters
 
 Written Arithmetic 33 
 
 of luck. (4) To require some particular form of analysis, only 
 to meet the idiosyncrasy of the teacher, is also a danger against 
 which we need to be on our guard. For example, always to 
 require a solution stated in one step, if possible, is a hobby that 
 some like to ride, because it seems to demand continued thought, 
 although it is entirely foreign to the plan that a common-sense 
 business man would adopt, and is not the form of reasoning 
 that we commonly take in mathematics. So to require that a 
 child should always take some unitary form of analysis, finding 
 in every case what one thing costs, may be the means of check- 
 ing the originality and dampening the ardor of some very 
 promising pupil. 
 
 In general, therefore, the teacher should see to it that there is 
 a reasonable amount of rapid, accurate solution, the " answer " 
 being the paramount object. He should also see that there is a 
 reasonable amount of written analysis, accurately stated, prefer- 
 ably in the convenient and terse form of steps, but not limited in 
 any notional way that would destroy originality or make a solu- 
 tion unnecessarily long. 
 
 In the marking of papers it should be born in mind that there 
 is only one test for a question involving a single operation. 
 Either the answer is right or it is wrong. If the problems 
 require some interpretation, a teacher may properly mark both 
 for operations and for method; that is, a pupil may perform 
 his operations correctly, but may have misinterpreted the mean- 
 ing of the problem. In that case some credit may properly be 
 given for the correct operation. In general, however, papers in 
 arithmetic should be marked, as they are in business, largely by 
 the accuracy of the result. In any single operation the work is 
 right or it is ivrong. A business man will not excuse a book- 
 keeper who writes $9250.75 instead of $90,250.75. Ony a zero 
 is missing, but it means a difference of over $80,000. If the 
 result is wrong, the paper is wrong. The converse of this state- 
 ment is not true, for the result may be right, and yet the paper 
 may be justly criticised for its slovenly appearance and the inac- 
 curacy of the forms used. Where a time limit has been set, and 
 a class has been given twenty minutes to solve as many problems 
 as possible, teachers must use their judgment as to marking 
 pupils who are naturally slow. If their work is accurate, and
 
 34 The Teaching of Arithmetic 
 
 they have done a reasonable number of examples, they are 
 entitled to credit and should receive commendation. 
 
 BIBLIOGRAPHY: The author's Teaching of Elementary Mathe- 
 matics, pp. 121-129; Handbook to Arithmetics, p. 8; Practical 
 Arithmetic, Boston, 1906, pp. 115, 159; Wentworth-Smith, Com- 
 plete Arithmetic, and Arithmetic, Book II, Boston, 1909 and 
 1911, p. 191.
 
 CHAPTER IX 
 CHILDREN'S ANALYSES 
 
 The questions of mental and written arithmetic lead naturally 
 to that of the analyses to be expected on the part of children. 
 What is their object, what should be .their nature? How exten- 
 sively should they be required? 
 
 As to the first, the only defensible object would seem to be 
 that through these analyses a child makes it clear that he under- 
 stands a particular problem or operation. That he acquires a 
 habit of formal statement that is helpful in other lines of work, 
 or that his memory is strengthened by learning set forms of 
 analysis, has been too often disproved to require argument. 
 To the extent that this analysis is really an explanation of his 
 process there is an unquestionable advantage, since it enables a 
 teacher to commend or improve the pupil's work. But how 
 often is this the case? Indeed, how often should it be expected 
 to be the case? Is it not the general experience that pupils too 
 often memorize their analyses, and that teachers commend glib 
 repetitions of their own words or those of the text-book, the 
 matter being so imperfectly comprehended by the child that he 
 is able to bear no questioning? 
 
 To take a concrete case, we occasionally hear some teacher 
 say that not a child in the class can explain why, in dividing by 
 a fraction, he inverts the divisor and multiplies. But why should 
 he explain it? And if he does, will he do any more than repeat 
 in a perfunctory way the analysis he learned from the book or 
 the teacher? It took the world thousands and thousands of 
 years to learn this process. It was a thousand years after 
 Euclid made his great geometry before it was used, and nearly 
 another thousand years elapsed before it appeared in a printed 
 book. This means that maturity of mind was required to de- 
 velop such a process, and still greater maturity was needed to 
 embody it in a text-book. 
 
 35
 
 36 The Teaching of Arithmetic 
 
 But does this mean that no explanations are to be given or 
 required? By no means. A child should know this process of 
 dividing, and he should learn it by a teacher's questioning; he 
 should thereby know that it is reasonable, and he should feel that 
 for the time he understands why he proceeds in this manner. 
 For that occasion he may be questioned as to all this, but that 
 he should long remember the " why " of it all, or that he should 
 be able, at any time that some teacher or supervisor thinks fit, 
 to give a lucid explanation of such a mature process is as un- 
 natural as it is unscientific. 
 
 So it is with the fundamental operations in general. There 
 is no good reason why a child should remember for any consid- 
 erable time an explanation for multiplying one integer by an- 
 other; it is sufficient that he learned the operation as a rational 
 one, and that he can perform it quickly and accurately as we can 
 or as any business man does. If he does give an explanation it 
 will usually be found to be merely a parrot-like repetition of the 
 teacher's or the text-book's words, without any apparent mental 
 content. 
 
 In the matter of the applied problems the case is different. 
 So long as a pupil does not blindly recite formal analyses, there 
 may be a good deal of value in his explanations. If allowed to 
 state his reasons in his own language, with limitations as to 
 tolerable English, he may acquire a habit of succinct and logical 
 statement that will help him in many other lines of expression. 
 This affords, moreover, a very good opportunity for the teacher's 
 commendation and advice, criticism in the best sense of the 
 term, the word too often being employed to signify mere fault- 
 finding. 
 
 My colleague, Professor Suzzallo, has properly called atten- 
 tion to the fact that the problem of teaching children to reason 
 in arithmetic is twofold: (i) " It is a matter of the ability to use 
 language; (2) It is a matter of good thinking." The former 
 has been confused with the latter by most teachers, it being felt 
 that if the child repeated the book language of reasoning he was 
 satisfying the demand for honest thinking. Genuine training 
 in reasoning is not this, however; it is a carefully thought out 
 process, beginning with problems involving only a single step, 
 and leading gradually to those involving two steps. This is a
 
 Children's Analyses 37 
 
 reasonable limit of primary work, problems involving three steps 
 being rather matters for the intermediate grades. 
 
 In all this work it should be borne in mind that there are 
 three things that are properly demanded at one time or another, 
 but not necessarily for each problem that is solved. These 
 three are: (i) to work rapidly and accurately; that is, to take 
 the shortest road to the answer, and to be certain the answer 
 is correct; (2) to put neatly on paper not merely the operation 
 but a brief explanation; (3) to give a brief analysis or oral 
 explanation. 
 
 For example, if 5 yd. of cloth cost $2.10, how much will 
 12 yd. cost? 
 
 (i) The number work: 
 
 .42 $.42 
 
 12 X $2.IO 12 
 
 5 84 
 
 42 
 $5.04 
 
 (2) The written explanation: 
 
 5 yd. cost $2.10. i yd. costs 1 / 5 of $2.10. 
 12 yd. cost 12 X V B of $2.10, or $5.04. 
 
 (3) The oral analysis: 
 
 Since 5 yd. cost $2.10, i yd. will cost Vs of $2.10, and 12 yd. 
 will cost 12 X 1 / 5 of $2.10, or $5.04. 
 
 Teachers will not need to call for all this work with every 
 example. Sometimes it will be necessary to emphasize (2) and 
 sometimes (3), but the important thing is that (i) should be 
 quickly and neatly, and, above all, accurately done. One of the 
 best ways to secure this accuracy and to avoid absurd answers 
 is to estimate the result in advance. The pupil should write 
 down this estimate and compare it with the answer, and if there 
 is a great difference look over his work again. 
 
 For example, if 5 yd. cost $2.10 we know 12 yd. will cost 
 nearly 2 l / 2 times as much, or somewhere near $5. When we 
 solve, if we find such a result as $50.40, we see at once that there 
 is a mistake, probably in the position of the decimal point. The 
 correct result is $5.04.
 
 38 The Teaching of Arithmetic 
 
 As an example of written work, involving both the computa- 
 tion and the analysis, the following may be considered: 
 
 A merchant bought 800 yd. of linen lawn at 67^ c. a yard, 
 and sold 725 yd. at 8oc. a yard, and the rest at a bargain sale 
 at 65c. a yard. Find his profit. 
 
 $0.67^ $0.65 $725 
 
 800 75 .80 
 
 400 325 $580.00 
 
 536 455 48.75 
 
 $540 $48.75 $628.75 
 
 540. 
 
 $88.75 
 The written analysis is as follows: 
 
 1. 800 X $0.67^ = $540, the cost. 
 
 2. 800 yd. 725 yd. = 75 yd., sold at bargain sale. 
 
 3. 75 X $0.65 = $48.75, received at bargain sale. 
 
 4. 725 X $0.80 = $580, received at regular sale. 
 
 5. $580 + $48.75 = $628.75, total receipts. 
 
 6. $628.75 - $540 = $88.75, Profit. 
 
 Of course this solution could easily be shortened, but for be- 
 ginners it is as well not to attempt too much brevity. 
 
 In conclusion it may be said that set forms of analysis seem 
 to be rather more harmful than helpful to children, but that 
 explanations in their own language may be the means of acquir- 
 ing valuable habits and of offering to the teacher the opportunity 
 for helpful suggestions. To acquire this power the child must 
 be initiated gradually, first in simple examples in one-step rea- 
 soning, and second in two-step reasoning. 
 
 BIBLIOGRAPHY : Young, p. 205 ; The author's Handbook to 
 Arithmetics, p. 9; Suzzallo, loc. cit.
 
 CHAPTER X 
 INTEREST AND EFFORT 
 
 There has of late years been a tendency throughout the country 
 to make arithmetic, as other subjects, more interesting to children. 
 What the real motive was it is hard to say, since it was probably 
 somewhat subconscious. Such statistical information as we have 
 shows arithmetic always to have been looked upon by children 
 as one of the most interesting subjects of the course, so that the 
 reason was not that it was relatively a dull study. Possibly the 
 desire was that the work of the teacher should become easier 
 through increased interest on the part of the pupils. But whatever 
 the reason it cannot be questioned that, other things being always 
 kept equal, there is a great gain in increasing the interest in any 
 kind of work. 
 
 There is, however, a general danger accompanying this effort 
 to increase interest. If this increase means that the subject is to 
 become anaemic, if it is not to require the same serious effort to 
 master it as heretofore, then it loses a considerable part of the 
 value that has generally been assigned to it. Moreover, through 
 this same cause it loses a considerable part of the very interest 
 that was expected to be fostered. Boys and girls do not like to 
 wrestle with infants or with infantile subjects, and unless a study 
 is suitably graded as to difficulty it will appeal in vain to the 
 interest, the vigorous attack, and the responsive mental effort of 
 the pupils. 
 
 Our lesson, therefore, is that we should do all in our power 
 to make arithmetic interesting and even attractive to the children, 
 but that we must not hope to attain this result by offering a sickly 
 substitute for the vigorous subject that has come down to us. 
 Unfortunately we have not been free from this fault of making 
 our arithmetic, and particularly our primary arithmetic, anaemic. 
 Foreign critics frequently comment upon this failing, and claim 
 with good reason that much of our work in the early grades 
 
 39
 
 4O The Teaching of Arithmetic 
 
 lacks vitality. Certain it is that in spite of many points of 
 superiority of the American school we do not at the end of eight 
 years bring our children as far as European experience would 
 seem to lead us to hope. 
 
 How can the interest in the applications of arithmetic be 
 aroused and maintained? The reply has already been made. 
 They must be real if they pretend to be so, they must relate when 
 possible to the child's daily environment, and they must reveal 
 the life of America to-day in such a way as to be broadly informa- 
 tional as well as mathematical. This can be accomplished with 
 no less demand for mental power than was required by the obso- 
 lete problems of our old-style books. There are, however, various 
 other channels through which we may pass to reach the required 
 end. For example, there are the number games for children of 
 the primary grades, games that have an interest that pleasantly 
 conceals the mental effort required, as tennis does the muscular 
 effort, but that accomplish the result efficiently. This subject is 
 considered later in this work. Then there are the problems of 
 heroic effort, then of mechanical effort, then of simple building, 
 and later of our national resources. These resources, correlating 
 as they do with geography, concern our supply of food, and 
 clothing, and home comforts; they touch our mines, our farms, 
 our transportation, and the great industries of our people. 
 
 As concrete illustrations of this type of problem the following 
 relating to some of the metals produced in this country may be 
 considered : 
 
 1. When 700 Ib. of copper was worth $73.50, what was 250 
 Ib. of the same quality worth? 
 
 2. A dealer sold 800 Ib. of nickel at 55 ct. a pound, thus gain- 
 ing 10% on the cost. How much did it cost him? 
 
 3. If a dealer buys 900 Ib. of zinc at $88 a short ton, at what 
 rate per pound must he sell it to gain 50%? 
 
 4. In one year silver averaged 54.98 ct. an ounce, which was 
 102.8% of the average for the preceding year. What was the 
 average then ? 
 
 5. Silver is sold by the troy ounce. This is what per cent 
 of the avoirdupois ounce (7000 gr. = 16 av. oz., 5760 gr. = 
 12 troy oz.) ?
 
 Interest and Effort 41 
 
 6. Quicksilver (mercury) is sold by the flask of 76*^ Ib. If 
 a dealer buys it at $38.25 a flask, at how much a pound must 
 he sell it to gain 20% ? 
 
 7. If a short ton of lead is worth $96, how much is 750 Ib. 
 worth? If a dealer bought it at this rate and sold it at 8 ct. a 
 pound, what was his per cent of profit? 
 
 .The interest in such topics is measurably greater than that in 
 equation of payments, or most problems in compound proportion, 
 or examples in the Vermont Rule of Partial Payments (a subject 
 that, however, naturally has its place in the curriculum of that 
 state). On the other hand the effort may be just as great as 
 we wish to make it. It is only a matter of complicating the 
 problem sufficiently, and using numbers and combinations of 
 proper difficulty, to make a modern problem about the coal indus- 
 try of Pennsylvania, or the silver output of Colorado, as hard as 
 any example in the arithmetics of fifty years ago. 
 
 We have, therefore, the following points that seem fair con- 
 clusions: (i) It is possible to bring our arithmetic work to a 
 higher plane of interest, through the game element and the appli- 
 cations ; (2) it is possible, with this, to keep the plane of effort as 
 high as we wish; (3) with the increased interest must necessarily 
 come an increase of power that is vital to the improvement of 
 our education. 
 
 BIBLIOGRAPHY : In the matter of modern problems consult the 
 author's arithmetics. In the matter of mathematical recreations 
 consult Ball, Mathematical Recreations, London, 1896; Schubert, 
 Mathematical Essays and Recreations, Chicago, 1899.
 
 CHAPTER XI 
 
 IMPROVEMENTS IN THE TECHNIQUE OF 
 ARITHMETIC 
 
 Nothing new goes into arithmetic without a protest, and so for 
 what goes out. Nevertheless there has been an evolution here 
 as everywhere else, and this evolution has made for the betterment 
 of the subject. To take a concrete illustration, the first printed 
 arithmetics had no symbols of operation. What we would write 
 as "4X5" was then written " 4 times 5," with the natural 
 variation of the word " times " according to the language em- 
 ployed. It was half a century later, and after the symbols -f- and 
 were invented, before was suggested, and some eighty years 
 after that before X was used, and a long time after that before 
 -r- appeared for division. It was several generations after these 
 were first used before they came into our school arithmetics for 
 the purposes that we use them to-day, and always with strong 
 protest on the part of those who wish to " let well enough alone." 
 It was argued that "4X5" was more abstract than " 4 times 5," 
 that it was hard because of the symbolism, and that it took arith- 
 metic from the written language and the customs of the common 
 people for whom it was of greatest use. Invented for algebra, 
 the conservatives said that all the symbols ought to remain there 
 and not seek to enter the field of arithmetic. This struggle of 
 symbolism seems strange to us to-day, when a child in the first 
 grade learns at least half a dozen signs of operation and relation, 
 and few would be found to advocate going back to the old cus- 
 tom. We are, however, face to face with similar questions and 
 many of the very ones who would argue to keep -4- (a symbol 
 practically unknown save in England and America, strange as 
 this may seem), are the ones who protest most vehemently against 
 letting x stand for a phrase that is too long to write conveniently. 
 But the question is the same. If we use -r- as a short-hand way 
 of writing " divided by," why should we not use x as a short- 
 
 42
 
 Improvements in the Technique of Arithmetic 43 
 
 hand way of writing " what the horse cost," or " the amount due 
 the first man," or any other phrase representing a quantity to be 
 sought, an " unknown quantity " ? Here, then, is one of the 
 improvements suggested by algebra to assist us in reasoning out 
 the solution of an arithmetical problem. That " I.KXF = $3300, 
 therefore x $3000 " is algebra instead of arithmetic, is no more 
 true than that " 4 X 5 " is algebra while " 4 times 5 " is arithmetic. 
 The symbol X was used only in algebra instead of arithmetic for 
 a century or so, as x was for over a century longer, but the 
 employment of each to assist in arithmetic does not make " a 
 solution by algebra." This illustration is brought forward as one 
 of the most prominent at the present time. It is impressed upon 
 me by numerous letters asking for " a solution by arithmetic 
 instead of by algebra" for some little problem that is made 
 clearer by the use of a single symbol in place of a phrase like 
 " the number of bushels," or " the cost of the farm." Teachers 
 should realize that they hereby show an ignorance that is hardly 
 pardonable at the present time, and that such improvements in 
 symbolism are a part of the natural development of the subject. 
 
 Of course there is the danger of overdoing all this. This has 
 often been seen, and is apparent to-day. For example, it is better 
 to train a child's eye to see that 4 and 5 are 9, by 
 putting the symbols in the form here given, the form 4 
 in which he will usually meet them in computation, 5 
 than to train it with the symbolism 4 -f 5 = 9, which 
 he rarely sees in practical work. On the other hand, 9 
 to neglect the latter entirely is to unfit him for reading 
 any save the simplest mathematics, and for expressing his solu- 
 tions in the condensed step-form necessary to allow the eye 
 quickly to grasp the reasoning. So with a symbol like x; we 
 may use it where there is not only no advantage in doing so, 
 but a positive disadvantage. It is the part of the text-book and 
 the teacher to suggest to the pupil the problems in which it should 
 be employed, and to furnish a reasonable amount of exercise in 
 the subject. 
 
 To consider a still more definite illustration, take this problem : 
 If some goods are sold at a profit of \2.y 2 % for $1,012.50, what 
 did they cost? Here we have several possible lines of attack, 
 among which are the following, (i) If there was a gain of i2 l / 2 %
 
 44 The Teaching of Arithmetic 
 
 they must have been sold for 112^ % of their cost. (But where 
 did that 100 come from?) And since $102}^ is H2y 2 % of the 
 cost, i% of the cost is T }^ (But how are we to read such a 
 fraction?) of $1,012^, and 100% (But why do we wish 100% ?) 
 is 100 times this. Such an explanation, while it could be learned 
 and recited, would be subject to questions like those inclosed in 
 the parentheses, and really would have very little reasoning in it. 
 
 (2) Let 1 00% stand for the cost. (But why 100% instead 
 of 200% or 50% or some other number?) Then 100% + i2 l / 2 % 
 will stand for the selling price. If 112^% = $1,012^ (But an 
 abstract number, 1.12^/2, cannot equal a concrete number, 
 $1,012^%) then i% = $1,012^ -r- 112*6 (Why? Why not mul- 
 tiply? Why not divide by H2y 2 % instead of 112^?), and 100% 
 (why not find 500%?) = 100 times as much, or $900. This 
 analysis is quite as superficial and unsatisfactory as the first. 
 
 (3) Let i = the cost. (But why not let 2 equal the cost? 
 and why take I instead of ioo%?) The rest of this solution 
 might follow the lines of (i) or (2), and in that case it would be 
 equally unsatisfactory. Like (2), it is a relic of the old and 
 long-since forgotten method of " False Position," the discarding 
 of which caused so many conservative teachers to feel that arith- 
 metic was losing its mental discipline. So it would be possible 
 to give a number of other plans of attack, some better than 
 the above, and some much worse. Consider, however, a single 
 other plan. 
 
 (4) Let c = the cost, a very natural thing to do since it is 
 the initial letter. Then the selling price is c + .\2.y 2 c. 
 
 But c + ,i2 J / 2 c = I.i2,y 2 c, and therefore I.i2y 2 c $1012.50. 
 Dividing equals by 1.12^2, = $900. 
 
 Now we are not troubled here about any 100%, or i%, or 
 letting i equal something that it cannot equal. As soon as we 
 think of \2.y 2 % as the same as 0.12%%, and know that c + .I2 l / 2 c 
 = i.i2jAc, just as c + l / 2 c = iy 2 c, the case is exceedingly simple. 
 
 Here, then, is one of the places in which the technique of 
 arithmetic has been greatly improved by the introduction of an 
 easy symbol from algebra, as it was years ago improved by the 
 introduction of the symbols of operation. No teacher who has 
 ever seriously tried to use this symbol has ever willingly aban- 
 doned it.
 
 Improvements in the Technique of Arithmetic 45 
 
 What has been said for the symbol x might be said for other 
 symbols if we needed them. It is of no particular consequence 
 that we use 4^ instead of V4 in arithmetic, because we do not 
 make much use of square root in arithmetic, but if we did the 
 more modern symbol would deserve a place for its influence in 
 later work, and the same may be said of the negative number, 
 the parenthesis, and other signs of algebra. 
 
 Outside of symbolism, however, there are certain improve- 
 ments in technique that demand our consideration. One relates 
 to subtraction, and in order to have it satisfactorily understood 
 it becomes necessary first to speak of it historically. .There 
 have been several successful plans for teaching subtraction ; each 
 has endured for a long period, and some of them are in use to- 
 day. Of the most prominent the following may be mentioned: 
 
 (1) The complementary method. Instead of taking 8 from 13 
 we may add 10 8 to 13 and then drop 10. 
 
 This depends on the relation 13 8=13+ 13 13 
 (10 8) 10, and since 10 8 is called the 8 2 
 complement of 8 (to the number 10), this is 
 known as the complementary method. It is 5 5 
 very old, appearing in a famous Hindu work 
 of the I2th century, in the first printed arithmetic (1478), and in 
 numerous other text-books. In a case like 452 348 the opera- 
 tion would be as follows : 2 + 2 = 4; since 10 must 
 452 be dropped, we may add i to the 4 instead; then 
 348 5 5 = 0. and 4 3=1. That is, the complemen- 
 tary idea need be used only when the minuend is 
 104 less than the subtrahend. This example is actually 
 taken from the first printed arithmetic. The plan is 
 the same as the one used in Pike's famous American arithmetic 
 a century ago, and some teachers still employ it. It is essentially 
 the one used when we employ co-logarithms in trigonometry. 
 
 (2) The borrowing and repaying plan, a name that we may 
 use for want of a better one. This may be illus- 
 trated by the annexed example, taken from the 6354 
 first great business arithmetic ever printed 2978 
 (Borghi's, of 1484). The operation is as follows: 
 
 8 from 14, 6; 8 from 15, 7; 10 from 13, 3; 3 from 3376 
 6, 3. This was the plan advocated most often by
 
 46 The Teaching of Arithmetic 
 
 the early printed arithmetics, and the expression " to borrow " 
 was a common one. It was known to the early Hindu arith- 
 meticians and in Constantinople before the invention of printing. 
 
 (3) Simple borrowing, to continue to use this old English 
 
 expression. In this method the computer says, " 7 
 42 from 12, 5; 2 from 3, I." This is probably the most 
 27 common plan in use to-day, and it has much to com- 
 
 mend it. It has a long history, appearing in the 
 15 oldest known manuscript on arithmetic in English, in 
 
 Spain in the I3th century, in Italy in the Middle Ages, 
 and in India still earlier. It has not, however, been nearly as 
 popular as the second plan, although more generally used in our 
 country to-day. 
 
 (4) The left-to-right method, a plan that has had a long history 
 in which many prominent advocates have appeared. It is more 
 adapted to the needs of a professional computer, however, than 
 to those of the average citizen, and may therefore be dismissed 
 with this mere mention. 
 
 (5) The addition, "making change," or "Austrian" method, 
 a vague way of naming several related plans. The efforts made 
 to adopt it in the Austrian schools, and the consequent notice 
 taken of it in Germany, have been the cause of its most inap- 
 propriate geographical name. As a definite method of 
 
 17 subtraction it is not as old as the others, although it 
 
 8 appears in the i6th century in Italy and has had 
 
 occasional prominent advocates since. It consists in 
 
 9 finding what number must be added to the subtrahend 
 to make the minuend. Thus in thinking of 17 8 we 
 
 think: "8 and 9 are 17," writing down the 9. If the numbers 
 are longer we may proceed in either of two ways, 
 
 as in the annexed example. Here we may say : " 8 423 
 
 and 5 are 13 ; 4 and 7 are n ; i and 2 are 3." Or we 148 
 
 may say : " 8 and 5 are 13 ; 5 and 7 are 12 ; 2 and 2 are 
 
 4." Of these two, one is as easily explained as the 275 
 other. The first might naturally be approached thus : 
 
 423 400 + 20 + 3 = 300 + 1 10 + 13 
 
 148 IPO -f- 40 + 8 = IPO +40+8 
 
 200 +70+5
 
 Improvements in the Technique of Arithmetic 47 
 
 The arrangement for the second would be as follows: 
 423 = 400 + 20 + 3 
 148 IOQ + 40 + 8 
 
 Since the difference will be the same if we add the same number 
 to both minuend and subtrahend, we will add 10 to each, and 
 then 100 to each, giving the following: 
 
 400+ 120 + 13 
 200+ 50+8 
 200 +70+5 
 
 Since the explanations are about equal in difficulty, we may 
 consider the sole question of rapidity in practical use, both as to 
 the method in general and as to these two sub-methods in 
 particular. 
 
 Is the general plan the best one? On the side of advantages 
 we have: (i)It is the common method of making change. If I 
 owe $7.65 and pay $10 the merchant finds the change and I 
 verify his work by saying : " 65C. and 5c. are 7oc., and 3oc. more 
 makes $i, and $2 more makes $10." That is, we find the differ- 
 ence by adding. This is familiar in all business and in the 
 school, and will remain so. It is, therefore, a natural plan to 
 use in all subtraction. (2) It avoids the necessity for learning 
 a separate subtraction table. Everything is referred at once to 
 the addition table, a table that unfortunately is not at present 
 known any too well. There is, therefore, an economy of time 
 and an increased efficiency in the very important subject of 
 addition. (3) The facts of addition being used so much more 
 often than those of subtraction, there is naturally an increase 
 in speed and certainty when we employ the addition instead 
 of the subtraction table. 
 
 On the other side, it is not desirable to change the customs 
 of the people unless there is a decided gain by so doing. 
 Parents who have been brought up on one plan, and who help 
 their children more or less in their lessons, do not easily adapt 
 themselves to a new one, and the result is a confusion in the 
 child's mind that is most unfortunate. The fair question, there- 
 fore, is, Is it worth the while to use the better method? 
 
 If we had a standard method known by all, this argument 
 would have much weight, but we have at least three rather com- 
 mon ones, with several variations, already in use in this country.
 
 48 The Teaching of Arithmetic 
 
 There is, therefore, bound to be more or less confusion. The 
 only thing for the school to do, then, is to teach the method that 
 will prove in the long run to be the most rapid and accurate, and 
 this seems a priori to be the " Austrian," although a scientific 
 investigation of the matter, on sufficient data, is desirable. And 
 of the two or three sub-methods, the second one described on 
 page 46 seems without doubt the better. 
 
 The question is, it should be repeated, not one of explanation, 
 since any one of the methods is easily explained. It is purely 
 one of practical utility, in which the teacher should divorce him- 
 self of all prejudice in favor of the method which was taught to 
 him and with which he is most familiar, a somewhat difficult 
 thing to do in a discussion of this kind. It should also be re- 
 marked . that it is of doubtful policy to attempt to change a 
 method that a child already knows and handles easily, inasmuch as 
 the difference in value is not great enough to warrant this course. 
 For beginners, however, the plan suggested is probably the best. 
 
 These two matters of improvement in technique have been 
 mentioned, not as so important in themselves, but as types of 
 various changes that are worthy of sympathetic consideration. 
 The placing of the quotient over the dividend in long division, 
 instead of at the right as was formerly done, so as to make more 
 clear the position of the decimal point; the multiplying of both 
 dividend and divisor, in the division of decimals, by such a power 
 of 10 as shall make the divisor an integer, thus avoiding the old 
 difficulty of determining the number of integral places in the 
 quotient ; the giving of the full form of an operation before the 
 abridged one, in explaining the process ; the writing of ratios in 
 the familiar form of fractions in the first steps in proportion; 
 the putting of the unknown quantity first instead of last in writ- 
 ing a proportion and the using of x to represent this quantity, 
 these are some of the other improvements in technique that are 
 getting into our books in these days and that should command 
 our interest. To make these more clear, the space not allowing 
 of elaborate discussion, an example of each is here given. Thus 
 in long division it is better to use the first of these forms than 
 the second, for the reason specified, that it makes clear the 
 position of the decimal point, a reason that holds equally true 
 for the second example given on page 49:
 
 Improvements in the Technique of Arithmetic 49 
 
 14.24 
 
 72)1025.28 72)1025.28(14.24 
 
 72 72 
 
 305 305 
 
 288 288 
 
 172 172 
 
 144 144 
 
 288 288 
 
 288 288 
 
 Similarly, the first of these forms is better than the second, the 
 problem being to divide 102.528 by 0.72 : 
 
 14^.4 
 
 72)10252.8 0.72)102.528(142.4 
 
 72 72 
 
 305 . 305 
 
 288 288 
 
 172 172 
 
 144 144 
 
 288 288 
 
 288 288 
 
 In the early stages the first of the following forms is better than 
 the second: 
 
 1424 1424 
 
 72)102528 72)102528 
 
 72000 = looo X 72 72 
 
 305 2 8 305 
 
 28800 = 400 X 72 288 
 
 1728 172 
 
 1440= 20X72 144 
 
 288 288 
 
 288 = 4 X 72 288
 
 5o The Teaching of Arithmetic 
 
 In introducing the idea of proportion it is better to begin with 
 known symbols, so as not to confuse the pupil too much. Thus 
 the first of these forms is better than the second in the early 
 stages : 
 
 x 39 
 
 91 : 39 :: 7 : (?) 
 
 7 9i 
 
 Indeed it is always better to use = than : :, and the latter is now 
 happily passing into oblivion in this country. 
 
 There are several more important questions for the future, in 
 relation to the technique of computation, but these can be men- 
 tioned only briefly. The first relates to limits of accuracy in re- 
 sults. This does not mean that the work may be inaccurate, 
 but that if we know the circumference of a circle only to two 
 decimal places we cannot from that find the diameter to three or 
 more decimal places. We express this by saying that the result 
 cannot be more accurate than the data. Suppose, for example, 
 we have made several measurements of the circumference of a 
 steel shaft, and their average is 7.57 in.; it is evidently useless 
 in dividing this by 3.1416 to carry the result beyond two decimal 
 places. Required, therefore, to divide so as to avoid unneces- 
 sary work. This is accomplished by what is known as con- 
 tracted division, thus: 
 
 241 
 
 3.1416)7.57 
 628 
 
 i 29 
 125 
 
 4 
 3 
 
 Now how much of this kind of work are the schools called upon 
 to teach? It is certainly not a thing that many people will need 
 to know, and, therefore, it is properly omitted from most text- 
 books to-day. In some localities, however, it might very pro- 
 perly be taught, and when our classes in physics require the 
 mathematics that they might properly demand, it may become
 
 Improvements in the Technique of Arithmetic 51 
 
 necessary to teach contracted multiplication and division in the 
 schools. 
 
 Another topic is logarithms. In all engineering computations 
 this labor-saving device is used, and the subject is easily taught. 
 What shall we do with it in arithmetic? So far as grade work 
 is concerned there is nothing to be done at present, because the 
 demand is not great. But we can hardly say what the future 
 may bring forth, and their use may become much more common 
 than we think if the teachers of physics and the advocates of 
 more mathematics in manual training stir up' the question of 
 greater facility in practical calculation. In the same way we 
 may yet see the slide rule (a simple instrument for computing 
 by machinery) find a place in business or technical courses in 
 the grades of the elementary school. At present our duty points 
 to a teacher's knowledge of all these matters of technique, and 
 a sympathetic awaiting of the time when some or all may de- 
 mand more serious attention on the part of the school.
 
 CERTAIN GREAT PRINCIPLES OF TEACHING 
 ARITHMETIC 
 
 Before considering the curriculum in arithmetic it is well to 
 devote a little attention to certain great principles that teachers 
 have as a whole agreed upon, in theory if not in practice. Some 
 of these have already been discussed in this article; others will 
 strike the reader as rather trite, which simply means that they are 
 generally accepted ; and others will not appeal to all. They will, 
 however, be found to be suggestive of the thought of leaders at 
 the present time, and they may, though stated dogmatically, 
 form the basis for profitable discussions by teachers. 
 
 1. Arithmetic is taught both for its usefulness in daily life 
 and for the training that it gives the mind in reasoning, in habits 
 of application, and in exactness of statement. 
 
 2. Most of the mental discipline of arithmetic can be secured 
 from those portions that may be called practical, and therefore 
 the practical side of arithmetic may safely be emphasized. 
 
 3. But in emphasizing this practical side we need to offer a 
 large amount of abstract work as well as concrete problems, skill 
 in either not necessarily signifying skill in the other. 
 
 4. In the concrete problems, whatever pretends to be genuine, 
 to represent practical questions of American life, should be so, 
 all obsolete business problems being replaced by modern ques- 
 tions. In particular, the daily industries of our people should 
 be drawn upon to the making of arithmetic interesting, informa- 
 tional, practical. 
 
 5. The topical arrangement of the curriculum has been per- 
 manently abandoned, and either a non-topical arithmetic should 
 accordingly be used, or some good modern topical arithmetic 
 should be adopted and selections should be carefully made so as 
 to offer a moderate " spiral " (" concentric-circle," " recurring 
 topic ") arrangement adapted to the growing powers of the child. 
 
 52
 
 Certain Great Principles of Teaching Arithmetic 53 
 
 6. No extreme of " method " should be adopted by any 
 teacher or school, but the best of every " method " should be 
 known as far as possible to all. To measure everything in sight, 
 to base all arithmetic on sticks, to go to extremes on number 
 charts, to put all the time on mental arithmetic, to have all 
 written work placed in steps, to get into any narrow rut what- 
 ever,- this is to fail of the best teaching and to narrow the hori- 
 zon of the children in our care. A good, usable text-book, broad 
 in its purpose, modern in its problems, and psychologically ar- 
 ranged, is one of the best balance-wheels on us all, and we should 
 depart from its sequence and methods only for reasons that have 
 been very carefully considered, while supplementing its good 
 features by all the problems with local color that we can find 
 time to use. 
 
 7. Mental (oral) arithmetic should play a part in every school 
 year, to the end that children should have not only an eye 
 training for numerical relations, but also an ear training and a 
 tongue training. The text-book may be expected to furnish a 
 considerable amount of the abstract work in this line, but the 
 concrete problems may well be correlated with local life and with 
 the other work of the class. 
 
 8. Children's analysis instead of being memorized should be 
 genuine statements of the reasons that prompt them to their 
 solutions. As to problems, the analysis should proceed gradu- 
 ally from one-step to two-step cases. As to operations, it is not 
 to be expected that children will long remember the reasons in- 
 volved; they should understand the process when presented 
 through a development by a series of simple questions ; but they 
 should not be expected to give a very elaborate explanation of 
 a topic like long division after the process is once understood. 
 
 9. Written arithmetic may at one time emphasize the rapid 
 securing of results, and at another the analysis of the problem. 
 Both are important, but in general the accurate result, rapidly 
 secured, is the great desideratum. To say that a child ought not 
 to work merely for the answer is a well-sounding epigram, but 
 if it is interpreted to mean that he may work in a slovenly 
 way, dawdling over his problem, and getting an answer that is 
 absurd, no amount of neatly written step-work can atone for his 
 mental laziness.
 
 54 The Teaching of Arithmetic 
 
 10. Arithmetic should be as interesting as any other subject 
 in school. To this end a teacher should know something of its 
 interesting story, should be familiar with its best applications to 
 local and national life, should know how to treat the mental 
 (oral) exercises in sprightly fashion, and should have a fair stock 
 of number recreations. 
 
 11. The improvements in the technique of arithmetic, includ- 
 ing the use of x and the later forms of operations, should be 
 sympathetically understood by teachers, to the end that the sub- 
 ject may not stagnate in our schools. 
 
 12. It is a matter of relatively little importance that we 
 present fractions, we will say, in this way or that, by sticks or 
 paper- folding or clay cubes or blocks; but it is a matter of great 
 importance that we present the subject in some concrete fashion, 
 to the end that the child does not proceed by arbitrary rules but 
 makes up his own directions, and that he is so guided that these 
 are the best that can be evolved at his age. What has been 
 rather pedantically called " heuristic teaching," in its. original 
 form as old as Socrates at least, should always be in the teacher's 
 mind to lead the child unconsciously to feel that he is the dis- 
 coverer, but to see to it that he is allowed to discover and to 
 fix in mind only what the world has found to be the best. The 
 carrying out of this policy, in arithmetic or any other subject, 
 is one of the essentials of good teaching. 
 
 These are a few of the larger principles that should guide the 
 teacher of arithmetic. The list might be extended, but these 
 suffice to show the spirit in which the subject should be ap- 
 proached. Minor principles will appear as we consider the work 
 of the various grades or school year. 
 
 BIBLIOGRAPHY: Smith, Teaching of Elementary Mathematics; 
 Young; The Teaching of Mathematics; De Morgan, On the 
 Study and Difficulties of Mathematics, Chicago, 1898; Clifford, 
 Common Sense of the Exact Sciences, 3d edition, New York, 
 1892.
 
 CHAPTER XIII 
 GENERAL SUBJECTS FOR EXPERIMENT 
 
 It is well, before leaving this general discussion, to consider 
 a few of the subjects for legitimate experiment in the teaching 
 of arithmetic, that might occupy the attention of schools of 
 observation or practice in connection with institutions for the 
 training of teachers. 
 
 (i) It is desirable to know just how far recreations in num- 
 ber can be used to advantage in teaching arithmetic. Of course 
 we have such games as bean bag, ring toss, and sometimes dom- 
 inoes and number games with cards, used in the school room. 
 This, however, is a mere beginning. There are many more 
 games that are usable for children. For example, more people 
 in the history of this world have learned elementary number 
 through dice than in the public school, and this is only one of 
 several widely used number games. It would be very easy to go 
 to a ridiculous and even dangerous extreme in this matter, and 
 a teacher who begins to work upon it will naturally tend to 
 do this, and will need a balance wheel upon his endeavors. 
 Nevertheless the work has never yet been done scientifically 
 and it ought to be undertaken. This is, however, only a small 
 part of the problem. There is the whole field of mathematical 
 recreations that must sometime be examined scientifically. We 
 have a large but undigested literature upon the subject, and no 
 one has ever yet studied it from the standpoint of the definite 
 needs of the grades. The result of such a study ought to add 
 greatly to the interest in arithmetic, without going to any ridicu- 
 lous and impractical extreme. Work there must always be in 
 arithmetic, and it ought to be good hard work, but there is no 
 reason why we should not let pupils see the amusements as well 
 as the other interesting phases of the subject. 
 
 In this matter of the play element the following brief list of 
 games available for primary number work may be of service. 
 
 55
 
 56 The Teaching of Arithmetic 
 
 It was prepared by Miss Julia Martin in connection with some 
 work undertaken for the State Department of Public Instruction 
 of Michigan: 
 
 (1) The Game of Tag. Every child in the room is given a number. 
 One child is the leader. He gives orally any number below a number that 
 has been designated by the teacher. All pupils who have numbers which 
 are factors of the given number must change seats. They may be tagged 
 while running or if they fail to run. 
 
 (2) The Game of "Simon says Thumbs Up." The teacher or one of 
 the pupils may act as leader. The whole class take a position with thumbs 
 up, and each pupil has a number. The leader says " Simon says 15." The 
 thumbs of numbers 3 and 5 (the factors) must go down. " Simon says 
 12." The thumbs of 2, 3, 4, and 6 must go down. Occasionally the leader 
 says " Simon says all thumbs up." This game may be used in addition 
 and subtraction drill work. 
 
 (3) The Guessing Game. This game is suitable for Grade II. The 
 materials needed are 20 counters for each pupil and the teacher, and a 
 slip of paper and a pencil for each. The teacher picks up a handful of 
 counters and says, " How many counters have I ? " The pupil guesses, and 
 if correct he gets the counters, and writes the number on a slip of paper. 
 If he fails to give the correct number he must give the teacher the differ- 
 ence between the number and his guess. He must write this number on 
 the paper. At the close of the game the pupils add the gains and losses, 
 and check their work by the counters left. 
 
 Miss Martin suggests the following additional list as valuable 
 in arithmetic : buzz ; fizz ; railroad station ; bean bag ; days of the 
 week; hop Scotch; school ball; bird catcher; marbles. To this 
 list may also be added certain card games of an arithmetical 
 nature published under the editorship of the author by the 
 Cincinnati Game Company. 
 
 (2) It is desirable definitely to map out the chief interests of 
 children from grade to grade, with a view to ascertaining the best 
 field for applied problems from year to year. We know these 
 interests in a general way, and we know the child's mind well 
 enough to judge of his arithmetical powers from year to year. 
 But we do not yet know these interests in the exact way that we 
 should know them. For example, when is the game element 
 strongest? When does the interest in the heroic become most 
 pronounced, and is the period the same for boys as for girls so 
 that we may use this information in problem work in a mixed 
 school? When does the interest become manifest in the food 
 supply of our country? When in the clothing supply? When
 
 General Subjects for Experiment 57 
 
 in transportation? When in the mines? When in manufactur- 
 ing? When in commercial life? We know all this in a general 
 way, but only so, not exactly, not as the result of any scientific 
 investigation. And when we do know this the rational appli- 
 cations of arithmetic from year to year will be so much better 
 understood that the subject will have an interest that is now 
 but feebly developed, and a value that we at present appreciate 
 only in part. 
 
 (3) We also need to know, statistically if possible, the result 
 upon a group of children of emphasizing the abstract problem; 
 upon another group of emphasizing the concrete; and upon a 
 third of leaving the two in about the balance that experience has 
 dictated. We have had some scientific investigation in this line, 
 but it has been very slight and it is therefore not conclusive. 
 To emphasize the concrete would be to diminish the number of 
 problems very greatly, and it would seem to give less exercise in 
 number relations, and it does seem to give less satisfactory re- 
 sults. On the other hand it may create interest in the work, 
 thereby increasing the power of impression of the number rela- 
 tions, so that because the child multiplies only V 25 as many times 
 it will not follow that he knows his subject only 1 / 23 as well. 
 At present the whole subject is in the domain of doctrinaire 
 argument; what is needed is a scientific investigation of the 
 problem by some school or some person who is unbiased in 
 the case. 
 
 (4) We are at present entirely unsettled upon the question of 
 time to be assigned to arithmetic. Two scientific investigations 
 have been made, but each is incomplete. It would seem that 
 excellence in arithmetic work is much less a function of the 
 time assigned to it than has formerly been supposed. Such an 
 investigation would probably require a number of years for its 
 satisfactory completion, but it might be undertaken in any 
 particular school system in a single year with some very helpful 
 results. 
 
 (5) We are quite uncertain as to the relative amount of time 
 to be devoted to oral and written arithmetic in our schools. 
 The wave of oral work, beginning with Pestalozzi and culminat- 
 ing in this country with Warren Colburn as mentioned in 
 Chapter VII, gradually subsided some years ago. What should
 
 58 The Teaching of Arithmetic 
 
 we be doing in the matter to-day? It is very easy to talk dog- 
 matically about it, but we need, if it is possible, a scientific 
 investigation as to the practical results of more " mental " arith- 
 metic, and of less. Would it be well to have the work much 
 more oral than at present? or would we gain by confining our 
 energies more to written work? Is there any scale by which 
 we can definitely measure this matter? and if so, what is it and 
 what are the results of the measurement? 
 
 (6) Just how far we are justified in departing from the old 
 plan of making the operations the basis for a course in arith- 
 metic, and of substituting therefor the applications? Shall we 
 ever be justified in giving up multiplication as a topic and in 
 substituting a chapter on housebuilding that shall bring in 
 multiplication as needed? Of course in the latter part of arith- 
 metic we put the application to the front, as in the early part 
 we put the operation there; but just where we should draw the 
 line, and how far should this latter plan encroach upon the 
 former? 
 
 To this list of general subjects for experiment my colleague 
 Professor Henry Suzzallo contributes a number of others, and 
 these make up the rest of this chapter. He remarks in the 
 first place that it is not sufficient that a new way or an old way 
 of teaching has succeeded, e. g., in the addition of fractions. 
 The test of the worth of a given method is not alone that it 
 gets a thing done efficiently ; it must get it done as economically 
 as possible. The method of most worth is the one that obtains 
 the efficient result with the least possible expenditure of energy. 
 The comparative worth of two methods must be investigated 
 under experimental conditions, with children of about the same 
 grade, age, and previous training, taught by teachers of fairly 
 even strength of personality, so that approximately the only 
 difference in the conditions of the trial is the difference in the 
 methods involved in the test. Finally the repetition of the ex- 
 periment in more than one school or system, reduces the danger 
 always present in assuming that a special group of children and 
 teachers is typical of school conditions in general. A sample 
 experiment will make the method of investigation clear, as 
 follows : 
 
 In the teaching of addition combinations it is the requirement
 
 General Subjects for Experiment 59 
 
 of certain courses of study that combinations and their reverses 
 be taught in association with each other. The teaching of " 3 
 and 2 are 5," should be followed at once by the learning of 
 " 2 and 3 are 5." In other courses of study or texts there 
 naay be quite an interval between the learning of these two com- 
 binations. In fact the student may learn each separately with- 
 out being conscious of any greater intimacy between these two 
 combinations than between any other two, e. g., " 6 and 3 are 
 9 " and " 4 and 2 are 6." Those who advocate the first method, 
 imply, if they do not expressly state, that learning a combina- 
 tion and its reverse simultaneously is more efficient than learning 
 them separately and unrelated. The problem for the investigator 
 is to determine whether or not this is true. 
 
 For example, in any large city school there may be two, 
 three, four, or more classes of one grade in which combinations 
 in addition are taught for the first time. In the case where there 
 are four classes, two could be set off against the remaining two, 
 with such judgment as to equalize number and quality of grades, 
 teachers' personalities, and other factors, in so far as they may 
 be equalized within such a limited range. All four classes could 
 then be given the same list of combinations to be learned, and 
 a common method of general procedure could be laid down, 
 by the experimenting principal, the only general difference in 
 procedure being that one group of classes would always learn 
 the reverses immediately following the original combination to 
 which it is related, while the other group would learn them in 
 an order that would separate the original combination and its 
 reverse. Thus : 
 
 Group I. 
 
 223243253435445 
 +232423524353454 
 
 455666777788899 
 Group II. 
 
 222323345345455 
 -r2 34354544222334 
 
 456677889567789
 
 6o The Teaching of Arithmetic 
 
 An equal amount of time having been spent in all the classes 
 up to a point where they have approximately covered the com- 
 plete list of combinations, a test could be given to determine 
 how far each individual had mastered these number facts. After 
 a lapse of a week or ten days another examination would show 
 how stable the mastery had been in each case. Any difference 
 between the group taught by one method and the group taught 
 by the other method as clearly shown by statistical interpreta- 
 tion would then tend to indicate the relative efficiency of the 
 two methods. 
 
 This being the method of experimentation, a few of the gen- 
 eral questions may now be considered. First, does the effective 
 Use of objective work demand a large amount of this work 
 with relatively young students and a smaller amount with rela- 
 tively old students? Or should the distribution of objective 
 work be determined by the fact of ignorance or immaturity in 
 any special phase of arithmetic regardless of the age of the stu- 
 dent ? To what extent does general arithmetical maturity require 
 less objective work in the first formal study of fractions than 
 in the first formal study of addition? 
 
 Do all of the fundamental combinations in any given field 
 (addition, multiplication, etc.) require objective development? 
 How many combinations need to be developed objectively 
 before the child will clearly know that succeeding combinations 
 given by the teacher authoritatively stand for real relations? 
 
 Furthermore, to what extent may the constant handling of 
 objects, pictures, and diagrams, and concrete imaging in gen- 
 eral, interfere with rapid abstract manipulation of numbers and 
 number combinations? 
 
 To what degree are the difficulties of children with arithmetic 
 problems due to a failure to understand underlying concrete 
 situations, becaus*e they do not understand the language by 
 which it is intended to convey them? What is the relative diffi- 
 culty in understanding the significance of a situation when the 
 presentation is (i) objective, (2) oral, and (3) written? Would 
 it be well to postpone the written presentation of problems until 
 a specific number of school years of language training have 
 been given ? 
 
 How far is it necessary to develop a special terminology for
 
 General Subjects for Experiment 61 
 
 school use in the subject of arithmetic, the terms being little 
 used in ordinary social relations? For example, consider the 
 case of the words multiplicand and dividend, the latter having 
 a radically different meaning in business life. In the case of 
 signs consider such semi-algebraic symbols as -T-, and even +. 
 If special signs are used in examples, to stand for the process 
 of calculation demanded by the situation (which might have 
 been expressed in concrete problem form) how wide and varied 
 should the language of problems describing situations be? If 
 + may be used in an example, while the same relation is ex- 
 pressed in a problem by such words and phrases as " added," 
 " and," " together," " how many," " altogether," how wide 
 should the latter vocabulary be? To what extent is a lack of 
 dealing with problems presented through varied language, ex- 
 planatory of the failure of children in problem tests given by an 
 outsider, another teacher, the principal or the superintendent, 
 when the children had always " done perfectly " the problems 
 that their own teacher gave them? 
 
 Should long forms of expressed calculations always precede 
 short forms? Taking the specific cases of division by a one- 
 figure divisor, or column addition involving several places, does 
 the fact that the long form is the one really used with a two- 
 figure divisor make the case different from addition of several 
 two-figure addends, where the long form is really not used in 
 business ? 
 
 To what extent are so called " aids " or " crutches " real helps 
 or final hindrances to efficient and rapid work, as in the 
 case where numbers are crossed out and others written in their 
 stead throughout the process of " borrowing " in column sub- 
 traction ? 
 
 Which types of problems are of most concrete interest to the 
 child, ( i ) those drawn from his own spontaneous play and work 
 life, or (2) those drawn from the facts of actual social life about 
 him? Are both essential in achieving the aims of arithmetic 
 teaching? If so is there any special law of advantageous usage 
 of each of the two types? Should problems from his own life 
 be used to introduce a field showing the necessity for learning 
 means of calculation, and those from social life be used for fur- 
 ther, later, and final application of the formal processes of cal-
 
 62 The Teaching of Arithmetic 
 
 dilation that have been mastered? Does it make a problem 
 concrete to the child merely because it is a concrete reality exist- 
 ing in the world? May not an imagined problem vividly within 
 the grasp of his own imagination, be more concrete in interesting 
 the pupil in its solution than one which actually exists in the 
 real world ? 
 
 Problems may be done by the pupil (i) silently ("mental 
 arithmetic"), (2) orally, (3) in written form on paper or black- 
 board, and (4) with a mixture of any two or all three of the 
 preceding methods. Which methods represent final forms in 
 which efficiency is demanded in ordinary life? Which forms 
 merely represent transitional means used by the teacher to keep 
 track of the workings of the child's mind? What is the proper 
 order and emphasis of these forms in the mastery of a single 
 new line of work, say in a problem where the child needs first 
 to divide and then to multiply? 
 
 To what extent do precise oral forms assist in the correct 
 analyses of problems, e. g., " If two apples cost 6 cents," etc. ? 
 To what extent do precise oral forms assist in the memorization 
 of combinations or manipulations, e. g., " 3 4's are 12," " put 
 down the three and ' carry ' the 2 " ? On the other hand, to what 
 extent do precise written arrangements of analyses assist the 
 child in carrying out a strictly logical mode of thinking? Take 
 the following case as an example: "3 pencils cost I5c. 
 i pencil costs 1 / 3 of I5c." etc. 
 
 Are certain algorisms more efficient and economical than 
 others? In which form should a pupil learn his addition com- 
 
 6 
 binations, 4 -f 6 = 10 or + 4? In the case of long division should 
 
 _IO 
 it be 45)6;836(/- or 45)67836? 
 
 Is it wise to use several different algorisms for a single 
 process, when it is possible to use but one? In other words 
 does not a multiplication of algorisms increase the amount of 
 memorization of forms required of the child? In division instead 
 of having three algorisms, one for the combinations 6-^-3 = 2, 
 another for short division 3)45735, and a third for long division 
 345)45735, would it be any better to have a similar form through-
 
 General Subjects for Experiment 63 
 
 out for easy identification of the three processes as fundamentally 
 one, thus 
 
 _2 15245 132 
 
 3)6, 3)45735, 345)45735 
 
 345 
 
 1123 
 1035 
 
 885 
 690 
 
 195 
 
 and in connection with all this what weight should be given to 
 Professor Smith's objection that the universal custom of the 
 business world does not recognize the first two forms of this 
 latter set? 
 
 Shall the attempt to get rapidity of calculation be preceded 
 by the attempt to get absolute accuracy, the quickening of the 
 work being left until certainty of command over combinations 
 is assured? Or shall rapidity in handling combinations and 
 manipulations be a parallel activity? Take for example the at- 
 tempt to have the children attack the successive combinations 
 in column addition with a definite rhythm, the teacher pointing 
 to each successive stage, or chorus work being utilized with the 
 quickest students setting the pace? 
 
 Does motor activity accompanying a process of memorization 
 of combinations require fewer repetitions than where no special 
 provision is made for motor activity? How far does it help 
 to repeat the numbers aloud ? to write them on paper ? to manipu- 
 late objects when the combination is being learned? 
 
 How far can rhythm be used in memorizing combinations or 
 tables? Does the use of rhythm decrease the number of repeti- 
 tions required for mastery? How much? 
 
 BIBLIOGRAPHY: On the general question consult the works of 
 Professor Young and the author. On the special question of 
 number games consult the following: 
 
 Johnson, G. E., Education by Plays and Games. Pedagogical Semi- 
 nary (Oct. 1894), Vol. III. This work contains a list of five hundred 
 games for children.
 
 64 The Teaching of Arithmetic 
 
 Johnson, G. E., Education by Plays and Games, Ginn and Company. 
 This book contains a suggestive course of plays and games, and these 
 games are correlated with the regular subjects of the curriculum. 
 
 Harper, C. A., One Hundred and Fifty Gymnastic Games, G. H. Ellis, 
 Boston. This and the two preceding are the most important. 
 
 Grey, Marion, Two Hundred Indoor and Outdoor Games, Freidenker 
 Publishing Company, Milwaukee. 
 
 Newell, W. W., Games and Songs of American Children. Harper and 
 Brothers, New York. 
 
 Kingsland, Mrs. Burton. The Book of Indoor and Outdoor Games, 
 Doubleday, Page & Co. 

 
 CHAPTER XIV 
 DETAILS FOR EXPERIMENT 
 
 Besides these subjects that may be designated as more or less 
 general, there are many details that demand investigation. These 
 are partly arithmetical and partly psychological in nature, and 
 belong quite as much in one field as another. So far as the 
 investigation itself is concerned, the trained psychologist is the 
 only one who could be expected to secure satisfactory results. 
 Some of these details have already been mentioned in this article, 
 and a dogmatic opinion has been expressed concerning them. 
 It is proper, however, to set them forth more at length for 
 the use of investigators. 
 
 At my request Professor Suzzallo, who has given the matter 
 much attention, has supplemented the suggestions given in the 
 preceding chapter by a further list of such special experiments 
 as occur to him, and has kindly permitted me to embody them 
 in this article. I therefore insert them without comment, it being 
 desirable that, in this place at least, they should appear with as 
 little expression of opinion as possible. This chapter XIV is, 
 therefore, to be considered as Professor Suzzallo's. 
 
 The quarrels that exist as to the proper methods of teaching 
 arithmetic are not merely theoretic. Every difference in practice 
 in the treatment of a subject in arithmetic implies a difference of 
 opinion, and consequently a controversy. It is in the settlement 
 of these practical controversies that careful experimental methods 
 can be of large service. It can scarcely be said that such an ap- 
 proximately scientific approach has been seriously attempted 
 even by educational theorists, and much less by school principals 
 and superintendents. That there are obvious reasons for this 
 fact goes without saying. The average teacher or principal is 
 too busy with other matters, which for the time being are 
 exceedingly urgent. But it must be equally obvious that pro- 
 vision for the investigation of teaching problems must be made 
 
 65
 
 66 The Teaching of Arithmetic 
 
 somewhere within the profession. Perhaps in the beginning, it 
 must be left to those scattered workers who have at once the 
 impulse and the opportunity to conduct investigations. In the 
 hope of being of assistance to such as these, the following list of 
 practical problems now existing in the teaching of primary arith- 
 metic has been prepared. The list is confined to the handling 
 of whole numbers in the first few grades, and is by no means 
 complete. It pretends merely to suggest certain differences 
 of practice, the relative value of which needs to be determined 
 by something more than mere opinion. 
 
 (1) Which is the best way to teach young children to count 
 serially from I to 100? To have them count by ones from the 
 beginning, extending the series as fast as the child can memorize 
 the same, without any conscious effort in the direction of show- 
 ing the child that the series repeats with a certain regularity 
 after twenty is passed? Or to have them memorize the names 
 in their order from one to thirty (by which time the regularity 
 is established as a basis) and then have them learn to count 
 by tens, later using the counting by ones and the counting by 
 tens as a double basis for learning to count serially from thirty 
 to one hundred? 
 
 (2) Assuming that oral counting leads mainly to the associa- 
 tion of a name (27) with a given position in a series of names 
 (between 26 and 28), how far is it advisable for a number to be 
 associated with a given idea of mass or grouping, as when the 
 device of two bundles of ten sticks each and seven individual 
 sticks is used to explain 27? Does the effort toward the asso- 
 ciation of concrete images and numbers ultimately interfere with 
 the rapid manipulation of figures in complex calculations ? How 
 far does the material in objective work need to be varied with 
 first-grade children (sticks, lentils, boys, etc.) so that the idea 
 associated with a number shall be abstract rather than the 
 image of any particular concrete thing or group of con- 
 crete things? 
 
 (3) How far is group counting (counting by 2's, 3*5, etc.) 
 really counting, that is, proceeding from one number to another 
 by an act of absolute memory (saying 3, 6, 9, etc., exactly as 
 one. says i, 2, 3, etc.) ? How far is it really a process of con- 
 secutive adding (3 and 3 are 6, 6 and 3 are 9, etc.) ? If it is a 
 mixture of both, where does one process end and the other
 
 Details for Experiment 67 
 
 begin? If group counting is really adding, should it not always 
 be classified with the work of addition, and placed so as to assist 
 it, rather than be operated independently as alleged counting? 
 How far is group counting as real counting desirable? How far 
 may it be used as another form of addition? In the latter case 
 should it precede or follow combination work in addition (6, 9, 
 12, etc., precede or follow 6 + 3 = 9, 9 + 3 = i 2 > etc.)? If 
 counting forward is an aid to addition, how far can counting 
 backward be an aid to subtraction? How far is real counting 
 backward (by sheer act of consecutive memory) of valid 
 social use? 
 
 (4) How far shall the three processes of (i) oral counting, 
 (2) reading of numbers, and (3) writing of numbers, be parallel 
 in the first year of formal arithmetic teaching? Should counting 
 precede reading, and reading precede writing of numbers ? How 
 far are they dependent upon each other? In relation to accom- 
 plishment in any one of these, when should the teaching of the 
 other begin? 
 
 (5) Why do young children who know their numbers up to 
 twenty write 16 correctly at first, and later, when they are sup- 
 posed to know their numbers to 100, write 16 as 61 ? Would 
 further and special drill on certain numbers of the series have 
 prevented this error? Which numbers require this special care? 
 Why do children sometimes say " five-teen ? " 
 
 (6) In teaching children to read and write numbers, how 
 far is it useful and how far is it confusing to have them know 
 the place names (unit of units, tens of units, hundreds of units, 
 etc.) ? Should such a classification be given to the child finally, 
 or not at all ? Is the so-called method of " group reading " 
 superior to the " place " method ? To the method of direct 
 memorization ? In the " group " method a child reads and writes 
 all his numbers as he would numbers of three figures or less, 
 naming them from the commas which mark off the groups of 
 three, as in 34,026, " 34 " = " thirty-four," " , " = " thousand," 
 " 026 " = " twenty-six." What are the special errors which are 
 peculiar to the " place " method ? What are the special errors 
 peculiar to the " group " method ? 
 
 (7) Are all numbers of from four to six places equally easy 
 to read and write? If not, what are the types representing 
 gradations of difficulty? Taking the following types:
 
 68 The Teaching of Arithmetic 
 
 4,000 In which are errors most frequent? When these 
 80,000 same figures appear, not in " thousands place " but 
 13,000 in " units place," would the order of difficulty 
 257,000 be the same or different? As in 1,000 
 900,000 i, 2 57 
 
 120,000 i,9 
 
 304,000 i, 1 20 
 
 1,304 
 1,013 
 1,004 
 
 i, 080? It will 
 
 be noted that there are seven types in the first list and eight in 
 the second, due to the introduction of ,000. Note also that 4, 
 becomes ,004 in the second list, and passes from the easiest to 
 next to the most difficult. Are such distinctions characteristic 
 of children's experiences with numbers? 
 
 How does some provision for equalizing drill in all types of 
 numbers minimize the unequal distribution of errors, as opposed 
 to the hit-and-miss methods of drilling from personal lists made 
 up by the teacher as he needs them? 
 
 According to the types enumerated above as a result of the 
 investigation of thousands of children's papers, would there 
 not be 56 (7X8 = 56) drill types for thousands, and 448 
 (7X8X8 = 448) for numbers in millions place? 
 
 (8) It is generally said that there are forty-five fundamental 
 combinations which are the basis of all work in addition. What 
 are the fundamental facts that are required as basic and which, 
 once learned, may be applied in new forms and situations over 
 and over again? 
 
 There are ten numbers, from o up to 9. Each of these may be 
 combined with itself and the nine others, thus making 100 combi- 
 nations, from + = up to 9 + 9 = 18. The 19 zero com- 
 binations are left out, leaving 81 combinations. Of the 81 
 remaining, 36 are reverses (2 + 7 = 9 ' 1S a reverse of 7 + 2 = 9). 
 Omitting these there are 45 combinations left as fundamental. 
 Is this procedure correct? 
 
 (9) How far does the learning of 7 + 2 = 9 also guarantee 
 the acquiring of its reverse, 2 + 7 = 9? Will the second be 
 known without further drill? With how many less repetitions
 
 Details for Experiment 69 
 
 will it be learned because the other combination is mastered? 
 Will the two combinations mentioned be learned with fewer repe- 
 titions when they are constantly learned together, as opposed to 
 being learned as separate individual combinations the relation of 
 which is not specially kept in mind? 
 
 (10) Is there a justification for saying that the zero combina- 
 tions (0 + 3 = 3) mav b e omitted as not being basic? The 
 contention is that they never occur in single combination. No 
 one says, " I have nothing and three, and adding them I have 
 three." In such a situation we merely count what we have, we 
 do not add our count to what we do not have, for we are not 
 conscious of the latter numerically. 
 
 But may not the zero combinations be necessary for their later 
 application in column addition ? 6 + 3 = 9 is used as 16 + 3= 19 
 and + 4 = 4 is used as 10 + 4 = 14. Is it true that all zeros 
 in column addition are ignored? 
 
 In four conceivable cases, o + 4 = 4 
 
 4 + 0= 4 
 10 + 4= 14 
 
 14 + o = 14 where the zero is 
 
 found in column addition, it may be said that in three the zero is 
 treated with one attitude ; it is ignored. In the case of 10 + 4=14 
 the zero is treated as part of quantity, and must be learned. Must 
 not the child know all the applied zero combinations from 
 10+2=12 up to 10 + 9=19, and must not these eight combi- 
 nations be provided somewhere in the child's instruction ? Which 
 then is the most economical and efficient way of teaching the zero 
 combinations mentioned ? To teach them as + 4 = 4 an d then 
 apply as 10 + 4= 14, or to teach as 10 + 4= 14 from the very 
 beginning? Experimentation ought to reveal the relative value 
 of the two methods. It ought to reveal the difference between 
 making some provision for them and making no formal provision. 
 
 (n) In the list of forty-five fundamental combinations, the 
 zero combinations were left out (when some should probably have 
 been left in) and the combinations with one (6+1=7) were 
 left in. Should they also have been left in? As no one adds o 
 to a number in a single combination in actual life, it might be 
 asked if we ever add one? We really count one more, not add. 
 When we have 6 and i more, do we not count 6, 7, nor add 6 + I
 
 70 The Teaching of Arithmetic 
 
 = 7. Counting is a more fundamental habit than adding, and it 
 is contended that when i is met in any column, the mind really 
 climbs the scale i, it does not group it as where 3 is met. If 
 this is so the children being able to count serially already, need 
 not learn one as an addition. This would omit 17 combinations. 
 Experimentation would show how far children taught the com- 
 binations with i were superior in column addition where I's 
 occurred, to children who had not had any training in com- 
 binations with ones. 
 
 (12) In actual instruction many teachers do not drill one type 
 of combination any more than another. The additional drill 
 comes later when the child fails or gets confused. Additional 
 
 drill is used as cure rather than as preven- 
 tion of mistake. Of the four types given, 
 4 + 5 = 9 which is the easiest for children ? Which 
 3 + 7 = 10 the hardest? If errors are more frequent in 
 9 + 6=15 some types than in others, is this due to 
 10 + 8 = 18 the innate difficulty of certain types or to 
 the methods of teaching them? Do chil- 
 dren add from large to small numbers 
 (9 + 5) more readily than from small to large numbers (5+9 
 -14)? 
 
 (13) Some courses of study require that a combination once 
 learned (5 + 7=12) be applied immediately to the higher 
 decades (15 + 7 = 22, 25 + 7 = 32, etc.). How much superior 
 in column addition is a class thus trained to one not so trained ? 
 Is it necessary to apply all combinations learned in this way? 
 May it not be that the general idea of application is soon acquired 
 with the first few combinations and that special drill is not 
 required thereafter? Are there certain combinations where 
 special drill must be insured always (5 +6 = eleven, 15 + 6 = 
 twenty-one) because the sound regularity is interfered with ? Or 
 may a strictly written presentation do away with the necessity 
 of special drill even here? 
 
 (14) Is there any increase of efficiency in drilling on combina- 
 tions in columns as soon as possible? As soon as the com- 
 binations that add up 7 are learned is there a special advantage 
 in immediately giving the child such columns in application as 
 the following:
 
 Details for Experiment 71 
 
 2 4 24 
 
 3 i 3i 
 
 2 2 22 
 
 7 7 77 
 
 (15) In some texts and courses of study the addition combina- 
 tions are presented in the order of the sizes of the 
 
 sums, thus 2+2 = 4, 2 + 3 = 5, etc. In others 6 
 the combinations are presented, regardless of the size 3 
 of the numbers involved, so as to immediately fit into 6 
 certain drill columns already prepared. Thus the an- 4 
 nexed column would require the following combina- 
 tions (beginning from the bottom), 4 + 6=10, 19 
 + 3 = 3, an d 3 + 6 = 9. What is the relative 
 worth of these two methods? 
 
 (16) In column addition, where carrying is involved, some 
 
 rationalize the process, and others teach it mechan- 
 
 23 ically as a mere bit of habituation. In the case 
 
 47 here given, some would add each column sepa- 
 
 36 rately, taking a second total of the partial sums. 
 
 Others would merely " put down the six and add 
 
 1 6 one to the next column," writing down only the 
 
 9 complete sum. Which is superior, in that it will 
 
 result in accurate and rapid column addition in 
 
 106 the shortest space of time? 
 
 The preceding treatment of addition will suggest 
 similar problems as more or less recurring in subtraction, e. g., 
 whether there is any gain in teaching 5 3 = 2 immediately 
 after learning that 5 2 = 3, etc. 
 
 (17) Is there any advantage in the so-called "Austrian" 
 method of subtraction by addition, over the method of subtracting 
 through specially learned subtraction combinations? If so, how 
 
 much considering that subtraction and addition 
 
 4 10 (i) mean different concrete situations, (2) 
 
 + 6 6 use a different written algorism, (3) use the 
 
 same oral form ("6 and 4 are 10"), and (4) 
 
 10 4 employ the same memorization (6 and 4 are 
 
 10) ? What extent of school energy is saved, 
 
 if any? Is the method of subtraction from the next digit in the
 
 72 The Teaching of Arithmetic 
 
 top number superior or inferior to the method of adding to the 
 next digit in the bottom number ? How is it when these methods 
 are applied to the addition-method of subtraction? to the " old " 
 subtraction-combination method ? 
 
 (18) Do children make fewer errors when they 
 
 are formally taught to make a preliminary inspec- 389 
 
 tion of subtraction examples before proceeding 421 
 to manipulate specific combinations? as when the 
 number cannot be subtracted; as when the 345 
 answer is zero. (See the two examples here 345 
 given.) 
 
 (19) Do children make fewer errors and manifest less con- 
 fusion where they are formally taught to handle the zero diffi- 
 culties prior to being confronted with them in column subtrac- 
 tion? As in the type cases given below: 
 
 (a) 867 (b) 867 (c) 867 (d) 870 
 467 400 32 650 
 
 400 467 835 220 
 Where borrowing from top? 
 
 (e) 128 (f) 602 (g) 612 (h) 612 
 
 76 237 318 308 
 
 52 365 294 304 
 Where adding to bottom? 
 
 (0 834 (j) 834 (k) 804 (1) 814 
 
 406 496 496 406 
 
 428 338 308 408 
 
 In subtraction, what preparation is needed in a command of 
 zero combinations to perform the column subtraction? (Note 
 each case given above.) Is there some general mode of handling 
 these zeros that will not require a mastery of it in connection 
 with each number it may be combined with? How does the 
 above apply to the combinations with ones? Where a one is 
 involved, is it merely counting downwards or backwards? Or 
 is the subtraction of I exactly like the subtraction of 3 or 4 or 
 any other number? 
 
 (20) What are the basic combinations required to perform 
 any given column subtraction? In what form may they be best
 
 Details for Experiment 73 
 
 mastered? Are zero subtractions (6 = 6) and subtractions 
 with one (61 = 5) to be included or omitted? Are the re- 
 verses to be taught as basic? (7 2 = 5 and 7 5 = 2.) Are 
 subtraction combinations of varying difficulty? Which are the 
 most difficult, as shown by children's errors? 
 
 (21) Will children have less difficulty, with fewer ensuing 
 errors, if they approach column subtraction through a series of 
 graded types of difficulty? Consider in the following: 
 
 (a) No borrowing. 
 
 1498 
 964 
 
 534 
 (b) Borrowing each time save the last. 
 
 8431 
 5987 
 
 2444 
 (c) Borrowing alternately. 
 
 8431 
 2917 
 
 5514 
 
 In what order should types (b) and (c) be given? 
 
 How rapidly may a child advance from two or three figures 
 to seven or eight? What new difficulties present themselves in 
 such an extension of figures? 
 
 (22) Are the first series of multiplication combinations best 
 presented (i) by the use of objects grouped and counted, or 
 (2) by the use of column addition? Or are these two methods 
 best used as supplementary to each other? Are the combinations 
 with zeros (6X0 = 0) and the combinations with ones 
 (6 X i = 6) best taught in the tables, or later, in con- 
 nection with their actual use in column multiplication? Is 
 there a gain in teaching the reverses in connection with the com- 
 binations to which they are related, exactly as with the addition 
 combinations, thus 6X3=18 immediately after 3X6=18? 
 What gradation of steps is most economical and efficient in 
 proceeding from combinations to their application in column 
 multiplication? Since partial products represent but stages in 
 calculation do they need to be understood as to their placing, or
 
 74 The Teaching of Arithmetic 
 
 should their placing be taught as a mechanical process through 
 habit formation? Is it economical to allow zeros to be recorded 
 which later will be abandoned? as in multiplying by 206? What 
 special drill on zero difficulties (and on the manipulation of ones) 
 is required in connection with their handling in column multipli- 
 cation? How can this be best provided? 
 
 (24) Is there any need for division tables of combinations? 
 May not the multiplication tables be used for division, precisely 
 as the addition combinations are used for subtraction? For 
 example, from " 3 2's are 6," may we not step to the case of 2)6? 
 " How many 2's are 6? " "3 2 ? s are 6." Here the identification 
 is through a common oral form of expression. Is there any need 
 to show that a specific written form or algorism in multiplication 
 is the equivalent of another one in division, since this is not done 
 in subtraction by addition? 
 
 (25) In column addition, column subtraction, and column 
 multiplication the fundamental combination is generally obvious 
 in the process of manipulation. Is the division combination 
 equally obvious in long and short division? Does this require 
 special treatment? 
 
 (26) Will children do long and short divisions more efficiently 
 if drill in division with a remainder (after 12-^-3 = 4, learning 
 13-4-3 = 4, with i remainder, and 14 -r- 3 = 4, with 2 remainder) 
 is inserted between the learning of the combinations and their 
 application to long and short divisions? Since the largest diffi- 
 culties seem to occur in connection with long division, and since 
 division by a one-figure divisor must precede division by a num- 
 ber of two figures, shall division by a one-figure number be first 
 taught in its complete form? Shall division by a one-figure 
 number be abridged to " short division " before or after the 
 development of division with a divisor of two figures? Does 
 the distinction between partition and measuring have any relation 
 whatsoever to skill in manipulation? What is the worth of the 
 distinction in interpreting problems and applying calculations? 
 
 (27) How many special types of zero difficulties need be antici- 
 pated and carefully drilled upon before children are allowed to 
 attack examples where zero difficulties are likely to confront 
 them? Is there any special order in which these zero difficulties 
 should be attacked for purposes of economical mastery? What
 
 Details for Experiment 75 
 
 special training in the handling of ones should be provided for, 
 especially if the one combinations in the tables are omitted? 
 
 (28) The above list deals entirely with investigations in teach- 
 ing which are mainly psychological. There is another large 
 series of investigations as to the materials required in the various 
 courses of study. These are sociological. In such investigations 
 one would make such inquiries as the following: What is the 
 demand for square root in ordinary business occupations ? What 
 types of fraction examples are called for frequently, and what 
 infrequently? Which types of reasoning combinations are most 
 used ? Do we " add and multiply '' within the same problem, 
 more frequently than we " multiply and add ? " Upon such social 
 investigations should the selection and the omission, the emphasis 
 and the subordination of specific topics be determined. Then 
 will our courses of study represent the highest social efficiency. 
 
 BIBLIOGRAPHY: Teachers should consult, on this topic, Pro- 
 fessor Suzzallo's work already cited.
 
 CHAPTER XV 
 THE WORK OF THE FIRST SCHOOL YEAR 1 
 
 The first question that naturally arises in connection with the 
 arithmetic of the first grade is as to whether or not the subject 
 has any place there at all. For several years past there has been 
 in this country a propaganda in favor of excluding it as a topic 
 from the first grade and even from the second. Like all such 
 efforts, the history of which is not generally known, the very 
 novelty of the suggestion, to many teachers, is sufficient to create 
 a following. It is well to consider briefly the reasons for and 
 against such a suggestion, and to attempt to weigh these reasons 
 fairly before attempting any decision. 
 
 In favor of having no arithmetic as such in the first grade it 
 is argued that the spirit of the kindergarten should extend farther, 
 perhaps even through all of the primary grades ; that number 
 work should come in wherever there is need for it, all learning 
 being made attractive and natural, and education appearing to 
 the child as a unit instead of being made up of scattered frag- 
 ments. Such a theory has much to commend it, not only in the 
 primary school but everywhere else. Opposed to it is the rather 
 widespread idea that most kindergarten work is superficial in 
 aim and unfortunate in result; that children who have had this 
 training are wanting in even the little seriousness of purpose that 
 they should have, that they have no power of application, that 
 they have been " coddled " mentally into a state that requires 
 constant amusement as the condition to doing anything. The 
 dispassionate onlooker in this old controversy probably feels that 
 there is truth in both lines of argument, and that mutual good 
 has been the result. Ancient education was a dreary thing, and 
 
 1 It is impossible in the space allowed to enter very fully into details 
 as to the work of the various grades. Teachers who desire such details 
 may consult the author's Handbook to Arithmetics (Boston, 1905). All 
 that can be done in this article is to give a brief survey of some of the 
 most important topics relating to the various grades. 
 
 7 6
 
 The Work of the First School Year 77 
 
 to the spirit of the kindergarten, although not to extreme 
 Frobelism, we are indebted for the brighter spirit of the modern 
 school. On the other hand, to make children self-reliant, inde- 
 pendent in thinking, conscious of working for a purpose, demands 
 more thought than seems to pervade the ordinary kindergarten. 
 
 Now as to arithmetic in the first grade: Shall we leave it to 
 the ordinary teacher to bring in incidentally such number work 
 as he wishes, or shall we lay down a definite amount of work to 
 be accomplished and assign a certain amount of time to it ? And 
 in answering these questions, are we bearing in mind the average 
 primary teacher throughout the whole country? Are we also 
 bearing in mind that arithmetic was never taught to children just 
 entering school until about a century ago, and that it was largely 
 due to Pestalozzi's influence that the subject was ever placed in 
 the first grade? When, therefore, we advocate having no arith- 
 metic in the first grade, we are going back a hundred years or 
 so, which may be all right, but which is not a new proposition 
 by any means. 
 
 Having thus laid a foundation for an answer to the question, 
 it is proper to proceed dogmatically, leaving the final reply to the 
 reader. Not to put arithmetic as a topic in the first grade is to 
 make sure that it will not be seriously or systematically taught 
 in nine-tenths of the schools of the country. The average teacher, 
 not in the cities merely but throughout the country generally, will 
 simply touch upon it in the most perfunctory way. Whatever of 
 scientific statistics we have show that this is true, and that 
 children so taught are not, when they enter the intermediate 
 grades, as well prepared in arithmetic as those who have studied 
 the subject as a topic from the first grade on. 
 
 Furthermore, while it is true that the essential part of arith- 
 metic can be taught in about three years, it cannot, for psycho- 
 logical reasons, be as well retained if taught for only a short 
 period. The individual needs prolonged experience with number 
 facts to impress them thoroughly on the mind. We can, for 
 example, teach the metric system in an hour to any one of fair 
 intelligence, but for one to retain it requires long experience 
 in its use. 
 
 But more important than all else is the consideration of the 
 child's tastes and needs. Has he such a taste for number as
 
 78 The Teaching of Arithmetic 
 
 shows him mentally capable of studying the subject at the age 
 of six, and are his needs such as to make it advisable for him 
 to do so ? There can be no doubt as to the answer. He takes as 
 much delight in counting and in other simple number work in the 
 first grade as in anything else that the school brings to him, and 
 he makes quite as much use of it in his games, his " playing 
 store," his simple purchases, his reading, and his understanding 
 of the conversation of the home and the playground, as he does 
 of anything else he learns. If we could be certain that in the 
 incidental teaching that is so often advocated he would have these 
 tastes and needs fully satisfied, then arithmetic as a topic might 
 be omitted from the first or any other grade; but since we are 
 pretty sure that this will not be accomplished in the average 
 school, then it is our duty to advocate a definite allotment of time 
 and of work to the subject in every grade from the first through 
 the eighth. 
 
 This being so, what should tnis allotment of work be? Of 
 course there is no general answer for the whole country. In 
 some schools there are many foreign born pupils who are unable 
 to speak English when they enter and therefore the first year's 
 work must be devoted largely to acquiring the language. In 
 other schools the children come from homes where they have 
 already been taught by governesses and are considerably advanced 
 over the average. In general, however, the course here laid 
 down may be considered a fair average for the ordinary American 
 school. 
 
 The Leading Mathematical Feature. The introduction to the 
 addition table, this being at the same time the simplest and the 
 most important operation in arithmetic. It is not advisable to 
 use a text-book in this year, on account of the children's in- 
 ability to read. 
 
 Number Space. It has been found best both from the stand- 
 point of mental ability and of needs of the children to set a 
 different limit to the numbers used in counting and in the opera- 
 tions. Children like to and need to count numbers that are larger 
 than those used in operations. For reading and writing numbers, 
 therefore, they may profitably go as far as 100, meeting these 
 numbers in the paging of books, the numbering of houses, the 
 playing of games, and the counting of various objects. For the
 
 The Work of the First School Year 79 
 
 operations, however, it is sufficient if they go as far as 12. 
 Indeed, 10 would make a good limit were it not for the fact that 
 in measuring they so often use 12 inches. 
 
 Addition. The addition tables should be learned at least as 
 far as sums of 10 or 12. Some prefer to go as far as 9 + 4 = 13, 
 but it is immaterial so long as the children know the table through 
 9's before the text-book is used, ordinarily the middle or the 
 end of Grade II. Appropriate combinations for the first year 
 may, therefore, be taken as follows: 
 
 123456789 
 I I I I I I I I I 
 
 23456789 10 
 12345678 
 
 22222222 
 
 3456789 10 
 
 1234567 
 
 3333333 
 
 4 5 6 7 8 9 10 
 
 123456 12345 
 444444 55555 
 
 56789 10 6789 10 
 
 1234 123 
 6666 777 
 
 7 8 9 10 8 9 10 
 
 12 I 
 
 88 9 
 
 9 10 10 
 
 This arrangement makes the sum the basis for selection. 
 Many prefer, however, to proceed to master the table of I's, 2's,
 
 8o The Teaching of Arithmetic 
 
 3's, and 4's, as mentioned above, thus giving the following com- 
 binations : 
 
 123456789 10 
 iiiiiiiiii 
 
 1234 
 
 2222 
 
 1234 
 
 3333 
 4567 
 
 1234 
 
 4444 
 
 5678 
 
 It is not a matter of great importance which of these two 
 arrangements is adopted in any given school system, at least so 
 far as we are able to judge from any scientific investigations thus 
 far made. The great thing is that the complete table shall be 
 known to 10 + 10 by the end of the second year. 
 
 Subtraction. Every fact learned in addition should, judging 
 from general experience, carry with it the inverse subtraction 
 case. That is, the question "3 + 2 equals what number ? " should 
 carry with it the questions " 3 + what number equals 5 ? " and 
 " 2 + what number equals 5 ? " or, if preferred, "5 3 equals 
 what number?" and "5 2 equals what number?" 
 
 Multiplication. Little attention should be given to this subject 
 in the first grade. The idea that 2 + 2 + 2 may be spoken of as 
 3 times 2, and the incidental use of the word " times " in other 
 simple number relations is desirable. 
 
 Division. Since multiplication is not taken as a topic, its 
 inverse (division) has no place, save as it appears in the fractions 
 mentioned below. 
 
 6 
 
 7 
 
 8 
 
 9 
 
 IO 
 
 ii 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 IO 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 7 
 
 8 
 
 9 
 
 IO 
 
 II 
 
 12 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 IO 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 8 
 
 9 
 
 IO 
 
 ii 
 
 12 
 
 13 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 IO 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 9 
 
 IO 
 
 ii 
 
 12 
 
 13 
 
 14
 
 The Work of the First School Year 81 
 
 Fractions. Children so often hear about the fractions l / 2 , %, 
 and y 3 , that these ideas and forms may profitably be introduced 
 at this time, although y 3 may be postponed to the next grade. 
 The statement that half the class may go to the blackboard, the 
 idea of % of a dollar, and that of l /$ of a yard, are all common in 
 the first year. In the introduction of these ideas and symbols it 
 is well to avoid extremes that will militate against the child's 
 future progress, such as the extreme of the ratio method, for 
 example. We should remember that a fraction, say y>, is com- 
 monly used in three distinct ways, and, that it is our duty to see 
 that, little by little, all these become familiar to the child. These 
 ways are as follows: (i) l / 2 of a single object, the most natural 
 idea of all, the breaking of an object into 2 equal parts; (2) ^ 
 as large, as where a 6-inch stick is l / 2 as long as a foot rule, 
 not half of it, but half as long as it is; this is essentially the 
 ratio notion, and it is necessary to the child's stock of knowledge, 
 but it is not necessary to make it hard by talking about ratios at 
 this time; (3) */2 of a group of objects, as in the case of l /2 of 
 ten children. 
 
 Denominate Numbers. Children in this grade should learn 
 the use of actual measures. They should know that 12 in. = i ft., 
 3 ft. = i yd., and should employ this knowledge in making meas- 
 urements. They should know the cent, 5-cent piece, dime, and 
 the dollar as 10 times (or even 100 cents), and should use toy 
 money in playing store. They should know the pint and quart, 
 and use these in measuring water or other convenient substance. 
 Other terms such as pound, week, minute, mile, and gallon may 
 be used incidentally, but they should not be, learned in tables, 
 at present. 
 
 Objects. It is important to use objects freely wherever they 
 assist in understanding number relations, but it is equally impor- 
 tant to abandon them as soon as they have served their purpose. 
 The continued use of any particular set of objects (blocks, disks, 
 measures, picture cards, etc.) is tiresome and narrowing. Pesta- 
 lozzi was wiser than many of his successors when he used 
 anything that came to hand to illustrate most of his number 
 work. To continue to use objects after they have ceased to be 
 necessary is like always encouraging a child to ride in a baby 
 carriage.
 
 82 The Teaching of Arithmetic 
 
 Symbols. It cannot be too strongly impressed upon teachers 
 that the symbols that children should visualize are those that they 
 will need in practical calculation. Thus it is much better to 
 drill upon the annexed forms than upon 
 6 9 9 6 + 3 = 9, 9 6 = 3, 9 3 = 6, since 
 + 3 6 3 the latter are never used in calculation. 
 , For ease in printing and writing, symbols 
 936 like 6 + 3 = 9 have their important 
 place, but the eye should become accus- 
 tomed to the perpendicular arrangement so as to catch number 
 combinations as it must do when we come to actual addition. 
 
 Technical Expressions. While it is proper to begin by reading 
 6 + 2 " six and two " and 8 6 " eight less six," the words 
 " plus " and " minus " should soon enter into the vocabulary of 
 the child as part of the technical language of the subject. It is 
 proper to call a cat a " pussy " for a while, and a horse a " pony," 
 but the time soon comes for " cat " and " horse," and so for 
 the technical expressions in arithmetic. 
 
 Nature of the Problems. In this grade problems of play, of 
 the simplest home purchases, and of interesting measures should 
 dominate. In general, for all grades, the oral problems should 
 have a local color, relating to real things that the children know 
 about. The building of a house near the school, the repairing of 
 a street, the cost of school supplies these and hundreds of simi- 
 lar ideas may properly suggest problems adaptable to every school 
 year. It is the business of the text-book in the grades where it 
 is used to furnish a large amount of suggestive written work, 
 but it can never furnish all the oral work needed nor can it 
 meet all local conditions. 
 
 As a specimen of the early work in this grade the following 
 oral exercise is submitted: 1 
 
 1. How many inches wide is the window pane? 
 
 2. How many feet long is your desk, and how many inches 
 
 over? 
 
 3. How many feet and inches from the floor to the bottom of 
 
 the blackboard? 
 
 1 These and other similar sets of problems used in this article are taken 
 from other works of the author.
 
 The Work of the First School Year 83 
 
 4. Stepping as you usually do in walking, find how many 
 
 paces in the length of the room. 
 
 5. How many paces wide do you think the room is? Pace 
 
 the width and see if you are right. 
 
 6. How tall do you think you are? Measure. How many 
 
 feet, and how many inches over? 
 
 7. How many inches from the lower left-hand corner of this 
 
 page to the upper right-hand corner? 
 
 8. How wide do you think the door is? Measure. How 
 
 many feet, and how many inches over? 
 
 Such problems suggest measurements of genuine interest to 
 the pupil, relating as they do to his immediate surroundings. 
 They allow for the actual handling of the measures and the form- 
 ing of reasonably accurate judgments concerning distances. 
 
 Abstract Computation. It is a serious error to neglect abstract 
 drill work in arithmetic. So far as scientific investigations have 
 shown, pupils who have been trained chiefly in concrete problems 
 to the exclusion of the abstract are not so well prepared as those 
 in whose training these two phases of arithmetic are fairly 
 balanced. Abstract work is quite as interesting as concrete; it 
 is a game, and all the joy of the game element in education may 
 be made to surround it. At the same time it is the most practical 
 part of arithmetic, since most of the numerical problems we meet 
 in life are simplicity itself so far as the reasoning goes; they 
 offer difficulties only in the mechanical calculations involved, and 
 constantly suggest to us our slowness and inaccuracy in the 
 abstract work of adding, multiplying, and the like. In the first 
 grade this work is largely but not wholly oral. 
 
 Forms. It is expected that children in this grade will become 
 familiar with the names of the common solids and polygons 
 needed in their work. For example, square, rectangle, triangle, 
 oblong, cube, sphere, cylinder, pyramid, prism, and similar forms 
 should be handled and their names should be known. Paper 
 cutting and folding is very helpful in the study of plane figures 
 and in the work with fractions, although like any other device, 
 it may be used to an extreme that is to be avoided. 
 
 The Time Limit. Even in the first grade, and still more in 
 the succeeding years, a time limit should be set on all number
 
 84 The Teaching of Arithmetic 
 
 work. The children should see how many questions they can 
 individually, or as a class, or as half of the class, answer in a 
 minute, or in some other period of time. Unless this is done, or 
 some similar plan is adopted, the tendency to dawdle over the 
 work will begin to crystallize into a habit, and computation will 
 take much more time than necessary. It is also to be observed 
 that, always within reasonable limits, rapid calculation contains 
 less errors than very slow work. The reason is apparent; we 
 concentrate our attention more completely, and other thoughts do 
 not take our minds from the numerical work. 
 
 BIBLIOGRAPHY : The author's Handbook to Arithmetics, p. 19 ; 
 C. A. McMurry, Special Method in Arithmetic. On paper fold- 
 ing consult Sundara Row's work, Geometric Paper Folding 
 (Open Court Publishing Co.), illustrated by photographs taken 
 by the author of the present work a few years ago. This work 
 is suggestive, although not adapted to grade work. Consult 
 also Wentworth-Smith, Stepping-Stones in Number, Boston, 
 1911.
 
 CHAPTER XVI 
 THE WORK OF THE SECOND SCHOOL YEAR 
 
 Whether or not arithmetic has a definite time allotment in the 
 first grade, it usually has one in the second, although some 
 teachers oppose it even there. The argument already advanced 
 holds the more strongly here, especially as, in many schools, 
 the child is quite prepared to use a text-book by the middle of 
 this year. 
 
 The Leading Mathematical Features. In schools of average 
 advancement, where the question of language is not as serious 
 as in some cities in the East, children in this grade may be 
 expected to complete the addition tables and to learn the multi- 
 plication tables to 10 X 5. 
 
 Number Space. Children will now take an interest in count- 
 ing to looo, first by units to 10, then by ID'S to 100, then com- 
 pletely to 100, then by loo's to 1000, and finally completely to 
 1000. Their operations may also be anywhere within this space, 
 although, of course, most of their results will involve only small 
 numbers. In the Roman notation the limit may be set at XII, 
 this sufficing for the reading of time and for the chapter num- 
 bers of their books. 
 
 Counting. Without going to an extreme in counting by 
 various numbers where no definite purpose is served, there is a 
 field in which counting is very advantageous. To count by 2's 
 from 2 to 10 and from I to n has the pleasure of any rhythmic 
 sequence and at the same time gives the addition table of 2's, and 
 the counting by 2's from 2 to 20 gives the corresponding multi- 
 plication table. Similarly, counting by 3's from 3 to 30 gives 
 the multiplication table of 3's, while the further counting from 
 i and 2 to 13 and 14 gives the different addition combinations. 
 The exercise is interesting to children, and the knowledge secured 
 in this way is more than one would at first think. 
 
 Addition. The tables should be completed during this year, 
 
 85
 
 86 The Teaching of Arithmetic 
 
 including the sums of any two one-figure numbers. There are 
 only 45 possible combinations of numbers below 10, viz.: I + I, 
 1+2, and so on to I + 9 ; 2+2, 2 + 3, and so on to 2 + 9 ; 
 3 + 3> 3 + 4> an ^ so on to 3 + 9 ; and similarly for the others 
 to 9 + 9, besides the zero combinations referred to earlier in 
 this paper. It is better, however, to continue the sums to 
 include 10, a simple matter but one that is often helpful. The 
 addition of numbers of two and even of three figures each may 
 be taken during this year, but not more than five or six in a 
 column should be used. 
 
 Subtraction. Subtraction may be carried far enough to include 
 numbers of three figures each. The method to be employed has 
 already been discussed in Chapter XI. In both addition and 
 subtraction there should be an effort to cultivate the habit of 
 rapidity, although never to the exclusion of accuracy. The time 
 limit on work, mentioned on page 83, should be employed in all 
 written work. In general in both addition and subtraction the 
 full form should be employed until it is thoroughly understood. 
 For example, in adding 247, 376, and 85, a problem that must 
 have been preceded by many simpler ones, it is well to use the 
 first of the following forms until the reasons are understood, and 
 then to adopt the second: 
 
 247 247 
 
 376 376 
 
 85 85 
 
 18 708 
 
 190 
 500 
 
 Likewise, if the addition or "Austrian " method is taken for 
 subtraction, it is better to begin a problem like 852 476 in the 
 full form, as follows: 
 
 852 = 800 +50 + 2 
 
 476 = 400 + 70 + 6 
 
 The difference between these is the same if we add 10 to each, 
 and also 100 to each, and we add them as follows, so that we can 
 easily subtract in each order: 
 
 800+ 150+ 12 
 
 500 +80+6 
 
 300 + 70 + 6 = 376
 
 The Work of the Second School Year 87 
 
 After this is understood we may proceed to the ordinary 
 arrangement. 
 
 Multiplication. The multiplication tables may be learned this 
 year as far as 10 X 5. Some schools go even as far as 10 X 10, 
 and others find it better to postpone all of this work until the 
 third grade. Products should be learned both ways, i. e., 5X6 
 and 6X5. There is a great advantage in reciting all tables 
 aloud, and even in chorus, since this leads to a tongue and ear 
 memory that powerfully aids the eye memory when the pupil 
 needs to recall a number fact. Counting enables the tables to be 
 developed in a rhythmic fashion that is pleasing to the ear, and 
 shows multiplication by integers to be merely an abridged addi- 
 tion, that is, that 3 + 3 + 3 + 3 is more briefly stated as 4X3. 
 
 Division. The multiplication table should carry with it the 
 division table. This need not be developed as a separate feature 
 but may be treated as the inverse of the multiplication table 
 exactly as subtraction is the inverse of addition. The fact that 
 4 X 6 = 24 should bring out the second direct fact that 6X4 
 = 24, and the two inverses, 24-1-6 = 4, and 24-^-4 = 6. These 
 inverses may be introduced in a way that is analogous to that 
 followed in subtraction. That is to say, after learning that 
 4 + 5 9 we ask, " What number added to 5 equals 9 ? " " What 
 number added to 4 makes 9 ? " Similarly, after 4 X 5 = 20 we 
 ask, " What number multiplied by 4 equals 20? " " What num- 
 ber multiplied by 5 equals 20?" These may then be expressed 
 as 20 *- 4 = 5, 20 -4- 5 = 4. 
 
 In division in this grade we also have an illustration of the 
 fact that the full form should precede the short one. A child 
 more easily grasps the idea of 36 -f- 3 if he sees the first of these 
 forms before he comes to use the second: 
 
 3)30 + 6 3)36 
 
 10+2 12 
 
 In the same way, when he comes to divide 36 by 2, it is better 
 to begin with the first of the following forms : 
 2)20+16 2)36 
 
 10+8 18 
 
 Teachers will find it better to write the quotient below the 
 dividend in short division, even though it is preferably written
 
 88 The Teaching of Arithmetic 
 
 above in the long process. There is no advantage in trying to 
 change the habit of the world on such a small matter. 1 
 
 Fractions. Children know the meaning of l /2, %, and often 
 of y$, on entering this grade. If y$ is not known it should be 
 introduced and ^, Vc> Vs mav a ^ so ^e added to the list at this 
 time, although many successful teachers prefer to postpone them 
 until Grade III. The use of objective work is imperative, and 
 it is better to take various simple materials than to confine one's 
 self to elaborate fraction disks or other similar devices. Every 
 school has cubes to work with, and the use of cubes, paper fold- 
 ing, paper cutting, and the common measures is recommended 
 as quite sufficient. 
 
 Denominate Numbers. The denominations already learned in 
 Grade I should be frequently used, and to them should be added 
 the relation between the ounce and pound; the pint, quart, and 
 gallon; the quart, peck, and bushel; the reading of time by the 
 clock, and the current dates. The idea of square measure (in 
 square inches) is introduced. All of this work should be done 
 with the measures actually in hand so far as this is possible. A 
 table of denominate numbers means very little unless accompanied 
 by the real measures. This will be felt by any American grade 
 teacher who teaches the metric system without the measures, and 
 who tries to think of his weight in kilos, his height in centi- 
 meters, and the distance to his home in kilometers. 
 
 Symbols. It has already been said that symbols like +, , 
 X, and -T- were invented for algebra and have only recently found 
 place as symbols of operation in arithmetic. 2 The desire to 
 employ them has led many teachers to use long chains of opera- 
 tions that are never seen in practical life and which, while serving 
 some purpose in oral work, are vicious as written exercises. For 
 example, 2+4-^-2 + 5X6-^-3 + 3 is a kind of work that should 
 never appear in the grades. Arithmetically it is easy enough, 
 and the answer is 17, but there is no use in puzzling a child 
 to remember which signs have the preference in such a chain. 
 This is a small technicality of algebra, of which the impor- 
 tance is much overrated even there, and it has no place in 
 
 1 See the author's Handbook, p. 27. 
 
 2 It is true that + and were first used in Widman's arithmetic of 
 1489, but not as symbols of operation. See my Kara Arithmetica.
 
 The Work of the Second School Year 89 
 
 the elementary school. With respect to the symbols 2 X $3 
 
 and $3X2 there is, however, a reasonable question, since there 
 
 is good authority for each. Modern usage favors the former 
 
 because we more naturally say " 2 times 3 dollars " 
 
 $3 than " 3 dollars multiplied by 2," and it is better to 
 
 2 read from left to right as in an ordinary sentence. It 
 
 should be repeated, however, that the forms which the 
 
 $6 child needs to visualize are not these but the one he 
 
 will meet in actual computation, as here shown. 
 
 Objects. It is here repeated, as essential to a discussion of 
 the work of the second grade, that objects are necessary in devel- 
 oping certain number relations, but that they should be discarded 
 as soon as the result is attained. Number facts must be memo- 
 rized by every one, and objects may become harmful if used 
 too often. 
 
 Nature of the Problems. This matter begins to assume con- 
 siderable importance in this grade, and it has been already dis- 
 cussed in Chapter IV. It may be said in general, however, that 
 several of our recent American arithmetics are making a serious 
 effort to improve the applications of the subject, adapting them 
 to the mental powers and to the environment of the pupils in- 
 stead of offering obsolete material of no practical value and of 
 little interest. 
 
 Necessity for Systematic Reviews. It is proper at this time to 
 call the attention of teachers to the matter of reviews, not those 
 that naturally occur from time to time during the year, but those 
 that should deeply concern every school at the close of one year 
 and at the opening of the next one. Any one who has ever had 
 much to do with the supervision of the grade work in arithmetic 
 is struck by the general complaint that children are never pre- 
 pared to enter any particular grade. Every teacher seems to feel 
 that the preceding teacher has imposed a poorly equipped lot of 
 children upon her own grade and that her problem is therefore 
 hopeless. Now if this were only an occasional complaint the 
 supervisor might well be worried, but he soon recognizes it as 
 part of the tradition of the school, and pays little attention to it 
 accordingly. What does it mean, however, and how should we 
 remedy the evil if evil there be? 
 
 If any teacher will himself learn, let us say, the logarithms of 
 the first fifty integers, between September and February, how
 
 po The Teaching of Arithmetic 
 
 many will he know in June? And if he knows them all in June 
 how many will he remember at the end of the summer vacation ? 
 And how will he feel, say about September 15, if some one 
 suddenly asks him to give the logarithm of 37 to six decimal 
 places, telling him, if he fails, that he must have been pretty 
 poorly taught the year before ? Now this is a fair illustration of 
 the mental position of a child with respect to the multiplication 
 table when he enters Grade IV. Psychologically it would be 
 strange if he could rapidly and accurately give every product 
 demanded ; his brain cells have clogged up or got disarranged or 
 gone through some similar transformation during his nine or 
 ten weeks of careless play. What, therefore, is the teacher's 
 duty? There are two things to do. First, at the close of each 
 school year, in June, there should be a thorough and systematic 
 review of those number facts and operations that are the funda- 
 mental features of the year's work. The teacher ought to be 
 satisfied that each child leaves the grade with such a mental 
 equipment as shall leave no chance of fair criticism. His respon- 
 sibility then ceases. Second, and even more important, at the 
 opening of each school year, in September, there should again 
 be a thorough and systematic review by the teacher in the next 
 grade, of these same features. But this review should be con- 
 ducted in the most sympathetic spirit. The teacher should be 
 surprised if the children have not forgotten much rather than 
 if they have failed to remember the facts perfectly. He should 
 think of his own fifty logarithms, for example, and the review 
 should be patiently and helpfully extended until the children's 
 arithmetical brain-cells resume their former state. After this 
 has been done in the spirit mentioned, and after the teacher has 
 gone into the next higher grade for a day to see 'how his own 
 pupils of the preceding year are standing the test, then he may 
 be justified in complaining, but not before. 
 
 It need hardly be mentioned that there are few more severe 
 tests of the ingenuity and patience of a teacher than are found in 
 these reviews. The " edge of interest " is already worn off in any 
 review, and it requires all the tact a teacher possesses to maintain 
 the enthusiasm of the pupils in such exercises. The result, how- 
 ever, is well worth the effort, and the school system that carries 
 out the plan will have less of complaint and more or sympathetic 
 cooperation than would at first be thought possible.
 
 CHAPTER XVII 
 THE WORK OF THE THIRD SCHOOL YEAR 
 
 The Preparation. Since the text-book is placed in the hands 
 of children during the latter part of the second school year or at 
 the opening of the third, it becomes particularly important to 
 have a systematic review of the work of Grades I and II at the 
 beginning of this year. The text-books usually provide for this, 
 and by their help these important things are accomplished: (i) 
 The children's memories are refreshed as to the essential fea- 
 tures of the preceding year's work, viz., the addition table, and 
 the multiplication table as far as the course of study may require. 
 (2) Children are " rounded up," brought to a certain somewhat 
 uniform standard, so that all can begin the serious use of the 
 text-book with approximately the same equipment. (3) The 
 superior capacity or the defect of the individual has an opportu- 
 nity to show itself early, allowing for such advancement or 
 special attention as the case demands. In other words the " lock- 
 step " can be broken without the usual delay. As to further argu- 
 ment for this autumnal review the reader may refer back to 
 Chapter XVI. 
 
 The Leading Mathematical Features. In this year rapid 
 written work is an important feature. The oral has predomi- 
 nated until now, but in Grade III the operations involve larger 
 numbers than before, and the child begins to acquire the habit 
 of writing his computations. Multiplication extends to two- 
 figure multipliers and long division is begun. The most useful 
 tables of denominate numbers are completed. 
 
 Number Space. It is usually considered sufficient if the child 
 understands numbers to 10,000 in this grade, although he may be 
 allowed to count by io,ooo's to 100,000 or even farther. Indeed, 
 as soon as he understands numbers to 1,000 he rather enjoys 
 showing his prowess by counting by i.ooo's and by writing large 
 numbers. Counting always extends far beyond the needs of
 
 92 The Teaching of Arithmetic 
 
 computation a law that is true to-day and has been true in the 
 historical development of all peoples. In the writing of Roman 
 numerals there is no particular object in going beyond C in the 
 first half-year, and M in the second half. It must be borne in 
 mind that we use the Roman forms chiefly in chapter or section 
 numbers, and less often in reading dates, so that all writing of 
 very large numbers by this system in an obsolete practice and a 
 waste of time. Indeed it is not strictly a Roman system any more, 
 so much have we changed the numerals from their early forms. 
 
 Counting. In this grade the counting of Grade II should be 
 continued, including the 6's, 7's, S's, 9's, and lo's, as a basis for 
 the multiplication tables and as a review of the addition combina- 
 tions. There is no need of counting beyond certain definite limits, 
 however. Thus in counting by 2's beginning with o, we have 
 o, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. This suffices for the multi- 
 plication table of 2's and even the last half of this is merely 
 a repetition of the first half with 10 added. 
 
 The Decimal Point. It becomes necessary in this grade to 
 write dollars and cents, and hence forms like $10.75, $ 2 5- IO > & n d 
 $32.02 are given. It is not necessary nor even desirable that the 
 children should know any of the theory of decimal fractions at 
 this time. The decimal point should be looked upon by them 
 simply as separating dollars and dimes, and it will give no trouble 
 unless the teacher confuses the class by the ever-present danger 
 of over-explaining. 
 
 Forms. It is usual in Grade III to review the simple geo- 
 metric forms already learned, such as the triangle, rectangle, 
 cylinder, and sphere. Formal definitions are, however, undesir- 
 able. The chief thing is that the child should use the names 
 correctly. Some little paper-folding may well be introduced as 
 a basis for simple square and cubic measure. 
 
 Square and Cubic Measure. The ideas of area (square inches 
 or square feet) and volume (cubic inches or cubic feet) may 
 enter into the work of this grade, although some successful 
 teachers prefer to introduce them in Grade IV, finishing this 
 work in Grade V. If introduced here, they are of course treated 
 objectively, usually with paper-folding, drawing, or inch cubes of 
 wood. There is hardly any trouble with this work unless the 
 teacher enlarges upon its difficulties. If there is accuracy of
 
 The Work of the Third School Year 93 
 
 language, spoken and written, from the beginning, this will con- 
 tinue; but if the teacher allows expressions like " 3 inches times 
 3 inches equals 9 square inches," instead of " 3 times 3 square 
 inches equals 9 square inches," there will be produced loose habits 
 of thought and expression that will lead to great trouble. 
 
 Devices for Fractions. It is still necessary in this grade to 
 make a good deal of use of objective work in treating fractions, 
 and to make the work largely oral dur- 
 ing the first half year. There is also an 2 3 4 5 
 advantage in using columns of figures 2345 
 
 like those here shown. Here it is very 2345 
 easy to see that ^ of 8 is two 2's, or 4 ; 2 3 4 5 
 that y 4 of 12 is 3 ; that ^4 of 16 is three 
 4's or 12, and that l / 2 of 20 is the same 8 12 16 20 
 as V* of 2O > or two 5's, or 10. From 
 the second arrangement it is easy to see 2 
 
 that 2 is y 2 of 4, y$ of 6, J4 * 8, and 2 2 
 
 Y B of 10; that 4 is ^ of 6, y 2 of 8; that 222 
 
 6 is 3/4 of 8 and 8 / 5 of 10, and so on. 2222 
 Devices of this kind add both to the in- 2222 
 terest in and clear comprehension of the 
 subject, and when not carried to an ex- 4 6 8 10 
 treme are valuable. 
 
 Addition. The 45 combinations of one-figure numbers should 
 be reviewed, and in the first half year oral work of the types of 
 20 + 30, 25 + 30 should be taken, to be followed in the second 
 half year by cases like 25 + 32 and 225 4- 32, where no " carry- 
 ing " is involved. Written work with four-figure numbers in- 
 cluding dollars and cents, should be given, but long 
 427 columns of figures should be avoided at present. 
 326 As already stated there is an advantage in intro- 
 
 452 ducing any difficulty in operation by using the com- 
 49 plete form. While, for example, the annexed prob- 
 lem in addition is not designed as an introduction 
 24 to the addition of three-figure numbers, it illustrates 
 130 what is meant by the complete form. The teacher 
 noo need have no fear that children cannot easily be 
 brought to use the abridged form ; " the line of least 
 1254 resistance " will bring that about, while on the
 
 94 The Teaching of Arithmetic 
 
 score of a clear understanding of the operation this complete 
 form is far superior to the other. It should also be mentioned 
 that the pupil should at this early stage be taught to recognize 
 his own liability to error and to do what every computer has to 
 do, add each column twice, in opposite directions, to be sure of 
 his result, to " check " it, as we say. 
 
 Subtraction. This subject has been sufficiently treated under 
 Chapter XI. The extent of the work is suggested by the work 
 in addition, and of the various methods the addition or "Aus- 
 trian " seems at present to be the best. 
 
 Multiplication. This, with division, constitutes the special 
 work of the year, addition and subtraction offering no essentially 
 new difficulties. In the first half year it is customary to com- 
 plete the tables through 10 X 10, and the products must be thor- 
 oughly memorized not merely in tabular form but when called 
 for in any order. The plan of carrying the tables to 12 X 12, 
 while necessary in England on account 
 s of the monetary system used there, has 
 
 generally been discarded in America, it 
 being felt that the time required for this 
 
 24 = 3 X 8 extra work could be better employed. In 
 
 270 = 3 X 90 the first half year multiplication may be 
 
 carried so far as to include three-figure 
 g Q4 = o x 208 multiplicands and one-figure multipliers, 
 
 208 and the work may at first be arranged 
 
 3 in the complete, and later in the com- 
 
 mon abridged form as here shown. 
 Since all such work is done in the class- 
 room where the teacher can supervise it, 
 
 there should be a time limit placed upon it, to the end that habits 
 of rapidity as well as of accuracy should be acquired. In the 
 
 second half year the work may usually 
 
 298 be extended to two-figure multipliers, in 
 
 43 which the complete form should again 
 
 precede the common abridgment, as 
 
 894 = 3 x 2 98 here shown. There is also introduced 
 
 11920 = 40X298 . ... ... ,. ,. 
 
 in this year such multiplications as 
 
 12814 = 43X298 that of $ 2 -75 by 7, thus preparing the 
 
 way for decimal fractions. The lat-
 
 The Work of the Third School Year 95 
 
 ter are not, however, treated in this grade, and the work should 
 not be made difficult by any unnecessary theorizing upon this 
 subject. 
 
 Division. In this year oral division by one-figure divisors 
 is introduced for such simple cases as 484 -j- 2, 484 -^-4, 481 -r- 2, 
 etc. Short division of numbers like 522 -r- 6, 
 should be introduced by some such form as 
 ) 5 22 the annexed. Such separations of the divi- 
 
 6)480 + 42 dend are made for the purpose of having the 
 
 8o+7 process seen in its simplest form, and teach- 
 
 ers should write problems of this kind on the 
 board often enough to make sure that the 
 process is understood. The children should not be required to 
 use this form, but should get to the practical work of division 
 as soon as possible. In the second half year the two-figure 
 
 divisor may be introduced, but 
 
 since the greatest difficulty in di- 
 
 75 vision consists in the estimating 
 
 2I ) I 575 of the successive quotient fig- 
 
 1470 = 70 X 21 ., . . c 
 
 ures, it is well to confine the 
 
 IO r divisors, for this year, to those 
 
 105 5 X 21 whose unit places are at first o, 
 
 then i, and finally 2. As an 
 
 <^Q .,- early form for long division, the 
 
 2 j)S7~77 annexed algorism is suggested. 
 
 6.30 = 21 X $0.30 The use of United States money 
 
 again brings in the decimal point 
 
 1-47 so naturally that the difficulty of 
 
 1.47 = 21 > decimal fractions is much dimin- 
 
 ished when that topic is reached. 
 
 The full form that may properly precede the common abridg- 
 ment is here set forth. 
 
 Scope of Work with Fractions. The pupil is now able to use 
 halves, thirds, fourths, fifths, sixths, and eighths, or, if not, this 
 work should be introduced at this time. Oral addition and sub- 
 traction of fractions with a common denominator. The reduc- 
 tion of halves to fourths, sixths, and eighths, and of thirds to 
 sixths, is introduced by means of objects, the objects being dis- 
 carded as soon as they have served their purpose. Fractional
 
 g6 The Teaching of Arithmetic 
 
 parts of numbers of three figures or less, these being selected so 
 as to be multiples of the denominator. 
 
 Denominate Numbers. Here as in other grades it is neces- 
 sary to review and frequently use the tables already learned. 
 The table of square units is often introduced and extended to 
 the square yard, although it may be postponed a year. The 
 gill is added to the table of liquid measure, and the table of 
 time is completed. Modern teaching finds it advisable to intro- 
 duce the units of measure only as rapidly as the child develops 
 the need for them and can therefore understand them. In all 
 cases it is desirable to have the measures where they can be 
 seen or in some other way appreciated. For example, when the 
 acre is introduced, somewhat later, a piece of land near the 
 school, approximately an acre in size, should be shown to the 
 class. In the same spirit they should see a ton of hay or a 
 ton of coal, a cord of wood where this is possible, a rod, a 
 gill measure, and so on. It is very important that the great 
 basal units used by our people should be visualized by the chil- 
 dren, so that bushel, mile, ton, etc., shall not be mere words. 
 
 Typical Problems. The following are suggested as two prac- 
 tical sets of problems, adapted to this grade, each telling a story 
 that may suggest other topics for original work by the class. 
 
 Oral Exercise 
 Some Home Meals 
 
 1. The coffee for our breakfast cost 6c., the potatoes 4c., the 
 meat 32c., and the bread 4c. How much did the bread and meat 
 cost? How much did all the food cost? 
 
 2. The oatmeal for a breakfast cost 8c., the milk 4C., the 
 fruit ioc., the rolls and butter 5c., and the eggs 8c. How much 
 did this food cost? 
 
 3. For a dinner the meat cost 3Oc., the vegetables 2oc., the 
 dessert 2oc., the coffee I5c., and the other food I5c. Find the 
 total cost. 
 
 4. The meals for a small family cost $1.70 on one day and 
 $2.20 on another day. How much did they cost for these two 
 days?
 
 The Work of the Third School Year 97 
 
 Written Exercise 
 
 The toad is one of man's best friends. One toad will keep a 
 garden of 800 sq. ft. free from harmful insects. 
 
 1. At this rate, how many toads would protect from insects 
 a garden 80 ft. wide and 100 feet long? 
 
 2. The eggs of 4 toads were counted and found to be 7547, 
 11,540, 7927, and 9536. How many were there in all? 
 
 3. If one out of 50 hatched, how many hatched? (Divide 
 all by 50.) If 715 of these were destroyed by other animals, 
 how many survived? 
 
 4. If each of these survivors destroys insects that would 
 cause $10 worth of damage, how much are they all worth to a 
 village? . 
 
 BIBLIOGRAPHY: The author's text-books on arithmetic, and 
 the Wentworth-Smith series, set forth his views in detail.
 
 CHAPTER XVIII 
 THE WORK OF THE FOURTH SCHOOL YEAR 
 
 The Leading Mathematical Features. In this the last year of 
 the primary grades it is well to feel that the essentials of arith- 
 metic have all been touched upon. It is, therefore, desirable to 
 review the four fundamental operations, extending the multipli- 
 cation and division work to include three-figure multipliers and 
 divisors. The common business fractions should also be in- 
 cluded, with simple operations as far as multiplication. 
 
 Number Space. In the first half year the numbers may extend 
 to 100,000, and in the second half year number names may be 
 given as high as a billion. The operations, however, should be 
 confined to the smaller numbers of such business as can be 
 appreciated by children of this age. 
 
 Counting. The prime object of the counting exercises, the 
 developing of the tables of addition and multiplication, has now 
 been accomplished, except when it is desired to carry the multi- 
 plication table to 12 X 12. In that case the counting may now 
 be continued by n's to 132 and by I2's to 144. -Otherwise the 
 only use of counting in this grade is for the purpose of review. 
 
 Addition and Subtraction. There should be much rapid oral 
 work with numbers like the following: 
 
 7 17 37 37 47 
 
 + 4 + 4 + 4+14+24 
 
 II 21 41 51 71 
 
 - 7 - 7 - 17 ~ 27 - 47 
 
 The written work should be undertaken with the aim of (i) 
 accuracy, secured by always checking the result; (2) rapidity, 
 secured by setting a time limit upon all work. Children should 
 by no means neglect this matter of checks, since it is used in all 
 the business world. Much; of the complaint of business men, 
 
 98
 
 The Work of the Fourth School Year 99 
 
 that boys from the schools are always inaccurate in arithmetic, 
 would be obviated if pupils were always required to check their 
 additions by adding in the opposite directions, and their other 
 results in some appropriate manner. In subtraction, for example, 
 if the result is obtained by the "Austrian " method it should be 
 checked by adding it to the subtrahend in the opposite direction. 
 
 Multiplication and Division. No new principles are involved 
 here, and the work of the preceding year is simply extended to 
 include larger numbers. In some schools the multiplication 
 table is extended to 12 X 12, although this is not important 
 enough for most people to make it worth the while. It is a good 
 plan, however, to learn all products less than 50, as 2 X 13, 
 3 X 15, 4X 12, and so on. since these are so often used in the 
 purchases of the household. Even a child ought to know the 
 cost of 2 Ib. of meat at 18 cents a pound, without using pencil 
 and paper. The practical checks on multiplication and division 
 are not advantageously discussed as early as this. 
 
 Fractions. Here as later the work in common fractions should 
 be confined to those needed in ordinary business, and at present 
 to those from ]/ 2 to %. Of course there is no objection to an 
 occasional example with denominators of two or three figures, 
 but the day of fractions like -$zfa is past, decimal fractions having 
 taken the place of all such forms. Children in this grade should 
 also know that $ l / 2 = 50 cents, and $>4 2 5 cents. The opera- 
 tions may extend as far as easy multiplications of an integer and 
 a fraction, two fractions, or an integer and a mixed number. 
 Unusual forms of operation, not practical in business, should not 
 be given, and the teacher should resist all temptation to depart 
 from this principle on any supposed ground of mental discipline. 
 
 Decimal Fractions. A brief introduction to this subject, based 
 on the work already given in United States money, may be al- 
 lowed in this grade, although the serious treatment of decimals 
 belongs later in the course. 
 
 Denominate Numbers. The tables needed in business life are 
 completed in this grade by adding that of land measure, and 
 completing long and cubic measure. In the work of adding and 
 subtracting compound numbers children should feel that there
 
 ioo The Teaching of Arithmetic 
 
 is no principle involved that is not found in integers. For ex- 
 ample, consider these two cases : 
 
 37 3 ft - 7 in - 3 lb - 7 oz. 
 
 25 2 ft. 5 in. 2 Ib. 5 oz. 
 
 62 6 ft. 5 Ib. 12 oz. 
 
 In the first, because 7 + 5 = 12, which is I ten and 2 units, the 
 I ten is added to the lo's. In the second, because 7 in. + 5. in. 
 12 in. or i ft., the I ft. is added to the feet. In the third, because 
 7. oz. + 5 oz. = 12 oz., which does not equal a pound, it is 
 written under ounces. In every case the principle is the same, 
 to add to the next order any units of that order that are found. 
 In general we use compound numbers of only two denomina- 
 tions, and it is on such numbers that we should lay the emphasis. 
 The use of numbers of four or five denominations is now 
 obsolete, and there is not enough disciplinary value in the sub- 
 ject to warrant using them instead of the numbers of actual 
 business. 
 
 A!s heretofore mentioned, there should be an effort to have 
 children visualize the standard measures of our country, such 
 as the acre, mile, ton, and bushel. 
 
 Teachers should be careful at this time that slovenly methods 
 of statement do not become habits. Such forms as the follow- 
 ing, for example, are inexcusable: 
 
 60 in. -f- 12 = 5 ft. 
 
 60 H- 12 = 5 ft. 
 
 60 in. H- 12 in. = 5 ft. 
 
 If we wish to reduce 60 in. to feet we have three correct forms, 
 any one of which is easily explained : 
 
 60 X Vi2 ft. = 5 ft. 
 
 60 in. -r- 12 in. = 5, the number of feet, 
 
 60 -T- 12 = 5, the number of feet. 
 
 If slovenly forms are allowed here they must be expected in all 
 subsequent grades, and they must be expected to lead to slovenly 
 thought in the treatment of all kinds of problems. 
 
 Review. At the close of the year there should be a review of 
 all the essential features of the work in the primary grades. 
 This requires skill on the part of the teacher lest it become stupid 
 and so wearisome as to lose its chief value. Original local
 
 The Work of the Fourth School Year 101 
 
 problems to test the children in the four fundamental operations 
 with integers and (as far as they have gone) with fractions, 
 will usually render the work interesting and will hold the 
 attention. 
 
 Nature of the Problems. Here as elsewhere the problems 
 should touch the children's interests and be adapted to their 
 mental abilities. The following may be taken as types: 
 
 Oral Exercise 
 
 1. Tell the cost of some kind of cloth. How much will 
 io l /2 yd. cost? 
 
 2. Tell the cost of a pair of shoes. How much will 2 pairs 
 cost? 
 
 3. If a man earns $3 for 10 hours' work, how many hours 
 must he work to earn enough to buy his daughter a pair of 
 shoes at $1.50? 
 
 4. How many hours must he work to earn enough to buy 
 a $6 suit of clothes for his son? 
 
 Written Exercise 
 
 1. Sarah's mother bought 4Y 5 yd. of cloth for a cloak, at 
 $1.25 a yard. What did she pay for it? 
 
 2. She also bought 3^ yd. of lining at SQC. a yard, and 
 4 l /4 yd. of braid at 2oc. a yard. How much did these cost? 
 
 3. She also bought 6 pearl buttons at $1.50 a dozen, and 2 
 spools of silk at 8c. a spool. How much did these cost? 
 
 4. The dressmaker charged $5 for making the cloak. What 
 did materials and making cost? 
 
 .5. John's mother bought 2^ yd. of goods for a coat, at $1.20 
 a yard, and 2^ yd. of lining at 48c. a yard. How much did 
 these cost? 
 
 6. She also bought a dozen buttons at 25c. a dozen, and 2 
 spools of silk at 8c. a spool, and paid $3 for making. How 
 much did the coat cost?
 
 CHAPTER XIX 
 THE WORK OF THE FIFTH SCHOOL YEAR 
 
 The Leading Mathematical Features. There should in this 
 year be a thorough review of the fundamental operations with 
 integers. This should be followed by the same operations with 
 the common fractions and denominate numbers of business. 
 Percentage may be begun, although in some places it is better 
 to postpone this until the following year. 
 
 Review. There is usually a new text-book begun in this 
 grade, and this, if properly arranged, offers plenty of material 
 for the review above mentioned, with numbers that are appro- 
 priately larger. Teachers should undertake this review in the 
 spirit and for the reason suggested in Chapter XVI. 
 
 Text-book. The new text-book begun in this grade will natur- 
 ally be topical in its arrangement, that is, each general topic 
 like percentage being treated once for all; or it will be on the 
 plan of recurring topics, a subject like percentage being met 
 two or three times. As has already been said, each of these 
 types has its advantages. If the school chooses one with recur- 
 ring topics that can probably be followed rather closely. If on 
 the other hand it adopts one arranged by topics there are two 
 courses open : ( i ) the teacher may select from the various chap- 
 ters such material as fits the course of study in use in the par- 
 ticular locality, a task of no great difficulty; (2) the book may 
 be followed closely, the pupils' work becoming purely topical. 
 We are apt to condemn the latter plan because it is old, but 
 perhaps on that very account it should be commended. The 
 world has used it, and used it successfully, and it has the merit 
 that it brings a feeling of mastery, a sense of thoroughness, and 
 a development of habit that is sometimes lacking with more 
 modern text-books. In general it may be said to depend upon 
 the school as to which type of book is the better, and as to 
 which plan of using the topical book is to be preferred. In a 
 
 102
 
 The Work of the Fifth School Year 103 
 
 school system with a reasonably permanent staff of teachers, 
 with adequate supervision, and with teachers' meetings that 
 allow classes to keep in touch with one another, the book with 
 recurring topics, or at least the course arranged on this plan, is 
 undoubtedly the better. It is more psychological and it allows 
 for a better grading of material. On the other hand where 
 teachers change frequently, as in rural schools, it is safer to use 
 the topical book and to follow it rather closely. In this and 
 the following chapters the arrangement by recurring topics is 
 followed, and any topical text-book can easily be adapted to the 
 sequence suggested. 
 
 Number Space, This number space is now unlimited, but 
 names beyond billion are of no particular importance. Large 
 numbers should always represent genuine American conditions. 
 It is better to perform several operations on the ordinary num- 
 bers of daily life than to perform one on an absurdly long 
 number; but on the other hand, a reasonable number of opera- 
 tions on large numbers that represent real business cases are 
 to be commended. 
 
 Addition. Larger numbers and longer columns may now be 
 used, but there is a limit to this matter. In general, the numbers 
 used by the average citizen are the ones to drill children upon. 
 Children should be encouraged to read columns as nearly as pos- 
 sible as they read a word. When we seek the word " book " 
 we do not think " b," " o," " o," " k," we think "book" with- 
 out any spelling ; so when we see the annexed column 
 we should not think, " 6 and 3 are 9, 9 and 3 are 5 
 12, 12 and 5 are 17," nor even " 6, 9, 12, 17," if we 3 
 can do better than this. Probably we cannot train 3 
 our eyes to see 17 at a glance, as we seek " book," 6 
 but it is well to encourage children to look at this 
 as 9 + 8, thinking of the 6 and 3 as 9, and the 3 
 and 5 as 8. But, however we think of such a column, we should 
 always check our result by adding in the reverse order. If 
 teachers do not think this necessary, let them add twenty sets 
 of say ten five-figure numbers each, working rapidly, and see 
 how many mistakes they themselves will make. 
 
 Subtraction. This subject has been sufficiently discussed on 
 page 45. The important matter is not now the explanation,
 
 IO4 The Teaching of Arithmetic 
 
 for the technique has already been learned; the operation, ac- 
 curately and rapidly performed, is the desideratum, the check 
 being of great importance in securing the essential accuracy. 
 
 Multiplication. It is now advisable to let the children know 
 some good, practical check on their work in multiplication, such 
 as computers actually use. Of the checks, the simplest is that 
 of " casting out 9's." 1 
 
 Division. The children are now old enough to understand 
 the two forms of division illustrated by the following: 
 
 $125. ^-$5 = 2$ 
 $125 -4- 25 = $5 
 
 There are no generally accepted names to distinguish these, 
 " measuring " and " partition " not meaning much to children. 
 It suffices that it is clear that there are these two forms, and to 
 see that we avoid such inaccuracies as $125 -r- $25 = 5 cows. 
 
 Factors and Multiples. This subject formerly played a. very 
 important part in arithmetic, when large fractions had to be 
 reduced to lower terms. With the introduction of the decimal 
 fraction about 1600, however, it lost much of its former im- 
 portance and need play but a small part in the arithmetic of 
 to-day. 2 
 
 Common Fractions. Some objective work will still be neces- 
 sary in treating common fractions but it should be dispensed 
 with as soon as possible and the material should not be of one 
 kind alone. In the operations children should not be required 
 to give very elaborate explanations, although they should see 
 clearly the reasons at the time they learn the processes. This 
 has been discussed already in Chapter IX and may, therefore, 
 be dismissed at this time. 
 
 Denominate Numbers. The operations with these numbers 
 should be a part of the work of the year, but only practical 
 cases should be taken. To divide a compound number of four 
 
 1 The explanation of this process is too long to be given here. The 
 reader may consult the author's Handbook, p. 57, Beman and Smith's 
 Higher Arithmetic, or the appendix to the Wentworth- Smith Complete 
 Arithmetic and the Arithmetic, Book III. 
 
 2 For a theoretical treatment of the subject from the advanced stand- 
 point, consult Beman and Smith's Higher Arithmetic.
 
 The Work of the Fifth School Year 105 
 
 denominations by another one of three, for example, consumes 
 time and patience to no worthy purpose. 
 
 How to Solve Problems. Inasmuch as the children now begin 
 to consider problems of more than two steps, it becomes neces- 
 sary to devote more attention to the methods of solving ex- 
 amples. The step form of analysis, therefore, has a legitimate 
 place in this year's work. If teachers hope for exactness of 
 thought they must insist upon accuracy of statement in these 
 written exercises. 
 
 Percentage and Decimals. The study of decimal fractions 
 may safely be undertaken in this grade, and this may be fol- 
 lowed, if desired, by an elementary treatment of percentage. If 
 at the outset children understand that 6% is only another way 
 of writing yf^ and 0.06, there will be but little difficulty in 
 introducing percentage. One important feature is the inter- 
 change of the per cent forms, decimal fractions, and common 
 fractions, as for example, in ]/$ =-r^= 0.25 = 2$%. It is 
 better not to introduce any formulas or rules in such work in 
 percentage as may be taken at this time, but to analyze each 
 problem as it arises. In the next school year it is allowable 
 to reverse this policy. 
 
 Discount. Of all the applications of percentage the most 
 common is 'discount, and it is at the same time the simplest. 
 This topic may, therefore, be introduced in this grade, the other 
 applications being reserved for the sixth year. 
 
 Nature of the Problems. The great industries of the country 
 may be taken up at this time as a profitable field for the applica- 
 tions of arithmetic. Children now begin to know enough geogra- 
 phy to permit of this wider view, and problems that relate to 
 their own country have an interest that the traditional ones about 
 the man who " owned a field of corn " lacked. Such problems 
 are not statistical to the extent that their data are to be memor- 
 ized, but they state real conditions instead of false ones. Of 
 problems suited to this grade the following are types relating 
 to the production of corn, one of the great food products of the 
 country. The greatest corn-producing states are Iowa, Illinois, 
 Nebraska, Missouri, Kansas, and Indiana, and such work may 
 be taken in connection with the study of the geography of these
 
 io6 The Teaching of Arithmetic 
 
 sections whenever this can be brought about without too great 
 change in the curriculum. 
 
 1. When this country produced 2,105,102,400 bu. of corn a 
 year, averaging 25 bu. to the acre, how many acres had we 
 in corn? 
 
 2. If 3 bu. of corn could then be bought for $i, what was 
 the total value of this yield of 2,105,102.400 bu? 
 
 3. When Iowa's annual product amounted to 305,800,000 bu., 
 this was how many times the 440,000 bu. produced by Maine? 
 
 4. To transport 1000 Ib. of corn from St. Louis to New 
 Orleans by river costs $i. How much will it cost to transport 
 1750 tons? 
 
 5. If the average value of corn for each of the 46,610 acres 
 given to it in Connecticut in a certain year was $21, and for each 
 of the 4,031,600 acres in Indiana $13, what was the entire value 
 of the corn crop of each state? 
 
 6. If the average annual corn crop per acre is 40 bu. in 
 Wisconsin, 36 bu. in Maine, 37 bu. in New Hampshire, 38 bu. 
 in Massachusetts, 38 bu. in Indiana, and 38 bu. in Iowa, find 
 the average by adding and dividing by 6.
 
 CHAPTER XX 
 THE WORK OF THE SIXTH SCHOOL YEAR 
 
 The Leading Mathematical Features. The leading features 
 of this year should be percentage and its applications, particu- 
 larly to discount, profit and loss, commission, and interest. 
 Ratio and simple proportion may also be included. 
 
 The General Solution of Problems. Since the work in per- 
 centage introduces the pupil to the problems of business, some of 
 which become rather intricate in the later school years, it is well 
 at this time to take up rather systematically questions of the solu- 
 tion of problems in arithmetic. To this end there should be con- 
 sidered exercise in analysis in general and in unitary analysis in 
 particular, and the equation may well begin to find place in the 
 mental equipment of*jjae* child. As to the matter of analysis no 
 question will be raised, but as to introducing the letter x some 
 teachers are In doubt. When, however, we come to consider that 
 it merely replaces- an awkward symbol that has long been used, 
 and makes the work much clearer, the objection cannot be main- 
 tained. For example, 2 + ( ?) = 7 is sometimes used as early 
 as the first schodl "y.ear ^.ttfis^ho wever, is only a complicated way 
 of writing 2 -K$-= 7, tr^e two meaning exactly the same thing. 
 Similarly, 4 : 7 =M2*: (?) is only an awkward way of writing 
 what is equivalent*? 
 
 x 7 
 
 12 \ 4 
 
 the latter being in every way simpler of understanding and 
 easier of solution. Theje are several classes of problem in per- 
 centage that are made clearer' by the use of this convenient x, 
 and its use is quite as arithmetical as algebraic. 
 
 Percentage. In this work special attention should be given 
 to the common per cents and fractions of business, such as 
 
 and so on. 
 107
 
 io8 The Teaching of Arithmetic 
 
 In the matter of solution, the x should be used in those inverse 
 cases where it makes the problem clearer. Such is the case of 
 finding the cost of goods that sell at 10% above cost, and sell for 
 $126.50. Here we have, if x represents the cost, 
 
 x + .\QX = $126.50 
 1. 10^ = $126. 50 
 
 X = $126.50 -4- 1. 10 
 #==$115 
 
 Other forms of solution might be used, but this is the most 
 satisfactory. 
 
 Discount. This being the first and most important of the 
 applications of percentage, considerable attention should be de- 
 voted to it. The case of several discounts may, however, be 
 postponed until the following year. 
 
 Profit and Loss on Purchases. This topic, so closely con- 
 nected with the business world with which the child is now 
 coming into closer contact, may claim to rank second in impor- 
 tance among the applications of percentage. The principles 
 involved are very simple, particularly if one allows the letter x 
 to throw light upon all inverse problems. The examples should 
 follow as closely as possible the common business customs of 
 the mercantile world. 
 
 Commission. This topic ranks possibly third in importance 
 among the applications of percentage. A considerable field of 
 applications exists, particularly in relation to the sending of farm 
 produce to the cities. The problems can, therefore, be made to 
 seem real to the children, whether they live in the country or 
 see farm products for sale in the city. 
 
 Interest. This subject may already have been met by the 
 children. It is now taken up and extended to more difficult 
 questions. Only real cases should, however, be considered. For 
 example, in this school year, at least, there is little advantage in 
 trying to find the capital, given the rate, time, and interest. It is 
 better to spend time in writing promissory notes and in comput- 
 ing the interest, than to put it on questions that seldom arise in 
 business life. If we wish more complicated problems they are 
 easily secured from genuine mercantile sources. 
 
 Ratio. This may be introduced this year or reserved for the 
 seventh grade. It was formerly introduced merely as an intro-
 
 The Work of the Sixth School Year 109 
 
 duction to proportion. It is easy, however, to see that it may be 
 of some i^se by itself, and teachers are advised to consider this 
 phase of tfce subject. We mix fertilizers on the farm in a given 
 ratio, we find ratios of attendance to absence in the school, and 
 the term is used in the same way in business life. 
 
 Proportion. This subject may also be delayed another year. 
 It has lost a good deal of its importance of late. A proportion 
 is, as shown on page 107, merely one method of writing a simple 
 equation, and with the letter x allowed in school, the equation 
 form is likely to replace that of proportion. When this is not 
 the case, ordinary analysis is likely to be substituted for pro- 
 portion. For example, consider this problem: If a shrub 4 ft. 
 high casts a shadow 6 ft. long at a time that a tree casts one 
 54 ft. long, how high is the tree? 
 
 Here we may write a proportion in the form, 6 ft. : 4 ft. = 
 54 ft.: (?), not attempting to explain it, but applying only an 
 arbitrary rule. This is the old plan. Or we may put the work 
 into equation form. 
 
 54 6 
 
 and deduce the rule for dividing the product of the means by the 
 given exteme. Or we may take the same equation and get our 
 result easily by multiplying these equals by 54, giving 
 
 * = 3 6 
 
 Or we may say : If a 6 ft. shadow is cast by a 4 ft. object, a I ft. 
 shadow would be cast by a */e ** object, and a 54 ft. shadow 
 would be cast by a 54 X 4 /e ft. object, or a 36 ft. object. 
 
 Of these plans the first is the most difficult to explain; the 
 rest are equally easy, and the third is the shortest. 
 
 Measures. The work in measures this year may be confined 
 to simple surfaces and solids, and may properly include practical 
 cases of house building, plastering, carpentering, and the like. 
 Here is a real field, interesting and profitable. Proportion leads 
 to exercises in similar figures, and this has some excellent appli- 
 cations in lumbering and in carpenter's work. 
 
 Nature of the Problems. In each succeeding year the prob- 
 lems now come to relate more and more to the industries of the
 
 no The Teaching of Arithmetic 
 
 people, and the range of applications becomes very great. The 
 farm child learns not only of his own surroundings but of the 
 great industries of the city, while to the city child the great story 
 of the soil and its products opens up a new world. The follow- 
 ing farm problems may be taken as types of the problems suited 
 to this grade: 
 
 1. A farmer puts 5 acres into celery, setting out 20,000 plants 
 to the acre. The yield being 1,500 doz. heads to the acre, what 
 is the ratio of the plants matured to the others? ' ... 
 
 2. He pays $95 an acre for seeds, fertilizers, labor, and other 
 expenses, and sells the crop at I5c. a dozen heads. What is his 
 profit on the 5 acres ?j? fciTO.oo 
 
 3. Another farmer tries setting out 30,000 plants to the acre, 
 but only 80% mature, and these are so small that he has to put 
 16 in a bunch to sell for a dozen, and then gets only i4c. a 
 bunch. His expenses are $100 an acre. At this rate what is 
 his profit on 5 acres ?f JTlTO.O* 
 
 4. A farmer has a 3O-acre meadow yielding i l /2 tons of hay 
 to the acre. If by spending $300 a year for fertilizers, he can 
 bring the yield to 4 tons to the acre, how much more will he make 
 a year, hay being worth $8 a ton? 
 
 5. A farmer reads that a good mixture of seed for his 
 meadow is, by weight, as follows: timothy 40%, redtop 40%, 
 red clover making up the rest. At 40 Ib. of seed to "the acre, 
 how many pounds of each should he sow? t to "T- I r> Jf-- t 
 
 6. The following is, by weight, a good mixture of seed for a 
 ~J pasture: Kentucky blue grass$25%, white clover \2,y 2 %; per- 
 
 ennial rye 28%%, red fescue $%, redtop 25%. At 32 Ib. to 
 the acre, how many pounds of each are used? 
 
 7. A cow weighing 1000 Ib. consumes the equivalent of 3>4 
 tons (2000 Ib. to the ton) of dry fodder a year; a loo-lb. sheep, 
 770 Ib. ; every ton of live pork, 12 tons ; and every ton of live 
 horseflesh, 8.4 tons. Each class of animals consumes what per 
 cent of its own weight of drv fodder a year? 
 
 T ^ 

 
 CHAPTER XXI 
 THE WORK OF THE SEVENTH SCHOOL YEAR 
 
 The Leading Mathematical Features. As in the preceding 
 grade, it is well to begin by a general review of the fundamental 
 processes from a higher standpoint than before. Ratio and pro- 
 portion are usually completed in this year, whether introduced 
 here for the first time or not, and the applications naturally cover 
 a broader field. Percentage is the leading topic of the year. 
 
 Our Numbers. The children are now ready to consider the 
 writing of numbers from a higher standpoint, to know something 
 of the interesting history of the numerals they use and of the 
 science of arithmetic that they are studying. The story of the 
 Roman numerals, 1 and that of the Arabic numerals make these 
 subjects seem more real at this time. The difference between 
 a uniform scale, as seen in our system of money, and a varying 
 one, as seen in the English system, should also be explained, 
 and the advantage of the former understood. The relation be- 
 tween integers, the various kinds of fractions, and compound 
 numbers, may now understandingly be taken up. 
 
 The Fundamental Operations. These may now be reviewed 
 in such way, that is by the introduction of such new material, 
 as to maintain the interest even in an old subject. This is par- 
 ticularly true if the teacher will now and then suggest such short 
 methods as may be found in most of the advanced arithmetics. 
 The check of casting out nines should now be used for all 
 products and quotients. It is simple, it takes but a moment, 
 and it checks most of the errors that are liable to arise. It 
 cannot be too much impressed upon teachers and pupils that 
 both are very liable to errors in all kinds of calculation, and 
 that they, like business computers, should always apply some 
 kind of check to every result obtained. Some teachers feel that 
 
 1 This is told in condensed form in the author's Handbook to Arithmetic. 
 See also the Bibliography at the close of Chapter I. 
 
 ill
 
 112 The Teaching of Arithmetic 
 
 the work should be so accurately done that checks should be 
 unnecessary. This is a good theory, but practically it will not 
 work even with the ones who advocate it. No good professional 
 computer would think of leaving his results without checking 
 them, and if a professional will not do this, why should we 
 expect a child to be so infallible as to do it? 
 
 Measures. All tables of measure in common use should be 
 reviewed in this year. If, in this review, some historical notes 
 are given on the origin of such measures as the yard, inch, foot, 
 mile, quart, gallon, and acre, the pupils will find the work taking 
 on a new interest. Teachers are advised that it is of little value 
 to memorize facts that will not be used in practical life. If we 
 wish to know the number of cubic inches in a bushel we may go 
 to an encyclopedia or a dictionary; it is surdy inadvisable to 
 burden our minds with such details. 
 
 Longitude and Time. This subject has greatly changed within 
 a few years. To-day most of the civilized world uses some 
 form of standard time. Therefore, our attention may properly 
 be confined to the geographical principle involved, to the prob- 
 lem of standard time, and to the question of longitude at sea. 
 Teachers are urged not to allow slovenly work in this subject 
 under the plea that bad forms bring true results in a shorter 
 time than good forms. This matter has been sufficiently dis- 
 cussed on page 30, and the chapter on longitude and time seems 
 to be one of the worst offenders in all arithmetic. A form 
 like 45 -r- 15 = 3 hrs. is false and serves to undo all of the good 
 to be derived from the topic. 
 
 Percentage. This topic, so vital in business life to-day, should 
 be touched upon several times in the elementary school. If 
 the work is sufficiently progressive the pupils will not find that 
 " the edge of interest " is worn off. In this year there should 
 be a good deal of oral work in the common per cents of 
 business, pupils coming to feel that pencil and paper are unneces- 
 sary in finding 12%%, 25%, 3^/3%, 50%, 66 2 / 3 %, and 75% of 
 ordinary numbers. As to the use of terms like " base," " rate," 
 " percentage," " amount," and " difference," there is little that can 
 be said in their favor. They were invented in the rule stage of 
 arithmetic, and have served their purpose. Of course, we need 
 " rate," it being a stock term of the business world. " Percent-
 
 The Work of the Seventh School Year 113 
 
 age " is, however, rather confusing than otherwise, ( i ) because 
 it is understood by the pupils as the name of the subject as a 
 whole, and (2) because the business world does not use it quite 
 as the school does. " Base " means so many things in mathe- 
 matics that its use is equally confusing, while of " amount " and 
 " difference " this is still more noticeably the case. On the 
 whole, therefore, it is as well not to use these terms, although 
 they are found in most of our leading books to-day because of 
 the demands of teachers. 
 
 It should also be remarked that, if the use of x is allowed, 
 there is no excuse for the old formulas of percentage. They 
 are nothing but condensed rules ; if they are not explained they 
 defeat part of the purpose of studying arithmetic; if they are 
 explained they are much harder than the equation form with 
 the single letter x. 
 
 It is well to bear constantly in mind, in the midst of the large 
 number of possible cases of percentage, that the two important 
 things in the subject are these: (i) to find some per cent of a 
 given number, and (2) to find what per cent one number is of 
 another. All the rest is relatively unimportant, and on these 
 two the emphasis should accordingly be laid. 
 
 Simple Interest. This is the leading application of percentage 
 in this year, and the attention of pupils should be concentrated 
 on the single problem of finding interest in practical cases. To 
 find the time, given the principal, rate, and interest, is of very 
 slight importance, and so for other similar cases ; but to find the 
 interest, that is the great point. 
 
 Ratio and Proportion. This work should, as stated in the 
 preceding grade, be confined largely to the treatment of practical 
 questions, and there are only a few where this subject can be 
 used to real advantage. These are chiefly related to similar 
 figures, although some other questions, like those of simple 
 physics, enter. Compound proportion has little reason to claim 
 a place in our schools to-day. If explained, the process is a 
 very hard one; if not, it is a useless one, since we now have 
 better methods of solving problems. 
 
 Nature of the Problems. With each succeeding school year 
 the children develop new interests and come nearer to the great 
 world that they are soon to enter. The range of topics is now
 
 114 The Teaching of Arithmetic 
 
 practically unlimited, and the opportunities for offering series of 
 related problems are excellent. As a type of such problems the 
 following may be given, appealing this time to the girls, who 
 are usually rather neglected in the matter of applied arithmetic : 
 
 Dressmaking Problems 
 Written Exercise 
 
 1. A dressmaker bought 16 yd. of velvet at $3 a yard, selling 
 9 yd. at a profit of i6 2 /^% and the rest at a rate of profit half 
 as great. What was the rate of gain on the whole? 
 
 2. She bought a 25~yd. box of chiffon velvet at $4 a yard, 
 with 10% off for cash, selling it at $4.35 a yard. What was her 
 gain per cent? 
 
 3. She bought a 75-yd. piece of silk skirt lining at 65c. a 
 yard. She sold 28 yd. at 9oc., 15 yd. at 95c., and the remainder, 
 at the close of the season, at /oc. What was her per cent of gain ? 
 
 4. She bought a 5o-yd. piece of silk waist lining at 75c. a 
 yard. She sold 12 yd. at $i and 10 yd. at 95c., but the remainder, 
 being kept in stock over the season, had to be sold at 65c. What 
 was her per cent of gain or loss? 
 
 5. She bought a 2O-yd. silk dress pattern at $2.10 a yard, 
 being allowed, as a dressmaker, a discount of $%, and 6% off 
 for cash. She charged her customer the marked price, $2.10. 
 What was her per cent of profit? 
 
 6. She charged her customer $25.50 for 3 yd. of Honiton 
 lace, which had cost her $7 a yard. What was her per cent of 
 profit ? 
 
 7. She charged her customer $2 for findings for the dress. 
 These consisted of 4 spools of silk at IDC. each, i spool of thread 
 at 5 c -> 3 yd- of featherbone at ioc., a card of hooks and eyes 
 at 8c., skirt braid i6c., plaiting 3oc., waist binding 3Oc., and 
 collar ioc. What was her gain per cent on the findings?
 
 CHAPTER XXII 
 
 The Leading Mathematical Features. The work this year is in 
 the line of business applications, including advanced mensuration. 
 
 Business Applications. The boy and girl should now begin 
 to feel that the world of business and of life is opening before 
 them. It should therefore be the duty of the school, even more 
 than in the preceding grades, to apply arithmetic to the genuine 
 problems of life, particularly with reference to the common occu- 
 pations of the people. 
 
 Banking. In banking, for example, we should not seek to 
 train accountants or bookkeepers or cashiers, but we should seek 
 to give a fair idea of the duties of these men in the ordinary 
 savings bank and bank of deposit. A girl, for example, needs 
 to know how to deposit money in a bank and how to draw checks 
 as well as a boy, and such operations should become as real 
 as the school can make them. School banks, with deposit slips, 
 checks, bank book, cashier, paying teller, and receiving teller, 
 should assist in this work. 
 
 Partial Payments. This subject has not the practical value 
 that it had when banks were not so numerous as now, and when 
 their machinery was not perfected. The old-style problem in 
 partial payments should therefore give place to the more prac- 
 tical cases found in our best modern books. 
 
 Partnership. This is another subject that has entirely changed 
 within a short time. The stock company (corporation) has 
 largely supplanted it, save in its simplest form. The work of 
 the schools should therefore be confined to this common form, 
 the obsolete ones being supplanted by work on corporations. 
 
 Simple Accounts. It is not worth while to teach an elaborate 
 form of bookkeeping to the average citizen. On the other hand 
 it is necessary that every one should know how to keep simple 
 accounts, and this work should be taken up in this year. It 
 should relate to the income and expenditures of daily life, in the 
 
 "5
 
 u6 The Teaching of Arithmetic 
 
 home, on the farm, or in the shop, rather than to the technical 
 needs of the merchant, the latter being part of the special train- 
 ing of the individual who enters this line of trade. 
 
 Exchange. Here again there has been a great change within 
 a few years. The form of time draft given in most of the old- 
 style arithmetics has given place either to sight drafts or to 
 another kind of time draft. Teachers should therefore be 
 particular to use only those types that the ordinary citizen meets 
 to-day, about which girls and boys alike should be informed. In 
 connection with this work a short talk upon the clearing house, 
 upon which any bank will gladly inform the teacher, will add 
 new interest. 
 
 The Metric System. This system might be taught much 
 earlier than the eighth school year, and there would be some 
 advantage in so doing. But when we consider that it is not yet 
 used practically by many Americans, it seems as well to postpone 
 it until this time. There are three chief reasons for teaching 
 it now: (i) General information requires us to know a system 
 that is used by a large part of the civilized world, excluding the 
 English-speaking portion; (2) it is used in all scientific labor- 
 atories in America; (3) our people should be sympathetic with 
 a system that is liable to replace our own before long in all 
 matters relating to our growing foreign trade; if we sell ma- 
 chines abroad, the measurements must be metric in most cases, 
 and to foster this trade many of our skilled workmen will 
 eventually need to use these instead of the awkward ones with 
 which we are familiar. 
 
 At the same time we must not go to an absurd extreme, but 
 must remember that our common system is the one that the 
 people use and that the children must know before all others. 
 In teaching the metric system the results will be poor unless the 
 children use the actual measures and come to visualize the basal 
 units as they should in their own system. 
 
 Taxes. This topic, like others of practical life, should be 
 treated from the standpoint of local conditions as far as possible. 
 It should include the question of tariff, and a few brief talks on 
 civics should make the whole question a real one for the 
 pupils. 
 
 Insurance. This subject has become so technical that all that 
 the schools can hope to do is to give a general conception of the
 
 The Work of the Eighth School Year 117 
 
 work of the various kinds of companies, and to confine the prob- 
 lems to the simplest practical cases that the people need to know 
 about. We should not attempt to enter upon the technicalities 
 of agent work, nor to do more than explain briefly some of the 
 common types of policy. 
 
 Corporations. As remarked under Partnership, the corpo- 
 ration has, for good or evil, replaced the individual in large 
 business ventures. Our schools must, therefore, adjust their 
 work to this change. Pupils should know what a corporation 
 is, its chief officials, how it is legally organized, what stocks and 
 bonds are, how dividends are declared and paid, and the legiti- 
 mate work of stock exchanges. On the other hand the schools 
 cannot be expected to teach the technicalities of the stock brok- 
 er's office, nor to supply information beyond that needed by the 
 general citizen. The newspaper stock reports furnish an excel- 
 lent basis for the practical problems that the case demands. 
 
 Powers and Roots. For purposes of mensuration square root 
 is necessary. Cube root may well be delayed until the pupil 
 studies algebra, because it has so few practical applications. 
 Even square root is more valuable as a bit of logic than as a 
 practical subject, since those who use it most employ tables. 
 The explanation, therefore, is even more important than the 
 technique of the work, and children of this age can easily com- 
 prehend it, either by the use of the diagram or by the formula, 
 the latter being quite easily understood by this time. 
 
 Mensuration. This work is now completed so far as the 
 needs of the average person are concerned. The teacher should 
 use simple models that can be made in the school room, as sug- 
 gested in the best arithmetics. It is not expected that strict 
 geometric demonstrations can be given, but it is entirely possible 
 to avoid arbitrary rules by giving enough objective work to make 
 the matter clear. It is not advisable to introduce work that is 
 not used in ordinary life, such as finding the volume of a frustum 
 of a cone, there being a sufficient amount of more important 
 work to occupy the time and attention of pupils. 
 
 Nature of the Problems. The problems should appeal to the 
 business needs that are soon to come to the children, and the 
 following are suggested as types: 
 
 I. A boy who has been working this year at $25 a month is 
 offered either an increase of 20% for next year or a salary of $7
 
 n8 The Teaching of Arithmetic 
 
 * 
 
 a week. Which will bring the more income, and how much 
 more per year? (Use 52 wk.) 
 
 2. A girl who has been working in a factory at $21.67 a 
 month is offered an increase of 10% where she is or a salary of 
 $5.60 per week elsewhere. Which will bring the more income, 
 and how much more per year? (Use 52 wk.) 
 
 3. A boy went to work at 9<Dc. a day. The second year his 
 wages were increased 20%, the third year they were 42c. a day 
 more than the second, and the fourth they were increased ZZY^%- 
 At 300 working days to the year, what was his total income for 
 each year? 
 
 4. A girl entering a trade school finds that graduates from 
 the dressmaking department receive on an average $4.60 a week 
 the first year; those from the millinery department, 5% less; 
 those from the embroidery department, 5% more than the dress- 
 makers ; and those from the operating department 66^3% as 
 much as the last two together. Find the average wages of each, 
 and tell which department the girl probably entered. (Use 52 wk.) 
 
 5. A girl leaving the public school finds she can enter a city 
 shop at a salary of $3 a week the first year, with 16%% more 
 the second year, and a 14* / 7 % increase the third year. Instead 
 of this she enters a trade school for a year, tuition free. She 
 then receives a salary of $5 a week the first year and 20% more 
 the second year. Counting 50 working weeks a year, how much 
 more does she receive in three years by the plan she follows 
 after leaving the public school than she would have received 
 without the trade-school training? 
 
 A Comparison of Eighth Grade Work. It is well known 
 that in Europe the specialization of schools is carried much 
 farther than has even been thought of here, or than seems pos- 
 sible or desirable in the future. To speak of the arithmetic of 
 these various forms of schools for foresters, builders, watch- 
 makers, barbers, and so on would therefore be unprofitable in 
 an article like the present. It will not, however, be out of place 
 to give an outline of the work in arithmetic in the eighth school 
 year in a girls' school in Munich, because this shows the ten- 
 dency at present, in one of the most progressive cities of Europe, 
 to have arithmetic touch the interests and needs of the people. 
 The work is as follows: 
 
 I. Simple domestic bookkeeping.
 
 The Work of the Eighth School Year 119 
 
 2. Calculation of the prices of foods, bought in large or in 
 small quantities, together with the question of discounts. 
 
 3. Cost of meals for the home. 
 
 4. Daily, monthly, and yearly supplies for the kitchen, to- 
 gether with the keeping of kitchen accounts. 
 
 5. Simple measurements as needed in the household. 
 
 6. Food values of different food stuffs as necessary for a 
 complete meal, with doubtless the application of ratio and per- 
 centage. 
 
 7. Cost of furnishing a kitchen. 
 
 8. Measurements of material, and the cost of buying, reno- 
 vating, and washing clothing made of various goods, as of 
 woolen or linen. 
 
 9. Relative cost of different systems of heating. 
 
 10. Relative cost of different systems of lighting. 
 
 11. Maintenance of the house, including questions of rent, 
 water, taxes, insurance, and interest on a mortgage. 
 
 12. Elementary commercial arithmetic, including such general 
 topics as percentage and its application in discount. 
 
 Such a course is highly to be commended. It meets the needs 
 of girls as we are not meeting them in our American schools. 
 Indeed it becomes a serious question if, in a subject like mathe- 
 matics, we are not bound to have separate classes for girls and 
 boys after the seventh grade and through our high school. 
 
 It may be well, also, to consider the work done in " Oberter- 
 tia " in a Prussian Gymnasium, corresponding in years to our 
 eighth grade. This correspondence is not exact in some respects, 
 because the Prussian school year is somewhat longer than ours, 
 but allowing three years before entering the lowest class (Sexta) 
 this becomes the eighth school year. There three hours a week 
 are allowed to mathematics, the arrangement allowing two to 
 geometry and one to algebra in one week, and two to algebra 
 and one to geometry the next week, and so on. The algebra 
 includes equations of the first degree with two unknown quan- 
 tities, and the geometry finishes the treatment of the circle, finding 
 the value of pi. The arithmetic work, as we call it, is practically 
 completed at the end of the sixth school year (in Quarta). 
 
 This gives an idea of what could be done in America if we 
 should care to set about it. As it is. the question is a fair one 
 if we would not be justified in materially reducing the arith-
 
 I2O The Teaching of Arithmetic 
 
 metic of Grade VII so as to include a considerable part of the 
 work of Grade VIII, thus allowing an elementary course in 
 algebra in that year. 
 
 Algebra. If algebra be introduced in Grade VIII, what is 
 the purpose and what should be its nature? Aside from the 
 general information thus given, and from the discipline that 
 comes from this or any other subject, there is a need that a few 
 years ago was hardly felt in this country. Boys are apt to leave 
 school after the work of Grade VIII is finished. They go into 
 the shops, into trade, into various occupations. Algebra was a 
 few years ago of no practical value to them, but to-day the 
 formula and the graph of a function are common features in our 
 trade journals. Here then is a suggestion to our schools. Why 
 should not elementary algebra be introduced by a study of form- 
 ulas, so that the simple algebraic expressions of our trade jour- 
 nals or our artisans' manuals can be read easily? Why should 
 we not introduce graphs of functions very early, not in compli- 
 cated forms but as used in the journals, the manuals, and the 
 workshop to-day? 
 
 The author's views as to details have been set forth in the 
 Wentworth-Smith Vocational Algebra, Boston, 1911, and in his 
 Algebra for Beginners, Boston, 1906. The whole question has 
 recently been discussed in a masterly little monograph written 
 by Mr. C. Godfrey, one of the present leaders in English edu- 
 cation, The Algebra Syllabus in the Secondary School, London, 
 1911. 
 
 As to the geometry, the work in mensuration in arithmetic 
 probably suffices for the present, although it is possible that we 
 may come to adopt the German plan of introducing the scientific 
 treatment of the subject into the elementary grades in the future. 
 
 Such are some of the problems in the teaching of arithmetic 
 to-day. Many are solved and many still await solution and are 
 occupying the attention of a large number of teachers. It is 
 with the hope of suggesting some of the larger problems that this 
 book is written, rather than with any desire to treat the minor 
 details that are sufficiently discussed in any good text-book. If 
 the work shall lead to sane experiment, to a conservative view of 
 the reforms to be accomplished, and to a sympathy with the 
 effort to improve the problems of arithmetic, it will have served 
 its purpose.
 
 
 SOUTHERN BRANCH 
 
 UNIVERSITY OF CALIFORNIA 
 LIBRARY 
 
 LOS ANGELES. CALIF,