THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 SOI 
 
 UNIVER rORNlA 
 
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 LIF.
 
 .The Teaching of Arithmetic 
 
 John C. .Stone
 
 
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 PREFACE 
 
 THIS book is a discussion of the aims and purposes of a 
 course in arithmetic and of the methods of presenting each 
 topic that should find a place in our elementary schools. 
 /-> Hence it is essentially a book for teachers, supervisors, and 
 those preparing to teach, and thus meets the needs of a text- 
 ; book for normal schools and teachers' reading circles. 
 
 The successful method of teaching any topic, however, 
 
 p depends upon a clear knowledge of the fundamental prin- 
 
 ^ ciples upon which the topic is based. Hence in the dis- 
 
 J^~ cussion of each topic, the fundamental principles involved 
 
 are fuHy presented in connection with the method of the 
 
 Classroom presentation of the topic ; in fact, they form a 
 
 very essential element of the so-called method of teaching. 
 
 Several of the chapters are summaries of lectures given 
 
 before teachers' associations and institutes and it has been 
 
 ^ the requests from such audiences for the printed lectures that 
 
 r^ has encouraged the author to bring them all together in 
 
 l<3 this form in the hope that they will have a wider influence 
 
 upon the teaching of arithmetic. 
 
 JOHN C. STONE. 
 
 April, 1918. 
 
 iii
 
 CONTENTS 
 
 PAOB 
 
 j* GENERAL PRINCIPLES AND SUGGESTIONS .... 1 
 
 - THE AIM OF A COURSE IN ARITHMETIC .... 8 
 
 COUNTING . . . 25, 
 
 THE PRIMARY FACTS OF ADDITION: WRITTEN ADDITION . 30 
 
 THE PRIMARY FACTS AND PROCESSES OF SUBTRACTION . 46 
 
 THE PRIMARY FACTS AND PROCESSES OF MULTIPLICATION . 52 
 
 THE PRIMARY FACTS AND PROCESSES OF DIVISION . . J61_ 
 
 _.,'.-THE USE OF GAMES IN -NUMBER WORK . . . .70 
 
 COMMON FRACTIONS . 95 
 
 DECIMAL FRACTIONS 117 
 
 PERCENTAGE AND ITS APPLICATIONS . . . . . 123 
 
 DENOMINATE NUMBERS AND MEASUREMENTS . . . 149 
 THE PURPOSES AND NATURE OF PROBLEMS . . . .163 
 
 THE ANALYSIS AND SOLUTION OF PROBLEMS . . . 174 
 
 PLANNING THE LESSON 191 
 
 THE COURSE OF STUDY IN ARITHMETIC .... 206 
 
 MEASURING RESULTS 221
 
 THE TEACHING OF ARITHMETIC 
 
 CHAPTER I 
 GENERAL PRINCIPLES AND SUGGESTIONS 
 
 ECONOMY AND EFFICIENCY 
 
 THE most important problems in the teaching of arith- 
 metic are those of economy and efficiency. How ; to put 
 the child in possession of the essential facts of the subject, 
 how to develop habits that lead to the economical and 
 efficient use of these facts in the real situations that arise 
 in everyday life, and how to accomplish this with a 
 minimum of waste are the problems that confront every 
 thoughtful teacher. 
 
 NUMBER WORK BEGUN Too EARLY 
 
 Pestalozzi, in Europe, and Colburn, in our own country, 
 are recognized as the leaders of the movement to bring 
 instruction in arithmetic down into the very lowest 
 grades and within the grasp of children. As an excuse 
 for this, Colburn says, in the preface of his First Lessons : 
 " As soon as children have the idea of more or less, and 
 the names of a few of the first numbers, they are able to 
 make small calculations. And this we see them do every 
 day about their playthings, and about the little affairs 
 
 1
 
 2 THE TEACHING OF ARITHMETIC 
 
 which they are called upon to attend to. * * * To 
 succeed in this (the science of numbers), however, it is 
 necessary rather to furnish occasions for them to exercise 
 their own skill in performing examples, rather than to give 
 them rules. * * * By following this mode, and making 
 the examples gradually increase in difficulty, experience 
 proves that at an early age children may be taught a 
 great variety of most useful combinations of numbers." 
 All observers of children will agree with Colburn that they 
 are able to make simple calculations about their play- 
 things and about their own little affairs. The work, 
 however, should stop here until wider experiences and 
 further needs demand greater knowledge of numbers. 
 
 iln the past we have begun number work entirely too 
 early and have forced adult applications of number be- 
 fore the needs, interests, and experiences of the child 
 Vwere ready for them. We have tried to force facts and 
 processes not needed in the natural activities of child- 
 hood, either in or out of school. Thus we have violated 
 every principle of economy in teaching in not finding the 
 child's needs and interests and teaching the right thing 
 at the right time. 
 
 THE CHILD'S NEED OF NUMBER 
 
 The child has certain needs of number during the first 
 year of his school life that should be met. But, in general, 
 these needs are confined to counting by ones, tens, and 
 fives to one hundred ; to reading numbers to 100 in order 
 to find a page in his book, or perhaps to 1000 in order to 
 find a house number if he lives in a city; to the few
 
 GENERAL PRINCIPLES AND SUGGESTIONS 3 
 
 simple combinations that he may need in his own little 
 affairs ; to the meaning of halves, thirds, and fourths of 
 single objects; to the Roman numerals to XII, in order 
 to tell time ; and to those common units of measure that 
 are needed in his home or school life. He has no natural 
 need of any of the written processes yet, and, in fact, he 
 makes figures with such conscious effort and knows 
 automatically so few facts that to give written work 
 before the third grade is to establish habits of computa- 
 tion that must be overcome later. 
 
 EDUCATIONAL PRINCIPLES INVOLVED 
 
 The economic mastery of the subject of arithmetic 
 demands that the teacher observe the principles that 
 knowledge to be real must be founded upon the actual 
 experiences of the individual learner; that knowledge to 
 be retained must be given an opportunity for use; and 
 that a necessary condition for learning is that the process 
 be self-actuated through motive or interest. Hence, the 
 subject must be made to minister to the child's needs 
 and kept within his actual experiences. It is when we 
 select material from the world of adults, dealing with 
 matters with which children have had no experiences, 
 and in which they have no interest, that we are asking 
 the impossible of children. 
 
 MORE DRILL WORK NEEDED 
 
 There are certain essential facts and processes that 
 must be made automatic. It is not sufficient that a child 
 knows that 7 and 8 are 15, but he must recall automatically
 
 4 THE TEACHING OF ARITHMETIC 
 
 the fact when seeing the figures or hearing the numbers 
 called, without consciously recalling the meaning of the 
 figures or the "how many" denoted by the numbers, 
 just as he sees letters and recalls words without recalling 
 the sounds represented by the letters. The fundamental 
 facts are clarified or rationalized by some concrete object 
 or by a problem of conditions very familiar to the child. 
 But when the meaning of the facts, processes, and terms 
 used has thus been made clear, next in order is drill with 
 abstract numbers until these facts or processes become 
 automatic. Sight drills must be used before mental 
 drills so as to furnish a mental image for the mental 
 work. Suppose there are placed upon the blackboard the 
 7 / 
 
 figures _8 . The child images this combination and 
 15 
 
 7 
 
 when later he sees 8_, he recalls 15. Then when he hears 
 "seven and eight," he has a mental picture of these forms 
 and can recall the s,um. Experiment shows that the 
 average person cannot grasp a group of objects larger 
 than four or five without analysis; that is, without 
 breaking them up into groups and adding the groups. 
 Then the idea of a number as large as 8 -f 7 = 15 is 
 not obtained through objects; that is, the fact is not 
 made automatic through seeing objects, but through see- 
 ing the figures or from hearing "eight and seven are 
 fifteen" until a mental image is formed. The objects, \ 
 then, are to make clear the meaning of a fact or process ; 
 but they do not help the memory to retain such a fact.
 
 GENERAL PRINCIPLES AND SUGGESTIONS 5 
 
 HABITUATION rs. RATIONALIZATION 
 
 The subject of arithmetic is taught very largely to-day 
 by inductive methods; that is, the child is led through 
 simple concrete illustrations and objective presentations 
 to discover his own facts, rules, and definitions ; but there 
 is danger that this phase of the work may receive undue 
 emphasis. It frequently happens that a teacher is able 
 to develop a subject very clearly and interestingly, and 
 yet her work may lack effectiveness, owing to her neglect 
 of drill work necessary to fix the facts and to give skill in ' 
 computation. 
 
 There should be a sharp discrimination between those 
 facts, the rationalization of which is of vital importance, 
 and those in which it makes but little, if any, real differ- 
 ence as far as the efficient use of the facts is concerned. 
 -Thus it is of vital importance that a pupil know the 
 meaning of addition and of such expressions as "five and 
 four are nine," etc. ; but it makes little, if any, difference 
 whether or not he knows why we " carry." The important 
 thing is that he has the proper habit of carrying. Like- 
 wise, in "long multiplication" he must have the habit 
 of putting the partial products where they belong ; but it 
 makes little, if any, difference, when first presented, 
 whether he knows why they are so placed. 
 
 When rationalization aids the memory in retaining a 
 fact, or when it renders the use of it more safe when 
 applied to problems, then rationalization should receive 
 proper emphasis; otherwise, it is of minor importance. 
 But in such cases as "carrying," "borrowing," and the
 
 6 THE TEACHING OF ARITHMETIC 
 
 correct placing of partial products, the proper habit is 
 the important thing and rationalization a very unim- 
 portant thing when the subject is first presented. On 
 the other hand, the rationalization of a rule in fractions 
 renders the use of it more safe and aids the memory in 
 retaining the "how." It is not uncommon to see gram- 
 mar school and high school pupils add f and f and get i 5 ^, 
 adding numerators and denominators. Proper rationaliza- 
 tion should have prevented such an error. 
 
 Likewise, all rules for mensuration are more easily 
 remembered if rationalized through an objective pres- 
 entation. 
 
 STANDARDS OF SPEED AND ACCURACY 
 
 In the past we have had no standards of skill in com- 
 putation. Each teacher has had her own standards. 
 Pupils have been marked and passed by these standards. 
 One teacher has emphasized the "how" to solve a prob- 
 lem, regardless of the time required to compute the result 
 or the accuracy of it. Others have emphasized merely 
 the power to compute. Recent tests show startling 
 variations in the work of pupils from schools reputed to 
 be of the very highest standing. Such tests as the 
 "Courtis tests," when given to enough pupils, may, by 
 taking a general average, show the average of what 
 schools are attaining by present methods; but they in 
 no way show what may be expected of pupils by properly 
 devised drills. From government statistics we may 
 find the average production of wheat or corn for a state 
 or the nation; but that does not show what should be
 
 GENERAL PRINCIPLES AND SUGGESTIONS 7 
 
 expected from improved methods of cultivation or the 
 most economical yield. If careful records are kept of 
 the progress made by some such methods of teaching 
 and system of drills as are suggested in this book, we 
 may ultimately know what should be expected of a pupil 
 in accuracy and rapidity in the various grades. If some 
 other method shows different results, we may thus judge 
 the efficiency of the method used and thus have a means 
 of judging the real value of various methods. At present, 
 however, a method is considered good or bad by a teacher 
 as it meets or fails to meet the particular whim of that 
 individual teacher.
 
 CHAPTER II 
 THE AIM OF A COURSE IN ARITHMETIC 
 
 IMPORTANCE or AN AIM 
 
 OUR methods of teaching, and the nature of the sub- 
 ject matter that we teach, depend very largely upon our 
 belief in what the subject may do for the learner; for, 
 if we believe that our pupils should obtain certain results 
 from our teaching, it naturally follows that we are going 
 to shape our instruction and select our material in the 
 way that seems to us best fitted to bring about those 
 results. 
 
 For example, if a teacher considers that the only pur- 
 pose of arithmetic is to train pupils to compute skillfully, 
 she will reduce her instruction to drills that will bring 
 about efficiency in computation. If problems are given 
 at all, they will be given merely in order to furnish an 
 occasion for computation, and their nature will not be 
 given serious attention. On the other hand, if she con- 
 siders that the purpose is "mental discipline," then she 
 will pay less attention to computation and to the me- 
 chanical side of the subject and spend a great part of the 
 time in developing and rationalizing the processes and 
 in selecting problems having just the proper complexity 
 to task the child's powers to discover what to do. The 
 
 8
 
 THE AIM OF A COURSE IN ARITHMETIC 9 
 
 problems will not be selected for any real purpose that 
 they can serve in the child's life, but they will be selected 
 to furnish a sort of mental gymnastic exercise. Or, she 
 may consider that the whole purpose of arithmetic is to 
 furnish a social insight into current business or industrial 
 practices. In this case, she will devote most of her time 
 to real problems that come up in life, regardless of the 
 mathematics employed, and spend much time in dis- 
 cussing the social phases of situations from which the 
 problems were made, disregarding drills and the accuracy 
 of the solutions. And thus we might go on with one 
 hypothetical case after another to show that a discussion 
 of the aims of a course in arithmetic becomes of funda- 
 mental importance in discussing methods of teaching. 
 
 TRADITIONAL VALUES OF THE STUDY OF ARITHMETIC 
 
 The values of arithmetic have varied with the devel- 
 opment of the race and have depended upon the civiliza- 
 tion of the people using the subject. The knowledge of 
 numbers must have been coeval with the human race. 
 With the notion of distinctness that this is separate 
 from that the notion of number must have had its 
 origin. One can scarcely carry on the simplest conver- 
 sation or make any communication with another without 
 a use of some knowledge of number. The savage on 
 returning from the hunt no doubt communicated to his 
 family or the tribe the amount of game he had seen or 
 killed, possibly by the use of his fingers or hands or by 
 pebbles or shells. To him, arithmetic was but a matter 
 of convenience. But when he began to barter skins or
 
 10 THE TEACHING OF ARITHMETIC 
 
 game for corn or trinkets and began to acquire property, 
 there grew up a necessity for a knowledge of the relation 
 of numbers a comparison of the number of one group 
 with that of another, or a comparison of their values. 
 His arithmetic now became a necessity it took on a 
 utilitarian value. 
 
 As the race advanced in civilization and schools for 
 the training of youths sprang up, arithmetic took on a 
 new meaning. It was given a disciplinary value as well 
 as a utilitarian value. But, just as our real need of 
 number has grown up slowly and naturally and has thus 
 gradually changed the utilitarian values of arithmetic, so 
 has our knowledge of the laws of mental growth changed 
 our thoughts about the disciplinary value of the subject. 
 
 Arithmetic was once taught to make one quick-witted 
 and enable him to see a point quickly and thus aid him 
 in argumentation. This concept of the value of arith- 
 metic led naturally to catch-questions and puzzles. 
 
 Later there grew up under the teaching of the so- 
 called "faculty" psychology the doctrine of "formal 
 discipline." This was a belief in the transfer value of 
 abilities developed by one study to all other lines of 
 thought, whether related or not. This led to studying 
 a subject in order to train the reason, or the memory, 
 or the imagination, etc. This doctrine, however, did 
 not result in any great change in the nature of the prob- 
 lems formerly given, but it merely gave a new reason for 
 retaining the type of problems already in use. But it 
 did make a great change in the methods of teaching. 
 There was a change from the old dogmatic rule method
 
 THE AIM OF A COURSE IN ARITHMETIC 11 
 
 to the demonstrated rule method, and later to the heuristic 
 or development method of presenting the facts and 
 processes. While a belief in the doctrine of "formal 
 discipline" undoubtedly resulted in improved methods 
 of teaching over those methods that preceded it, it never- 
 theless placed undue emphasis upon the rationalization 
 of the facts and processes, particularly in the lower grades, 
 and it caused many obsolete topics and processes to be 
 retained in the schools long after they could be justified 
 upon practical or social grounds. 
 
 DOCTRINE OF FORMAL DISCIPLINE DENIED 
 
 A few years ago psychologists began to doubt the doc- 
 trine of formal discipline and to deny the transfer of abilities 
 gained in one field of work to unrelated fields. By many 
 teachers this new attitude of psychologists was entirely 
 misunderstood. Much careless teaching and indifference 
 toward the subject of arithmetic resulted. A teacher 
 recently said to the writer that her class in arithmetic 
 was very poor, and then added, "But if they are poor in 
 anything, I suppose that it had better be in arithmetic 
 than in anything else." Then she asked, "Hasn't arith- 
 metic been taught in the past for nothing but discipline, 
 and now hasn't it been found that there is no such thing ? " 
 It is just such an attitude as this, resulting from a mis- 
 understanding of recent educational discussion, that has 
 seriously affected the teaching of arithmetic. 
 
 Our modern psychology views our mental life as made 
 up of specialized^abilitles .nH j^jijjjLgpjiargitp ffl.rnlt.ies, 
 each trained by some special kind of study. The result
 
 12 THE TEACHING OF ARITHMETIC 
 
 is that we now train for habits of accuracy, of attention, 
 of analysis, of comparison^ of judgment, etc., each in its 
 own field. It follows, then, that each topic taught must 
 have some bearing upon the life of the individual as a 
 sort of socializing factor in order to justify its place in 
 the curriculum. 
 
 THE PRACTICAL VALUES OF ARITHMETIC 
 
 As would naturally appear from the discussion of the 
 last topic, the emphasis in arithmetic is now placed upon 
 the practical values of the subject. That does not mean 
 that all the older aims, as discipline, pleasure, culture, 
 and preparatory values, are now wholly ignored ; but it 
 does mean that the emphasis has shifted from these to 
 the practical in the broad sense of that term. But in so 
 doing, whatever claims of recognition any of the older 
 values have may be taken care of when teaching the 
 subject from the practical standpoint. 
 
 But, like the term "formal discipline," the term "prac- 
 tical values" is very much misunderstood and misused. 
 Some would view the practical side as relating to nothing 
 but the ability to do the computing necessary in one's 
 vocation. Others think that it relates to fitting one for 
 some specialized vocation, usually a clerkship in some 
 commercial pursuit. And there are still others that 
 consider the practical as referring to any real problem 
 met in any of the world's varied activities. Many of 
 the problems that have found their way into the schools 
 in recent years under the guise of "practical" are just as 
 impractical and far more ridiculous than some of the
 
 THE AIM OF A COURSE IN ARITHMETIC 13 
 
 problems of the past which were given for recreation 
 and mental gymnastics. 
 
 But, by a more careful analysis of the practical values, 
 from the standpoint of the general user of arithmetic hi 
 society rather than the needs of the man in some specialized 
 vocation, it is easily seen that the following things are 
 practical to the average person in any walk of life : 
 
 1. Efficiency in computation. 
 
 2. A social insight into business and industrial practices 
 that will enable one to interpret references to such practices 
 met in general reading or in social and business intercourse. 
 
 3. Power to express and to interpret the numerical 
 expressions of the quantitative relations that come within 
 our experiences. 
 
 4. The habit of seeing such relations, particularly those 
 that are vital to our welfare. 
 
 While to a degree these four abilities are of importance 
 to all, their values vary with the different users of the 
 subject. 
 
 EFFICIENCY IN COMPUTATION 
 
 Computation is not an end in itself, but a means to an 
 end. Just as ability to form the letters of the alphabet 
 is necessary if we are to express our thoughts in writing, 
 but useless unless we have thoughts to express, so ability 
 to compute is of no value unless we know what process 
 to apply to a problem, for there is no relation between 
 ability to compute and ability to reason out what process 
 to apply to a given problem. This is sometimes mis- 
 understood, and drill hi pure computation is overem-
 
 14 THE TEACHING OF ARITHMETIC 
 
 phasized in the thought that we are thus teaching arith- 
 metic. 
 
 Since mechanical ability, like computation, is more 
 easily measured than other results of teaching, this phase 
 is likely to be overemphasized, particularly in those 
 schools where "standard tests" are too closely relied upon 
 in measuring the work of the school. However, there is 
 need for accuracy and reasonable speed in using the four 
 fundamental processes in whole numbers, fractions, and 
 decimals ; skill in mental calculations ; and ability to see 
 approximate results without a pencil. 
 
 The absolute control of the decimal point must be 
 developed. The lack of such a control is the cause of 
 numerous errors. Recently I saw a class attempt to 
 find the cost of 2.9 cubic feet of gas which they had used 
 hi an experiment, knowing that the price was $.90 per 
 1000 cubic feet. The class was divided into about three 
 equal groups giving 26.1^f, 2.61 ff, and .261 respectively 
 as the answer. 
 
 Efficiency in computation, however, requires more 
 than a mere automatic control of number facts and 
 processes. Through a knowledge of the fundamental 
 principles involved, pupils of the upper grammar grades 
 should have the power to see relations that will save 
 figures and even whole processes. Thus, if one is to find 
 what per cent a gain of $2,346,275 is of the total sales 
 of $86,724,342, he should be able to see that not half of 
 the figures given need to be used in the calculation. Or, 
 if he is to find the interest at 6 per cent by multiplying 
 the principal by the number of .days, pointing off three
 
 THE AIM OF A COURSE IN ARITHMETIC 15 
 
 more decimal places, and dividing by six a very 
 common method he should see that, if the number of 
 days is a multiple of six, the division may be saved and 
 the result found by multiplying i>y one sixth of the num- 
 ber of days, and pointing off three more places. 
 
 SOCIAL INSIGHT INTO BUSINESS AND INDUSTRIAL 
 PRACTICES 
 
 The proper study of arithmetic should develop an ability 
 to interpret and comprehend the quantitative problems 
 that arise in the world's varied activities. Whether one ' 
 is to be actively engaged in a commercial or industrial 
 vocation or not, in his daily reading and in intercourse 
 with others he will meet references to many of the prob- 
 lems of commercial and industrial activities, and he must 
 be able to interpret them if he is to take an important 
 place in society. So the problems of arithmetic are now 
 taking on a much more real nature than those of the past 
 did. In a discussion of borrowing, loaning, and invest- 
 ing money, of taxes, insurance, etc., the indirect prob- 
 lems of the past are fast giving way to those problems 
 that one actually needs to ask himself in any of these 
 phases of social life. 
 
 For some students, the power to interpret and compre- 
 hend such problems has no further practical value than 
 the enrichment of their understanding of the life about 
 them. But to a vast majority of the students, there is 
 a more direct vocational value; for, in our present com- 
 mercial and industrial competition and struggle, other 
 things being equal, the one having the keenest sense of
 
 16 THE TEACHING OP ARITHMETIC 
 
 number relations has by far the greatest chance of 
 success. 
 
 POWER TO EXPRESS AND INTERPRET QUANTITATIVE 
 RELATIONS 
 
 The relations between numbers representing magnitudes 
 are either differences or ratios. Of the two means of 
 expressing relations, the ratio, expressed as a fraction 
 or a per cent, gives the most definite or vital idea or 
 picture. For example, to say that in 1915 the world's 
 production of cotton was 14,126,500 bales, of which the 
 United States produced all but 2,934,700 bales, creates 
 no mental picture and means but little to us. But, to 
 give a more definite picture, we say that the United States 
 produced about four fifths of the world's crop of cotton 
 in 1915. 
 
 The task of developing power to see and to interpret 
 quantitative relations is a most difficult one. It can only 
 be done by making a constant application to the quantita- 
 tive aspects of life, ^ While many feel that we have lost 
 a means of developing such power by discarding the kind 
 of problems found in the older types of "mental" and 
 "intellectual" arithmetic, it seems reasonable that a 
 closer correlation of arithmetic with the actual quantitative 
 situations that arise in other school activities and in the 
 everyday life of the child will come more nearly develop- 
 ing an ability that is of real use in interpreting the quan- 
 titative phases of life and in doing the world's work than 
 can possibly come through artificial problems made up 
 merely for mental gymnastics.
 
 THE AIM OF A COURSE IN ARITHMETIC 17 
 
 ELIMINATION OF TOPICS 
 
 A discussion of the values of a course in arithmetic 
 naturally leads to a discussion of what is worth while in 
 the course and what may be eliminated as not con- 
 tributing to any of the purposes for which arithmetic 
 is taught. While there can be no definite statement 
 made that will meet every case, since much depends upon 
 the individual user of arithmetic, but few, if any, in the 
 ordinary walks of life will find much use for the following : 
 
 1. The greatest common divisor. The only application 
 of the greatest common divisor that is found in elementary 
 arithmetic is its use in reducing a fraction to its lowest 
 terms. But, if the greatest common divisor is to be found 
 by the factoring method, it will be more economical to divide 
 out the common factors as they are found than to take 
 their product and then divide both terms by this product. 
 
 2. Addition and subtraction of fractions with large or 
 unusual denominators. On account of the nearly uni- 
 versal use of decimals, the addition and subtraction of 
 fractions, except the very simplest ones, rarely ever occurs 
 in practical work. Even those fractions and mixed 
 numbers that are added and subtracted usually arise 
 from expressing some number as 5 bu. 3 pk., 3 Ib. 7 oz., 
 etc., in terms of a single unit as 5f bu., 3^ Ib., etc. 
 
 The fractions ' with larger terms, used to express a 
 ratio, are never added. Such ratios, however, are usually 
 expressed decimally. 
 
 3. The least common multiple. In elementary arith- 
 metic, the only application of the least common multiple
 
 18 THE TEACHING OF ARITHMETIC 
 
 of two or more numbers is its use in reducing fractions 
 to common denominators. So, if only the fractions found 
 in the practical problems of everyday life are given, the 
 least common multiple is not needed, for the common 
 denominators can be found more quickly by inspection. 
 
 4. The more complex forms of complex fractions. It 
 has become common among writers in recent years to 
 recommend unreservedly the elimination of all complex 
 fractions. However, a careful examination into the 
 nature and use of the complex fraction causes a modifica- 
 tion of such a recommendation. The relations wanted 
 among fractions or mixed numbers are quite as common 
 as those among whole numbers; and a complex fraction 
 is but a ratio or an expressed division between two frac- 
 tions or mixed numbers or a whole number and a fraction 
 or a mixed number. Hence, the following ratios, ex- 
 pressed as fractions, are as apt to occur as those between 
 whole numbers, namely : 
 
 i 5i 1 2i M 
 
 ? 8i' 2*' 3i' 5' f 
 
 It may be urged that, since these are but special forms 
 of expressing a division, why call them fractions, and 
 why not simplify them as in ordinary division of fractions. 
 But perhaps by writing them in this form and applying 
 the principle that Multiplying both terms of a fraction by 
 the same number does not alter the value of the fraction is 
 the most economical way of simplifying such ratios. 
 Thus, at sight, the above "complex fractions" become: 
 
 & ii, &, f, TS, and ^.
 
 THE AIM OF A COURSE IN ARITHMETIC 19 
 
 5. Obsolete tables and those used in specialized vocations. 
 Under these tables would be included apothecaries' 
 weight, troy weight, surveyors' measure, folding of paper, 
 foreign money, and also the gill, furlong, rood, dram, 
 quarter in avoirdupois weight, etc. 
 
 6. Impractical reduction in denominate numbers. 
 Reductions of more than one or two steps rarely, if ever, 
 occur. One might have occasion to reduce feet to inches 
 or inches to feet, but there is not apt to arise a need of 
 changing miles to inches or inches to miles. 
 
 7. Addition, subtraction, multiplication, and division 
 of compound denominate numbers. In practical prob- 
 lems, a measure is expressed as a whole number, a mixed 
 number, or a decimal in one unit instead of as a compound 
 denominate number, before it is added, subtracted, mul- 
 tiplied, or divided. Hence, these topics need not receive 
 any attention in the schoolroom. 
 
 8. The present type of inverse problems in fractions and 
 percentage. Most of the inverse or indirect problems 
 given in fractions or percentage could not meet a real 
 need of any one. They are a sort of "hide-and-go-seek" 
 kind given for "exercises in analysis." Thus, in the prob- 
 lem : "If Mr. A sold a suit for $24, thereby making 20 
 per cent of the cost, find the cost," the answer cannot 
 meet a real need of Mr. A, for he must have known the 
 cost before he could have furnished the data of the 
 problem. 
 
 It may occur, however, that Mr. A has a suit of which 
 he knows the cost and wishes to know a selling price that 
 will iyield a certain per cent of itself. This would involve
 
 20 THE TEACHING OF ARITHMETIC 
 
 the indirect type of problem. Usually, however, com- 
 petition determines the price. 
 
 There are perhaps sufficient reasons for bringing up 
 the inverse or indirect type of problems in a high school 
 course; but there is much more profitable work for the 
 grammar school pupil, at least before the eighth grade, 
 than even the real applications of the indirect problem. 
 
 9. The various short methods of finding interest. The 
 one who has to compute interest often, as a clerk in a 
 bank, makes use of a book of tables. One who computes 
 interest but infrequently is apt to forget any of the 
 various short methods; but, if he knows the meaning of 
 interest, he cannot forget the general method, that is, 
 that the principal multiplied by the rate, and that product 
 multiplied by the time expressed as a fraction of a year, 
 gives the interest. 
 
 10. All inverse problems of interest. The inverse or 
 indirect problems of interest do not occur in practical 
 life, and do not make more clear the meaning of interest, 
 hence they can be justified only "for analysis." But 
 that justification only is not sufficient, for there are plenty 
 of problems that furnish sufficient drill hi analysis while 
 also serving a more important purpose. 
 
 11. Partial payments. The subject is not of sufficient 
 importance to justify it from any of the purposes of 
 arithmetic. 
 
 12. Annual interest. This cannot be justified from 
 the standpoint either of practical uses or of social insight. 
 
 13. Undue emphasis upon the discounting of interest- 
 bearing notes. Beyond knowing the meaning of Bank
 
 THE AIM OF A COURSE IN ARITHMETIC 21 
 
 Discount that is, that when a bank collects the interest 
 in advance, this interest is called Bank Discount drill 
 in finding discount is not worth while. 
 
 14. True discount. As a business custom, this is obso- 
 lete. Yet, it is given in many of the textbooks still in use. 
 
 15. Partnership. This does not lead to an insight 
 into business practices, for it is obsolete except in the 
 smaller businesses; and the problems usually given lead 
 to an erroneous notion of the division of profits even in 
 a partnership business. 
 
 16. Proportion as a general method of solving problems. 
 
 Other methods of analysis and solution are not only 
 shorter but develop greater power to see and express 
 quantitative relations. 
 
 17. Foreign and domestic exchange. Except the knowl- 
 edge of how indebtedness may be canceled by check or 
 draft instead of by an actual transfer of money, the sub- 
 ject is of no real practical value to the general student of 
 arithmetic. 
 
 18. The measurement of uncommon areas and volumes. 
 
 The measurement of such areas and volumes as those 
 of trapezoids, frustums, wedges, spheres, etc., are so un- 
 common in practical everyday life that the method of 
 finding them is soon forgotten. 
 
 19. Square root and the Pythagorean theorem. One of 
 these topics is usually given as an excuse for giving the 
 other. jWhile they have but little practical value to the 
 general user of arithmetic, reference to them is yet so 
 common in general reading that a mere acquaintance 
 with what they mean may justify a brief discussion of
 
 22 THE TEACHING OF ARITHMETIC 
 
 them. It is better, however, to delay these topics until 
 the high school course. 
 
 20. The metric system. Unless the metric system is 
 going to be needed in science, its teaching cannot be 
 justified. It is sometimes urged as a means of bringing 
 about a general use of the system, but it is difficult to 
 show its advantages in an appealing way to grammar 
 school pupils. The teaching of the subject should be 
 delayed, then, until it is met in science. 
 
 THE ESSENTIAL WORK IN ARITHMETIC 
 
 There are those who seem inclined to think that the 
 elimination of these twenty topics means a greatly re- 
 duced time to be given to arithmetic, but this is not the 
 case. On every hand, one hears from the business world 
 the complaint that grammar school, high school, and 
 college students know but little arithmetic ; that is, that 
 the arithmetic of the schools does not "function" in 
 actual practice. On every hand teachers are either 
 asking for or suggesting a remedy. No panacea has yet 
 been given. But it seems that the following suggestions 
 may help bring about better results : 
 
 1. Develop better habits of accuracy. From the very 
 beginning of the work, pupils should be given methods of 
 checking their work, and all work should be carefully 
 checked ; also all work handed in should be 100 per cent 
 accurate in computation. The same standard of speed 
 cannot be expected of all pupils, but all can work at a 
 piece of computation until they are assured of its ac- 
 curacy, just as an accountant does.
 
 THE AIM OF A COURSE IN ARITHMETIC 23 
 
 2. Develop reasonable speed. There is some danger 
 that too great an effort is made to get a fixed speed. In 
 fact, too conscious haste upon the part of the pupil may 
 lead to injurious results. The purpose of speed is largely 
 to save time, so we must ask if the time saved is worth 
 the effort to attain it. 
 
 3. Encourage short methods. Pupils should under- 
 stand fundamental principles that will lead to a saving 
 of figures and processes. This is a slow and gradual 
 growth that comes about with greater maturity and a 
 greater insight into fundamental principles and numerical 
 relations. 
 
 4. Emphasize oral work. In life, one uses much more 
 oral than written arithmetic. Pupils should learn to 
 work without a pencil and to give either exact answers or 
 very close approximations to all types of problems that 
 arise in everyday life. 
 
 5. Habituate rather than rationalize those facts and 
 processes of frequent recurrence. When a fact or process 
 is used without variation and arises very frequently in 
 the life of the average student of arithmetic, drill should 
 continue until the recalling of the fact or process becomes 
 a habit. But the reason for the fact or process may well 
 be delayed or omitted entirely. Thus, the primary facts 
 and the fundamental processes with whole numbers must 
 be made automatic, but to discuss why we "carry" or 
 why we "borrow" is unnecessary when the work is first 
 taken up. 
 
 6. Avoid stereotyped solutions. The solution of a 
 problem should never be an act of memory but always
 
 24 THE TEACHING OF ARITHMETIC 
 
 a process of reasoning. Otherwise, the pupil is not 
 gaining power to see relations but merely developing skill 
 in juggling figures. 
 
 7. Use the quantitative situations that arise in all other 
 activities in or out of school. Every process as learned 
 should find an immediate use in answering some real 
 inquiry of the pupil about the quantitative relations that 
 are vital to his interests.
 
 CHAPTER III 
 COUNTING 
 
 ROTE AND RATIONAL COUNTING 
 
 THE idea of number is coeval with the human race. 
 With the ability to distinguish differentia that this 
 is not that originated the concept of number. But 
 the first concept is the "how many" idea of number, 
 the "how much." The basis for the number facts is 
 counting. Hence, the first step in the teaching of num- 
 bers is counting, first by rote, then rationally. In other 
 words, the child first gets the order or sequence of the 
 numbers and is able to say, "one, two, three, four, five, 
 six," etc., without being able to recognize any of these 
 numbers of things. Then from objects he learns the 
 meaning of these names; that is, the counting becomes 
 rational. 
 
 When pupils have trouble in getting the order of the 
 numbers fixed, or when they have developed a wrong 
 order, as "one, two, three, five, eight, nine," etc., rote 
 counting is often taught or corrected through the use of 
 little rhymes, as : 
 
 One, two, three, four, five, 
 
 I caught a hare alive ; 
 Six, seven, eight, nine, ten, 
 
 I let hun go again. 
 25
 
 26 THE TEACHING OF ARITHMETIC 
 
 And, 
 
 One, two, Five, six, 
 
 Buckle my shoe ; Pick up sticks ; 
 
 Three, four, Seven, eight, 
 
 Shut the door ; Lay them straight ; 
 
 Nine, ten, 
 
 I see a fat hen. 
 Or, 
 
 One little, two little, three little Indians, 
 Four little, five little, six little Indians, 
 Seven little, eight little, nine little Indians, 
 Ten little Indian boys. 
 
 When a child can call the names of the numbers in 
 order, the next step is to teach rational counting ; that is, 
 through counting objects about the room, the child must 
 see the meaning of the names he has learned. 
 
 In having a pupil count objects, the teacher should 
 take care that he doesn't get the idea that the second ob- 
 ject is "two," the third one "three," etc. This is avoided 
 by presenting groups and raising the question of "how 
 many" and then finding it by counting. Thus, present- 
 ing three things as I I I ask, "How many?" The 
 child counts, "One, two, three," and says, "There are 
 three of them." Presenting four things, ask, "How 
 many ? " " one, two, three, four, there are four of 
 them," etc. Make use of objects in the room. Ask how 
 many windows, how many children, how many pictures, 
 how many books, etc., until the children can apply their
 
 COUNTING 
 
 27 
 
 rote counting to finding the "how many" of the things 
 about them. Use objects in which the children are inter- 
 ested, not those made for the occasion, as splints, beads, etc. 
 
 THE NAMES OF THE FIGURES 
 
 Before teaching any of the primary combinations, the 
 pupil should recognize the names and meanings of the 
 figures. Thus, he must associate the figure with the 
 name and with the objects, as I I I I I, five, 5; 
 I ! I, six, 6; etc. The teacher will devise many interesting 
 ways of presenting this so as to fix these relations. One 
 very enjoyable way to fix them is through a little " match- 
 ing game." To play this game, take thirty cards ten 
 of them containing the figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
 (one figure on each card) ; ten others containing the 
 names, one, two, three, etc.; and ten others containing 
 objects as dots arranged as on domino cards. These 
 cards are distributed among the pupils and, as a pupil 
 is named, he runs to the front of the room and shows his 
 card. The other pupils "match" the card, by coming 
 up and standing in line. Thus, if a pupil shows "six," 
 it is matched by 6 and III, shown below. 
 
 In this and other ways, a pupil is soon able to associate 
 the figures with the names and with the things which they 
 represent.
 
 28 
 
 THE TEACHING OF ARITHMETIC 
 
 3 
 
 13 
 
 4 
 
 14 
 
 5 
 
 15 
 
 6 
 
 16 
 
 7 
 
 17 
 
 8 
 
 18 
 
 9 
 
 19 
 
 COUNTING TO TWENTY AND TO ONE HUNDRED 
 
 Since our number system is based upon a scale of ten, 
 counting by ones to ten is the first unit of instruction in 
 counting. This is followed by counting to 
 twenty. 
 
 Counting from 10 to 20 may be accom- 
 plished at the same time that the written 
 forms are given. In this way, the child 
 sees the similarity between 3 and 13, 4 and 
 14, etc. At the same time, he gets the 
 sound "three," "thirteen"; "four," "four- 
 teen" ; etc., and sees that by adding "teen" 
 the order is like the order he already knows. 
 
 Eleven and twelve must be taught separately, and 
 there are no sound-aids to help the memory as in "teens." 
 
 The next step is counting by tens to one hundred. 
 This is easily accomplished by calling the child's attention 
 to the similarity in sound between "two" 
 and "twenty"; "three" and "thirty"; 
 "four" and "forty"; "five" and "fifty"; 
 etc. Thus he sees that it is very much like 
 counting by ones to ten if the suffix ty is 
 annexed to each of the number names 
 from two to nine. This, too, may be ac- 
 complished by using the written forms as 
 in the "teens." 
 
 The third step is the filling of each dec- 
 ade with the one, two, three, etc., as twenty-one, twenty- 
 two, twenty-three, etc. 
 
 2 
 
 20 
 
 3 
 
 30 
 
 4 
 
 40 
 
 5 
 
 50 
 
 6 
 
 60 
 
 7 
 
 70 
 
 8 
 
 80 
 
 9 
 
 90
 
 COUNTING 29 
 
 READING AND WRITING NUMBERS 
 
 When a child is first taught to read and write num- 
 bers, the place-value feature of our system of writing 
 numbers should not be emphasized even if it is taught 
 at all. He should be taught to read and write numbers 
 as he reads and writes words. Thus, while writing 36, 
 say, "this is thirty-six," etc. The child thus sees that 
 the 3 suggests the thirty and the 6 the six, and easily reads 
 any two-figured number. In the same way he is taught 
 to read hundreds, thousands, etc. The real significance 
 of "place value" should be reserved until the child is 
 more mature and until such knowledge is needed hi order 
 to understand the why of the written processes.
 
 CHAPTER IV 
 
 THE PRIMARY FACTS OF ADDITION: WRITTEN 
 ADDITION 
 
 ASIDE from a few primary facts that have been picked 
 up incidentally during the work in counting in the first 
 year, the work of teaching the primary facts seems to be 
 more economically done by teaching but one process at' 
 a time, rather than by teaching all processes about a given 
 number at once, as in the Grube plan. While teachers 
 differ as to the best order in which to teach the primary 
 facts of addition, whether to teach the tables of ones, 
 twos, threes, etc., or to teach all of the addition facts 
 about a certain number, the order is much less important 
 than the nature of the drill work that is needed to make 
 the facts automatic. However, the order is important 
 enough to merit serious thought and discussion. 
 
 THE FIRST GROUP OF PRIMARY FACTS 
 
 There are forty-five possible combinations of two one- 
 figured numbers. These are called "the forty-five pri- 
 mary facts of addition." 
 
 The first group of these facts to be thoroughly fixed, 
 whatever order is used, consists of the twenty-five com- 
 binations whose sums do not exceed ten. This is not 
 
 30
 
 THE PRIMARY FACTS OF ADDITION 31 
 
 only the logical and psychological division of the facts, 
 but an economic one in the matter of learning them. 
 The most economic order seems to be to follow the tables 
 in the ones and twos at least; for, when a child can 
 count rationally by ones, he really knows nine of the 
 first twenty-five facts; that is, he sees that to add one 
 is to call the next number in the order of counting. How- 
 ever, some drill in calling these sums when seeing the 
 written figures is necessary. 
 
 When any new sum is learned, the pupil should see 
 it written. This written form should be placed before 
 him for days until it leaves a mental picture. Thus, when 
 the ones are presented, the pupil should see : 
 
 12345678 9 
 
 !!!!!!!!_L 
 
 2"34~5678910 
 
 This should leave such a mental image that when seeing 
 
 123456789 
 
 1III1IIII 
 
 he can automatically name, or write, the sum. Each 
 combination learned must be recognized instantly when 
 written in either of the two possible forms shown below : 
 
 31 41 51 61 71 
 1 3 1 4 1 5 1 6 1 7 
 
 In order to see the meaning of addition, the pupil 
 should find the first group of twenty-five facts, through 
 counting objects. Thus, in the table of twos, 2+2,
 
 32 THE TEACHING OF ARITHMETIC 
 
 3+2, 4+2, etc., to find 4+2, he should take four 
 objects, then picking up two others, count /'four, five, 
 
 4 
 
 six" and write 2_. In this way he should discover for 
 
 6 
 himself the facts of each table up to 5+5. 
 
 THE USE OF OBJECTS 
 
 Objects are used to fix the meaning of addition, not to 
 fix the facts themselves; that is, through this objective 
 discovery of the facts, the pupil sees clearly what is meant 
 by "three and four are seven," and is able to use these 
 facts in applications to problems within the range of his 
 experiences. When these facts have thus been rational- 
 ized, the objects have served their purpose and their use 
 should be discarded. In the drills that follow, if a pupil 
 miscalls a fact, the right result should be shown or told 
 him at once. Nothing is gained by having him find the 
 result again through counting. In fact, to have him do 
 so is to establish a "counting habit" that must be broken 
 up later. If 6+3 is given incorrectly, 6+3 = 9 should 
 be shown at once. 
 
 THE ORDER AND NATURE OF DRILLS 
 
 The drills must build up a mental imagery in the 
 child's mind. Hence, the first drills should be sight 
 drills; that is, drills in which the child sees the figures. 
 In these drills the figures should be written in column 
 form as ^iey are to appear later in written work. 
 
 These sight drills form a mental imagery so that in
 
 THE PRIMARY FACTS OF ADDITION 33 
 
 pure mental drills, drills in which the figures are not seen, 
 the child calls results from this jnental picture. There 
 are some children, however, who learn the combinations 
 more easily by hearing and repeating the facts as "five 
 and three are eight"; hence, this kind of drill should 
 also find a place, but not as important a place as the sight 
 drill. 
 
 THE IMPORTANCE AND EXTENT OF DRILLS 
 
 The pupil must have automatic control of all the 
 primary facts and all the related facts needed in the 
 fundamental processes. It is not sufficient that he knows 
 the meaning of "four and five" and can find it through 
 counting, or even recall it after a moment's time ; but he 
 must know it automatically. Just as he calls the words 
 in a reading lesson without consciously recalling the 
 letters of each word, so he must be able to call sums, 
 products, etc., without being conscious of the numbers 
 represented by the figures. 
 
 The drill work necessary to attain this need not be 
 dull and repressing in the least. It may be made the 
 most enjoyable part of the work. An appeal to the play 
 instinct, the use of scoring games, flash cards, charts, 
 number downs, races, playing store, and various games 
 stimulate a keen interest in the work. This subject is 
 discussed in Chapter VIII. 
 
 PREPARING FOR THE SECOND GROUP OF PRIMARY FACTS 
 
 Before proceeding beyond ten, all the subtraction facts 
 corresponding to the first group of addition facts must
 
 34 THE TEACHING OF ARITHMETIC 
 
 be made automatic. The methods and devices used in 
 teaching the subtraction facts are given in Chapter V. 
 
 The success of the method given below for teaching 
 the remaining twenty facts of addition requires automatic 
 control of the following subtraction facts : 
 
 10 10 98877665544332 
 _i_.!!-?!!^!21^^2J_ 
 
 This will be evident from the method shown below. 
 
 The next step in the preparation is to bring out the 
 meaning of "teen." That is, show the pupils that 
 thirteen, fourteen, fifteen, etc., mean three and ten, four 
 and ten, five and ten, etc. 
 
 Let the pupils then tell at sight 
 
 10 10 10 10 10 10 10 
 _9 _i J- J> _! _i _?. 
 
 Let 10+1 = 11 and 10+2 = 12 follow this work, since 
 the sums are not called "oneteen" and "twoteen," but 
 eleven and twelve. 
 
 DEVELOPMENT OF THE REMAINING TWENTY FACTS 
 
 By the plan here presented, the pupil makes the remain- 
 ing tables much more easily than through counting, and 
 by a method much more useful to him as a means of re- 
 taining the facts. In fact, to find through counting that 
 eight and nine are seventeen in no way aids the memory 
 in retaining this fact. The child is unable to recognize 
 the number of objects in such large groups, so the finding 
 of the fact in this way has no advantage whatever in fur-
 
 THE PRIMARY FACTS OF ADDITION 
 
 35 
 
 nishing a mental imagery to aid the memory. The purpose 
 of finding the first group through counting was not to 
 help fix the facts, but to make clear the meaning of 
 addition. But addition should be thoroughly clarified 
 by this tune and the objects should have served their 
 purpose. 
 
 Begin the remaining twenty facts with the "nines." 
 Write upon the blackboard the following : 
 
 Now take some one t of these and, by the use of objects, 
 show that one less than the number added to nine is the 
 number of "teen" in the sum. Thus: 
 
 Through drill, train the pupils to form the habit of 
 looking at the number to be added to 9 and thinking 
 "one less" as the number of "teen." Thus pointing to 
 the numbers written below, say "Add to nine." 
 
 57648 
 
 As the teacher points to them, the pupils subtract 
 one from each and say : "fourteen," "sixteen," "fifteen," 
 "thirteen," "seventeen." The pupil is now ready to 
 drill upon the tables as written above. 
 
 Take up the table of "eights" in the same way showing
 
 36 THE TEACHING OF ARITHMETIC 
 
 that two less than the number added to 8 shows the num- 
 ber of " teen " in the sum. t 
 
 Using charts, flash cards, games, playing store, etc., 
 as in the first group, drill until the "eights" and "nines" 
 can be given automatically. 
 
 They are the following combinations : 
 
 9998898 
 
 1 1 2 Z 2 
 
 8989899 
 
 It will be found that these fourteen facts, usually 
 the most difficult to teach, are as easily fixed as the 
 simple facts of the first group. 
 
 There are yet six facts to teach. They are : 
 
 777766 
 
 _! Ji JL _! JL JL 
 
 14 13 12 11 12 11 
 
 
 
 These may be given upon cards and the pupil asked to 
 find through counting that they are correct. The chief 
 thing, however, is not the finding that they are correct, 
 but the fixing of them as mental pictures. 
 
 ADDING ZEROS 
 
 While to add zero is to add nothing at all, and while 
 the combinations with zero are not among the " forty-five 
 primary facts" of addition, some practice in calling such
 
 THE PRIMARY FACTS OF ADDITION 
 
 37 
 
 combinations is necessary in order to give an automatic 
 control of them. Such combinations, however, need 
 not receive as much tune as the other facts. 
 
 A COMPLETE CHART OF PRIMARY ADDITION FACTS 
 
 The child needs drill upon all the possible ways of 
 writing the figures including the zeros. Thus, there are 
 "one hundred facts" instead of the so-called "forty-five 
 facts" when zeros are used. In the following table, the 
 easiest group is given first and the hardest last. Do not 
 spend as much tune, then, with the first part of the table 
 as with the last. 
 
 
 
 
 
 4 
 
 1 
 
 6 
 
 
 
 
 
 7 
 
 
 
 3 
 
 9 
 
 5 
 
 2 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 3 
 
 
 
 
 
 
 
 8 
 
 ft 
 
 1 
 
 
 
 5 
 
 
 
 
 
 . 
 
 
 
 4 
 
 
 
 6 
 
 9 
 
 2 
 
 7 
 
 1 
 
 1 
 
 1 
 
 4 
 
 1 
 
 7 
 
 6 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 
 1 
 
 8 
 
 4 
 
 3 
 
 5 
 
 5 
 
 9 
 
 2 
 
 8 
 
 1 
 
 5 
 
 1 
 
 1 
 
 3 
 
 2 
 
 9 
 
 6 
 
 7 
 
 1 
 
 3 
 
 4 
 
 1 
 
 4 
 
 1 
 
 5 
 
 1 
 
 1 
 
 2 
 
 3 
 
 2 
 
 4 
 
 3 
 
 8 
 
 2 
 
 6 
 
 7 
 
 4 
 
 3 
 
 5 
 
 7 
 
 2 
 
 2 
 
 5 
 
 6 
 
 3 
 
 2 
 
 3 
 
 3 
 
 4 
 
 6 
 
 2 
 
 5 
 
 2 
 
 5 
 
 3 
 
 2 
 
 3 
 
 2 
 
 2 
 
 3 
 
 4 
 
 7 
 
 3 
 
 2 
 
 6 
 
 8 
 
 7 
 
 9 
 
 4 
 
 6 
 
 4 
 
 4 
 
 8 
 
 7 
 
 9 
 
 4 
 
 3 
 
 6 
 
 2 
 
 8 
 
 3 
 
 4 
 
 9 
 
 7 
 
 8 
 
 5 
 
 9 
 
 2 
 
 7 
 
 6 
 
 6 
 
 8 
 
 3 
 
 7 
 
 4 
 
 5 
 
 9 
 
 4 
 
 9 
 
 4 
 
 8 
 
 9 
 
 3 
 
 4 
 
 8 
 
 4 
 
 9 
 
 6 
 
 9 
 
 5 
 
 7 
 
 6 
 
 9 
 
 5 
 
 8 
 
 9 
 
 5 
 
 7 
 
 7 
 
 8 
 
 6 
 
 8 
 
 9 
 
 7 
 
 5 
 
 6 
 
 8 
 
 -7 
 
 7 
 
 7 
 
 8 
 
 8 
 
 6 
 
 6 
 
 5 
 
 8 
 
 9 
 
 5 
 
 6 
 
 7 
 
 5 
 
 6 
 
 5 
 
 9 
 
 8 
 
 9 
 
 9 
 
 Such a chart as this should be made for permanent use 
 and also a hundred flash cards containing the same num- 
 bers should be made. Daily drill from these for a few 
 minutes should be kept up long after they are "learned" 
 and when other facts or processes are being developed.
 
 38 THE TEACHING OF ARITHMETIC 
 
 DRILLS THAT PREPARE FOR WRITTEN WORK 
 
 Ability to give these primary facts automatically is 
 not the only ability required in adding a column. Thus, 
 to add the column in the margin, if we add down, 
 9+6 is the only primary combination. The next 9 
 is 15+8, the next is 23+4, and the next is 27+7. 6 
 Thus it is plain that to add columns the pupil must 8 
 be able to add mentally (without seeing the figures) 4 
 a two-figured number to a one-figured number. T_ 
 While this is closely related to the primary facts, 
 special drill upon such combinations is needed. Such 
 drills are sometimes called "adding by endings." Thus 
 8 added to 9 gives 17, a number ending in 7 ; so 8 added 
 to any number ending in 9 gives a number in the next 
 decade ending in 7.- While there should be some sight 
 work from charts of such combinations as 
 
 7 27 47 37 97 67 17 57 
 
 9 29 19 59 89 39 79 49 
 66666666 etc. 
 
 for all of the^rimary combinations, a great deal of pure 
 mental work should be done in which the teacher should 
 point to any one of the nine digits 
 
 123456789 
 
 and say, "Add 17," "Add 26," "Add 58," "Add 43," 
 for all numbers from 10 to 100.
 
 THE PRIMARY FACTS OF ADDITION 39 
 
 ADDING SINGLE COLUMNS 
 
 Before pupils are required to add two or more columns, 
 they should have some skill in adding single columns, 
 for adding numbers of two or more places requires, in 
 general, carrying and also requires a greater tax upon 
 attention; that is, longer attention before a break of 
 taking up a new exercise. 
 
 A great deal of the drill work should be sight drill from 
 charts, so that the effort of recording the figures in the 
 answers will not detract from the attention needed for 
 the addition. It is because pupils are plunged from the 
 table of primary facts, not yet made automatic, directly 
 into written addition, that they develop, slow, inaccurate 
 habits ; and, instead of really adding, they find the sums 
 through counting. When a pupil is slow and inaccurate 
 in written work or has the "counting habit," there has 
 not been sufficient drill in developing one or more of the 
 three fundamental abilities : (1) automatic control of 
 the forty-five primary facts; (2) automatic control of 
 adding mentally a two-figured number to a one-figured 
 number ; and (3) the habit of reading a sum from a single 
 column. 
 
 WRITTEN WORK IN ADDITION 
 
 There is a question among teachers as to whether the 
 written processes should be rationalized or whether the 
 pupil should merely form the correct habit of doing them. 
 Since the fundamental processes are always performed 
 in the same way and arise so often in all future work,
 
 40 THE TEACHING OF ARITHMETIC 
 
 rationalization need not be urged as a help to the memory. 
 Also, the efficient use of the processes in the solution of 
 problems depends upon a rationalization of the meanings 
 of the processes, not upon the rationalization of modes of 
 procedure carrying, borrowing, etc. hence, it seems 
 that there is little argument for rationalization. Upon 
 this one principle all will agree, viz. : The important and 
 essential thing in the fundamental processes is that the 
 pupils form correct habits in the modes of procedure; 
 that is, habituation, and not rationalization, is of chief 
 importance. 
 
 There are pupils, however, to whom the work is made 
 more interesting through rationalization and there are 
 those who always want some authority for doing what 
 they do. Rationalization need not take much time. 
 Even if all of the class do not get clearly the "why," 
 the objective presentation, or some method of rationaliza- 
 tion of a process, satisfies the pupils that there is a reason- 
 able basis for proceeding with the work as we do, and to 
 some it makes the work more interesting. 
 
 METHOD OF DEVELOPMENT 
 
 Before the process of addition can be rationalized, 
 the decimal-place-value feature of our notation must be 
 understood. This can be shown by splints, toothpicks, 
 or something of this nature. First write upon the board 
 two or three two-place numbers as 24, 35, 46. Have the 
 pupils count as many splints by ones. Then count ten 
 and put a rubber band around them, then another ten 
 in the same way, and so on until less than ten are left.
 
 THE PRIMARY FACTS OF ADDITION 41 
 
 If 24 is the number counted, the pupil finds 2 tens and 
 4 ones left. If 35, then he finds 3 tens and 5 ones. Thus 
 he sees that the 2 of 24 stands for tens and the 4 for ones, 
 etc. Now, to test the understanding of the pupils as to 
 what you have presented, have a pupil hold up any num- 
 ber (up to ten) of tens, and a number of ones and have 
 the rest of the class write down the number shown. Thus, 
 if three tens and six are shown, the pupils write 36. In 
 this way it is easy to show the decimal-place-value prin- 
 ciple of our notation. 
 
 Now there are two principles involved in adding num- 
 bers of two or more places. First, only like numbers can 
 be added. Hence, the numbers are so written that the 
 ones will all be in one column, the tens in another, etc., 
 and each column is added separately because it is made 
 up of like things. Second, when the sum of any column 
 is greater than ten, it is reduced to units of a higher order, 
 and those of the higher order added to the next column. 
 Hence, the first problems should include but the first of 
 these two principles in order to introduce but one diffi- 
 culty at a time. However, it will be necessary to take 
 but a few problems in which there is no carrying, for it 
 does not take long to form the habit of adding a column 
 at a time and putting the result under the column added. 
 The more difficult habit to fix is that of working from right 
 to left, for in all other work the child works from left to 
 right. 
 
 Then, to rationalize the process of addition, suppose 
 you have written the problem given in the margin. With 
 splints or toothpicks bound up in bundles of ten each
 
 42 THE TEACHING OF ARITHMETIC 
 
 and with loose ones, have a pupil hold before the class 
 
 2 tens and 6 ones, and another pupil, 4 tens and 
 
 3 ones, to represent the 26 and 43. Then let the 26 
 class tell how many both pupils have, and thus see 43 
 that the 69 stands for the 6 tens and 9 ones held 69 
 by the two pupils. This need not be presented 
 objectively more than two or three tunes. The presenta- 
 tion should then be followed with a few days' drill in 
 adding two or three numbers of two or three places in 
 which there is no carrying. All drill should be under 
 the direct supervision of the teacher so as to insure the 
 forming of proper habits. As soon as pupils automatically 
 work from right to left that is, add ones first, then 
 tens, etc. they are able to take up the next class of 
 exercises. 
 
 This is done by taking up some such problem as the 
 one in the margin and proceeding as before. Here 
 the class sees that there are 15 ones. Have them 38 1 
 separate the 15 ones into 1 ten and 5 ones. Have 47 
 the class see, then, that instead of 7 tens and 15 85 
 ones, we have 8 tens and 5 ones. But few objec- 
 tive illustrations are needed to rationalize the carrying 
 process. Drill should then follow until the process is fixed. 
 It is a waste of time to present the work objectively ex- 
 cept when it is first taken up. Do not require pupils to 
 "explain" their problems daily by the use of splints. 
 They have seen that there is a rational basis for carrying, 
 and that is sufficient. In fact, even this much of the 
 process of rationalization is unnecessary to the efficient 
 use of the subject.
 
 THE PRIMARY FACTS OF ADDITION 43 
 
 CHECKING THE WORK 
 
 The best time to develop the habit of checking the 
 computations is at the time the processes are first taught. 
 The pupils should feel that when a process is performed 
 but once the work is but half done. By reviewing it in 
 some way he must become convinced of its accuracy. 
 
 In addition, it is best to form the habit of first adding 
 in a given direction and checking by adding in the opposite 
 direction. This brings up different combina- 
 tions. Thus, in the exercise in the margin, by 78 
 adding down, the combinations are 15, 23, 27; 47 
 and 9, 13, 22, 25. By adding up, they are 12, 19, 98 
 27 ; and 5, 14, 18, 25. Until the pupil is pretty _34 
 sure of the result, the first result should be kept 257 
 on a bit of "scratch paper" until it has been 
 verified by the check. 
 
 DRILLS IN WRITTEN WORK 
 
 Skill comes through much practice. Besides drill in 
 adding numbers of two or more places, sight and mental 
 drill in the primary facts, adding by endings, and add- 
 ing single columns should be continued. But little of 
 the drill work should come from problems. There is a 
 means of securing greater interest than that gained through 
 problems, however real they are. It is the interest that 
 comes to the pupil from the knowledge that he is develop- 
 ing greater skill daily. In order to show this clearly, 
 exercises of the same weight should be used as a test 
 very frequently ; that is, exercises in adding four or five
 
 44 THE TEACHING OF ARITHMETIC 
 
 numbers of two or three figures each should be given two 
 or three times a week, throughout the term, and a record 
 kept of the average number tried by the class and the 
 average number that are correct. Graphs of these results 
 kept in sight will prove an inspiration to the class, for, if 
 the drilling is done properly, there will be a marked im- 
 provement shown during a month or a term. Individuals 
 should also keep their own records in the same way. 
 
 Not only will pupils be interested to see the graph 
 of then* progress gradually rise, but they will also be 
 interested in watching the graph of accuracy approach 
 the graph of the number of exercises tried. 
 
 It will be found that the incentive from trying to beat 
 one's former record and a desire to see the "graph of 
 progress" rise will have a greater appeal to the pupils 
 of the fourth grade and above than to the pupils of the 
 lower grades. 
 
 Such a record as that described above will also serve 
 another very important purpose. It will enable the 
 teachers to standardize their work. Through the average 
 of a number of such records, one will have a standard 
 by which he may know what a child of a certain grade 
 should do. In other words, when a child can perform but 
 a certain number of exercises in a certain time with a 
 certain per cent of accuracy, it is clear that in the par- 
 ticular subject he belongs in the third, fourth, fifth, etc., 
 grade. 
 
 Another device to secure interest in classroom drill 
 in written work is to have the pupils call out the order 
 in which they finish their work. Thus, the first pupil
 
 THE PRIMARY FACTS OF ADDITION 45 
 
 to finish an exercise calls "one" and writes "1" above 
 his work; the next one calls "two" and writes "2" above 
 his work ; and so on. When all are done >and the answers 
 given, all who have wrong answers draw a line through 
 their numbers. At the end of the period, if one has a 
 record like 1, 1, 3, 4, it shows both speed and accuracy. 
 If one has 1$, 20, 10, Z0, etc., it shows the teacher at a 
 glance that the pupil is both slow and inaccurate. The 
 bright pupils like such a contest, and the dull ones work 
 hard when they know such a test is to be given once or 
 twice a week. 
 
 The same device may be used in any kind of written 
 work throughout all grades. 
 
 THE NATURE AND USE OF PROBLEMS 
 
 The problems of the primary grades are given chiefly 
 for one of two purposes. They are given either to clarify 
 the processes or to furnish a motive for, or interest in, 
 the work. Hence, they must be very concrete and real 
 to the pupil and must be those to which he might wish 
 to know the answer. While enough problems should be 
 used to show a need of arithmetic, when the purpose of 
 the instruction is to develop skill in computation, pure 
 abstract drills should be used. It is much easier, too, 
 to make well-graded drills that will emphasize just the 
 facts or processes needed than to get the same drills 
 through problems, for a problem should present real 
 conditions. Hence, the size of the numbers is regulated 
 by the nature of the problems, and such drills are not in 
 general as well graded as the pure abstract exercises.
 
 CHAPTER V 
 
 THE PRIMARY FACTS AND PROCESSES OF 
 SUBTRACTION 
 
 THE primary facts of subtraction come directly from 
 those of addition. Thus, when a child can tell you that 
 3 and 5 are 8, the question, " 3 and how many are S ? " 
 comes not as a new fact, but from the addition fact already 
 known. The first drills should be answers to such ques- 
 tions as the one above. In sight drill they are written 
 as follows : 
 
 *34**46 * * 7 
 
 6 * * ,5 6 * * 8 6 * etc. 
 
 978'787710109 
 
 The pupil gives the missing number that, with the 
 given one, makes the sum below. 
 
 By subtraction we find the answer to three types of 
 questions illustrated by : 
 
 (a) 5 apples taken from 8 apples leave how many 
 apples ? 
 
 (6) How many apples added to 5 apples make 8 
 apples? 
 
 (c) If one plate contains 8 apples and another 5 apples, 
 how many more in the first plate ? 
 
 46
 
 THE PRIMARY FACTS OF SUBTRACTION 47 
 
 That is, Subtraction is the process of taking one number 
 from another to find how many remain; of finding what 
 number must be added to a given number to make a 
 given sum; and of finding the difference between two 
 numbers. 
 
 While all these meanings should be shown objectively, 
 the pupil must see clearly that the primary facts come 
 from addition. 
 
 The nature of the drill work upon the primary facts 
 depends upon the method to be used in written work. 
 If the "addition method" is to be used, the thought in 
 the drills for 14-9 = 5 will be "9 and 5 are 14"; while, 
 if the "taking-away method" is to be used, the thought 
 will be "9 from 14 leaves 5." 
 
 THE ADDITION METHOD 
 
 In deciding upon a method to use, the questions asked 
 are: "By which method is there the least liability of 
 error?" "Which method is more easily presented?" 
 and, "By which method can the work be done most 
 rapidly ? ' ' There are advocates of the " addition method ' ' 
 who would answer, "The addition method," to each of 
 the above questions. However, there do not seem to be 
 available data to warrant such a reply. But that there 
 are strong points in favor of the method, one must recog- 
 nize. Thus, it follows so closely the work of addition 
 that much of the skill in addition is carried over to sub- 
 traction. It is also more nearly the method used by 
 clerks in counting out change. 
 
 The way in which to present the addition method is first,
 
 48 THE TEACHING OF ARITHMETIC 
 
 to take a problem in which each figure of the subtrahend 
 represents a number less than the corresponding 
 number in the minuend. Thus, to present the 48- 
 problem in the margin, the pupil must have ac- 23 
 quired the idea from the drills upon the primary 25 
 
 986 
 facts as, 4 3 2 etc., that the top number is the sum of 
 
 554 
 
 the other two. Like "a corn on his chin" of Riley's Man 
 in the Moon, which is " a dimple turned over, you know," 
 so subtraction is "addition turned over, you know." 
 Then the thoughts in the above problem are : 3 and 5 
 are 8, write 5 ; 2 and 2 are 4, write 2. Since the tables 
 should have been made practically automatic before 
 taking up the written work, there is practically nothing 
 new to teach, for the pupil has the habit of beginning at 
 the right-hand column and of considering one column at 
 a time, which was formed in his work in addition. The 
 "why" of this need not be shown. However, if addition 
 was rationalized, the reason for this follows; but the 
 work may be presented objectively if it is thought that 
 such a presentation will answer the curious or make the 
 work more interesting. Remember that the important 
 thing is to establish proper habits of procedure and 
 to develop skill in doing the work accurately and 
 rapidly. 
 
 To develop the second problem, the one in which the 
 minuend has digits less than the corresponding digits 
 of the subtrahend, review those primary facts of sub- 
 traction whose minuends are two-figured numbers, as :
 
 Call attention to the fact that in these, the number 
 below is less than the one directly above. 
 
 Then present some such example as the one in the mar- 
 gin. Call attention to the 4 being less than 6, so it must 
 be the 4 of 14, which is the sum of 6 and some number, 
 as in the table above. So ask "6 and what make 54 
 14?" Write the answer, 8, below the 6. Now, 26 
 since 54 is the sum of 26 and some number, the 4 is 28 
 the 4 from 14, the sum of 6 and 8 ; hence, the 1 (ten) 
 of 14 was carried to 2 of the given addend. So the next 
 question is " 3 and what make 5 ? " 
 
 There are two things, then, for the pupil to 
 observe : 
 
 (1) When the number above is less than the one below 
 it, picture it as the right-hand figure of some number of 
 "teen." 
 
 (2) In such cases, one is always added to the next higher 
 digit of the addend (subtrahend). 
 
 In the second example in the margin we think 8 521 
 and 3 are 11; write 3. 1 (carried from 11) and .6 268 
 and 5 are 12; write 5. 1 (carried from 12) and 2 253 
 and 2 are 5 ; write 2. 
 
 If "why" arises, show that it is merely the carrying 
 that was done in addition, for the top number is the sum 
 of the given number and the number found. 
 
 In presenting this, or any other process, the process 
 should first be presented very carefully by the teacher
 
 50 THE TEACHING OF ARITHMETIC 
 
 and enough examples taken to show fully the "how." 
 Then the pupils should be sent to the blackboard where 
 the first work can be done directly under the eye of the 
 teacher in order that she may detect any mistake in using 
 the process before that mistake becomes a habit. If 
 seat work or home work is not given until every pupil 
 knows how to perform the process, then the only errors 
 made will be the accidental errors. The purpose of further 
 drill, seat work, home work, etc., is to develop greater 
 skill in handling the problems. 
 
 THE TAKING-AWAY METHOD 
 
 The method in most common use in the schools in this 
 country is the " taking-away " method of subtraction. 
 There is no trouble in presenting the first class of 
 problems ; namely, those in which each number in the 48 
 subtrahend is less than the corresponding number in 16 
 the minuend. Thus, in the problem in the margin, 32 
 the pupil thinks 6 from 8 leaves 2 ; write 2. 1 from 
 4 leaves 3 ; write 3. "'Since the pupil knows the tables 
 automatically before taking up written work, this prob- 
 lem presents nothing new, for the habit of subtracting 
 one column at a time and of working from right to left 
 follows the habit established in addition and is not really 
 a new habit here. After a few problems to see that these 
 habits are fixed, further drill need not be ,given before 
 taking up the next class of problems; that is, those' 
 in which the so-called "borrowing" is necessary. The 
 pupil is very apt to ask why here, so the teacher may
 
 THE PRIMARY FACJS OF SUBTRACTION 51 
 
 well present one or two examples' objectively. Thus, to 
 present the problem in the margin objectively, show 
 62 as 6 bundles of tens and 2 ones, using toothpicks 62 
 or splints. Now ask a pupil to take away 7 ones. 27 
 He sees that, before he can do this, he must have 35 
 more than 2 ones. Therefore, have him take one 
 bundle of the 6 tens; take off the band and put the 10 
 splints with the ones, thus giving 12 ones. Now taking 
 away 7 leaves 5; write 5. Now ask him to take away 
 2 tens from the remaining 5 tens, leaving 3; write 3. 
 After a few objective presentations and a few exercises 
 without objects, the pupils should be sent to the black- 
 board and should work under the direct eye of the teacher 
 until they can work with small numbers with but slight 
 danger of error. Increase the difficulty of the exercises 
 very gradually. Avoid exercises in which it is necessary 
 to take one from the next higher order when the figure 
 of that order is zero, as in 502 136, until the pupils 
 handle the other simpler exercises well. Next take up 
 such an exercise having zeros in the minuend, as a special 
 lesson. Thus, in the problem above, show that one is 
 taken from 5, making 10 tens; and that one of the tens 
 is then taken, leaving 9. This class of exercise presents 
 special difficulty, as may readily be seen, and requires 
 special drill. This difficulty, however, does not occur 
 in the use of the addition method.
 
 CHAPTER VI 
 
 THE TABLES DEVELOPED FROM ADDITION 
 
 MULTIPLYING by a whole number is only a short way 
 of finding the result of adding a number of equal num- 
 bers. Thus, 7X345 is only a short way of writing and 
 finding 345+345+345+345+345+345+345. Hence, it 
 would seem that the most logical and psychological method 
 of developing the primary facts is through addition, and 
 experience shows this to be the most economic and effi- 
 cient method. 
 
 Begin by selecting all the doubles from the forty-five 
 facts of addition. Show the class the following drill 
 cards used in addition : 
 
 123456789 
 
 The results, of course, are known. Now remove the 
 cards and ask the class to tell you how many of each 
 number they saw on a card and to tell you the result. 
 The reply will naturally be, "two ones are two," "two 
 twos are four," "two threes are six," "two fours are 
 
 52
 
 53 
 
 eight," etc. Hence, the only new thing in the "two 
 times" table is the notation. That is, 2X1 =2, 2X2=4, 
 2X3 = 6, etc., which should be read, "two ones are two," 
 and not "two times one equals two.") Always use lan- 
 guage that gives the clearest picture of the real meaning 
 of the process. 
 
 Before taking up multiplication, the pupils should 
 have developed some skill in column addition. They 
 should at least be able to add single columns of four or 
 five numbers. Then to develop the "three times " table, 
 meaning three of each number have been added, have the 
 class write down three of each of the nine digits as follows 
 and find the sum. Thus : 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 _9 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 When the pupil sees what a time-saver this new process 
 of multiplication is, he will have a motive for learning 
 the tables and how to use them. Thus, to find 
 the sum in the margin, let him see that by remember- 468 
 ing that three 8's are 24, he does not need to add. 4Q8 
 Also, show him that the second method in the 46g 
 margin is a shorter way to write the same thing, 
 and that he now thinks three 8's, three 6's, and 468 
 three 4's, instead of seeing them, as in addition. JX3 
 Thus, by this method of development the written 
 work can be taken up with each table, thereby forming
 
 54 THE TEACHING OF ARITHMETIC 
 
 a strong motive for learning this new process and also 
 furnishing much needed drill in fixing the facts. 
 
 After each table is developed by addition, it is written 
 down as multiplication, thus : 
 
 3X1=3 3X4 = 12 3X7 = 21 
 
 3X2 = 6 3X5 = 15 3X8 = 24 
 
 3X3 = 9 3X6 = 18 3x9 = 27 
 
 These are read "three 1's," "three 2's," "three 3's," etc., 
 for this is just what the pupil sees in the development. 
 In many textbooks the tables are written : 
 
 1X3 3X3 5X3 7X3 9X3 
 2X3 4X3 6X3 8X3 10X3 
 
 This is because the tables were developed through count- 
 ing by 3 and thus the notation accurately describes the 
 thing the pupil saw in the development. This is the 
 table of "threes" and not the "three tunes " table, which 
 was developed through addition. 
 
 After a few tables have been found, the word "times" 
 may be used, for the expression will now have a meaning 
 when the pupil knows that 3X5 means that three 5's 
 have been added. 
 
 This method of development is to make clear the 
 meaning of multiplication, that it is a short process to 
 save adding equal addends. Hence, it is not important 
 that all of the tables be found by the pupil. Perhaps 
 the finding of the tables through the "five times" table 
 is sufficient to fix the full meaning and use of multipli-
 
 THE PRIMARY FACTS OF MULTIPLICATION 55 
 
 cation. If the child's interest seems to lag in the 
 making of the tables after the first four or five 
 tables have been found, the "nine times" table 
 may well be taken up next as follows : The pupil 
 should have been shown objectively or otherwise 
 from the first that 2X3 = 3X2, 3X5 = 5X3, etc. 
 Thus in the diagram in the margin, in rows there are 
 5X3, while in columns there are' 3X5. 
 
 Now, since 2X9 = 18, 9X2 = 18; since 3X9=27, 9X 
 3 = 27; since 4 X 9 = 36, 9 X 4 = 36. These are known from 
 the tables already learned. Now, point out that in these 
 three products in the "nine times" table, the tens' digit 
 of each product is just one less than the number multi- 
 plied by 9, and that in every case the sum of the digits 
 in the product is 9. Now tell the class that this is always 
 so, and hence to find 9 times any number, as 9X6, it is 
 merely necessary to write one less than 6 for the tens' 
 digit, that is, 5, and for the ones' digit write a number that 
 with 5 makes 9. That is, 9 X 6 = 54, 9 X 8 = 72, 9 X 5 = 45, 
 etc. 
 
 Now, if the 3 times, 4 times, 5 times, and 9 times tables 
 have been developed, but six new facts remain. They 
 are : 6x6, 6X7, 6X8, 7x7, 7x8, and 8X8. It is better 
 to give these facts to the child than to have him find them 
 through addition; or the facts may be given and the 
 pupils may prove them by addition. This will be a test 
 as to whether or not they really understand the meaning 
 of multiplication. 
 
 Drill must be continued on the tables until all the facts 
 can be recalled automatically.
 
 56 THE TEACHING OF ARITHMETIC 
 
 A SECOND METHOD OF DEVELOPMENT 
 
 Many of the older textbooks and some of the newer 
 ones develop the tables through counting by 2's, 3's, 
 4's, 5's, etc. The plan is to learn to count by 2's, 3's, 
 etc., then build up the tables as follows : 
 
 
 
 
 
 
 
 2 
 
 2 
 2 
 
 2 
 2 
 2 
 
 
 
 
 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 
 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 While, of course, counting by 2's is adding 2's, yet to the 
 child this does not seem to be addition as in the former 
 method, and does not leave as clear a meaning in the 
 child's mind of what multiplication really is. More- 
 over, the learning of a single table does not fit the pupil 
 at once to take up written work as is the case in the 
 former method. Thus, the pupil could not use the 
 above table to find 2x768. Or, if the 3's were learned 
 in that way, he could not find 3 times any number, as 
 3X645, from this table. There is a further danger, too, 
 that pupils may rely upon counting to find the products 
 in future work rather than fix the facts through 
 memory.
 
 THE PRIMARY FACTS OF MULTIPLICATION 57 
 
 Observe, also, that to use the method given here, the 
 tables are not written in the same order as when de- 
 veloped by the first method. They would be written: 
 1X2, 2X2, 3X2, 4X2, 5X2, 6X2, etc. The difference 
 in the two methods of development accounts for the two 
 ways of writing the tables. 
 
 This method of development, then, is not recommended. 
 
 DRILLS THAT PREPARE FOR WRITTEN WORK 
 
 Just as in written addition it was seen that the mere 
 learning of the tables did not fit the pupil for successful 
 written work, so the tables alone are not sufficient for 
 successful written work in multiplication. Thus, to 
 find 3X485, the first product is but 3X5, but the next 
 one is 3X8+1, and the next 3X4+2. 
 
 So there should be drill upon announcing any product 
 plus any one-figured number. A simple device for such 
 drill is a chart made as follows : 
 
 To use the chart, merely point 
 to any number whatever in each 
 column, going from left to right. 
 Then let the pupil give the product 
 of the first two plus the third. It 
 will be seen that no possible com- 
 bination in any written work can 
 occur that cannot be drilled upon 
 from such a chart. A five-minute 
 drill daily upon such a chart will aid 
 very greatly in efficient written work. 
 
 1 
 
 
 9 
 
 1 
 
 2 
 
 ^ 
 
 8 
 
 2 
 
 3 
 
 * 
 
 7 
 
 3 
 
 4 
 
 
 6 
 
 4 
 
 5 
 
 X 
 
 5+' 
 
 5 
 
 6 
 
 
 4 
 
 6 
 
 7 
 
 
 3 
 
 7 
 
 8 
 
 
 2 
 
 8 
 
 9 
 
 
 1 
 
 9
 
 58 THE TEACHING OF ARITHMETIC 
 
 WRITTEN MULTIPLICATION 
 
 It was shown above that written multiplication by 
 multipliers of one figure should be taken up when each 
 table is developed in order to furnish a motive for 
 learning the tables and to give further drill that 
 is needed to fix them. When a pupil sees 
 that instead of a long problem in addition, 
 
 as in the left-hand margin, the same result 
 
 475 , , , , u . % .. . ,, 3325 
 
 may be found by multiplication, as m the 
 
 right-hand margin, he not only understands what 
 multiplication means, but sees a very strong 
 
 3325 , 
 
 motive tor learning it. 
 
 In presenting written multiplication, it is necessary 
 merely to point out that, instead of seeing the number of 
 equal addends, we think them; that is, instead of seeing 
 seven 5's as in addition, we think seven 5's are 35; in- 
 stead of seeing seven 7's, we think seven 7's are 49, and 
 3 to carry are 52; instead of seeing seven 4's, we think 
 seven 4's are 28 and 5 are 33. This is the only explana- 
 tion necessary when the tables are taught by the first 
 method given here. 
 
 WHEN THE MULTIPLIER HAS Two OB MORE FIGURES 
 
 Before written multiplication by two-figured multi- 
 pliers is taken up, the pupil should understand the effect 
 of annexing a zero to a whole number. He should never 
 be allowed to multiply by 10 by writing down the multiplier 
 beneath the multiplicand, as is so often seen even in our 
 best schools. The pupil should be shown that annexing
 
 THE PRIMARY FACTS OF MULTIPLICATION 59 
 
 a zero to a whole number has the effect of moving each 
 digit one order higher, and thus multiplies each digit by 
 10. Thus in 36, by annexing a zero making 360, the 
 6 ones become 6 tens, and the 3 tens become 3 hundreds. 
 If place value has not been taught up to this time, it 
 must be taught here if the pupil is to see why annexing a 
 zero multiplies by 10. This may be followed by showing 
 that to multiply by any number of tens, as 40, first multiply 
 by 4 and then annex a zero to the product, which multi- 
 plies the result by 10. These two facts that annexing 
 a zero multiplies an integer by ten, and that any integer 
 is multiplied by a multiple of ten, by multiplying it by 
 the multiple and annexing a zero to the product are 
 preliminary to taking up two-figured multipliers. 
 
 Pupils in the third or fourth grade, where this work is 
 first taken up, are too young to follow a full rationaliza- 
 tion of the process. In order that the pupil may satisfy 
 his curiosity as to the reason for performing the work 
 as he does, the teacher may ask some such questions as 
 the following : " Three 8's and two more 8's are how 
 many 8's?" "Six 5's and three more 5's are how many 
 5's?" Then, "Five 83's and twenty more 83's are how 
 many83's?" The class should reply, " Twenty-five 83 
 83's." The teacher may then say, "We will find 25 
 5X83 and then 20X83 and add the results, and 415 
 that will give us 25X83." As the work is per- 166 
 formed by the teacher upon the blackboard, the 2075 
 pupils giving the products as she records them, she 
 should point out that the first product (415) is 5X83. 
 Now in finding 20X83, the teacher gets from the class
 
 60 THE TEACHING OF ARITHMETIC 
 
 that 20 is 10 times 2, and says, "Then if I multiply 83 
 by 2 and place the first result in tens' place, that will be 
 multiplying the result by 10, so in that way I multiply 
 by 10X2 or 20." It will be observed that this is a very 
 imperfect rationalization, but it will be found that it is 
 enough of the "reason why" to satisfy the curious and 
 is perhaps as much as the class can comprehend, if not 
 more. 
 
 As in all cases, the work is first presented several tunes 
 by the teacher, then the class works at the blackboard 
 under her direct supervision until the process is mastered 
 and before any home or seat work is given.
 
 CHAPTER VII 
 
 THE PRIMARY FACTS AND PROCESSES OF 
 DIVISION 
 
 THE TABLES 
 
 THE primary facts of diyision follow at once from the 
 multiplication fafets. Thus, ask such questions as, 
 "Thfee 5's are how many?" .(Answer, 15.) "How many 
 5's hi 15?" "Four,6Ts are how many?" "How many 
 6's in 24 ? " " *Three 7 y s- are how many ? " " How many 
 7'sin21?" 
 
 Notice that thege are the questions that should bring 
 out an answer from the child's mental picture 8 
 
 of multiplication. Thus, 4X8 means four 4v o_ 8 
 8's written and added, as in the margin. So, 8 
 
 when the question, "How many 8's in 32 ?" is 
 asked, the answer very naturally 'is "Four." 32 
 
 The notations should then be given by saying, "This 
 is the way we write the fact that there are four 
 8's in 32." As a drill to fix the meaning of the 8)32, 
 notations, have a large miscellaneous number of 4 
 the facts with answers written as : 
 
 3)21^ 6)30 8)56 5)45 7)42 4)28 6)24 
 7 5 7 9 6 7 4, etc. 
 
 61
 
 THE TEACHING OF ARITHMETIC 
 
 and ask the class to read them. They should say, "There 
 are seven 3's in 21 " ; "There are five 6's in 30" ; "There 
 are seven 8's in 56"; etc. Do not allow the expression 
 "goes into," so often heard, and so often pronounced, 
 "gus-sin-tu." 
 
 When the notation is understood, have the pupils 
 write out all the division tables in this way from their 
 knowledge of multiplication. 
 
 It will be observed that the meaning of the division 
 here presented is measurement; that is, the quotient shows 
 how many times the dividend contains the divisor. When 
 these are well known, the partition phase of division may 
 be presented. This phase is shown as follows : 
 
 
 
 
 5 
 
 6 
 
 8 
 
 
 
 3 
 
 2 
 
 4 
 
 5 
 
 6 
 
 8 
 
 7 
 
 6 
 
 3 
 
 2 
 
 4 
 
 5 
 
 6 
 
 8 
 
 7 
 
 6 
 
 3 
 
 2 
 
 4 
 
 5 
 
 6 
 
 8 
 
 7 
 
 6 
 
 9 
 
 6 
 
 12 
 
 20 
 
 24 
 
 32 
 
 21 
 
 18, etc. 
 
 Write upon the blackboard several sets of equal addends 
 as shown above. Now, if it has not been done before, 
 show the class, dividing up some object as you do so, 
 that one of the three equal parts of anything is one third 
 of it ; one of the four, one fourth of it ; one of the five, 
 me fifth of it, etc., and thus through the rhythm of it, 
 they can at once tell you the name of one of any number 
 of equal parts of anything. 
 
 Now, say, "What is one of the three equal numbers 
 that make 9?" (Answer, 3.) Then, "What part of 9 
 shall we call 3?" (Answer, one third of it.) "What is
 
 THE PRIMARY FACTS OF DIVISION 63 
 
 one of the four equal numbers that make 20 ? " (Answer, 
 5.) " What then shall we call 5 ? " (Answer, one fourth 
 of 20.) 
 
 These questions are first asked about a large number 
 of facts written upon the blackboard in addition form. 
 Later they are asked and answered from the mental 
 picture of the multiplication facts. 
 
 THE NOTATION OF UNIT FKACTIONS 
 
 Without any further discussion of fractions, except 
 the limited meaning and use shown above, the written 
 notations may be given : 
 
 Thus, one sixth is written 
 
 one seventh is written y 
 one eighth is written %, etc. 
 
 The pupil is now ready to write the facts that he has just 
 given orally as follows : 
 
 i of 6 = 2; i of 8=2; of 15 = 3; i of 24 = 4, etc. 
 
 The pupil should be shown that the first form of ex- 
 pressing division may be interpreted as "a part "of " 
 
 A\A 
 
 when the divisor is an abstract number. Thus, may 
 
 2 
 
 be interpreted either as "There are two 3's in 6," or 
 "One third of 6 is 2." 
 
 These are the two most important forms of expressing 
 division, but for the purpose of brief written analysis 
 the form 6-5-3 = 2 should be given.
 
 64 
 
 THE TEACHING OF ARITHMETIC 
 
 Since the form for written work is the first one given 
 ( J, this is the form that should be used in all sight 
 
 drills. It may be interpreted as either the "measure- 
 ment" or the "partition" type of division. 
 
 A mere knowledge of the division tables is not sufficient 
 for successful written work. Thus, to find 3)162, the 
 first division requires not one of the primary facts, but 
 the fact that there are five 3's and 1 in 16. As a result, 
 it is seen that not only is drill upon the tables needed, but 
 the pupil must drill upon every possible dividend from 
 one times the divisor, to ten times the divisor. Thus, 
 if "the sixes" are being studied, a drill is needed upon 
 some such chart as the following, including all numbers 
 from 6 to 60 inclusive : 
 
 ABCDEFGHIJK 
 
 1 
 
 20 
 
 38, 
 
 60 
 
 29 
 
 7 
 
 49 
 
 42 
 
 11 
 
 36 
 
 66 
 
 24 
 
 2 
 
 17 
 
 48 
 
 6 
 
 67 
 
 34 
 
 16 
 
 23 
 
 53 
 
 66 
 
 15 
 
 60 
 
 3 
 
 40 
 
 9 
 
 "41 
 
 10 
 
 46 
 
 37 
 
 62 
 
 33 
 
 18 
 
 59 
 
 19 
 
 4 
 
 28 
 
 39 
 
 30 
 
 61 
 
 22 
 
 26 
 
 13 
 
 63 
 
 46 
 
 12 
 
 44 
 
 5 
 
 47 
 
 21 
 
 8 
 
 27 
 
 36 
 
 14 
 
 43 
 
 31 
 
 26 
 
 64 
 
 32 
 
 In drilling upon the table, for example in column B, 
 the pupil says, " 6 and 2 remaining ; 8 ; 1 and 3 remain- 
 ing; 6 and 3 remaining; 3 and 3 remaining." A few 
 minutes daily from such a chart will greatly aid in secur- 
 ing rapid and accurate written work in short division.
 
 THE PRIMARY FACTS OF DIVISION 65 
 
 THE FIRST WRITTEN DIVISION 
 
 The first written work may be taken up along with 
 the tables and should proceed very gradually from the 
 tables themselves. Thus, when the pupil writes 
 
 ' - , etc., it is a very easy step to get him to 
 43 6 
 
 2)86, 2)48, 2)128 
 take up such examples as '-* L ' , etc., where 
 
 T:O ^TC O4 
 
 there are no remainders after each division. The oral 
 drills prepared the pupils to think 5-j-2 = 2 and 1 remain- 
 ing; 9-i-2=4 and 1 remaining; 7-7-2 = 3 and 1 remaining, 
 etc. Therefore, it is an easy step to show that the 1 
 remaining, used with the number that follows, is the nex 
 number to divide. 
 
 If it seems wise to rationalize the work more fully, 
 it may be done objectively. For example, to show that 
 52-f-2=26, show 52 as five bundles of ten splints and 
 two ones. Now, if the five tens are arranged in two equal 
 groups, there are 2 tens in each group and 1 ten remaining. 
 Now, this ten and 2 make 12 ones. Dividing 12 into 
 two equal groups, there are 6 in each. So there are 2 
 tens and 6, or 26, in each of the two parts. 
 
 It will be seen, however, that this explanation is really 
 based upon the "partition idea" of division, and not upon 
 "measurement." One could not ask children, "How 
 many tunes do 5 tens contain 2?" and expect an answer. 
 So, if the work is to be presented objectively, more oral 
 drill upon the partition phase of the subject is necessary. 
 Thus, hi giving 5-J-2, 7--2, 9-7-2, ll-v-2, etc., the pupil
 
 66 THE TEACHING OF ARITHMETIC 
 
 says, "One half of 5 is 2 and 1 undivided," "One half of 
 7 is 3 and 1 undivided," etc. 
 
 In drilling upon the table given on page 64 the pupil 
 thinks "One sixth of 20 is 3 and 2 undivided"; "One 
 sixth of 17 is 2 and 5 undivided," etc. ; and not "There 
 are three 6's and 2 remaining in 20 " ; " There are two 
 6's and 5 remaining in 17," etc. 
 
 It seems unwise, however, to take up the matter of 
 rationalization with pupils of the third grade unless they 
 are exceptional children. It seems better to begin with 
 very easy work as given above, and to show how to do 
 the work and then, as the work becomes easy through 
 drill, to progress to more difficult exercises. 
 
 LONG DIVISION 
 
 The pupil has been performing short division for nearly 
 a year before taking up long division. By writing down 
 all the work that was done mentally, the pupil should 
 be shown the longer form. 
 
 Thus : 268 
 
 9)2412 9)2412 
 
 268 18^ 
 
 61 
 
 54 
 
 72 
 72 
 
 As this is shown, question the pupil as to what he 
 thought in short division and write it down in long division. 
 Thus, in short division he thinks, "There are two 9's in
 
 THE PRIMARY FACTS OF DIVISION 67 
 
 24 and 6 remaining, for 2X9 = 18, and 24-18 = 6." In 
 long division this is written down. 
 
 The only reason that pupils find long division difficult 
 is because they have trouble in estimating the quotient 
 figures. This, then, is the feature upon which they should 
 be drilled. However, as a preparation for long division, 
 it is well to give a few lessons in which the divisors are 
 multiples of from 2 to 9 times some power of ten, the 
 work to be done by dividing both dividend and divisor 
 by that power of ten and by using short division. 
 
 Thus: 3000)24000 400)1700 etc. 
 
 8 4; 100 rem. 
 
 The teacher should recognize that it is much easier 
 to estimate the quotient figures from some divisors than 
 from others and that some quotients are much more 
 easily determined than others. Thus, the easiest divisors 
 are those that are nearly some multiple of 10 as 61, 71, 
 81, etc. ; or 59, 69, 79, etc., the child thinking as he uses 
 them, "About 60, about 70, about 80," etc. The easiest 
 quotients to determine are those that fall about midway 
 between two consecutive multiples of 10, as 65 (midway 
 between 60 and 70), 74, 75, 76, 84, 85, 86, 54, 55, 
 56, etc. 
 
 It is seen then that, as to difficulty, there can be at 
 least three grades of long division exercises. The divi- 
 dends should be made by selecting the divisors and quo- 
 tients wanted. The types are 61X75; 59X69; 64X59. 
 The first of these will have an easy divisor (61) and an 
 easy quotient (75); the second an easy divisor (59),
 
 68 THE TEACHING OF ARITHMETIC 
 
 but a hard quotient (69) ; and the third, both a hard 
 divisor (64) and a hard quotient (59). 
 
 76 
 
 Thus, to find 6156-7-81, as shown by fol- 81)6156 
 
 lowing the work in the margin, the only con- 567 
 
 sideration is 61-f-8 = 7 and 5 remaining; 486 
 
 48-^8 = 6. 486 
 
 In 4209 -5- 61, as may be observed, the consideration is 
 not merely 42-7-6 = 7, but a further observation 69 
 
 that the next digit (0) of the dividend does 61)4209 
 not contain the next digit (1) of the divisor 7 366 
 times, and hence the quotient digit cannot be 549 
 
 larger than 6. 549 
 
 While in 2124-J-36, the pupil who makes the same kind 
 of estimate as he made above will observe that 21-r-3 = 7 
 and try 6. But he must be shown that, before 59 
 
 writing down 6 in the quotient, he should 36)2124 
 think 3X6 = 18; 21-18 = 3; this 3 with the 2 180 
 makes 32; now, 32 does not contain 6 six 324 
 
 times, hence the quotient cannot be larger 324 
 
 than 5. 
 
 These three illustrations should make it clear that 
 there are at least three grades of exercises as to diffi- 
 culty, even in exercises containing the same number of 
 figures. 
 
 A large number of each class of exercises should be 
 worked out and charts prepared as follows :
 
 THE PRIMARY FACTS OF DIVISION 69 
 
 Type I, 
 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 F 
 
 G 
 
 H 
 
 1 
 
 2025 
 
 2916 
 
 3645 
 
 4293 
 
 3807 
 
 4374 
 
 6103 
 
 5265 
 
 > -J-81 
 
 2 
 
 6885 
 
 5994 
 
 6966 
 
 5427 
 
 6804 
 
 7695 
 
 6075 
 
 7614 
 
 3 
 
 3726 
 
 6156 
 
 7857 
 
 5184 
 
 4617 
 
 7723 
 
 7533 
 
 5913 
 
 TypeH 
 
 4 
 
 5551 
 
 4819 
 
 3111 
 
 4941 
 
 3599 
 
 5978 
 
 2562 
 
 5429 
 
 > -4-61 
 
 5 
 6 
 
 5002 
 
 4758 
 
 3782 
 
 3538 
 
 4209 
 
 3172 
 
 4148 
 
 2379 
 
 2501 
 
 3721 
 
 4392 
 
 2928 
 
 5612 
 
 2989 
 
 2867 
 
 4331 
 
 TypeHI 
 
 7 
 8 
 9 
 
 4788 
 
 7068 
 
 5544 
 
 4104 
 
 2052 
 
 5548 
 
 7372 
 
 5092 
 
 > -^76 
 
 4028 
 
 6308 
 
 3496 
 
 5776 
 
 4864 
 
 2888 
 
 6536 
 
 6612 
 
 2736 
 
 5624 
 
 7296 
 
 2812 
 
 2128 
 
 4332 
 
 3572 
 
 3268 
 
 In type I, it is easy to estimate the quotient figure, for 
 dividing the number represented by the first two figures 
 of the dividend by the first number in the divisor always 
 gives the correct quotient figure. This is not always the 
 case in type II. A mental multiplication, however, will 
 show whether or not this result follows. But in type III, 
 more careful consideration is required. 
 
 There are two uses of such drill charts : (1) They may 
 be used for "sight work" in which the pupil will give the 
 first quotient figure only ; and (2) they may be used for 
 the usual written work.
 
 CHAPTER VIII 
 THE USE OF GAMES IN NUMBER WORK 
 
 IT is clearly recognized by all who have given any 
 thought to the subject, that a very necessary condition 
 for true learning is that the process be self-actuated 
 through motive or interest. If the thing taught a child 
 can be made to meet his needs or interests, he finds but 
 little trouble in learning it. Hence, a problem of the 
 teacher is to find a use to the child for the number facts 
 that she must teach him. A number problem has no\ 
 great interest to a child unless the answer to it serves 
 some purpose meets some personal need or appeals 
 to his curiosity. The range of problems that have any 
 real interest to a pupil learning the primary facts is so 
 limited that they are not sufficient to hold the interest 
 long. 
 
 Play is the chief occupation of most children of the 
 lower grades, hence the appeal through the play instinct 
 games, dramatized vocations, etc. is a very strong 
 one, and may be used to very great advantage in the 
 early number work. 
 
 The games may be grouped as to use as : (1) school- 
 room games,; (2) playground games; and (3) home 
 games. 
 
 79
 
 THE USE OF GAMES IN NUMBER WORK 71 
 
 The schoolroom games must be those through which 
 the teacher may get a maximum amount of number 
 combinations before the class with as little of the time 
 taken up by the game as possible. The games must be 
 so selected as to have all pupils silently attending to the 
 number combinations when not actually reciting. 
 
 The school ground and home games are more to furnish 
 a motive for learning the work presented in class and, 
 hence, the number work should in no way overshadow 
 the recreative elements of the game. But such games 
 serve to show the pupil that what he learns in school may 
 be made use of out of school ; and, by turning them into 
 "make-believe" games, they may be used in the class- 
 room drills, as will be shown in the games that follow. 
 
 As to then* nature, both the in-school and out-of-school 
 games may be grouped under three general heads : (1) scor- 
 ing game's; (2) imaginative or "make-believe" games; 
 and (3) games of motor activity. 
 
 SCHOOLROOM GAMES 
 SCORING GAMES 
 
 The scoring games are perhapB mmt familiar to 
 all. They include Bean Bag, Rin|H J Ten Pins, and 
 such standard games as may be founom any toy shop. 
 Yet, as found in the toy shop, they bring in but few of the 
 combinations. 
 
 The essentials of a scoring game are : (1) that so little 
 skill is required to score that every "throw" will give a 
 number for a combination; and (2) that the targets in-
 
 72 
 
 THE TEACHING OF ARITHMETIC 
 
 elude all the nine digits. Below are targets that may be 
 used in some of the games that follow : 
 
 / 
 
 7 
 
 4 
 
 5 
 
 9 
 
 6 
 
 J 
 
 8 
 
 2 
 
 There are many tossing, bowling, and shooting games 
 that may be easily devised by any resourceful teacher. 
 The tossing, bowling, or shooting, however, does not 
 have to be carried on to any great extent in the school- 
 room. Any of the following scoring games may be used 
 to score a few times, then the game turned into a " make- 
 believe" game. Thus, the teacher may point to any 
 number on such targets as these drawn above and pre- 
 tend that they are the scores made by pupils, thus motivat- 
 ing the work through "make-believe" target practice. 
 
 THROWING THE ARROW 
 
 A target about two feet across, like any of those shown, 
 may be made 0! aBt board and covered with burlap to 
 hide the marks Wm the shots, and hung against the 
 wall. . The pupils throw an arrow made as follows : 
 take a large pin or a slim finishing nail sharpened to a 
 point, a small cork about one inch long and three fourths 
 of an inch in diameter, and three feathers from five to 
 seven inches long, and make an arrow as shown in the 
 picture. This can be thrown with great accuracy.
 
 THE USE OF GAMES IN NUMBER WORK 73 
 
 Let each child throw twice and have the sum or product, 
 depending upon the drill, of the numbers "hit if he calls 
 them correctly. After a few throws, the .teaCher may 
 merely point to two numbers 
 on the target, asking a pupil 
 to name the score. If correct, 
 it is added to his other scores. 
 After a pupil can perform 
 written addition, he may add his scores every five turns 
 or chances and thus the same game may motivate the 
 written addition. 
 
 BEAN BAG 
 
 Take a large cardboard, or a light board, and make 
 nine holes in it, numbering them from 1 to 9 inclusive. 
 Leaning it against a support of some kind, the children 
 toss small bean bags through the holes, scoring as hi the 
 arrow game. 
 
 HOOK IT 
 
 A board with nine hooks on it, numbered from 1 to 9 
 inclusive, is hung against the wall. The children try to 
 ring the hooks by tossing soft rubber rings. 
 
 BRIDGE-BOARD MARBLE GAME 
 
 Cut nine semicircular archways about three inches in 
 diameter from one edge of a board. Then support the 
 board edgewise on the floor so that these arches are next 
 the floor. Number the archways from 1 to 9 and let 
 the children roll marbles through the openings, scoring 
 as in the preceding games.
 
 74 THE TEACHING OF ARITHMETIC 
 
 BOWLING WITH MARBLES 
 
 A target may be marked off on the floor and a marble 
 rolled into it, the child scoring the numbers upon which 
 the marble stops. 
 
 These are but a few suggestions as to the possibilities 
 with scoring games. The methods of scoring may easily 
 be varied. 
 
 IMAGINATIVE OR "MAKE-BELIEVE" GAMES 
 
 These include the dramatization of various vocations, 
 as "playing store," "the mailman," "the milkman," 
 and also the pretended playing of some scoring game, or 
 other game, by use of diagrams drawn upon the black- 
 board. 
 
 There is no end to imaginative games that may be 
 made by the teacher. The following will show how a re- 
 sourceful teacher used a game, which she called " Playing 
 Fireman," for several weeks without any loss of interest, 
 and how the game suggested itself to her. The fire de- 
 partment had just passed the school ground during inter- 
 mission and the children were greatly interested in 
 watching it. When the period for number work came, 
 the teacher asked how many would like to play "Fire- 
 man." Of course, all wanted to play. She stepped to 
 the blackboard and drew two ladders leading to the top 
 story of a burning building. Between the rungs of the 
 ladders she placed the number combinations upon which 
 she wished to drill. She said, "Now we will play that
 
 THE USE OF GAMES IN NUMBER WORK. 75 
 
 the one who can run up one ladder and down the other 
 without making an error has rescued some one from the 
 burning building; but those who make a mistake have 
 fallen down and so are not good firemen." 
 
 After a delightful period of "Playing Fireman" it 
 was suggested that two permanent companies should 
 be organized and that each company should have a fire 
 chief. The pupils said that the strongest and boldest 
 quickest and most accurate should be chosen chief ; 
 but, of course, each child hoped that he might be the one 
 chosen for that office. Then followed more drill for a 
 few days before having a contest to determine who should 
 be the chiefs. At last, two were chosen. The chiefs 
 wanted to choose then: own companies. As a result, 
 more drill was needed in order that the chiefs might see 
 whom they wished to choose. The first day resulted in 
 choosing but two or three for each side. Finally, all 
 were chosen. The next step was for each chief to train 
 his men for a special exhibition of skill. A day was fixed 
 for the exhibition. Then each man in each company 
 gave all the combinations. The time required and the 
 errors made were noted in order to tell which company 
 won the prize. 
 
 Then there was a big fire in a down-town district and 
 whole blocks were aflame and many ladders were put up 
 to rescue people. Each team was greatly interested in 
 seeing who could save the most people in a given 
 time. 
 
 This is discussed in detail as being somewhat typical 
 of what may be done with many such number games.
 
 76 ' THE TEACHING OF ARITHMETIC 
 
 .SIMON SAYS "THUMBS UP" 
 
 It is often desirable to have a single pupil recite all of 
 a group of facts. Thus, a teacher may wish to have a 
 pupil give several or all of the facts of addition or multi- 
 plication. But, unless she can in some way get the 
 attention of the whole class and get all to attend silently 
 to the combinations, the time of all but the one reciting 
 is wasted. The following game shows how a resourceful 
 teacher got the attention of the whole class in such a drill 
 without any loss of time and without disturbance or extra 
 material. 
 
 She called the game, " Simon Says * Thumbs Up.' ''' One 
 pupil stood before the class and showed the number cards. 
 Another stood and gave the combinations. The class 
 sat with "thumbs up." When a mistake was made, all 
 thumbs were to turn down. The last to turn thumbs 
 down stood to give the combinations, and the first to do 
 so had the privilege of showing the cards. 
 
 SURPRISE PARTY 
 
 In using toy money and making change a good plan 
 is to have a "Surprise Party." Ask the children how 
 many would like to go to a surprise party. Of course, 
 all want to go. Then ask what each would like to take 
 and write his answer on the board, as apples, bananas, 
 oranges, candy, nuts, etc. Next, appoint a clerk to step 
 to the board and write the prices at which the articles 
 can be bought at the store. Give the children toy money 
 and let them buy the things of the clerk. Beads, blocks,
 
 THE USE OF GAMES IN NUMBER WORK 77 
 
 chalk, etc., may represent the articles. If the clerk makes 
 a mistake, discharge him and let some one else take his 
 place. 
 
 PLAYING STORE 
 
 This game may be adapted to any of the primary grades 
 and resembles somewhat the "Surprise Party." A table 
 at the front of the room may serve as a counter of the 
 store. Let the children determine the nature of the 
 things to be bought and sold fruit, flowers, toys, sport- 
 ing goods, furniture, groceries, dolls, or candy. If varied 
 frequently, the interest is keener. Things made in 
 manual training or handwork periods may be used to 
 form the stock. Charts may be used in place of actual 
 objects. Added interest is gained by having toy money, 
 toy scales, and a toy telephone. The children ask the 
 prices of what they want, then determine how much they x 
 will buy. They should calculate the amount of their 
 purchases so as to be able to detect errors on the part of 
 the storekeeper. 
 
 LUNCH ROOM 
 
 One child is the keeper of a make-believe lunch room. 
 The other children pretend to buy their lunches from him. 
 A chart showing the cost of each article of food should be 
 posted. 
 
 SOLDIERS 
 
 The children form a company and choose their captain. 
 The captain forms his men into ranks and files. He 
 then drills them with number combinations. The answers 
 should be given promptly, accurately, and with decision,
 
 78 THE TEACHING OF ARITHMETIC 
 
 as becomes a soldier. Should they be given lazily or 
 incorrectly, that soldier loses his rank, or may even be 
 sent to barracks (out of the game). Sometimes two 
 companies may combat with each other, the company 
 giving most combinations correctly being victorious and 
 taking the others captive or capturing their standard. 
 
 KING OF THE CASTLE 
 
 A child, the king, sits on a chair in the front of the 
 room. The other children try to dethrone the king by 
 giving him number combinations which he cannot answer. 
 As long as he gives correct answers, he remains on his 
 throne in the castle ; but, when he fails, the child who gave 
 the combination becomes king in his place. 
 
 TOM TIDDLER'S GROUND 
 
 A square on the floor represents "Tom Tiddler's 
 Ground," in the center of which Tom is drawn. Some 
 one asks, " Who will dare to go on Tom Tiddler's ground ? " 
 A boy in the class stands in front of the room to repre- 
 sent Tom. The child who dares to go on Tom Tiddler's 
 ground is then questioned by Tom, who asks various 
 number combinations. If correct answers are given 
 three times, the child may stay on his ground without 
 flanger; but if he misses, he is chased out by Tom and 
 some one else tries. 
 
 HORSE RACE 
 
 Lay before each child a flash card, as 6+4. Give a 
 quick review to be sure each child knows his own card
 
 THE USE OF GAMES IN NUMBER WORK 79 
 
 correctly. Choose as many children for horses as there 
 are aisles. The number cards are the hurdles, and the 
 horses all start together. The horses giving correct 
 combinations have surmounted the hurdles and pass on 
 down the aisle. If a horse makes a mistake, the child 
 holding that combination stretches his hand across the 
 aisle. This barrier cannot be passed until the correct 
 combination is given. The first horse to give all the com- 
 binations correctly in his aisle wins the race. 
 
 ELEVATORS 
 
 Write a series of combinations as in the margin. 4 
 To go up, add. To come down, subtract. Each 2 
 combination represents a Jbor. If an incorrect 3 
 answer is given, the elevator stops. 2 
 
 5 
 
 2 
 JACK AND THE BEAN STALK 
 
 After telling the story say to the children, "Let us 
 play that we are going to climb the bean-stalk ladder that 
 Jack climbed, only we are going to climb it in a different 
 way." Make a ladder of cardboard so that all the class 
 can easily see it, or draw one on the board. In the spaces 
 write number combinations in addition, subtraction, or 
 multiplication. The one who can go up and down 
 the ladder most rapidly (by giving the answers to the 
 combinations) and without stumbling or slipping (making 
 a mistake) can be Jack. The one who makes no mis- 
 takes and takes the shortest time to go up and down the
 
 80 THE TEACHING OF ARITHMETIC 
 
 ladder makes the best Jack. A hard sum to overcome 
 may be given for the giant and the one who can conquer 
 this big giant is really " Jack-the-Giant-Killer." 
 
 PICKING APPLES 
 
 Draw an apple tree on the board with apples on it. 
 Each apple has on its side a number combination. The 
 children pick apples by giving the combinations. The 
 one who picks the most apples wins the game. 
 
 SAVING THE CHEW 
 
 A sinking ship is drawn on the board. Half of the deck 
 of the ship is in the water. Combinations as 8+2, 2X7, 
 82, 10+2, etc., are placed on the mast and on every 
 part of the ship which is above water, to represent the 
 crew of the ship waiting to be rescued. Then as many 
 boats as there are pupils are drawn. Each child owns a 
 boat and his initial is placed on it. Then, the one who 
 saves the most of the crew that is, tells the greatest 
 number of combinations is captain. For each man 
 rescued, the child gets a flag which he places in his boat. 
 
 PLAYING SOLDIEES 
 
 Divide the class into two armies, A and B. Let them 
 choose captains. The captain of army A asks a soldier 
 of army B a combination. If the soldier cannot reply 
 correctly, army B loses a soldier and army A gets him. 
 If he replies correctly, however, he has the privilege of 
 asking a soldier of army A a combination. The army
 
 THE USE OF GAMES IN NUMBER WORK 81 
 
 having the most soldiers at the end of a stated tune wins 
 the battle. This game may be used in subtraction, 
 addition, multiplication, or division. 
 
 A HOME-RUN 
 
 Divide the class into two baseball teams. Draw a 
 baseball diamond upon the blackboard and write three 
 or four numbers at each base, as in the figure. Some 
 number to be added or to be 
 used as a multiplier is written 
 in the pitcher's place. The 
 pupil makes a home-run by 
 giving all the answers (sums 
 or products) as the teacher, 
 starting from the home plate, fi 
 points to some number at 
 each base. Thus, to use the 
 game to multiply by 8, the 
 teacher may point to 6, 9, 7, 
 and 5, the pupil calling 48, 
 72, 56, 40, which constitutes a home-run. If a mistake 
 is made, say at the second base, the umpire calls "out 
 on second." The game is to see which team gets the 
 most home-runs. 
 
 A GTJESSING GAME 
 
 L 
 
 As a drill in addition, the teacher may say, "I am 
 thinking of two numbers which added make 10." The 
 pupils guess all of the combinations of numbers which 
 added make 10, and when the last combination is given,
 
 82 THE TEACHING OF ARITHMETIC 
 
 the teacher may call that the right one. The same drill 
 may be used in multiplication. For example, the teacher 
 may say, "I am thinking of two numbers whose product 
 is 24." The pupils will guess two 12's, three 8's, six 4's. 
 Or, the teacher may say, "I am thinking of the 'nine 
 times' table." A child will ask, "Are you thinking of 
 81?" The teacher will say, "No, I am not thinking of 
 9X9." When the correct answer is given, the guesser 
 may suggest the table and answer the questions, and 
 thus it may be carried out between two children, the 
 teacher acting as umpire. 
 
 STORY GAMES 
 
 The stories with which the children are familiar, such 
 as "Clytie," "Snowwhite," "Sleeping Beauty," etc., 
 may be utilized in addition, subtraction, and multiplica- 
 tion drills. For example, in "Sleeping Beauty" a picture 
 of the schoolhouse, thorns, huge walls, and a castle may 
 be drawn on the board. Several combinations may be 
 placed on each and a boy may start for jthe castle. If he 
 succeeds in getting into the castle, having overcome all 
 obstacles by giving the correct combinations on the various 
 obstacles, he is dubbed knight and some emblem is pinned 
 upon the lapel of his coat to signify as much. If he fails, 
 some one else may try. Competition may be gained by 
 having several such series on various boards, and several 
 children trying to gain the castle first. The one who 
 gives the combinations soonest is, of course, the honored 
 one. 
 
 The children may take a ride with Clytie in her wonder-
 
 THE USE OF GAMES IN NUMBER WORK 83 
 
 ful seashell carriage, or visit the seven dwarfs in their 
 mountains of gold. If so, the dwarfs or mermaids re- 
 spectively may hold large cards on which are combinations, 
 and these must be given before the child is allowed to 
 arrive and visit the dwarfs' cave or Clytie's beautiful 
 home. The children are enthusiastic over these simple 
 devices. 
 
 These examples will serve to show the possibilities 
 along this line of schoolroom games. A teacher inter- 
 ested in devising games to motivate her work will find 
 suggestions on every hand. 
 
 GAMES OF MOTOR ACTIVITY 
 
 In schoolrooms devoted to a single grade, so that 
 moving about does not disturb others, there are many 
 games in which there is some movement ; yet the amount 
 of number work involved is great enough to meet the 
 requirements of the classroom. 
 
 O-U-T SPELLS OUT 
 
 The pupils may be arranged in a circle or remain in 
 their seats. The first child gives some combination, as 
 "5 and 7." The next in turn gives the sum and some 
 other number, as "12 and 3." The next may say, "15 
 and 8." The game continues in this way, the first num- 
 ber given each tiDaebfiiftg^tbe,8um of the last two numbers 
 given. For the fist mistake 'a child makes he scores the 
 letter O ; for the second, It ; and for the third, T. He is 
 then out of the game.
 
 84 THE TEACHING OF ARITHMETIC 
 
 BASKETBALL 
 
 Large numbers are pinned on each child. The chil- 
 dren stand in a circle about one child in the center who 
 tosses a large ball to some one in the circle, at the same 
 time calling out some number. If the child in the circle 
 catches the ball and gives the correct sum, he exchanges 
 places with the one in the center and is privileged to toss 
 the ball. If he fails to catch the ball or gives the wrong 
 sum, he remains in the circle. 
 
 A COLUMN RELAY RACE 
 
 Divide the class into two teams. Give each a large 
 card containing a number less than ten. The children 
 stand in line. The leader of each team calls out his num- 
 ber and shows his card. The next in turn shows his num- 
 ber and adds it to the first, giving the sum. The next 
 shows his number and adds it to the sum already given, 
 and so on to the end of the line. The line completing 
 the addition first wins the game if the sum is correct. 
 
 The numbers of one team should duplicate those of 
 the other but should be arranged in different order. The 
 sums will then be the same. 
 
 A NUMBER RACE 
 
 Divide the class into two teams The Reds and The 
 Blues. Write several numbers not larger than eighteen 
 upon the blackboard, as 12, 17, 15, 13, and 18. Let the 
 leader of each team go to the board. The remaining
 
 THE USE OF GAMES IN NUMBER WORK 85 
 
 pupils in turn, alternating from one side to the other, 
 call some combination, the sum of which is on the board. 
 Thus, a pupil from the Blues calls "7 and 8." The two 
 pupils at the board see who can first cross out 15. After 
 each has called a combination, the scores are taken. If 
 the leader from the Reds has won 8 tunes and the one 
 from the Blues has won 6 tunes, the score is 2 for the 
 Reds. If a pupil calls a combination whose sum is not 
 on the board, that takes one from the score of his side. 
 
 A BLACKBOARD RELAY RACE 
 
 Before the class meets, the teacher places all the com- 
 binations to be drilled upon on the board, beginning at 
 the ends of the board and working toward the center, at 
 which a goal is drawn. The same combinations are on 
 either side of the goal but written in a different order. 
 The class is divided into two teams. At a given signal, 
 the leader of each team runs to the board and records the 
 result of the first combination, returns and gives his chalk 
 to the next in line, who records the next result, giving 
 his chalk to the next in order, and so on until the goal is 
 reached. 
 
 OLD WITCH 
 
 Have an old witch, mother, and children. Give the 
 children numbers such as 15, 14, 16, etc. The old witch 
 comes to the door and says, "I want a child." "Is it 
 8 and 7?" asks the mother. "No, it is not 15," says the 
 witch. "Is it 7 and 7?" asks the mother. "Yes, it is 
 14," says the witch. "You must catch me then," says
 
 86 THE TEACHING OF ARITHMETIC 
 
 14, and he runs. If the old witch catches him, he is hers ; 
 if not, he is free. If the child does not know the combina- 
 tion and is not ready to run, he belongs to the old witch. 
 The teacher may take the place of the mother, so that 
 new combinations may be given. Addition, subtraction, 
 and multiplication combinations may be used. 
 
 SHEPHERD AND WOLF 
 
 One child is the shepherd and one is the wolf, and all 
 the rest are sheep. The shepherd stands at one "end of 
 the room, the sheep at the other, and the wolf between 
 the two. The shepherd has some number, as 5. The 
 sheep each have numbers as 1, 2, 3, 4, 5, etc. The wolf 
 gives a number, as 12. When the wolf says 12, the num- 
 ber that, added to the shepherd's number 5, will make 
 12 i.e. 7 must run across the room. If the wolf 
 says 8, then number 3 runs, etc. If a sheep fails to run 
 when he should, he belongs to the wolf, who takes him to 
 his den. If he runs and is not caught, he may go back 
 to his former place. 
 
 BEAST, BIRD, AND FISHES 
 
 Children stand in the circle with one child in the 
 center. The one in the center points quickly to a child 
 in the circle, giving him a combination to answer. Then 
 the child in the center counts 1, 2, 3, 4, 5. The one 
 pointed to must give the answer before 5 is counted or 
 he is "out" of the game.
 
 THE USE OF GAMES IN NUMBER WORK 87 
 
 WHO TAPS 
 
 One child is outside of a circle of children. He taps 
 a child on the back. "Who taps?" demands the one 
 tapped. The answer given is a combination. Then the 
 other must reply with the answer to the combination or 
 else go outside of the circle. 
 
 CATCH THE BALL 
 
 The children form two lines. Each has a number 
 pinned on in full view. Some one tosses a ball, or a bean 
 bag, to some one on the other side, who gives the sum 
 (product or difference) of his number and that of the 
 pitcher. If correct, he tosses it back to some one on the 
 first side, who answers in the same way. When a mis- 
 take is made, the one making it drops out of the game 
 and the ball or bag goes to the head of the opposite side 
 to toss. 
 
 NUMBER BASKET UPSET 
 
 The children sit in a circle with numbers pinned on 
 them. Some one stands in the center and gives a com- 
 bination three times in succession, as "7 and 2, 7 and 2, 
 7 and 2." The child with 9 must say "9" before the 
 one in the center has finished. If he is not quick enough, 
 he forfeits his seat to the child in the center and then 
 gives the combinations. The one in the center may at 
 any time say, "Number basket upset," and every child 
 must change his seat, the one in the center also trying 
 to get a seat. There will be one child left without a chair 
 who must be in the center and give the combinations.
 
 88 THE TEACHING OF ARITHMETIC 
 
 GRUNT 
 
 Each child is given a number, and all but one form a 
 circle. One child in the center is blindfolded. He has 
 a pointer or small rod. The children march around him 
 in a circle. When the one in the center taps the floor 
 with the rod, all stop. He touches some one in the circle, 
 who takes one end of the rod and gives the sum of his 
 number and the number of the one who is blindfolded. 
 The one blindfolded then "guesses" the number of the 
 one touched, which he does, of course, by subtracting 
 his number from the sum given. If correct, the one 
 touched is blindfolded. 
 
 A JUMPING GAME 
 
 Two children "turn" a rope and the others stand in 
 line ready to jump the rope in turn. Each child is to 
 jump as many times as he can, counting by some number, 
 as 5, each tune he jumps. Whoever counts farthest wins. 
 If he makes a mistake in counting, he stops jumping. 
 This gives practice in adding equal addends. 
 
 BALL AND HOOP 
 
 A hoop is suspended in the room. The object of the 
 game is to throw a ball through the hoop. The players 
 are divided into sides and take turns alternately. Each 
 time a child throws the ball through the ring, the captain 
 of the opposing team challenges him with some number 
 combination. If he can answer it, he scores two for his
 
 THE USE OF GAMES IN NUMBER WORK 89 
 
 team ; but, if he cannot answer it correctly, he loses one 
 point for his team. 
 
 HOT BUTTER BEANS 
 
 Instead of hunting for a bean, as in the real game, 
 let the children hunt for a numbered block. Let one 
 child hide the block. When ready, the others hunt. 
 The hider says "hot" or "cold" according to the near- 
 ness of the children to the block. The child who finds 
 the block adds the number to his score if he can call the 
 combination quickly. He then hides a block. The 
 number on the block is changed each time. Let the 
 final score be 100, and the child which gets it first wins 
 the game. 
 
 FENCING 
 
 The children stand in two lines facing each other. 
 Each child must be opposite an opponent. Then some 
 number is decided upon, as twelve. One child gives a 
 number, such as seven, and his opponent is required to 
 give the number which must be added to make twelve. 
 This is done all along the line. Whenever a child misses, 
 the mistake is recorded against him. The mistakes are 
 counted after a given time, and the side which has made 
 the fewest mistakes wins. 
 
 OUT-OF-SCHOOL GAMES 
 
 Most of the games played by children out of school 
 may be adapted to number games without taking away 
 the recreative elements of the games. These games
 
 90 THE TEACHING OF ARITHMETIC 
 
 should not be given as a requirement, nor should they 
 involve enough number work to task the children's 
 abilities. The purpose of them is not drill in number 
 work, but they are given to motivate the classroom work 
 by showing a use of the things they are learning. The 
 following will suggest ways of using the well-known games. 
 
 Fox AND GopSE 
 
 Large numbers are pinned on each child. Choose one 
 child for the fox and one for a goose. The other children 
 stand in pairs about the playground with linked arms. 
 The fox tries to catch the goose. In order to escape from 
 the fox, the goose may link arms with any one of any 
 couple, first calling out the sum of the numbers of the 
 couple. The pupil with whom the goose did not link 
 arms then becomes goose. If the goose calls the wrong 
 sum, she is not safe and must run to another couple and 
 try again, or be caught. 
 
 PUSSY WANTS A CORNER 
 
 All children are numbered with large visible numbers, 
 each two children having the same number, and all but 
 one has a corner (a base of some sort). The child with- 
 out a corner goes from one to another saying, "Pussy 
 wants a corner." The one to whom it is said asks, "What 
 corner?" The reply is some two numbers whose sum 
 (or product) is represented by the children. The two 
 children having the sum (or product) exchange places. 
 The child without a corner tries to get one of the vacant 
 places. Thus, if he answers "7 and 8," the two 15's
 
 THE USE OF GAMES IN NUMBER WORK 91 
 
 exchange places. If he can get one of the places, the one 
 left becomes "pussy." 
 
 SQUAT TAG 
 
 Pin a visible number on each child. Let one child 
 be chosen as "it." He chases some one of the other 
 children. To prevent being caught, the one who is 
 chased squats down, giving the sum (or product) of the 
 number pinned on "it" and his own number. If the sum 
 (or product) is incorrect, he may be caught and then 
 becomes "it." 
 
 Fox A,NB GEESE 
 
 Divide the class into foxes and geese. Give the foxes 
 a certain amount of space and geese the same amount. 
 Geese fly over into the territory occupied by the foxes. 
 If a goose is caught there, she must answer a combination. 
 If she answers correctly, she may go free; if not, she 
 must be a fox. Foxes may visit the home of geese and 
 take away with them all the geese which cannot answer 
 correctly the combinations put to them. The aim of the 
 geese is not to be caught; or, if caught, to answer cor- 
 rectly the combination given so that they may go free. 
 
 TAP THE RABBIT 
 
 All the children but one, who is to be the leader, form 
 in a ring. Each child has a number pinned upon him. 
 The leader runs around the ring and taps some one in 
 the ring and calls a number. The one tapped gives the 
 combination of his number with the number called, before 
 he starts around the ring in the opposite direction. The
 
 92 THE TEACHING OF ARITHMETIC 
 
 one reaching the vacant place first stays in the ring ; the 
 other is leader. This game may be played with addition, 
 multiplication, or subtraction combinations. 
 
 THREE DEEP 
 
 Have two rings of children, one child in front of the 
 other. Have two children outside of the ring, one pur- 
 sued by the other. All have numbers pinned on front 
 and back of them. The one pursued runs and stands in 
 front of a child in the ring, who quickly gives the sum of 
 his number and the number of the child who stood in 
 front of him and immediately chases the former pursuer. 
 If a child is caught, he must become the pursuer. 
 
 DROP THE HANDKERCHIEF 
 
 The children form a ring. A number is placed in the 
 center of the ring. One child runs around the outside 
 of the ring with a handkerchief. He throws it behind 
 some one in the ring and calls some number less than the 
 one in the ring. The child behind whom it is thrown 
 quickly subtracts the number called from the number in 
 the ring, and starts around the ring in the same direction. 
 If he overtakes the one who dropped the handkerchief 
 before he reaches the vacant place, he then drops the 
 handkerchief. If he does not, he is out of the game. 
 
 HANDS UP 
 
 The children form a ring around the room or on the 
 playground. A number is pinned on each child. A 
 catcher moves around inside of the ring. The children
 
 THE USE OF GAMES IN NUMBER WORK 93 
 
 tantalizingly hold up both hands above their heads, 
 tempting the catcher to tap them while the hands are 
 raised. If he can succeed in touching any of them while 
 their hands are raised, and at the same time can give the 
 combination of his number and that of the one caught, 
 he takes the place of the one caught, who then becomes 
 catcher. 
 
 HOP AND SHAKE HANDS 
 
 A ring is formed and the children have numbers pinned 
 on them. One person is "it." He runs around the 
 ring, taps some one on the back, and calls some number. 
 Before the one tapped can run, he must add his number 
 to the number called and give the sum. After he gives 
 the sum, he runs in the opposite direction to meet the 
 tapper. When they meet, they shake hands and then, 
 hopping on one foot, they try to see which one can get 
 in the vacant place first. 
 
 WITCH'S CIRCLE 
 
 One child is the witch. All the children run across 
 the line and, if the -witch catches a child, she places him 
 in a small circle containing a number combination. If 
 he can answer the combination in the small circle as soon 
 as he is placed in it, he may go free ; if not, he must stay 
 until the old witch releases him. 
 
 IN THE POOL 
 
 There is one child at each corner of a square, making 
 in all four players called "strikers." The other players
 
 94 THE TEACHING OF ARITHMETIC 
 
 stand in the middle of the square called "the pool." 
 The players in the pool have numbers pinned on them. 
 This game may be used in multiplication or addition. 
 Let us suppose that it is the 4's in multiplication. The 
 four players at the corners throw a ball from one to 
 another. When they catch the ball, they try to touch one 
 of the children in the pool with it. If that child's num- 
 ber is 6 and he at once calls "24," he may stay in the 
 pool. If he makes a mistake, it is the "striker's" place 
 to correct him. If the "striker" does not do this, some 
 one in the pool may call out the answer. The child who 
 calls the answer may then take the "striker's" place. 
 The object of the game is to see who can stay "striker" 
 the longest. 
 
 HAVE You SEEN MY SHEEP 
 
 The children form a circle, one child outside being the 
 shepherd. The shepherd taps some one on the back, 
 saying, "Have you seen my sheep?" The one tapped 
 will reply, "No, how many pounds did he weigh?" The 
 shepherd will reply with some such combination as 3x5, 
 or 8+9, or 10 4, and then start to run around the circle. 
 The one tapped must give the answer to the combination 
 before he can run after the shepherd, who tries to get 
 around to the vacant place before he is caught. If the 
 shepherd is not caught, the one tapped becomes shepherd. 
 If the one tapped cannot answer his combination cor- 
 rectly, he must take his place in the center of the circle.
 
 CHAPTER IX 
 COMMON FRACTIONS 
 
 SINCE a little objective work in fractions is done in 
 the lower grades, before a systematic study of the subject 
 is taken up in the fifth and sixth grades, it seems best .to 
 break up the discussion into two parts, the first relating 
 to the work of the primary grades, and the second to the 
 work of the intermediate grades. 
 
 I. EARLY WORK IN FRACTIONS 
 
 The work done in fractions in the first four grades 
 is preparatory to the written work with symbols in the 
 fifth grade. A clear notion of fractions cannot be obtained 
 from the symbols. The first work must be objective. 
 By the use of objects, diagrams, etc., the child sees a half, 
 a fourth, or any fractional unit as an individual unit, just 
 as a quart, or a foot. He adds, subtracts, multiplies, or 
 divides a number of these units just as he would a num- 
 ber of integral units. Let him get the names of these 
 fractional units by having him notice the similarity of 
 sound between the number of equal divisions and the 
 names of the parts. Thus, one of four equal parts is a 
 fourth; one of five equal parts, a, fifth; one of six, a sixth; 
 etc. 
 
 95
 
 96 THE TEACHING OF ARITHMETIC 
 
 First study the fractions in related groups as halves 
 and fourths, then halves, fourths, and eighths, etc., and 
 learn the relations among them. This should be done 
 objectively so that a child sees that a half equals two 
 fourths, just as he sees that a quart equals two 
 pints, etc. 
 
 If the work is objectively presented, the child is able 
 to use intelligently the common fractions before know- 
 ing the notation for them. Thus, he knows that a pint 
 is half of a quart, that a quart is a fourth of a gallon, etc. 
 When a fractional unit is studied, it should not only be 
 introduced by some real need of such a relation, but 
 followed by real applications. If halves and fourths are 
 studied, there are many real applications. Thus, if milk 
 is 10 cents per quart, how much is a pint worth ? If cider 
 is 20 cents per gallon, how much is a quart worth? If 
 walnuts are 80 cents a bushel, how much is a peck worth ? 
 In this way, through handling objects, folding paper, 
 making diagrams, etc., the pupil comes to think of a 
 fractional unit as a concrete thing, just as an integral 
 unit. Thus he gets the first of the three steps in the 
 proper teaching of fractions, viz. : (1) the meaning, (2) the 
 notation, and (3) the manipulation. 
 
 THE NOTATION OF A FRACTION 
 
 It is very important that the pupil should get the full 
 meaning of the notation of a fraction, for all the manipula- 
 tions depend upon the notation. It is not so important 
 for him -to get the names "numerator" and "denominator" 
 when the notation is first presented; in fact, it may be
 
 'COMMON FRACTIONS 97 
 
 better to avoid using these terms until after the function 
 of the terms is well understood. 
 
 In developing the notation, take some object or dia- 
 gram, and divide it into a number of equal parts, say 4. 
 Then have the children point out some number of these 
 fourths, as 3. Now write 3 on the board and ask if that 
 alone tells us that there are three fourths. Show 3 of 
 some other unit as dollars, cents, feet, etc., and let the 
 pupils see that the 3 shows "how many" and that some 
 other sign is needed to show what they are, as $3, 3^, 3 ft., 
 etc. Then show that a sign is needed to show what the 
 3 fourths are, and that the sign is a 4 written below the 
 3. Thus in the 3 shows "how many" and the 4 shows 
 what they are. 
 
 To test the pupils' understanding of the notation, write 
 several fractions and ask of each fraction, "How many 
 things?" "What are they?" as : 
 
 8527537 3 + 
 
 ?> > ~s> ~s> r, -5, -$> TG, etc. 
 
 This notation is the basis of all the processes and must 
 be thoroughly understood. But when it is understood, 
 the manipulations follow from those of integers. 
 
 THE FUNDAMENTAL PROCESSES IN FRACTIONS 
 
 In teaching the processes they should be related to 
 those already known. Thus, just as $3-f-$l=$4, so 
 f +=! Just as 3 ft. +8 in. cannot be added until 
 changed to a "like unit," so |+i cannot be added until 
 changed to a like unit. In the same way, subtraction 
 follows that of integers.
 
 98 THE TEACHING OF ARITHMETIC 
 
 Just as 3X$5 means $5+$5+$5, or $15, so 3Xf = 
 f +f +f = ; or 3 times 5 of any unit is 15 of that unit. 
 
 When we speak of multiplying by a fraction, we are 
 giving a new meaning to multiplication. The process is 
 based upon partition, not upon addition, as was the case 
 when multiplying by an integer. But just as f of an 
 object is found by first dividing the object into 4 equal 
 parts, then taking 3 of them; or f of a group, as 16, 
 is found by dividing 16 into 4 equal groups, then taking 
 3 of the groups ; so to find f of a fraction means the same 
 thing.X Thus, f of f "means that we first divide each of 
 the three fifths into 4 equal parts, making three twentieths, 
 then take 3 of them, making ^. . That is, f of f = \y. 
 
 Also, just as $8-i-$2 = 4 means that $8 will contain 
 $2 four times, so f-j-f = 4 means that f will contain f 
 four times. Likewise, 8 of any unit will contain 2 of that 
 unit four times. 
 
 And, just as one cannot tell how many times 2 ft. will 
 contain 8 in. without first knowing how many inches in 
 2 ft., so one cannot tell how many times one fraction will 
 contain another until they are both expressed in the same 
 unit. Thus, the first work in the division of fractions is 
 based upon the measurement phase of division of integers. 
 
 H. FRACTIONS COMPLETED 
 
 OBJECTIVE PRESENTATION OF FRACTIONS 
 
 It was shown in the first part of this chapter that frac- 
 tions are studied objectively in the lower grades and that 
 pupils gain naturally and informally one of the several
 
 COMMON FRACTIONS 99 
 
 ideas involved in fractions. This is, that a fraction is one 
 or more of the equal parts into which some whole has been 
 divided. This notion is the basis upon which the funda- 
 mental processes are built. 
 
 In the lower grades and through the use of objects, the 
 child also begins to get one of the real uses of fractions, 
 viz., its use as a means of expressing a relation. He 
 knows that, if he has 4 marbles and loses one half of them, 
 he will have but 2 left; and also that, if he has 4 
 marbles and loses 2 of them, he has lost half of them. 
 This use of a fraction to express a relation comes slowly 
 and gradually to him through such related questions 
 as, "If you have 6^ and spend half of them, how much 
 will you spend?" and, "If you have 6 and spend 3j, 
 what part of your money do you spend?" 
 
 * 
 THE IDEAS INVOLVED IN A FRACTION 
 
 There are three ideas involved in a fraction : (1) a 
 fraction is one or more of the equal parts of a whole ; (2) 
 a fraction is an indicated division; and (3) a fraction is 
 an expression of the ratio of one quantity to another. 
 The development of the fundamental processes from the 
 corresponding processes with whole numbers depends 
 upon the first of these ideas. Much of the pupil's later 
 work depends upon the second and third ideas, and these 
 must be firmly fixed in his mind before the completion 
 of the subject ; but they need not be taken up until after 
 the development of the fundamental processes with frac- 
 tions addition, subtraction, multiplication, and division.
 
 100 THE TEACHING OF ARITHMETIC \ 
 
 A FRACTION AS A PART OF A WHOLE 
 
 First through dividing objects, folding paper, using 
 diagrams upon the blackboard, or with similar illustra- 
 tions, get the pupil to see the principle involved in naming 
 the fractional unit. That is, have him see that, if a thing 
 is divided into four equal parts, each part is a fourth of 
 the whole; if in fine equal parts, each is & fifth; if in six, 
 each is a sixth; etc. Next have him count the fourths, 
 fifths, sixths, etc., just as he would count any unit, and 
 thus f, $, $-, etc., will become just as real numbers of things 
 as 3 feet, 4 quarts, and 5 bushels are. 
 
 It is now important that the pupil get the correct 
 idea of the notation from this point of view of a fraction 
 as a part of a whole. When he points out a number of 
 fractional units as 3 or 4 of them, have him see that just 
 as in three dollars or three feet, 3 only tells how many, 
 and that, to express the whole idea, the sign of the unit 
 is necessary as $3 f and 3 ft. ; so in fractions three fourths 
 is written f, the 3 denoting the number of things and the 
 4 telling what they are, just as "$" or "ft." does. 
 
 A little drill upon such fractions as , , f, f, -fa, etc., 
 asking, "How many?" pointing to the numerator; then, 
 "What are they?" pointing to the denominator, will 
 soon fix in mind the function of the two terms of a frac- 
 tion, a thing much more important than to know their 
 names. 
 
 Knowing the meaning of the terms, however, the names 
 may be so given as to make it easy to remember them. 
 Thus, writing some fractions on the blackboard, the
 
 COMMON FRACTIONS 101 
 
 pupil will tell you that the upper term tells "how many" 
 or the number of things. It is then the numberer or numer- 
 ator, and through the similarity in sound he fixes the 
 name and its meaning. He then tells you the lower 
 term tells what the number of things denoted by the 
 upper term are. Get him to see then that it names the 
 unit or is the namer and hence is called the denominator. 
 The pupil can remember this if the teacher will call his 
 attention to the fact that when we name a man to run for 
 office we nominate him, and that we are " the nominators " ; 
 and likewise the term that names the fraction is the 
 de-nominator. 
 
 A FRACTION AS AN INDICATED DIVISION 
 
 It is essential in changing common fractions to deci- 
 mals, and in other work that follows, that a child get a 
 fixed impression of a fraction as an indicated division^ 
 but it is hard to give him a satisfying basis for this notion. 
 It is as well to make no attempt to give such an idea until 
 after the processes have been mastered. But, having 
 mastered the processes, he knows that finding % of any- 
 thing means to divide it into 4 parts. So i of 8 means 
 8-:- 4. He also knows that of 1 is -j, and that of 8 
 is eight times as large, and hence is f . From this and 
 similar examples get him to see the truth of the following 
 statements : 
 
 iof 8= |, and = 8-5-4; 
 ^ of 6= f , and = 6^3; 
 of 15=^, and =15-5-5.
 
 102 THE TEACHING OF ARITHMETIC 
 
 Hence, 
 
 f= 84-4; 
 
 *- 6-4-3; 
 
 ^=15-J-5. 
 
 Follow this by having the pupils express such forms as 
 3-5-4, 5-f-7, 3-7-8, 2-r-5, etc., as fractions. 
 
 A FRACTION AS A RATIO 
 
 This conception is a slow growth that comes from using 
 a fraction to express the relations of simple objects from 
 the very first conception of a fraction. The pupil may 
 get much help, however, from the meaning of the flota- 
 tion of a fraction. Thus, is 5 of the 8 equal parts; 
 that is, it is 5 of 8. So, if the pupil has eaten 5 of 8 apples, 
 he has eaten "5 of 8" or f of them. If he has 11 marbles 
 and loses 8, he has left 3 of 11 or -n> And thus he comes 
 gradually to see that the ratio of one number to another 
 is the fraction whose numerator is the first number and 
 whose denominator is the second. 
 
 Objectively the pupil may see that the ratio idea is 
 consistent with his first idea of a fraction as one or more of 
 the equal parts of a whole. 
 
 I 
 
 Thus, if we consider 9 things, say, as the whole, then each 
 is one of the nine equal parts that made the whole, or 
 of the whole. And 7 of them is of the whole, and thus 
 the ratio of 7 to 9 is $ ; that is, 7 is % of 9.
 
 COMMON FRACTIONS 103 
 
 ADDITION OF FRACTIONS 
 
 After the nature and notation of a fraction is under- 
 stood, the addition of fractions presents nothing new; 
 that is, the work has the same meaning and follows the 
 same fundamental law, that only like things can be added, 
 which the pupil has had in whole numbers. The impor- 
 tant thing is that the work be carefully graded so as to 
 introduce but one form at a tune and that the pupil 
 sees that it is like the addition he already knows. The 
 gradation should be : (1) two fractions whose units are 
 alike ; (2) two fractions whose units are unlike, but where 
 one can be changed to the other ; (3) two fractions whose 
 units are unlike and a new unit must be found to which 
 both can be changed. ' 
 
 ADDING FRACTIONS WITH LIKE UNITS 
 
 The pupil knows that the upper term, or numerator, 
 shows the number of things considered and that the lower 
 term, or denominator, shows what they are. So just 
 as $2+$3 = $5, or 2 ft. +3 ft. = 5 ft., so 2 of any unit +3 
 of a like unit =5 of that unit. Hence, f+f=f, or f+f 
 = f, etc. And thus he should see clearly that when the 
 units are alike, the numerators only are added for they 
 are the number of things that are combined. The de- 
 nominator merely shows what they are as do the abbre- 
 viations and signs in denominate numbers. 
 
 If the work is presented in this way, pupils will never 
 attempt to add two fractions by adding numerators and 
 denominators, a mistake often seen in grammar and high 
 schools.
 
 104 
 
 THE TEACHING OF ARITHMETIC 
 
 ADDING FRACTIONS WITH RELATED DENOMINATORS 
 
 The pupil knows from his work in whole numbers and 
 with denominate whole numbers that 2 bu.+3 pk. does 
 not equal 5 bu. or 5 pk. ; that, before he can combine 
 them into one number, they must both be expressed in 
 like units, and that he must express them as 8 pk.+3 pk. 
 before he can add them. So in i+f, he knows that, 
 although he has 1 thing + 3 things, he cannot call them 
 4 of either, for 1 of them is a fourth while the other 3 are 
 eighths. So, before they can be added, the i must be 
 expressed as f. Then f+ != 
 
 It will be noticed that this class of addition calls for a 
 knowledge of the relations of certain related fractional 
 units. While a study of the idea of a fraction as one or 
 more of the equal parts of a whole through the use of ob- 
 jects has no doubt shown the pupil certain relations, this 
 class of addition furnishes a motive for further comparison, 
 and the pupil should now become as familiar with the 
 relations of those units likely to occur in life as he is with 
 the relation of the units in the various tables of denominate 
 numbers. This he does through the use of objects and 
 diagrams. 
 
 Thus, a diagram like the following shows him the rela- 
 tion of halves, fourths, and eighths. 
 
 a 
 
 y, 
 
 X 
 
 1 A 
 
 l /4 
 
 % 
 
 % 
 
 y* 
 
 
 
 H 
 
 y s 
 
 % 
 
 % 
 
 %
 
 COMMON FRACTIONS 
 
 105 
 
 The following shows the relation of thirds, sixths, and 
 ninths. 
 
 >/3 
 
 >/3 
 
 1 4 
 
 % 
 
 */6 
 
 % 
 
 I A 
 
 H 
 
 Vo 
 
 % 
 
 % 
 
 y 9 
 
 % 
 
 */ 9 
 
 'A 
 
 % 
 
 */9 
 
 % 
 
 ADDING FRACTIONS WITH UNRELATED UNITS 
 
 The adding of such fractions asf and^ brings up the 
 auxiliary process of changing fractions to like units or to 
 "a common denominator." Such a problem as this, 
 then, should be presented in order to furnish an ami or 
 motive for such a process. 
 
 THE LEAST COMMON DENOMINATOR 
 
 Using the problem presented; namely, f+f, let us 
 find a unit to which both 3ds and 4ths can be changed. 
 
 By diagrams show that if 3ds are divided into 2 equal 
 parts we have 6ths; if into 3, we have 9ths; if into 4, 
 we have 12ths ; etc. Have the pupil see that the denomi- 
 nators of the fractions to which 3ds can be changed are 
 the successive multiples of 3 and thus introduce the term
 
 106 THE TEACHING OF ARITHMETIC 
 
 multiple. If that is always so, then 4ths can be changed 
 to 8ths, 12ths, 16ths, 20ths, etc. Let him see from the 
 diagram that that is just what he would have. 
 
 He is now led to see that the denominators to which 
 3ds and 4ths can be changed will contain both 3 and 4. 
 He should now be able to find by inspection the unit 
 to which two or more fractional units can be changed. 
 
 REDUCING FRACTIONS TO NEW UNITS 
 
 Using the diagrams above, lead the pupil to see that 
 when 3ds are changed to 6ths, while the units are half 
 as large, there are twice as many of them; changed to 
 9ths, they are one third as large but three times as many ; 
 etc. Thus, f = $ = = & = T$, etc., which may be found 
 mechanically by multiplying both terms by the same 
 number. 
 
 Here it is well to call attention to the use he will have 
 of the truth illustrated here that both terms of a frac- 
 tion may be multiplied or divided by the same number 
 without changing the value represented by the fraction. 
 
 Now, to change f and f to like units, ask, "What 
 denominator will contain both 3 and 4 ? " " By what must 
 3 be multiplied to give 12?" "Then by what must 
 each term of f be multiplied to change it to 12ths?" 
 
 and then ask similar questions about f. 
 
 REDUCING IMPROPER FRACTIONS 
 When adding the fractions proposed above, we have 
 
 ^. Now the pupil will observe that 
 is not a fraction in the sense that he has thought of a
 
 COMMON FRACTIONS 107 
 
 fraction, for in dividing a whole into 12 parts to get 12ths 
 there are but 12 of them in any whole. So this may be 
 called an improper fraction, since 12 of the 17 twelfths 
 will make a whole and 5 of them will remain. Hence, 
 T! =!& Unless the pupil has come to the second no- 
 tion of a fraction, that it is an expressed division, he sees 
 no meaning in " dividing the numerator by the denomi- 
 nator" as a method of reduction. The natural develop- 
 ment from his first idea of a fraction is that, since the 
 denominator shows the number of parts into which the 
 whole was divided, it is the number of parts that it takes 
 to make a whole. Thus, in ^, it takes 8 of the 8ths to 
 make a whole ; so there will be as many wholes as there 
 are 8's in 25, or 3 of them and 1 remaining. So ^ = 3. 
 Through such a thought as this, he sees that mechani- 
 cally the answer is obtained by "dividing the numerator 
 by the denominator," the rule usually given. 
 
 FORMS OF ADDING MIXED NUMBERS 
 
 There are several forms of adding mixed numbers. 
 The two most common are given below : 
 
 Form A Form B 
 
 24 
 
 _ 10 3 
 
 16|=16H 
 
 17* 
 
 3 
 18 
 20 
 
 41 
 
 Form A is an uneconomical procedure for it requires 
 rewriting the whole numbers and the common denominator
 
 108 THE TEACHING OF ARITHMETIC 
 
 with each fraction. In form B the common denominator 
 is written above and the numerators are written in a form 
 easily added. 
 
 ADDING SPECIAL FRACTIONS 
 
 The pupils should always be alert for special combi- 
 nations that will save work. There are two types of exer- 
 cises in the addition of fractions that should be noticed. 
 These two types are shown by the following examples : 
 
 First type: +t+|+f+i 
 
 Instead of changing all to 8ths, the pupil should collect 
 those whose units are alike. So f+i = l; f+t=l; 
 and, hence, the sum =2. 
 
 Second type: i+i. 
 
 Since 4 and 5 have no common factor and since the nu- 
 merators are each 1, the new numerators will be 5 and 4 
 respectively. Hence, the sum is -fo which is " the sum of 
 the denominators over their product." 
 
 SUBTRACTION OF FRACTIONS 
 
 As in addition of fractions, the pupil should see that 
 the meaning and the process of subtraction are the same as 
 in whole numbers. The gradation is not so important as 
 in addition, for all the principles needed have already 
 been taught. Yet, the grading is exactly the same as in 
 addition except that the passing from one type to another 
 may be done more quickly. The grading is : 
 
 (a) Like units, asf f = f=i; 
 
 (6) Related units, as f i = f f = f; 
 
 (c) Unrelated units, as f i = & TS = &.
 
 COMMON FRACTIONS 109 
 
 FORMS OF WORK IN MIXED NUMBERS 
 
 When the fraction in the minuend is greater than the 
 one in the subtrahend, they are subtracted as in pure 
 fractions. With pupils in the lower grades the work may 
 be written down as in form B under addition. With 
 mature pupils the change to like units and the subtrac- 
 tion may be done mentally and the numerators of the 
 like fractions not written. That is, 
 
 46| 46} 
 
 13! . not 
 
 8 
 3 
 
 5 
 
 When the fraction in the minuend is smaller than the 
 one in the subtrahend, work is saved if the fraction in 
 the subtrahend is subtracted from 1 and the result added 
 to the fraction of the minuend. Thus, in 
 
 we think 
 
 Now we may use 1 f +i = !+i= : Tj, instead of 1-J f = 
 
 SUBTRACTING SPECIAL FRACTIONS 
 
 When the numerators are each 1 and the denominators 
 have no common factor, the result may be written down 
 at once. Thus ! ! = i%, for the new numerators are 5 
 and 2 respectively and the common denominator is 10; 
 that is, the result is "the difference of the denominators 
 over their product."
 
 110 
 
 THE TEACHING OF ARITHMETIC 
 
 MULTIPLICATION OF FRACTIONS , 
 
 MULTIPLYING A FRACTION BY A WHOLE NUMBER 
 
 The pupil should be shown clearly that multiplying 
 by a whole number and multiplying by a fraction have 
 very different meanings. He has learned that multiply- 
 ing by a whole number is a means of saving addition' when 
 the addends are alike. So, just as 3X$2 = $6, 3X2 ft.= 
 6 ft., and 3X2 of any unit =6 of that unit, so 3Xf = $, 
 3 X = $, etc.; that is, the numerator being the number 
 of units under consideration is the number that is multi- 
 plied. The denominator is merely written to show what 
 the units are, as were the " $ " and " ft." in the examples 
 shown above. 
 
 MULTIPLYING A FRACTION BY A FRACTION 
 
 To find the product of fX| means to find f of $. But 
 to find f of anything always means the same. It means 
 
 */ 
 
 A 
 
 to divide the thing into 3 equal parts and then take 2 of 
 them. Now, 5ths divided into 3 equal parts gives 15ths,
 
 COMMON FRACTIONS 111 
 
 so i of = A- Hence, f of f=2 X A = & Hence, it 
 is seen that the product of two fractions is found by tak- 
 ing the product of the numerators for the numerator of 
 the product, and the product of the denominators for the 
 denominator of the product. The above product may be 
 shown objectively. 
 
 MULTIPLYING BY A MIXED NUMBER 
 
 To multiply by a mixed number will involve multiply- 
 ing a whole number by a fraction. As seen above, to 
 multiply anything by a fraction requires both a division 
 and a multiplication. But, just as 4^2 X3 gives the same 
 result as 4X3-:- 2 (that is, the order may be changed), 
 the multiplication by the numerator may take place 
 before the division. Thus, 13f X25 is found as in the 
 margin. 
 
 Enough drill should be given to fix the habit 25 
 of putting the partial products in the right 13f 
 place. Some pupils will put the 5 of the second 4)75 
 product under the 1 of the first. 18f 
 
 Where both multiplier and multiplicand con- 75 
 tain fractions and the whole numbers are small, 25 
 work may be saved by changing both to improper 343f 
 fractions. Thus, 3iXl2f = 
 
 The pupil who has understood the two meanings of 
 division of whole numbers will have no difficulty in see- 
 ing the corresponding meanings when applied to fractions, 
 which to him are now a sort of denominate number.
 
 112 
 
 THE TEACHING OF ARITHMETIC 
 
 DIVIDING A FRACTION BY A WHOLE NUMBER 
 
 Any division by an abstract whole number may be inter- 
 preted4i8 the partition idea of division; that is, as a divi- 
 sion into parts. Just as $8 -j-2 = $4, 8 ft. ^2= 4 ft,, and 
 8 of any unit -i-2=4 of that unit, so f-7-2 = $, f-j-2=^, 
 etc. 
 
 */ 
 
 /4 
 
 Likewise, from the same meaning of dividing into 
 parts, each fractional unit may be divided into parts 
 as shown above. So, when the numerator cannot be 
 divided without a remainder, the denominator may be 
 multiplied by the divisor. 
 
 This last fact is also shown from the same meaning; 
 that is, to divide by 2 is to find half of; by 3, to find a 
 third of; by 4, to find a fourth of; etc. Thus, |-r-5 = 
 
 DIVIDING A FRACTION BY A FRACTION 
 
 To divide by a fraction cannot mean partition, for a 
 thing cannot be divided in f equal parts. Such an ex* 
 pression has no meaning. A thing can only be divided
 
 COMMON FRACTIONS 113 
 
 into 2 or 3 or 4 or some whole number of equal parts. 
 Hence, this type of division is the measuring idea of divi- 
 sion; that is, it is finding how many times the dividend 
 will contain the divisor. Just as 8 qt.-?-2 qt. = 4, or 
 8 ft. -5- 2 ft. =4, so 8 of any unit will contain 2 of that 
 unit 4 times. Then, f-r-f=4, f-r-f = 4, etc.; that is, 
 just as in any denominate number, the units must be alike, 
 then the numbers of them are the numbers used in the 
 actual division. 
 
 DIVIDING BY INVERTING THE DIVISOR 
 
 There are several ways of developing the in verted di- 
 visor method of division. Perhaps the simplest method 
 is to analyze the work done in changing to like units. 
 Thus, take the example given above of f-5-f. To get 
 like units, both terms of the dividend were multiplied by 
 the denominator of the divisor. Then both terms of the 
 divisor were multiplied by the denominator of the divi- 
 dend. But when thus reduced, only the new numerators, 
 2X4-4-3X3, were actually used in the division. This 
 
 2X4 
 
 quotient expressed as a fraction is - , which is the 
 
 0X0 
 
 product of f Xi- But this is the product of the dividend 
 by the divisor inverted. 
 
 Another method in quite common use is to develop the 
 process from the two known facts that multiplying the 
 divisor divides the quotient, and dividing the divisor
 
 114 THE TEACHING OF ARITHMETIC 
 
 multiplies the quotient; and that with the divisor re-. 
 maining constant, multiplying or dividing the dividend 
 multiplies or divides the quotient by the same number. 
 With these two principles hi mind, the development is as 
 follows : 
 
 i+i-ij 
 
 1*1=4X1=4; 
 
 From several examples like this, the pupil gets the follow- 
 ing generalization, that the quotient of one divided by any 
 fraction is the fraction inverted. The second step is as 
 follows : 
 
 Since I-^T=& 
 
 Then i+f-fX* = t. 
 
 That is, to* divide by a fraction, invert the divisor and 
 multiply. 
 
 DIVIDING A MIXED NUMBER BY A WHOLE NUMBER 
 
 While no new principles are involved, the method of 
 dividing a mixed number by a whole number needs to be 
 taken up. Thus, to divide 3896f by 7, we have 
 
 7)3896f 
 
 556, 4f rem. 
 
 4f^-7 = |XY=f- Hence, quotient = 556| 
 
 That is, the division is performed as in whole numbers 
 until a remainder less than the divisor is found and then 
 this is treated as any fraction divided by a whole number.
 
 COMMON FRACTIONS 115 
 
 DIVIDING A MIXED NUMBER BY A MIXED NUMBER 
 
 There are two ways of proceeding. When the whole 
 numbers are small, the mixed numbers may be changed 
 to improper fractions. When the whole numbers are 
 large and the terms of the fractions small, time is saved 
 by multiplying both dividend and divisor by the least 
 common multiple of the denominators and thus reducing 
 them to whole numbers. Thus, 
 
 21 . 03 5 S/__ 25 . 
 TS~ <% ?XTS ^7 
 
 And 345i-r-16f 
 4 4 
 
 Or, a third method may be used in such a problem as 
 468Tf-*-3-|. Multiplying both dividend and divisor by 
 2, we have 937^ -=-7. 
 
 7)937jf 
 133, 
 
 COMPLEX FRACTIONS 
 
 A complex fraction is merely an indicated division, 
 expressed in the form of a fraction instead of by the di- 
 vision sign (-i-), where one or both of the terms (dividend 
 and divisor) are fractions or mixed numbers. Such ex- 
 pressions arise in expressing the work to be done in the 
 solution of a problem or in dealing with a remainder as 
 shown in some of the above divisions. Such fractions, 
 then, involve nothing new and are simplified by perform-
 
 116 THE TEACHING OF ARITHMETIC 
 
 ing the indicated divisions, or by multiplying both terms 
 of the complex fraction by the least common denominator 
 of the fractions composing the terms. Thus, 
 
 6J = 13 J5_ = 39 
 7* 2 22 44 
 
 
 7* 7iX6 44 
 
 Among the topics often urged for elimination is the 
 topic of "complex fractions," but there seems to be no 
 more reason for this, when a complex fraction is considered 
 as a mere form of expressing division, than to urge that 
 no division must ever be expressed as a fraction. The 
 thing to be urged is that, as in all work, no more complex 
 exercises be given for pure computation than may arise 
 in solving the problems that arise in doing the ordinary 
 life work.
 
 CHAPTER X 
 
 OURS is a decimal-place-value system of notation. Ten 
 units of any order make one Unit of the next higher order. 
 Or, expressing the same thing in another way, a unit in any 
 order has one tenth of the value of a unit in the next higher 
 order. The notation, then, of a decimal fraction should be 
 considered as an extension of our notation to the right of 
 ones' place. Then the first order to the right of ones' place 
 is tenths; the next is tenths of tenths or hundredths ; the 
 next, tenths of hundredths or thousandths; etc. 
 
 This notation should be presented by taking some 
 number all of whose digits are alike and asking the pupil 
 the value represented by each digit and by having him 
 compare their values. Thus, in $1111, ask what each of 
 the four 1's represents. Next compare one with another 
 and bring out the fact that each represents one tenth of 
 the value of the one to the left of it. Now, calling any 
 digit a certain unit, get the pupil to see that he then knows 
 what each unit must be, since each has one tenth the value 
 of the one to the left of it. 
 
 Next take some number as 3467 and have some pupil 
 read it. Now suppose that the 4 is in ones' place; get 
 the pupil to see that all the other places are known and 
 that 3 must then be in tens' place, that the unit of the 
 
 117
 
 118 THE TEACHING OF ARITHMETIC 
 
 place occupied by 6 must be one tenth of one or tenths, 
 and that 7 must be in hundredths' place, for a tenth of a 
 tenth is a hundredth. And thus introduce the use of 
 the decimal point to locate ones' place. 
 
 Then practice in reading and writing decimals, and 
 mixed decimals should follow until the pupil visualizes 
 the position of each figure as a number is read. 
 
 Have pupils see clearly that each zero written be- 
 tween a decimal point and a digit that follows divides 
 the number by 10 for it moves the digit to a lower order. 
 Thus if .5 is changed to .05 or .005, it is changed to a 
 number ^ and y^y as large, respectively. 
 
 But if a zero is annexed to a decimal, it does not change 
 its value, for it does not change the order of any of the 
 digits. Thus, .5 may be changed to .500, which is still 
 5 tenths and no hundredths and no thousandths, without 
 changing its value. 
 
 ADDITION AND SUBTRACTION OF DECIMALS 
 
 The pupil who understands the fundamental principle 
 of addition and subtraction that only like units can 
 be added or subtracted and who understands the 
 notation of a decimal will see that the decimal points 
 must come under each other in order to bring like units 
 in the same columns. Hence, these two processes present 
 nothing new and introduce no difficulties. 
 
 MULTIPLICATION OF DECIMALS 
 
 The only difficulty arising in the multiplication of 
 decimals not already encountered in whole numbers is
 
 DECIMAL FRACTIONS 119 
 
 the pointing off of the product. This is easily overcome 
 by proper gradation. The work should be rationalized, 
 showing how it follows naturally from the meanings of 
 multiplication by a whole number and by a fraction, 
 already discussed. 
 
 MULTIPLYING A DECIMAL BY A WHOLE NUMBER 
 
 Multiplying by a whole number is always an abridged 
 way of finding the sum of a number of equal addends. 
 Hence, 4X7.38 means 7.38+7.38+7.38+7.38 or 29.52. 
 Thus, it is seen that multiplying a decimal by a whole 
 number must give a product with the same number of 
 decimal places as there were in the multiplicand. 
 
 MULTIPLYING A DECIMAL BY A DECIMAL 
 
 Since to multiply by any fraction is to perform both 
 a division and a multiplication, dividing a decimal by 
 10, 100, 1000, etc., must be understood before developing 
 this phase of multiplication. This division goes back to 
 the meaning of the notation of a decimal. Thus, 5.34 is 
 but one tenth as large as 53.4, for each digit occupies an 
 order one lower in 5.34 than in 53.4. Likewise 5.34 is one 
 hundredth as large as 534. for each digit occupies a place 
 two orders lower. Thus, it is seen that every order to 
 the left to which the decimal point is moved brings each 
 digit down to the next lower order and thus divides the 
 number by 10. 
 
 .4X34.2 means 4 times one tenth of 34.2. This re- 
 quires moving the point one place to the left and multiply- 
 ing the result by 4. But the multiplication may take
 
 120 THE TEACHING OF ARITHMETIC 
 
 place first. Hence, by a few such illustrations, it is easily 
 seen that the numbers are multiplied as if they are whole 
 numbers and then as many decimals pointed off in the 
 product as there are in both factors. 
 
 DIVISION OF DECIMALS 
 
 The new difficulty arising in division of decimals is the 
 pointing off of the quotient, and this is easily developed 
 by the proper gradation. The first problem should be 
 the division by a whole number, before division by a 
 decimal is considered. 
 
 DIVISOR A WHOLE NUMBER 
 
 Division by an abstract whole number may always be 
 interpreted as the partition idea of division, which gives 
 a quotient of the same unit as the dividend. Thus, just 
 as $8* 2 = $4, 8 ft.H-2 = 4 ft, f^2 = |; so .8-5-2.4, 
 .08 4- 2 = .04, etc. 
 
 With this understanding of the meaning of division, 
 the pupil who has been taught from the first to write 
 each digit of the quotient directly over the 
 right-hand digit of the partial dividend, 34J5 
 
 from which it was obtained, will experience 53)1833.8 
 no new difficulty whatever. Thus, to divide 159 
 1833.8 by 53, the pupil sees that, since the 243 
 
 first dividend was 183 tens, the quotient was 212 
 
 tens, the next quotient ones, the next tenths, 31 8 
 
 and that the decimal point in the quotient 31 8 
 
 must come directly over the decimal point in 
 the dividend in order to show this.
 
 DECIMAL FRACTIONS 121 
 
 DIVISOR A DECIMAL 
 
 The pointing off in this case is more easily developed 
 by changing to a new problem whose divisor is a whole 
 number and thus make the pointing depend _ fi 
 
 upon the partition idea as in the preceding 09x1709 
 case. This will require, first, that the pupils - ' 
 
 see that a thing could not be divided into, say, TQ9 
 3.2 equal parts; and, next, that multiplying 1 
 
 both dividend and divisor by the same num- 
 
 ber does not change the quotient. The first may be 
 shown objectively and the second by such illustrations 
 as 6-r-2 = 3; 60*20 = 3; 600-r- 200 = 3, etc., and thus 
 that 17,92* 3.2 = 179.2-*- 32. In the actual work it is 
 not desirable to erase the original decimal points. They 
 may be needed in checking up the original problem. 
 But the new position in the dividend may be marked by 
 a caret to show the position of the point in the quotient. 
 
 The pupil should see that, when the dividend has fewer 
 places than the divisor, as many or more places may be 
 created by bringing down zeros ; for this does not change 
 the value, but merely records the fact that no units of 
 those orders exist. 
 
 CHANGING COMMON FRACTIONS TO DECIMALS 
 
 It is important that the pupil be able to change a com- 
 mon fraction or a ratio to a decimal fraction, but this 
 requires no new fact or process. It comes from consider- 
 ing a fraction or ratio as an expressed division and then 
 performing the division.
 
 122 THE TEACHING OF ARITHMETIC 
 
 There are common fractions with special denominators 
 that are more easily changed by first changing to a com- 
 mon fraction whose denominator is a power of ten. Thus, 
 
 THE APPLICATION OF DECIMALS 
 
 As in all subjects, the subject of decimals should be 
 used to meet the pupil's needs if possible. However, 
 his own personal needs rarely ever require a knowledge 
 of decimals, but statistics and data met in other school 
 work and in general reading will no doubt involve decimals 
 if. the work is not given too early. The sixth grade 
 seems the most suitable place in the curriculum for the 
 subject. Then such relations as the circumference of a 
 circle to its diameter, the amount of rainfall, the crop's 
 yield per acre, the digestive nutriments of stock feed and 
 of human food, problems involving time and speed, the 
 various constituents of fertilizers, all these furnish a 
 motive for the work in decimals. The use in the subject 
 of percentage, however, is the most common use that 
 the pupil will find for the subject; and, hence, a study 
 of decimals should be followed up by the general dis- 
 cussion of percentage.
 
 CHAPTER XI 
 PERCENTAGE AND ITS APPLICATIONS 
 
 IN the past, both textbooks and teachers have pre- 
 sented the subject of percentage as if it were some new 
 topic with its own rules and processes, involving new 
 mathematical principles. The subject was not only 
 presented as a new phase in the development of arith- 
 metic, but it was subdivided into "cases" each with its 
 own rule and formula. 
 
 This method of treatment has not entirely passed. We 
 still find textbooks in use that treat the subject in this 
 way ; and many of the older teachers, brought up by such 
 a method, still persist in giving it to their pupils as it was 
 given them. 
 
 But such teaching- is rapidly passing. The pupil is 
 led to see that the subject is merely a continuation of 
 the study of fractions, a new language for an old and well- 
 known idea, and that this change to a new language does 
 not lead to new processes or to new principles. 
 
 THE NOTION AND NOTATION OF PER CENT 
 
 The term per cent must be seen to be a special name for 
 a special fraction, one whose unit is hundredths. 
 
 To introduce the need of the new term, pupils may be 
 asked such questions as, " If you solve 3 out of 4 of your 
 problems, what part of them do you solve?" "If in an 
 
 123
 
 124 THE TEACHING OF ARITHMETIC 
 
 orchard 7 out of 8 of the trees are apple trees, what part 
 are apple trees?" "If a man in business makes $15 out 
 of every $100 he receives for goods, what part of his 
 receipts is profit?" etc. 
 
 Lead the pupil to see that it is much easier to say that 
 a merchant makes "15 per cent" than to say that he 
 makes "$15 out of $100," or to say that he makes "15 
 hundredths," and that the term "hundredths" is rarely 
 ever used in business, but that the term per cent, which 
 means the same, is used instead. The term is from the 
 Latin per (out of) and centum (a hundred). So 7 per 
 cent, written 7%, means 7 out of 100, or 7 hundredths. 
 
 THE THREE PROBLEMS OF PERCENTAGE 
 
 All problems arising under percentage and its applica- 
 tions fall under three general types: (1) to find a per 
 cent of a number; (2) to find what per cent one number 
 is of another; and (3) to find a number when some per 
 cent of it is known. Examples of these are: (1) Find 
 25% of 300 ; (2) 75 is what per cent of 300 ? and (3) 75 is 
 25% of what number ? 
 
 The first two of these problems are by far the most 
 common in business practice, but the third has a place, 
 as will be shown later. 
 
 To FIND A PER CENT OF A NUMBER 
 
 When the pupil understands the meaning of the nota- 
 tion for per cent, this class of problems gives no difficulty, 
 for he is familiar with the process involved. He knows 
 that .06X350 means .06 of 350, or that .17 of 285 means 

 
 PERCENTAGE AND ITS APPLICATIONS 125 
 
 .17X285; etc. So to find 18% of 365 means .18X365. 
 The only new thing, then, is the notation and the ability 
 to express any per cent as a decimal. When the number 
 of per cent is an integer of but one or two figures, as 6% 
 or 35%, the pupil has no trouble in the translation to 
 hundredths ; but when the per cent is a fraction, common 
 or decimal, or when it is a whole number larger than 100, 
 the pupil often has difficulty in expressing it as hun- 
 dredths. Examples of such are: f%, .4%, 250%, etc. 
 Errors in changing such per cents to decimals are often 
 made by normal school students and by teachers. 
 
 These need not give much trouble if properly pre- 
 sented. .4% means 4 tenths of one hundredth, hence 
 4 thousandths or .004. The pupil need not go through 
 this rationalizing process for each, but should visualize the 
 expression .4% as "no tenths or hundredths, but 4 in the 
 next lower order " ; hence, .4% = .004, .38% = .0038, 1 .2% = 
 .012, etc., observing that, since % is a symbol for hundredths, 
 when it is removed two more decimal places must be used. 
 
 When this type is fixed, it is easy to present those 
 per cents expressed as common fractions. Thus, ^% = 
 .5% = .005; f% = .75% = .0075; f% = .375% = .00375. 
 
 The pupil will easily understand the meaning of those 
 per cents larger than 100 if they are first expressed in 
 the common fraction form. Thus, 
 
 And thus he sees that, when removing the per cent sign, 
 two more decimal places must be pointed off. So 375% = 
 3.75; 560% = 5.60 = 5.6. 
 It is imperative that the pupil can express any per cent
 
 126 THE TEACHING OF ARITHMETIC 
 
 as a decimal before he can find a given per cent of a num- 
 ber; for to find 1|% of $3800 is to find .0175X$3800; 
 to find 175% of 396 is to find 1.75X396. 
 
 To find a per cent of a number, then, no new rule or 
 formula should be used; but the per cent should be 
 changed to a decimal form and used as a multiplier. 
 
 To FIND WHAT PER CENT ONE NUMBER Is OF ANOTHER 
 
 In this problem we are simply asking for the relation 
 of two numbers when expressed as hundredths or per 
 cent. This involves the ratio idea of a fraction ; that is, 
 the use of a fraction to express a relation, and the ability 
 to change a fraction to a decimal. A review, then, of 
 these two problems should be given to" see that the pupil 
 has the necessary ability to solve this problem of per- 
 centage. Thus, if a merchant made $24 from a sale of 
 $96, he made $24 out of $96 or ff of the receipts. f = 
 .25 = 25%. And, in general, to find what per cent one' 
 number, as 325, is of another, as 468, proceed as follows : 
 
 The steps are : 
 
 1. Express the relation as a .6944 = 69.44% 
 fraction. 468)325.00 
 
 2. Change the fraction to a 2808 
 decimal. 44 20 
 
 3. Express the decimal in 42 12 
 terms of per cent. 2 080 
 
 This second problem is often 1 872 
 
 treated as an indirect problem, 208 
 
 the inverse of the first problem. Thus, to find the result 
 in the problem given above, the reasoning is that 468
 
 PERCENTAGE AND ITS APPLICATIONS 127 
 
 multiplied by the answer, if known, would give 325. 
 So the statement is made : 
 
 3X468 = 325, 
 z = 325 -*- 468 = .6944 = 69.44%. 
 
 But this method takes away the relation idea expressed 
 by per cent, is more difficult to understand, and is not 
 related to the processes and principles already known, 
 and is not to be recommended. 
 
 It will be observed that this problem requires the 
 ability to express any decimal as a per cent, the inverse 
 of the ability needed in the first problem. So some 
 practice in expressing such relations as the following is 
 needed : 
 
 .26 =26% 1.25 =125% 
 
 .07 =7% .004 =.4% 
 
 .065 = 6.5% 1 .645 = 164.5% 
 
 To FIND A NUMBER WHEN A PER CENT OF IT Is KNOWN 
 
 This type, usually called the indirect problem of per- 
 centage, occurs less often than the other two. It is quite 
 commonly urged in current articles on the teaching of 
 arithmetic that this type of problem be omitted alto- 
 gether, and the recommendation would be well founded 
 if there were no other applications of it than those given 
 in most textbooks. The usual type given is, "If a mer- 
 chant sold a suit for $24, thereby making 20% of the 
 cost, find the cost." Now, this is not a problem that 
 meets the needs of any one. The answer, if the problem 
 is real, must have been known before the problem could
 
 128 THE TEACHING OF ARITHMETIC 
 
 have been made. This is a sort of "hide-and-go-seek" 
 problem given for "analysis" or "mental discipline" 
 reasons no longer sufficient when so many real problems 
 are to be found on every hand that meet the needs of 
 man in doing the world's work. 
 
 The indirect problem, however, frequently arises hi 
 actual business practice. Thus, a manufacturer, knowing 
 the cost of an article and the average overhead charges, 
 etc., based upon the sales, knows that the very minimum 
 price at which the article must be listed must give a 
 certain per cent of profit reckoned on the sales. 
 
 Thus, if it costs $10.26 to make an article, at what 
 must it be sold to give a profit of 24% of the selling price ? 
 
 Since the cost and profit make up the selling price, if 
 24% of it is profit, the remaining 76% of it must be the 
 cost. So, if the selling price of it were known and mul- 
 tiplied by .76, the result would be the cost, or $10.26. 
 Hence, the relation : 
 
 .76 X selling price = $10.26. 
 
 Here we have the product of two factors given ($10.26) 
 and one of the factors (.76) to find the other (selling price). 
 Hence, it is a case of division from the definition of divi- 
 sion. The solution then is : 
 
 $13.50 
 .76)$10.26 A 
 76 
 266 
 228 
 380
 
 PERCENTAGE AND ITS APPLICATIONS 129 
 
 Another real application of the same problem would be 
 to find a catalogue price that must be put upon an article 
 in order to discount it at a per cent and yet give a fixed 
 net price. Thus, a merchant may know that $4.80 is 
 the minimum price at which he can sell an article, but he 
 must put a price upon it from which he can allow his 
 customer a discount of 20% and then get $4.80. 
 
 Since he gets but 80% of the list price, the known 
 relation is .8 X list price = $4.80; hence, the list price is 
 $4.80 -5- .8 or $6. 
 
 In treating this type, then, the problems should be 
 those of the business world and not those now found in 
 most textbooks, whose only excuse for being there is that 
 they serve as a sort of mental gymnastics. 
 
 THE APPLICATIONS OF PERCENTAGE 
 
 The problems given under the topic of percentage are 
 for two general purposes: (1) to develop the ability to 
 see, express, and interpret the relationships between any 
 magnitudes in terms of per cent as encountered in general 
 reading about increases, decreases, discounts, etc. ; and 
 (2) to give a social insight into certain phases of com- 
 mercial life needed for a proper understanding of ref- 
 erences met in general reading and in conversation. 
 
 ABILITY TO EXPRESS AND INTERPRET RELATIONSHIPS 
 
 IN PER CENT 
 
 This ability is developed through problems of a general 
 nature relating to magnitudes of all kinds familiar to the 
 pupil and not confined to money transactions as in buy-
 
 130 THE TEACHING OF ARITHMETIC 
 
 ing, selling, loaning, etc. So often a pupil gets a notion 
 that per cent refers to money as "cents on a dollar," or 
 some such false notion. 
 
 Hypothetical problems, both direct and indirect, may 
 be used for such development as long as they do not 
 misrepresent facts or give a wrong notion of real practices 
 or conditions. 
 
 All three problems may be brought out from the same 
 data ; as, " The population of a certain city was 390,300 
 in 1910 and 468,360 in 1915. (a) Find the rate of in- 
 crease in 5 years. (6) At the same rate of increase, what 
 will the population be in 1920? (c) If the population 
 increased at the same rate from 1905 to 1910, what must 
 it have been in 1905?" 
 
 BUSINESS APPLICATIONS OF PERCENTAGE 
 
 The commercial problems given should be more for 
 the purpose of giving a social insight into current practices 
 in order that one may understand references to them met 
 in general reading and in conversation, rather than to 
 fit for a certain vocation. And yet the methods used 
 should be those used in business and not some stereotyped 
 forms of analysis and procedure so often found in the 
 schoolroom. The problems, too, must be those met in 
 actual business and made from data fairly true to life 
 in order to give true information and not misinformation. 
 
 COMMISSION 
 
 As a fee for services, salesmen of almost all lines of 
 goods get a per cent of their sales, sometimes in addition
 
 PERCENTAGE AND ITS APPLICATIONS 131 
 
 to a fixed salary and sometimes as their only fee for their 
 work. This fee is called their commission. Buyers also 
 usually work on the commission or commission plus salary 
 basis. This application, then, is only the first problem 
 of percentage and requires multiplication only. Ex- 
 amples of commission are found on every hand. The 
 real estate agent gets a per cent of his sales as a com- 
 mission, the agents who canvass from house to house, 
 insurance agents, and salesmen all get a per cent of their 
 sales. 
 
 The pupil should get clearly this meaning of commis- 
 sion, which is the meaning he needs in order to interpret 
 references met in general reading or conversation. This 
 may be followed by the work of the commission merchant 
 who sells farm produce, grain, cotton, pork, etc. 
 
 Indirect problems found in most of the books of a 
 decade or so ago should be entirely omitted, as they give 
 a false notion of business transactions. The following 
 is a type of such problems : An agent received $1200 
 with which to buy goods after deducting a 10% commission 
 for buying. Find the commission and the amount spent 
 for goods. 
 
 This gives a wrong notion of business practices, for a 
 business firm would not send its agent a fixed amount from 
 which to buy goods after taking out his fee for services. 
 
 DISCOUNT 
 
 In general conversation and reading, the term discount 
 is used to denote a deduction from a former price. Thus, 
 a boy may sell his bicycle at a discount upon what he paid
 
 132 THE TEACHING OF ARITHMETIC 
 
 for it because he has used it a season. A store may give 
 a discount on goods soiled or out of season. A dealer 
 may allow a discount for a cash payment from the price 
 at which the goods are sold on time. The teacher should 
 find such familiar uses of the term as the pupil meets in 
 his everyday life. The chief problem here, as in commis- 
 sion, is the first problem of percentage to find a per 
 cent of a number. A question, however, may arise as to 
 what per cent a certain fixed discount is. For example, 
 a table marked at $30 may sell for $24 during a special 
 sale. What per cent of discount was given ? 
 
 The indirect problem can answer no real need, for it 
 answers no question that would arise either in the mind 
 of the buyer or the seller, and hence it should not be given 
 under the subject of discounts. v 
 
 Commercial discount is a more specialized term and 
 refers to transactions between the retailer and the whole- 
 saler. But the term has a general interest to the average 
 person and should be considered. Thus, it is a custom 
 among certain classes of wholesalers and manufacturers 
 to publish catalogues with a fixed list price of their goods 
 and then allow a deduction to the retailer handling their 
 kinds of goods. This is what is known as commercial 
 discount, in distinction from the general use of the term. 
 
 Usually the list price, or catalogue price, remains the 
 same for long periods; but, when the market changes, 
 new discounts are made. If the market price increases, 
 a smaller discount is allowed; but, if the market price 
 decreases, the discount is increased. The increase in 
 discount is usually made by stating a new discount to be
 
 PERCENTAGE AND ITS APPLICATIONS 133 
 
 applied to the previous net price. When two or more 
 discounts to be deducted in this way are allowed, they 
 are called successive discounts. 
 
 PROFIT AND Loss 
 
 In the commercial \forld the subject of profit and loss 
 is one of the important applications of percentage. But 
 for the average person, whether engaged in business or 
 not, the terms are so much used that the subject becomes 
 one of general importance. 
 
 This application of percentage may properly bring in 
 all three of the problems of percentage. Thus, a busi- 
 ness man may be confronted with any one of the following 
 questions : 
 
 (a) An article cost me $48. At what price must I sell 
 it to make a gross profit of 20% of the cost ? 
 
 Solution. $48+20% of $48 = $57.60. 
 
 (6) An article cost me $48 and I sold it for $57.60. 
 What per cent of the cost was my gross profit? What 
 per cent of the selling price ? 
 
 Solution. $57.60 -$48 = $9.60, gross profit ; 
 9.60 -r- $48 = 20%, rate on cost ; 
 9.60 -T- $57.60 = 16f%, rate on sales. 
 
 (c) An article cost me $48. At what price must I sell 
 it to make 16f% of the selling price? 
 
 Solution. Since the gain is of the selling price, the 
 cost is of it. Hence, of the selling price = $48. The 
 whole of it=X$48 = $57.60.
 
 134 THE TEACHING OF ARITHMETIC 
 
 It must be remembered that, in teaching this or any 
 other practical application of arithmetic, the purpose is 
 to give accurate information so that one may properly 
 interpret business methods and practices. Thus, in 
 teaching profit and loss one should not teach that loss or 
 gain is always reckoned on the cost a thing taught 
 in nearly all textbooks for it is not true. There is no 
 uniform agreement among business men as to what should 
 be used as the basis in reckoning the rate of profit. Some 
 reckon the profit on the net cost, some on the delivered 
 cost, and others upon the selling price. No confusion 
 arises, however, if the basis is stated. But the pupil 
 must be led to see that the mere statement that a mer- 
 chant made 25% has no meaning unless the basis upon 
 which it was reckoned is stated. Thus, one should say, 
 "he made 25% of the prime or net cost," "25% of the 
 delivered cost," or "25% of the sales." Most textbooks 
 make this error. That is, they still give such problems 
 as, "A merchant paid $24 for an article. For how much 
 must he sell it to gain 20% ? " This problem is not definite. 
 If one interprets the gain as reckoned on the cost, the 
 solution is: 1. 20 X $24 = $28.80; if it is reckoned upon 
 the selling price, the solution is: $24-7- .80 = $30. So it 
 should be clear that, if a per cent is to have a definite 
 meaning, it must be followed by a phrase showing the 
 basis upon which it is reckoned. 
 
 SIMPLE INTEREST 
 
 The topic of simple interest is so common in conversa- 
 tion and in general reading that it is one of the earliest
 
 PERCENTAGE AND ITS APPLICATIONS 135 
 
 applications of percentage to be discussed. It presents 
 no new difficulty and the time element is the only feature 
 not already understood. The topic is taken for informa- 
 tion rather than to develop skill in calculating interest 
 or to develop mathematical power. So it is not advisable 
 to develop special methods of computation but rather to 
 teach a general method that follows from the meaning of 
 interest. Hence, the pupil should understand that interest 
 is money paid for the use of money or for an accommoda- 
 tion on an unpaid debt ; and that it is reckoned as a per 
 cent of the debt for a year's use of it, even though the 
 interest is collected semiannually, quarterly, or oftener. 
 Thus, the interest of $600 at 6% is $36 for a whole year. 
 But for 6 months it would be but half as much, or $18, 
 and for but one month it would be $3. In this way it 
 is easily shown that the interest for one year multiplied 
 by the time expressed as a part of a year gives the interest 
 for the given tune. 
 
 When the interest is for some simple fractional part 
 of a year, as to find the interest of $1200 at 5% for 4 
 months, the work may be arranged as follows : 
 
 The interest for 1 year being 5% of $1200 
 is $60. But for 4 months the interest would $1200 
 be but i of $60. .05 
 
 Where the fractional part of a year is not : 
 
 $20 00 
 so simple, all the work to be done should first 
 
 be written down and all common factors can- 
 celed. Thus, to find the interest on $1500 at 6% for 86 
 days, the work should be :
 
 136 THE TEACHING OF ARITHMETIC 
 
 The only problem of arithmetic that concerns the 
 person who is borrowing or loaning money is how to find 
 the interest. Hence, the first general problem of per- 
 centage is the only one of the three that is used in interest. 
 The three possible indirect problems still given by some 
 textbooks "for analysis" should have no place in the 
 course, as they never occur in life and detract from the 
 real purpose of teaching the subject, which is to develop 
 a social insight into methods of borrowing and loaning 
 money. 
 
 More important than the mere ability to find interest 
 is a knowledge of current rates of interest, the form and 
 meaning of a promissory note, the methods of securing 
 payment, how often the interest is usually collected, etc. 
 While such knowledge is found in most textbooks, those 
 in use may have been written many years ago and some 
 changes may have taken place. So it creates more 
 interest and insures accurate local information if a com- 
 mittee from the class can go to some bank and get the 
 information and bring it fresh to class. 
 
 Those problems of the textbook giving rates not in 
 use and with the time element more than one year should 
 not be given, for they give wrong notions as to the price 
 paid for money and also as to the time when interest must 
 be paid. Have pupils see that a man may give a note for
 
 PERCENTAGE AND ITS APPLICATIONS 137 
 
 a loan and secure it properly, say by a mortgage on real 
 estate, and not pay the debt for years ; but that the in- 
 terest must be kept paid up as it falls due, which is at 
 least yearly if not more often, as semiannually, this 
 being specified in the note. 
 
 SHORT METHODS OF COMPUTING INTEREST 
 
 In any business in which much computation of interest 
 is required, as in banks, trust companies, or life insurance 
 offices, use is made of interest tables which greatly lessen 
 the labor of computation. A person outside of such a 
 vocation has so little need of computing interest that it 
 seems unwise to teach more than the general method to 
 pupils in the grammar school. 
 
 By the general method is meant: time X rate X 
 principal = interest, whether a year's interest is first found 
 and that result multiplied by the number of years, or 
 when all the work to be done is written and common 
 factors canceled. 
 
 Since most textbooks give several short methods, two 
 are given here. They follow from first solving the prob- 
 lem by the general method, and thus they help the pupil 
 to see how special cases may lead to special methods. 
 
 Thus, to find the interest of $1750 at 6% for 119 days : 
 
 By the general method, ^X-^-X$1750= 119x$175 
 
 100 6000 
 
 60 
 
 Interpreting the result, we see the work to be done may 
 be expressed as follows : To find the interest of any
 
 138 THE TEACHING OF ARITHMETIC 
 
 principal at 6% for any number of days, multiply the 
 principal by the number of days, point off three more 
 decimal places, and divide by six. 
 
 Illustration. Find the interest of $850 $850 
 at 6% for 70 days. The work is given in 70 
 
 the margin. 6)$59.5QO 
 
 Had the number of days been 60, then all $9.92 
 factors would have canceled except the prin- 
 cipal divided by 100. Thus, to find the interest of $1350 
 at 6% for 60 days, the general method is : 
 
 100 
 
 From this the pupil may observe that to find the interest 
 of any principal at 6% for 60 days, point off two more 
 decimal places in the principal. 
 
 COMPOUND INTEREST 
 
 When the interest due at the end of any interval is 
 added to the debt and thus draws interest for the next 
 succeeding interval, as in a savings bank, the interest 
 that accrues is called compound interest. 
 
 The subject is of chief importance to large investors, 
 as building and loan associations, life insurance companies, 
 banking corporations, etc., who wish to compute the final 
 incomes from reinvesting all interest as it falls due. 
 Such computations are made by the use of compound 
 interest tables.
 
 PERCENTAGE AND ITS APPLICATIONS 139 
 
 In teaching the subject the teacher may use it as the 
 basis of a discussion of thrift, the nature and importance 
 of the savings bank, etc. It is always interesting to see 
 how rapidly small regular deposits increase at even a 
 small rate of interest. Thus, $1 per week for 40 years 
 at 5%, compounded annually, will amount to about 
 $6600. And $1 per week for only 10 years at 5%, com- 
 pounded annually, will amount to nearly $700. 
 
 INSURANCE 
 
 The subject of insurance, both property insurance and 
 health and life insurance, is such an important factor in 
 modern life that it should be one of the applications in- 
 cluded under percentage, although the term per cent is 
 not very largely used in discussing insurance. The subject 
 is studied, however, for its informational value rather 
 than for the arithmetical problem involved. Canceled 
 or specimen policies should be brought before the class 
 and the subject thus made as real and vital as possible. 
 
 FIRE INSURANCE 
 
 While there are various kinds of property insurance, 
 as tornado, lightning, burglary, live stock, marine, plate 
 glass, transit, etc., the subject of fire insurance is the 
 most common kind in most communities and may be 
 taken up as a type of all. 
 
 Fire insurance is an agreement by a fire insurance 
 company to indemnify the insured against actual losses 
 from accidental fires. The "loss by fire" includes any
 
 140 THE TEACHING OF ARITHMETIC 
 
 damage resulting from chemicals or water used in ex- 
 tinguishing the fire. 
 
 The form of policy is regulated by the laws of the various 
 states. It is a fixed form of agreement to do or not to do 
 certain things for a fixed consideration called the premium. 
 
 To some policies are attached riders granting certain 
 privileges or making certain restrictions. Among the 
 standard riders the most important is the average or co- 
 insurance clause. This is an agreement on the part of 
 the insured to carry a certain amount of insurance upon 
 his property or, failing to do this, to become a coinsurer 
 with the company for whatever amount his insurance 
 lacks of the amount agreed upon. Under a coinsurance 
 clause, then, the insurance company pays only such a 
 part of any loss as the amount of their policy bears to the 
 amount of insurance agreed upon. Thus, if a man ac- 
 cepts an 80% coinsurance clause as a part of his contract 
 upon property valued at $10,000, he agrees to carry 
 $8000 of insurance. If he carries but $6000 and a loss 
 occurs, he can collect but three fourths of it from the 
 insurance company. 
 
 The rates of insurance depend upon the nature of the 
 risk. Upon a building they depend very largely upon : 
 (1) the location ; (2) the construction ; (3) the occupancy ; 
 (4) the exposure; and (5) whether or not there is co- 
 insurance. The pupils should find the rates upon build- 
 ings in various parts of the community and attempt to 
 justify the different rates. Details of local conditions 
 should be learned from some agent in the neighborhood 
 and the problems made to meet these conditions.
 
 PERCENTAGE AND ITS APPLICATIONS 141 
 
 LIFE INSURANCE 
 
 About the only arithmetical problem of life insurance 
 that a pupil in the grades is able to handle is to find the 
 difference in the amounts left the beneficiary by life 
 insurance and the amounts of the premiums at simple 
 or compound interest, knowing the number of premiums 
 that have been paid. But he can understand the nature 
 of the three general forms of policies and .also the ele- 
 ments that make up the premium. 
 
 The three general forms of policies are: (1) ordinary 
 life ; (2) limited life ; and (3) endowment. 
 
 In the ordinary life policy, the premiums are paid 
 during the life of the insured, and the insurance company 
 agrees to pay a fixed sum to the beneficiary at the death 
 of the insured. 
 
 In the limited life policy, the premiums are paid for 
 a fixed number of years, but the guarantee in the policy 
 is not paid until the death of the insured. 
 
 In an endowment policy, the insurance company agrees 
 to pay the insured the full amount of the policy if he sur- 
 vives beyond a specified date, or to pay the amount as 
 above if he dies before that date. 
 
 The premium that the insured pays is composed of 
 three items: (1) mortality cost; (2) reserve; and (3) ex- 
 pense loading. The first two of these items form the net 
 premium and are determined by a given mortality table. 
 The American Mortality Tables are compiled from the 
 experience of the New York Mutual Life Insurance 
 Company and are the tables prescribed by statute in
 
 142 THE TEACHING OF ARITHMETIC 
 
 most of the states as the basis upon which the reserve of 
 life insurance companies shall be computed. 
 
 The mortality cost is the amount necessary to collect 
 at the beginning of each year to pay the death claims of 
 the current year as determined by the mortality tables. 
 
 The reserve element of a premium is the amount set 
 aside from each premium at a given rate of compound 
 interest, usually 3% or 3^%, which will amount to the 
 face of the policy in a given time. So the company's 
 risk at any time is only the difference between the face 
 of the policy and the amount of the reserve. 
 
 The expense loading, which is from 15% to 25% of the 
 net premium, is added to meet the expenses of manage- 
 ment. 
 
 While there are other important things to consider in 
 life insurance, the things discussed here are sufficient to 
 show a pupil the types of policies and the things for which 
 the premiums must be collected. The problems of the 
 subject are too difficult for a pupil in the grammar school. 
 The subject, then, is for information and not for the 
 mathematics involved. 
 
 TAXES 
 
 The subject of taxes, again, is taken for its informa- 
 tional value, not its mathematical problems. The pupil 
 should understand for what purposes taxes are levied, 
 the way in which the expenses to run a town, county, or 
 state are equitably distributed, and the current rates of 
 taxation. Since the method of levying and collecting 
 taxes varies in different states, the local problems and 

 
 PERCENTAGE AND ITS APPLICATIONS 143 
 
 rates are to receive the chief emphasis. In such a study 
 of the local methods, the pupil should become acquainted 
 with such terms as assessment, valuation, assessor, col- 
 lector, board of review, board of equalization, and delin- 
 quent taxes. 
 
 The tax rates are usually expressed as mills on a dollar, 
 dollars on a hundred dollars, or dollars on a thousand 
 dollars. The general method of finding the rate is to 
 divide the amount to be raised by the assessed valuation 
 of the property. 
 
 To facilitate computation, a tax table is usually pre- 
 pared giving the taxes on sums from $10 to $99, and then 
 the tax on other sums is found by properly pointing off 
 and by addition. The pupils may no doubt be able to 
 get a copy of such a table at the office of the assessor of 
 taxes. 
 
 The pupil should see the difference between collecting 
 taxes to meet the expenses of the city, county, and state, 
 and the method of collecting those for the expenses of 
 the National Government. This will lead to a study of 
 duties and customs and the income tax. Here again the 
 mathematics involved is of less importance than the in- 
 formation about the general expenses of the government, 
 where they go, and how they are met. 
 
 BANKING 
 
 Pupils should know the function of a modern bank 
 that it is an institution where money is deposited for 
 safe-keeping and from which it may be withdrawn when 
 wanted ; and also that it is a place where money is loaned
 
 144 THE TEACHING OF ARITHMETIC 
 
 on personal or other security. They should know, too, 
 that most of the money that a bank handles is that of its 
 depositors, and that it is from loaning these deposits that 
 a bank earns money. 
 
 In a commercial sense, banks are classified as banks 
 of deposit, banks of discount, and banks of circulation. All 
 commercial banks exercise the first two of these functions, 
 and national banks the third. The function of a bank 
 of deposit is to receive money on deposit and pay the 
 checks drawn upon these deposits. The function of a 
 bank of discount is to loan money on promissory notes. 
 The function of a bank of circulation is to issue its own 
 promissory notes, which are used as currency. To make 
 this last clear, take before the class some bank note 
 issued by some national bank or federal reserve bank, 
 the only institutions allowed to issue them. 
 
 The pupils should also understand the method of 
 opening an account with a bank, the making out of a 
 ticket of deposit, making out and indorsing a check, and 
 the purpose of having a check, that is to be sent out of 
 town, certified by the bank upon which it is drawn, and 
 the transmitting of money by means of a bank draft. 
 All this is vitalized by having pupils gather the informa- 
 tion from a local bank. 
 
 BANK DISCOUNT 
 
 The subject of bank discount need not present any new 
 difficulties. It is only a method of collecting interest. 
 When interest is collected in advance, it is called bank 
 discount. Thus, if one borrows $300 at a bank for 90
 
 PERCENTAGE AND ITS APPLICATIONS 145 
 
 days, the cashier computes the Interest at the bank's rate, 
 which is quite generally 6%, and deducts it from the 300 
 or face of the note. At 6% the interest is $4.50, leaving 
 $295.50, called the proceeds. When the interest is col- 
 lected in advance, a non-interest-bearing note is given 
 for the loan, for only the $300 is paid when the note is due, 
 not $300 "with interest." Since one paying interest in, 
 advance is paying upon the maturity value of the note, 
 it follows that, in discounting an interest-bearing note at 
 a bank, the bank would reckon the interest on the amount 
 of the note when due and not upon the face of the note. 
 Thus, if a note of $1000 bearing 5% interest is to run one 
 year, it is worth $1050 when due. So, if it is sold at a 
 bank 60 days before it is due, the bank will compute the 
 interest on $1050 for 60 days, and not upon $1000. The 
 interest (bank discount), then, is $10.50, leaving the 
 proceeds of $1039.50. 
 
 STOCK INVESTMENTS 
 
 In the past, the applications of percentage have been 
 hard for the pupil in the grammar school for two reasons : 
 (1) more time has been spent upon the mathematical 
 side than upon the social side, hence the pupil has not had 
 a concrete background for the arithmetical side of the 
 work; and (2) the indirect problem, never arising in 
 business, has been featured for "analysis." If teachers 
 and textbooks would give merely the problems needed to 
 interpret the references met in general reading and con- 
 versation, the work would not offer great difficulties. 
 
 Interest in the stock market is so general that the
 
 146 THE TEACHING OF ARITHMETIC 
 
 subject becomes one of the important applications, 
 but for its information and not for the mathematics 
 involved. 
 
 There was a time when the owner of a business took hi 
 one or more partners when he wished to increase the busi- 
 ness beyond his own power to supply the necessary capital. 
 But such partnerships often proved unsatisfactory. Each 
 partner was usually held responsible for the acts of any 
 one of the partners. Each change of partnership required 
 new adjustments. As business enlarges, these and many 
 other difficulties arise. 
 
 Organizing business under a corporation or company 
 does away with many of the objections of the partner- 
 ship arrangement. Some of the advantages of a corpora- 
 tion are: (1) one may become but a small investor in 
 several corporations, giving no personal attention to the 
 business, instead of having to make a large investment 
 and giving the business more or less attention as in the 
 case of partnership; (2) one may transfer his ownership 
 in a corporation without affecting the business; (3) the 
 business can more easily expand its capital ; and (4) the 
 plan enables the hiring of experts to run the business 
 rather than having it run directly by the partners. 
 
 In organizing a corporation a special number of persons 
 interested in starting a business of some sort make ap- 
 plication to the Secretary of State of their own or some 
 other state, giving the proposed name of the company, 
 the place and nature of the business they propose to carry 
 on, the amount of capital and the number of shares into 
 which they propose to divide it, and any other informa-
 
 PERCENTAGE AND ITS APPLICATIONS 147 
 
 tion required by the law of the state in which they make 
 the application. 
 
 The Secretary of State files their application and grants 
 them a permit to sell the capital stock. 
 
 When the legal amount has been sold, the buyers of 
 the stock adopt a set of by-laws and elect a board of di- 
 rectors who in turn elect a president, secretary, and 
 treasurer, and other officers from their number. A 
 record of this, showing that the law has been complied 
 with, is then filed with the Secretary of State, who issues 
 a charter, which is an instrument defining the powers, 
 rights, and duties of the corporation. 
 
 The capital with which the company organizes divided 
 by the number of shares into which it has seemed de- 
 sirable to divide it gives the par value of a share. The 
 par value, then, is no indication of the real value of the 
 stock, but serves to show what part of the business is 
 owned by the holder of a certificate. Thus, if the $100,000 
 capital of a corporation is divided into 1000 equal shares, 
 the size of each is $100. So a person owns one thousandth 
 of the business for each share that he owns. This simpli- 
 fies the distribution of the earnings. 
 
 The real or market value of stock is what it can be 
 bought or sold for in open market. Chief among the 
 factors affecting the market price of stock are : (1) the 
 real or prospective earning power of the corporation; 
 and (2) the confidence of the buying public, or lack of it, 
 in the general stability of the enterprise. When the real 
 or anticipated earnings are small, the price is low ; when 
 large, the price is high.
 
 148 THE TEACHING OF ARITHMETIC 
 
 The earnings of a corporation distributed among its 
 stockholders are called the dividends. They are declared as 
 a per cent of the capital of the corporation, and each stock- 
 holder gets that per cent of the par value of the stock 
 that he owns. Thus, if a corporation, whose capital is 
 $200,000, is to distribute $24,000 in profits, the directors 
 declare a 12% dividend and the holder of each $100 share 
 gets $12. Stock of this kind, whose dividend depends 
 upon the earnings of the corporation, is called common 
 stock. There is another kind of stock sometimes issued 
 called preferred stock, in which the rate of dividend 
 paid is named in the certificate. These dividends, then, 
 become an obligation of the corporation and are deducted 
 from the gross earnings before a dividend can be declared 
 upon the common stock. 
 
 There are two types of buyers of stock: the investor 
 who buys and holds for the dividends he will receive, 
 just as he loans money for the interest or buys a house 
 for the rent; and the speculator who buys, expecting 
 the price to advance so that he can sell at a profit. 
 
 The problems that naturally arise are very simple 
 when one understands the terms involved. They are: 
 (1) to find how much is lost or gained when buying stock 
 at one price and selling it at another; and (2) to find 
 what per cent of its market value stock paying a certain 
 dividend is earning. These are the only problems with 
 which the general reader or the buyer is concerned.
 
 CHAPTER XII 
 DENOMINATE NUMBERS AND MEASUREMENTS 
 
 I. NATURE AND PRINCIPLES OF DENOMINATE 
 NUMBERS 
 
 FOEMERLY the subject of denominate numbers was 
 taught as a distinct topic. Now the work is distributed 
 throughout the grades. The tables are taught as need 
 for them arises in the child's activities in or out of school. 
 A single unit from a table may be taught alone as need 
 for that unit arises without teaching its relation to the 
 other units of the table. Thus, the child may recognize 
 a foot in length and use the measure in finding length 
 without knowing that 12 inches make 1 foot or that 3 
 feet make 1 yard. 
 
 To get a practical knowledge of the tables or the various 
 units, the pupil must see and handle all the common 
 measures. From these units he should make his own 
 table. Thus through measuring he discovers that 2 pt. = 
 1 qt. and that 4 qt. = 1 gal. ; or that 12 in. = 1 ft. and that 
 3 ft. = 1 yd. It is, of course, unnecessary and impractical 
 to have all of a table found in this way where a very large 
 number is required, as finding how many feet or yards 
 in a mile. After the relation of the various units of a 
 table has been found, through measurement, the tables 
 
 149
 
 150 THE TEACHING OF ARITHMETIC 
 
 should be memorized and used in problems; but just as 
 much of a table as is needed at the time should be learned. 
 Thus, inches, feet, and yards are needed long before rods 
 and miles are needed ; ounces and pounds before tons ; 
 pints and quarts before gallons or barrels ; etc. 
 
 All obsolete units of any table and all tables belonging 
 exclusively to a technical education should be entirely 
 omitted. 
 
 REDUCTION OF DENOMINATE NUMBERS 
 
 The reduction of denominate numbers involves no new 
 principles. If a pupil understands the meaning of multi- 
 plication and division and when to apply these processes 
 to simple one-step problems, he should have no difficulty 
 in reducing units from one denomination to another when 
 need for such a reduction occurs in any practical applica- 
 tion. 
 
 The old plan of having each table of the weights and 
 measures followed by a lot of exercises for reductions, 
 apart from any industrial or commercial problem, is fast 
 disappearing, and the reductions are now being made 
 when need for them arises in some problem of everyday 
 life. 
 
 Such reductions, either descending or ascending, usually 
 involve one step, or one step and the addition of a lower 
 unit, or one division and a remainder. 
 
 Such reductions embody the following operations : 
 
 (1) Reduce 5 ft. 8 in. to inches. 
 
 5ft.= 5Xl2 in. = 60 in.; 60 in.+8 in. = 68in.
 
 DENOMINATE NUMBERS 151 
 
 (2) Reduce 68 in. to feet or to feet and inches. 
 1 in. = -&ft; 68 in. = 68X^r ft. = 5| ft. 
 
 Or, 68 in. -T- 12 in. = 5, plus remainder of 8 in. 
 not divided. Hence, 68 in. = 5 ft. 8 in. 
 
 That is, the last solution is the measuring problem 
 of division. The quotient 5 shows the number of 12 in. 
 in 68 in. ; and, since each 12 in. is a foot, 5 is the number 
 of feet. 
 
 Avoid such erroneous statements as : 
 
 5 ft. 12)68 in. 
 12 5 ft. 8 in. 
 
 60 in. or 
 
 _8 in. 12 in.)68 in. 
 68 in. 5 ft. 8 in. 
 
 ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION 
 
 The four fundamental processes of arithmetic need not 
 be taught as topics and processes of denominate numbers 
 to any great extent. But little, if any, use of such pro- 
 cesses occurs in the everyday problems of life. In ordinary 
 measurement, results are expressed by use of fractions 
 in terms of but one unit, instead of two or more before 
 adding, subtracting, multiplying, or dividing them. 
 Thus, instead of adding or subtracting 16 ft. 8 in. and 
 12 ft. 6 in., one would add or subtract 16f ft. and 12^- ft. 
 Instead of multiplying 2 bu. 3 pk., one would multiply 
 2| bu. or 2.75 bu. Instead of dividing 12 ft. 10 in., one 
 would divide 12f ft. or 12.83 ft., etc.
 
 152 THE TEACHING OF ARITHMETIC 
 
 H. MENSURATION 
 
 If properly presented, the interpretative value of the 
 subject of mensuration is very great. A large number 
 of terms and concepts met in a wide variety of practical 
 activities is acquired. These include the ideas of dis- 
 tance, area, and volume as applied in scale drawing, in 
 computing costs in paving streets and laying sidewalks, 
 in measuring farms and computing productions, in measur- 
 ing bins for grains, coal, etc., in computing costs of ex- 
 cavations; in fact on every hand are found applications 
 of these fundamental facts of mensuration. And because 
 there is such a rich field of applications, opening up such 
 a possibility of problems of great variety, the topic, like 
 many other topics of arithmetic, is often carried beyond 
 its real need as a tool with which we interpret the physical 
 world about us. 
 
 But to have a broad interpretative value to the pupil, 
 he must be led to formulate his own laws and rules and to 
 find an application for them in the life about him. It is 
 not uncommon to find pupils computing the cost of side- 
 walks of given dimensions as problems found in their 
 textbooks and yet who have no idea of the cost of the 
 walk in front of their own homes or the school, or who are 
 unable even to go out and take the measures and find the 
 cost of such a walk. This sort of teaching is valueless 
 as a means of developing ability to interpret life about 
 them. 
 
 It is only when the subject is taught through having 
 the pupil make his own measurements, constructions,
 
 DENOMINATE NUMBERS 153 
 
 drawings, etc., and applying the knowledge gained to out- 
 of-school measurements that the subject becomes of real 
 value. 
 
 THE AKEA OF A RECTANGLE 
 
 The pupil must see that to measure anything is simply 
 to apply some standard unit and see how many times it 
 is contained ; and thus to measure the area of a rectangle 
 is to see how many times it contains a chosen square 
 whose side is some linear unit. 
 
 Through actual construction the pupil should draw a 
 rectangle whose sides are some integral number of units 
 and then lay it off into square units and count them to 
 find the area in order to see clearly what is meant by the 
 measurement of a rectangle. Through drawing to a 
 scale and the use of squared paper 
 upon which to represent rectangles, 
 he should become familiar with such 
 facts as : the number of squares in a 
 row along any side is the same as the 
 number of linear units in that side; 
 that there are as many rows, all alike, as there are 
 units in the other dimension; and thus he should see 
 that the total number of square units is the product of 
 the number of linear units in the two dimensions. Thus, 
 the area of a rectangle 4 inches by 3 inches is 4X3 square 
 inches. 
 
 Properly developed, pupils will never get the erroneous 
 notion that "inches multiplied by inches give square 
 inches," a thing still taught in many schools. Such a
 
 154 THE TEACHING OF ARITHMETIC 
 
 statement would seem as ridiculous to them as " 3 horses X 
 4 horses = 12 square horses." 
 
 Following the development of the rule or law, pupils 
 should apply their knowledge by measuring actual areas 
 about them. In finding costs, get local prices so that 
 when the cost of plastering a wall, laying a floor, construct- 
 ing a sidewalk, paving a street, sodding a lawn, or roofing 
 a house is computed the pupil will see about how much 
 of the thing measured can be bought for a certain price ; 
 that is, the pupil should know what the sidewalk in front 
 of his home or in front of the school would cost. He 
 should have some idea of the cost to plaster a wall of the 
 schoolroom, or to place the metal ceiling if there is one, 
 or the cost of the blackboards. It is through such work 
 as this that the topic has a real social value. 
 
 THE AREA OF A PARALLELOGRAM 
 
 Through construction, the pupil must see that a gen- 
 eral parallelogram whose angles are not right angles can- 
 not be directly laid off into square niches as in the case of 
 the rectangle. Then impress upon him the fact that the 
 rectangle, being the only figure that can be laid off into 
 squares, is taken first and used as a basis, and that all 
 other figures are measured by finding their relation to 
 some rectangle. 
 
 The first step, then, in measuring a parallelogram is 
 to reduce it to a rectangle. The teacher's blackboard 
 drawings are not conclusive enough for the child. The 
 part which she removes by erasing and the part which 
 she places in another position by drawing are not the
 
 DENOMINATE NUMBERS 
 
 155 
 
 same parts. From her blackboard presentation of the 
 method of transforming a parallelogram into an equivalent 
 rectangle, the pupil should construct a parallelogram 
 from cardboard, actually cut off a part, and, by rearrang- 
 ing this part, transform the given parallelogram into a 
 rectangle. He need not divide this new rectangle into 
 square inches, for he can deduce 
 the rule from that for finding 
 the area of a rectangle, for he 
 has now discovered that the 
 area of a parallelogram is the 
 same as that of a rectangle 
 having the same dimensions. 
 His attention should be called 
 to the fact that the perpendicu- 
 lar distance between the base and opposite side becomes 
 the altitude of the rectangle and is thus the altitude of 
 the parallelogram ; otherwise he may associate the two 
 adjacent sides of the parallelogram with its dimensions. 
 
 The pupils may not be able to find as many examples 
 of this figure as of the other four plane areas. It is given 
 largely as a basis for the others in order to save convert- 
 ing them into rectangles. 
 
 THE AEEA OF A TRIANGLE 
 
 As in the parallelogram, pupils will see that a triangle 
 cannot be divided directly into squares as in the case of 
 a rectangle. After having thus raised the question of 
 how to measure a triangle, the teacher should have each 
 pupil cut two triangles of the same form and size from
 
 156 THE TEACHING OF ARITHMETIC 
 
 cardboard. But all children should not make triangles of 
 the same size. The teacher should draw various forms 
 and sizes on the blackboard and let pupils use their own 
 pleasure in drawing any shape they wish. Then have 
 each child place his two triangles together so as to form a 
 
 parallelogram and thus discover 
 that a triangle is just half as 
 large as a parallelogram having 
 the same dimensions. From this 
 fact, he makes his rule. The 
 examples of this kind of figure are more numerous. Have 
 hun find areas to be measured that are triangular in form, 
 take the measurements himself, and find the areas. 
 
 THE AREA OF A TRAPEZOID 
 
 Since the pupil should be shown how to find an area 
 by dividing it up into figures that he has studied, it is 
 hardly necessary to take up the measurement of a trape- 
 zoid for any practical purpose ; yet it is required by most 
 
 courses of study and 
 
 is so easy to present 
 that the teaching of 
 it is not open to seri- 
 ous criticism. As in 
 
 the triangle, have the pupils each make two trapezoids 
 just alike and place them so as to form a parallelogram. 
 They thus see that, since it took two trapezoids to form 
 the parallelogram, a trapezoid is half as large as a parallelo- 
 gram having a base equal to the sum of the bases of the 
 trapezoid and having the same altitude as the trapezoid.
 
 DENOMINATE NUMBERS 
 
 157 
 
 As with all other figures, have the pupils look for 
 examples of such figures, measure them, and find the areas. 
 
 THE CIRCUMFERENCE AND AREA OF A CIRCLE 
 The pupil does not have to accept the facts and rules 
 for the measurement of a circle upon the authority of 
 textbook or teacher any more than he did the four pre- 
 ceding areas. The rules and relations are as readily dem- 
 onstrated objectively, and the pupil easily discovers 
 them for himself. 
 
 In finding the relation of the circumference to the 
 diameter, use as large circular objects as possible. The 
 pupils may be directed to measure the circumference 
 and diameter of some objects found at home, as the din- 
 ing room or reading room tables, and bring the data to 
 school. If the measurements have all been carefully 
 taken, the ratios of circumference to diameter in all cases 
 will be nearly enough 87- to satisfy the class that the 
 ratio is constant. The more exact ratio, 7r = 3.1416, 
 should be given and used where great accuracy is required. 
 
 A B 
 
 To discover the area, have pupils each cut from card- 
 board circles of various sizes, not too small, and divide 
 them into at least sixteen equal parts as in figure A.
 
 158 THE TEACHING OF ARITHMETIC 
 
 Have them take half of these parts and arrange them as 
 in the lower half of figure B, and then take the other half 
 and fit them into these, completing figure B, and thus 
 discover that the area of a circle is equal to that of a 
 parallelogram whose base is half the circumference and 
 whose altitude is the radius. Then, A=%CxR. But 
 C=TrD, hence %C=irR. Then, A=irRxR=7rIP. 
 Find all the local applications possible. 
 
 THE VOLUME OF RECTANGULAR SOLIDS 
 
 This is the fundamental volume from which the meas- 
 urement of other volumes is deduced. The same plan 
 of comparing the volume with some standard unit, sug- 
 gested hi the measurement of a rectangle, should be fol- 
 lowed here. Or, instead of dividing a solid into cubic 
 units, a box whose inside measure is some integral number 
 of units may be filled with cubic units. The pupil should 
 see that the number of cubic units along any dimension, 
 as the length, is the number of linear units in the length ; 
 the number of such rows in the width is the number of 
 linear units in the width; and that the number of such 
 layers in the whole volume is the number of linear units 
 in the height. Thus, if there are 6 inches in the length, 
 there will be 6 cu. in. in a row along the length. If 4 
 inches wide, then there will be 4X6 cu. in. in a layer; 
 and, if 5 inches deep, there will be in the whole volume 
 5X4X6 cu. in. or 120 cu. in. If the rule is developed in 
 this way, the pupil will not use such erroneous statements 
 as 5 in.X4 in.X6 in. = 120 cu. in. 
 
 Follow up the development by local problems, as ex-
 
 DENOMINATE NUMBERS 
 
 159 
 
 cavations for cellars, capacity in tons of coal bins, in 
 bushels of grain bins, the air space of the room, the snow 
 removed from walks, or hi any grading or constructing 
 work that is being done in the neighborhood. 
 
 OTHER VOLUMES 
 
 The illustrations already given serve to emphasize 
 the method of teaching measurement. Any good text- 
 book gives diagrams to illustrate the development. But 
 with children, diagrams of the textbook are not sufficient. 
 The objects illustrated by the diagrams should be used. 
 
 GRAPHIC REPRESENTATION OF STATISTICS 
 
 The relations between magnitudes is more easily grasped 
 by most people if expressed by graphs rather than by 
 figures to represent the number of units. For this reason 
 newspapers, magazines, trade journals, and all kinds of 
 circulars and pamphlets make very frequent use of the
 
 160 
 
 THE TEACHING OF ARITHMETIC 
 
 graph where magnitudes are shown in comparison with 
 other magnitudes or with their former values. Thus, the 
 graph on the previous page is reproduced from an adver- 
 tisement of the American Sugar Refining Company. 
 
 There are three general forms in use depending upon 
 the nature of the comparisons to be made. 
 
 In comparing the productions, wealth, population, etc., 
 of different states or countries, straight lines are generally 
 used. Thus, the comparative production of corn in the 
 following states, 1917, could be represented graphically 
 as follows : 
 
 Illinois. 
 444,843,000 bu. 
 
 Iowa. 
 411,656,000 bu. 
 
 Missouri. 
 263,463,000 bu. 
 
 Indiana. 
 208,522,000 bu. 
 
 Ohio. 
 162,859,000 bu. 
 
 Kentucky. 
 126,859,000 bu. 
 
 Kansas. 
 121,097,000 bu. 
 
 Tennessee. 
 117,273,000 bu. 
 
 For the construction of such graphs, squared paper is 
 very convenient. Thus, the following comparison of the 
 wheat crop of 1917 in the five leading wheat states may 
 be shown as follows : 
 
 Minn. 
 N. Dak. 
 8. Dak. 
 Wash. 
 Mont.
 
 DENOMINATE NUMBERS 
 
 161 
 
 To show the relative changes of the same magnitude 
 through a given period of time, both the straight line and 
 the broken line graphs are used. Thus, the variation of 
 the September prices of hogs ranging over a period of 
 eight years may be represented in either of the following 
 ways: 
 
 10 
 
 8. 
 
 II I 
 
 II I 
 
 II I 
 
 II I 
 I I I 
 
 1910 '11 '12 '13 '14 '16 '18 '17 
 
 13 
 
 10 
 
 1910 '11 '12 ,'13 '14 '15 '16 '17 
 
 To show the relation of a part to the whole, a circle is 
 more often used. Thus, the following graph shows how 
 the rent was apportioned on each floor of three stores 
 during a recent year. It is a portion of a chart given in 
 The System, December, 1917. (See page 162.) 
 
 Pupils should be encouraged to bring in graphs found in 
 general reading and interpret them. It will be found
 
 162 
 
 THE TEACHING OF ARITHMETIC 
 
 that there is no rigid adherence to a certain form for a 
 certain type of comparison, but that the uses shown here 
 are general. 
 
 Enough work in the construction of the three kinds 
 shown here to readily interpret those found in general 
 reading should be done if time permits.
 
 CHAPTER XIII 
 THE PURPOSES AND NATURE OF PROBLEMS 
 
 THE great advances made in the teaching of arithmetic 
 during recent years have come very largely through an 
 improvement in the nature of the problems. This im- 
 provement has followed from the broader conception of 
 the purposes of a problem discussed in this chapter, for 
 a problem is no longer considered a mere tool to train the 
 mind. 
 
 In the lower grades the problems differ greatly hi their 
 nature and purpose from the problems of the upper grades. 
 For that reason problems will be discussed under two 
 general divisions : the problems of the primary grades, 
 and the problems of the upper grammar grades. 
 
 THE PROBLEMS OF THE PRIMARY GRADES 
 
 While the child in the first four or five grades may 
 get several things as by-products of his study of a problem, 
 such as the habit of looking upon the quantitative side 
 of life, an appreciation of the value of arithmetic in doing 
 the world's work, something of a social insight into cer- 
 tain phases of life, etc., the chief purposes of the problem 
 during these years are: (1) to clarify or rationalize the 
 
 163
 
 164 THE TEACHING OF ARITHMETIC 
 
 facts and processes; and (2) to motivate the pure drill 
 work in the fundamental processes. This leads, then, to 
 an inquiry into the nature of problems that will do this. 
 
 CLARIFYING A PROCESS 
 
 We are beginning to realize that the child's own thought 
 and activity should dominate classroom instruction ; that 
 all knowledge, to be real, must be based upon the real 
 experiences of the individual learner ; and that arithmet- 
 ical ideas must have a background of mental images which 
 are the outgrowth of real experiences, in order to exist. 
 Hence, all facts and processes should be presented ob- 
 jectively or through very familiar situations. Through 
 such teaching the facts have a real background of vivid 
 mental images and the pupil is able to use them in the 
 situations that arise in his real life. The chief source of 
 failure of a pupil to apply his arithmetical facts is that 
 they have been put before him as mere verbal facts with- 
 out any concrete basis. 
 
 When a problem, however, is given to clarify a fact or 
 process, it must be made from real objects seen and handled 
 by the class, from the pictures of such objects, or from 
 experiences that are very familiar to the pupils. 
 
 It is through developing the facts of addition, multi- 
 plication, etc., through such concrete situations, that the 
 pupil gets the real meaning of them and thus knows how 
 to use them in new situations that arise. If the first 
 problems are made up about real things that the child 
 sees, then next about pictures that call up real objects, 
 and finally about things that he easily images, he grows
 
 PURPOSES AND NATURE OF PROBLEMS 165 
 
 into the real method of approaching the more difficult 
 problems that he will meet later in the course. 
 
 MOTIVATING DRILL WORK 
 
 Aside from rationalizing a fact or process, the problems 
 given during the early years of a child's school life are 
 given very largely to furnish a motive for, or an interest 
 in, adding, subtracting, multiplying, and dividing. Such 
 problems should require little, if any, thought in order 
 'to discover the process to be used. 
 
 To motivate the drills, however, the problems will 
 have to be more attractive than mere drill work from 
 abstract numbers. This will require that the answer is 
 really wanted because it meets some personal need of 
 the pupil, adds in some way to his pleasure, or satisfies 
 his curiosity. 
 
 There are at least five general types of problems that 
 may be made to meet these conditions : 
 
 (1) Those meeting some personal question ; as, "If you 
 are saving money to buy a coaster costing $5.75 and now 
 have $3.90, how much more will you need?" 
 
 (2) Those having to do with some home affair; as, 
 "If your father uses 12 tons of coal each winter, costing 
 $7.25 per ton, how much does he have to spend for coal ?" 
 
 (3) Those about some neighborhood activity, as costs 
 of keeping up the parks, playgrounds, schools, etc. 
 
 (4) Those that appeal to his civic pride, as those com- 
 paring the growth of his town or state with some other, 
 or those about some industry in which his town or state 
 excels.
 
 166 THE TEACHING OF ARITHMETIC 
 
 (5) Problems in which the answer simply appeals to 
 his curiosity, as the number of things, such as movie 
 tickets, baseball bats, etc., that could be bought for the 
 money that it costs to run the government for a day, or 
 with the wealth of some of the well-known men of wealth. 
 These furnish larger numbers than those of the real 
 problems discussed under the first four heads. 
 
 THE WORDING OF A PROBLEM 
 
 In the primary grades, as has just been pointed out, 
 the child's present interest is the fundamental factor to 
 be considered in selecting or making problems. The 
 ultimate use of the subject in adult activities has little, 
 if any, place in such a selection in the lower grades. That 
 a child may grow into the true method of solving a problem 
 by comparing the magnitudes involved and thus discov- 
 ering whether they are to be added, subtracted, multi- 
 plied, or divided, the early problems should be "story 
 problems" about objects present, then about objects 
 owned by the children or familiar to them. 
 
 Not only should problems be about familiar objects, 
 but they should be problems that the child might very 
 probably ask. Thus, "Who weighs more, you or I?" 
 would be a problem that a child would be much more 
 likely to ask than, " How much do you and I both weigh ? " 
 
 In the early years, the interest is determined much 
 more by the wording of the problem than by the numbers 
 and processes involved. Thus, a "story problem" such 
 as, "Mary picked 8 pink roses this morning. She is 
 going to leave 5 at home for her mother and take the rest
 
 PURPOSES AND NATURE OF PROBLEMS 167 
 
 to her teacher. How many can she take to her teacher ? " 
 is much more interesting to a primary pupil than the same 
 problem stated as, "A farmer had 8 sheep and 5 of them 
 died. How many were left?" or worse yet, "Eight 
 horses less 5 horses are how many horses?" 
 
 It would be much more interesting and do much more 
 toward developing a proper method of solving problems 
 to give the problem, "If you had put 9 little rabbits in 
 a pen and, when you went to feed them, found that there 
 were but 6, how many were gone ? " than to give, " A man 
 had 9 cows and sold 6 of them ; how many were left ? " 
 
 It is much better to say, "Frank got 50 Saturday Eve- 
 ning Posts this week and sold all but 4 of them ; how many 
 did he sell ?" than to say, "A farmer had 50 acres of wheat 
 and he harvested all but 4 acres ; how many acres has he 
 harvested?" 
 
 In making problems, then, the teacher must assure her- 
 self that the objects are very real to the children and the 
 problem is so woven into a story that they get the correct 
 mental picture that she describes. 
 
 THE PROBLEMS OF THE UPPER GRAMMAR GRADES 
 
 There are some who seem to consider that the develop- 
 ment of ability to compute is the final end of a course in 
 arithmetic. But skill in computation is but the means 
 to an end. Those who would try to meet the demands 
 of the business world for a better product of the schools 
 in the subject of arithmetic by merely emphasizing drill 
 in computation are following a course about as inadequate 
 for the purpose of developing a number sense or a mathe-
 
 168 THE TEACHING OF ARITHMETIC 
 
 matical type of thought as a training that confines itself 
 to forming the letters of the alphabet would be as a final 
 preparation for the career of journalism. 
 The final aims in a course in arithmetic are : 
 
 (1) To develop power in the student to see and to ex- 
 press the quantitative relations that exist among the mag- 
 nitudes that come within his experience, and to inter- 
 pret the numerical expressions of such relationships ; 
 
 (2) To develop in the student the habit of seeing such 
 relationships, especially those vital to his present or future 
 welfare; and 
 
 (3) To give the student a social insight into current 
 business and industrial practices through which he can 
 interpret references, met in general reading and in con- 
 versation, to the world's activities. 
 
 These three aims, of course, come through the problem 
 side of arithmetic. 
 
 POWER TO SEE RELATIONS 
 
 The task of developing power to see and to use quanti- 
 tative relations is a most difficult one. When we think 
 we are developing such power we are often merely storing 
 the memory with rules, forms, and processes. If the 
 wording is changed, pupils are often at sea as to how to 
 proceed. To develop power, the problems and the word- 
 ing must be so varied that the solution has to depend 
 upon a rational analysis of conditions and not be a mere 
 act of memory. This power can never come through 
 blind adherence to rules and stereotyped forms of solu- 
 tion made to fit artificial types of problems.
 
 PURPOSES AND NATURE OF PROBLEMS 169 
 
 The first requirement of a problem to meet this pur- 
 pose is that it be concrete to the pupil. Before he can 
 analyze it and thus discover the steps of the solution, he 
 must have a very clear comprehension of the facts an 
 accurate mental image of conditions. 
 
 The problems must not only be very concrete, but they 
 must suitably task the pupil's powers. However real a 
 problem is, a pupil is much more interested in it if it is 
 difficult enough to require some thought in order to deter- 
 mine what to do. A pupil soon tires of long lists of prob- 
 lems all solved alike where he gets no exercise except in 
 computation. Teachers will find that without thought- 
 provoking problems interest will soon lag. 
 
 While problems must always be concrete, they need 
 not always be real. In the development of this power 
 to see quantitative relations, the hypothetical problem 
 still has its place. These problems, however, must never 
 be such as to give wrong ideas of business practices. 
 One does not have to resort to the old "hare and hound" 
 type of problems of the past nor develop wrong notions 
 of business practice to get problems that furnish valuable 
 training of power to see and express relations. The data 
 may be worth while and the problems those that may 
 arise in life. For example, if one is trying to develop the 
 power to see and to interpret the relationships expressed 
 by per cent, he may begin with a certain bit of data and 
 bring in all the various relations without giving any wrong 
 impression of business practices whatever. For example, 
 let it be given that it cost $38.45 to manufacture a certain 
 gasoline engine, and the cost to sell it amounted to $15.38
 
 170 THE TEACHING OF ARITHMETIC 
 
 more. Now, the following questions may be asked of an 
 advanced class : 
 
 1. The cost to sell was what per cent of the cost to 
 manufacture ? 
 
 2. The cost to sell was what per cent of the total cost 
 to manufacture and sell ? 
 
 3. At what must it be sold to give a profit of 25% above 
 the cost to manufacture and sell ? 
 
 4. At what must it be sold to give a profit of 25% of 
 the selling price ? 
 
 5. At what price would it have to be listed in order to 
 allow a discount of 40% from the list price and still give 
 a profit of 20% of the total cost to manufacture and sell ? 
 
 6. At what price must it be listed in order to discount 
 the list price 33% and 10% and still make a profit of 
 20% of the actual selling price ? 
 
 And thus one may go on making up a lot of questions 
 of increasing difficulty to meet the maturity of the pupil's 
 powers, that have great value in bringing out the rela- 
 tions expressed by per cent. While one hears a great 
 amount of criticism of any but "real" problems, it ought 
 to be clear to any teacher that such criticism is not based 
 upon a careful consideration of the aims to be attained. 
 
 THE HABIT OF SEEING RELATIONS 
 
 One of the greatest sources of waste in so much of our 
 teaching is that pupils are not so taught that they can 
 apply the knowledge gained in the classroom to the work 
 found outside of the classroom. There should be a con- 
 stant utilization of outside experiences to clarify and
 
 PURPOSES AND NATURE OF PROBLEMS 171 
 
 rationalize the work of the schoolroom, and likewise the 
 facts learned in the schoolroom should constantly be 
 used to interpret conditions met without the school- 
 room. Every subject in the textbook should be supple- 
 mented by "community problems" made from data 
 gathered by the teacher and pupils from the immediate 
 community in which they live. It is only through mak- 
 ing such problems that pupils develop the habit of look- 
 ing upon the quantitative side of life. Too often the 
 problems of the text mean no more to them than mere 
 assignments in order to see if they can get the answers. 
 However, with the improved type of problems that are 
 beginning to find their way into our newer texts, a 
 teacher can do a great deal toward developing proper 
 habits by making the right use of them. But the pupil 
 must be taught to picture the situation and to ask him- 
 self if the conditions of the problem meet local condi- 
 tions, whether the answer is a reasonable one, etc. 
 
 Problems of thrift are among the most useful in the 
 development of the habit of seeing those quantitative 
 relations vital to our welfare. Thus, such questions 
 may be : How much is saved by buying enough potatoes 
 for the winter direct from the farm at digging time in- 
 stead of buying them by the peck from the grocer during 
 the winter? How much is saved by buying canned goods 
 by the case instead of by the can? How much is saved 
 by buying material and making your own dress instead 
 of buying it ready made? Will it pay better to rent a 
 house or to own one? Will it pay in your town to build 
 a house for rent ? About how much could you expect to
 
 172 THE TEACHING OF ARITHMETIC 
 
 make net upon your investment? Has the price upon 
 vacant lots in your town gone up so that the one holding 
 them for a certain time made or lost? In fact, there is 
 no end to good problems that may be made from the 
 material found in any community. 
 
 PROBLEMS THAT CONTRIBUTE TO SOCIAL INSIGHT 
 
 As the pupil's social world expands, he is carried from 
 problems dealing with his own personal affairs and local 
 situations to those of more general interest. However, 
 the teacher must have a much broader concept of such 
 problems than that they are to meet the demands of the 
 commercial and industrial world. They should be prob- 
 lems that contribute to the pupil's interpretation of the 
 social world in which he lives that contribute to " a 
 broadly socialized and modern culture," not a mere voca- 
 tional training. That is, a study of arithmetic should 
 enable one to interpret references met in general reading 
 and in social intercourse to all the common affairs of life, 
 as taxes, insurance, interest, stocks, bonds, notes, mort- 
 gages, and other business forms, and to any references to 
 measurements and relations, to the statistical relations 
 represented by graphs, and to any other phases of the 
 subject through which he can better interpret the quanti- 
 tative side of his environment. 
 
 To do this, the problems must be real, true to present- 
 day conditions, and not fictitious problems about real 
 things. But however "real" they are, they must be full 
 of human interest to the child. The following problem 
 given in a recent textbook for third grade pupils may be
 
 PURPOSSS AND NATURE OF PROBLEMS 173 
 
 real, but it lacks any element that makes it interesting to 
 a child of nine years. The problem is : " If a 200 pound 
 sack of fertilizer contains 8 pounds of nitrogen, 10 pounds 
 of potash, and 16 pounds of phosphoric acid, how many 
 pounds of these plant foods are there in a sack?" 
 
 Simply because a problem is "real" to one in some 
 specialized vocation, it does not follow that it necessarily 
 contributes to the social insight of the pupil. Unless the 
 problem is one likely to come up in interpreting some 
 phase of life which is of interest to the student either at 
 the time or at a probable future time, it is useless from this 
 standpoint. 
 
 It is not to be understood that a problem meets but 
 one of the three aims of a problem at a time. A good 
 problem may, and often does, meet all three at once. 
 The problems to meet the last condition discussed will 
 in general meet this one. The necessary conditions are 
 that the problems answer some question arising from 
 some social issue, and that the data be real and of enough 
 interest to warrant their use. Such problems are as varied 
 as the activities of life. Those of greatest value are the 
 ones having to do with industrial and commercial terms 
 and practices, but they are not all included in these topics. 
 They may relate to all the various productions needed 
 for the world's comfort the amount, the value, the 
 means and cost of the distribution of them, and so forth.
 
 CHAPTER XIV 
 
 DIFFICULTIES IN TEACHING 
 
 IT is the experience of teachers that to teach a pupil 
 how to interpret and solve a problem is a much more 
 difficult task than to develop rules and train for skill in 
 computation. And it is not strange that this is so. The 
 formal side the methods of computation is merely a 
 machine which the pupil learns to operate, and it works 
 the same upon all occasions and under all conditions. 
 But the solution of a problem is an application of judg- 
 ment. The problem must be analyzed, the conditions 
 must be carefully studied, the magnitudes wanted must 
 be compared with those given, and an act of judgment 
 tells what processes are to be used. The solution of a 
 problem cannot be done by rote, as the process of addi- 
 tion or multiplication can be done. The solution can be 
 discovered only through a contribution from the experi- 
 ences of the individual and an act of judgment. And, 
 hence, if the necessary experiences are lacking, the prob- 
 lem cannot be solved. 
 
 It is obvious, then, that no rule can be laid down that 
 will teach a pupil how to solve a problem ; but a discus- 
 sion of the sources of failure may prove profitable to the 
 teacher. 
 
 174
 
 ANALYSIS AND SOLUTION OF PROBLEMS 175 
 
 When a problem is difficult to any large number of a 
 class, the teacher cannot lay the blame to the dullness of 
 the children. A large per cent of our pupils are capable, 
 normal children. Then we must look for the trouble 
 either in the material we are using or in our method of 
 presenting it. If we examine the material found in our 
 textbooks, we shall find much that may be very justly 
 condemned. While there has been a very marked im- 
 provement in our textbooks in recent years, many of the 
 problems still have but little meaning and but little appeal 
 to children. Many of the problems designed for the pri- 
 mary grades are taken from the adult's world and not the 
 child's world. They are about business, manufacturing, 
 transportation, and various industries with which chil- 
 dren have no vital contact, instead of problems about 
 the child's play, his daily activities about the home, or 
 his constructive activities. In so far as textbooks have 
 poorly chosen their problem material, they have contrib- 
 uted directly to the trouble in question and hence are 
 at fault. But the material in the textbook is not the only 
 available material. A resourceful, wide-awake teacher 
 who understands the child's interests and needs and who 
 understands the fundamental principles of teaching will 
 be able to obtain admirable results in spite of a poor 
 book. 
 
 A FUNDAMENTAL PRINCIPLE OF TEACHING 
 
 At the very root of all successful teaching lies the 
 fundamental principle that all knowledge possessed by
 
 176 THE TEACHING OF ARITHMETIC 
 
 the individual to be real and permanent must be grounded 
 upon and developed out of his real experiences. 
 
 And thus all arithmetical ideas must have a back- 
 ground of mental images in order to exist. These mental 
 images in turn must be the outgrowths of some form of 
 experience. Henca, the real ideas which constitute the 
 child's arithmetical knowledge must be developed through 
 the individual experiences of the learner which have been 
 provided for either in or out of school. The failure to 
 provide these clear mental images, which the statement 
 of a problem should call up in the child's mind, is perhaps 
 the chief source of all failure to solve a problem. 
 
 CONCRETE PROBLEMS 
 
 Another way of stating this source of failure to solve 
 a problem is to say that the problem was not concrete to 
 the pupil ; that is, he was not able to form a clear mental 
 picture of the situation described. Unless such a picture 
 can be formed, the pupil is unable to make a comparison 
 of what is wanted with what is given in order to form 
 a judgment of what to do to obtain the desired result. 
 
 When a pupil fails to form such an image, it may be 
 that he has not read the problem carefully. Or he may 
 have read it carefully and yet, through lack of experi- 
 ence in the situation described, he may not be able to form 
 a picture of the situation. Recently, a little girl in the 
 fourth grade told me that her pencil cost five cents. I 
 asked her how much two such pencils would cost. She 
 told me at once, of course, and then she reminded me that 
 she was in the fourth grade and that such problems were
 
 ANALYSIS AND SOLUTION OF PROBLEMS 177 
 
 too easy for her. So I said, "Well, try this one: If a 
 meter of silk costs five francs, how much will two meters 
 cost?" She could not tell me. She said, "I don't know 
 what a meter is and I don't know what francs are, so of 
 course I can't solve it." 
 
 Some months ago I was trying to explain to a young 
 teacher that the quotient of the length of a row, divided 
 by the distance between plants, did not give the number 
 of plants in the row; but that the quotient was merely 
 the number of divisions into which the plants divided 
 the row, and hence there must be one more plant than 
 division. After I had exhausted my stock of word pic- 
 tures, I took a pencil and began a diagram, when the 
 teacher to whom I was explaining the problem exclaimed : 
 "Oh, I see it. It is just like hanging handkerchiefs on a 
 clothesline. It takes one more clothespin than there 
 are handkerchiefs." I had finally called up some expe- 
 rience in her own We through which the problem became 
 concrete to her. 
 
 Sometimes the largeness of the magnitudes involved 
 takes away the concreteness of the situation, and the 
 mind sees but the empty figures without any concrete 
 basis back of them. I recently saw a student attempt 
 to find the capacity of a bin, having been given the di- 
 mensions and also the statement that one cubic foot was 
 equal to .8 of a bushel. After correctly finding the num- 
 ber of cubic feet in the bin, he divided by .8 to find the 
 number of bushels. But the teacher said, " What part 
 of a bushel does one cubic foot equal? Then what does 
 2 cubic feet equal? What does 3 cubic feet equal?"
 
 178 THE TEACHING OF ARITHMETIC 
 
 These were correctly answered ; then the pupil saw clearly 
 that he should multiply .8 of a bushel by the number 
 of cubic feet in the volume in order to get the capacity in 
 bushels. 
 
 The only true aid that the teacher can give is to help 
 the pupil translate the problem into his own experiences. 
 Power to do this cannot be given by any book upon 
 "method," but depends upon the personality, versatility, 
 and experiences of the teacher. Teachers need to get as 
 wide a contact as possible with that side of life to which 
 arithmetic is applied in order to give practical and con- 
 crete applications to every topic of the subject. It is 
 only through such knowledge that the most efficient work 
 can be done in the application side of the subject. 
 
 BOOK PROBLEMS vs. LOCAL PROBLEMS 
 
 The problems of any textbook must of necessity be 
 designed for pupils of all communities. They must draw 
 upon the experiences of all children alike. They cannot 
 take into account the different environments of children 
 of different localities. Evidently, then, all problems 
 cannot be equally concrete to all children. Hence, it 
 follows that the textbook should be supplemented by 
 problems taken from the child's own local environment. 
 These problems are not only more concrete to him but, 
 if dealing with affairs in which he is interested, they fur- 
 nish a much stronger motive for learning arithmetic than 
 do those of the book. These local problems may deal 
 with personal or family purchases, with the child's con- 
 structive activities, with local stores and markets, econ-
 
 ANALYSIS AND SOLUTION OF PROBLEMS 179 
 
 omy in buying in quantities, local improvements, farm 
 crops, etc. The data for these problems should be gath- 
 ered by both teacher and pupils, and much of the problem 
 making should be done by the pupils themselves. In 
 this way arithmetic provides a much better equipment 
 for life than can be obtained from the textbook alone, 
 for the problems encountered in real life are not in general 
 formulated for us; but from known data the individual 
 must formulate his own problem. In so far as possible, 
 the school should provide conditions for incidental prob- 
 lem work in the various school activities. These prob- 
 lems, however, must be "unavoidable" problems that 
 arise and must be solved. In other words, they should 
 not be problems that are assigned merely in order to have 
 a problem that will give drill in certain processes, but the 
 problems must be those in which it is necessary to know 
 the answer and not problems giving results for which there 
 is no need or interest. The making of "number stories" 
 by the children is a beginning in this important phase of 
 the problem work. 
 
 OTHER SOURCES OF FAILURE 
 
 While no doubt the chief source of failure to solve 
 a problem is the lack of a clear-cut mental picture of the 
 situation described, there are other causes that should 
 be discussed. If a teacher will list the reasons why a 
 pupil fails to get the correct answers to assigned problems, 
 she will probably find that the failures will fall under the 
 following general heads : 
 
 1. The problem was not concrete.
 
 180 THE TEACHING OF ARITHMETIC 
 
 2. The computation was inaccurate. 
 
 3. Approximations were made before the computation 
 was completed. 
 
 4. The meaning of the fundamental processes was not 
 understood. 
 
 5. The author of the problem assumed facts that were 
 not known by the pupils. 
 
 6. The problem was too difficult ; that is, the relations 
 existing among the data were too complex. In other 
 words, the pupil lacked the mental power to reason out 
 what processes to use. 
 
 INACCURATE COMPUTATION 
 
 The lack of concreteness has already been discussed. 
 Perhaps errors in computation are as frequent a cause 
 of wrong results as lack of concreteness. And yet this 
 source of error may be practically eliminated. From the 
 very first step in the teaching of the written processes, 
 the pupil should be taught to check his work. He should 
 feel from the first that when a computation has been 
 performed but once it is but half done. It must be re- 
 viewed or checked in some way to insure its accuracy. 
 This is done in every vocation. However small the 
 amount, a clerk always checks every sale-slip. To make 
 one error out of a thousand, a clerk would lose his posi- 
 tion. Yet a pupil is allowed to hand in work without 
 checking it and, if he gets but two out of ten answers 
 wrong, he is given "80%" or "good." Some schools 
 are taking as their motto : " One hundred per cent accu- 
 racy in all computation is our aim." This does not mean
 
 ANALYSIS AND SOLUTION OF PROBLEMS 181 
 
 that pupils are never expected to make a mistake but 
 that, just as in business life, their computations are to 
 be checked until they know when work is turned in that 
 it is 100% accurate. 
 
 But this habit cannot be developed as long as teachers 
 call the work "good" when two out of every ten answers 
 are wrong. Higher standards will have to be required 
 in order to furnish a motive for checking the computation. 
 
 ERROR IN APPROXIMATING RESULTS 
 
 Even if all computation has been carefully checked, 
 a pupil may fail through making an approximation before 
 the computation is completed. Pupils should be taught 
 to see clearly just what effect an approximation during 
 the computation is going to have on the final result. 
 Recently a pupil was "just sure the answer book is wrong" 
 to the following problem : " How many cubic feet of water 
 can a V-shaped gutter discharge (flowing full) in a day if 
 it is 8 inches deep and 16 inches wide at the top, and the 
 water flows a foot a second?" 
 
 He had reasoned correctly and his computation was 
 correct as far as he had carried it. He saw that the 
 problem was to find the volume of a triangular prism 
 whose base was the cross-section of the gutter and whose 
 height was 60X60X24 feet. His error came from first 
 reducing the cross-sectional area (64 square inches) to 
 a decimal part of a square foot "true to thousandths" 
 before multiplying by 86,400. He did not realize, how- 
 ever, that the error in approximating the third decimal 
 was multiplied 86,400 times in his answer.
 
 182 THE TEACHING OF ARITHMETIC 
 
 Pupils should be taught that, when a solution consists 
 of multiplications and divisions only, all work should 
 first be indicated, then all common factors canceled from 
 both dividend and divisor, and all multiplications in the 
 dividend performed before any of the division is done. 
 Had this been done, the problem given above could have 
 been solved without any use of a pencil except in the 
 cancellation. 
 
 FORECASTING RESULTS 
 
 After a pupil has "thought out" what to do in the solu- 
 tion of a problem, he should be required to estimate the 
 result. Such an estimate serves as a check upon absurd 
 errors in computation, such as misplacing the decimal 
 point, failing to multiply by all factors, etc. 
 
 But the habit of making close approximations has a 
 very much more important use than merely to help check 
 up the work of computation. In practical life, a mere 
 approximation of a result often suffices. So, if one is 
 trained to make rather close approximations without a 
 pencil, he is getting a training in a very practical phase 
 of the subject power to see the world about him from 
 the quantitative standpoint. 
 
 MEANING OF THE PROCESSES NOT KNOWN 
 
 It sometimes occurs that a pupil does not know what 
 process to apply because he does not know what the pro- 
 cesses mean. I once asked a little girl in the third grade 
 the question, "Five threes make how many?" She 
 looked at me in bewilderment and then asked, "Do you
 
 ANALYSIS AND SOLUTION OF PROBLEMS 183 
 
 mean and or times or less ? " She would no doubt have had 
 trouble in applying her facts to the solution of problems. 
 ,Not understanding the meaning of a process, or not 
 seeing just what it means when applied to a given situa- 
 tion, often leads to error. In the problem, "How many 
 plants 8 inches apart can be set in a row which is to be 
 20 feet long from the first to last plant ? " the pupil usually 
 divides 240 inches by 8 inches and answers "30 plants," 
 not seeing what the division actually means that it 
 is finding how many times the row will contain an 8-inch 
 measure and that, from the number of times it is contained, 
 the number of plants must be found by visualizing the 
 act of placing a plant at each end of the measure the first 
 time it is applied and then one plant for each new space 
 measured off. 
 
 In the problem, " From a sheet of Manila board 22 inches 
 by 28 inches, how many cards 8 inches by 11 inches can 
 be cut?" the pupil often answers "7." He jumps at 
 the conclusion that it is one area divided by the other and 
 does not see that this is not the same as actually laying off 
 the sheet into cards, in which case he would find that he 
 could get but six. 
 
 NECESSARY FACTS NOT KNOWN 
 
 Sometimes a pupil has the correct picture of the situa- 
 tion described in a problem, but lacks a knowledge of 
 the facts upon which the solution depends. This failure 
 occurs most frequently in the problems of mensuration 
 where the pupil does not know how to find certain areas 
 or volumes. Thus, I recently saw a pupil attempt the
 
 184 THE TEACHING OF ARITHMETIC 
 
 solution of the following problem: "The cross-section of 
 a railroad tunnel 132 yards long is in the form of a 16-ft. 
 square surmounted by a semicircle. Find the number 
 of cubic yards of earth removed in its excavation." The 
 pupil stepped to the blackboard and sketched accurately 
 the picture. She stated that the problem was to find the 
 volume of a prism and half that of a cylinder and add 
 them. She then failed in the solution because she tried 
 to find the volume by multiplying the perimeter of the 
 cross-section by the length of the tunnel. In a case of 
 this kind, there is nothing to do but re-teach these funda- 
 mental rules. It is not sufficient to ask the pupil to 
 "review the rules for to-morrow," but the rules should 
 be again presented as concretely as possible. Pure 
 memorization of facts without the mental image of 
 personal experiences as a basis is almost valueless. If 
 all rules of mensuration and all other facts of arithmetic 
 are presented objectively, they are much more easily 
 retained than when learned by rote. The vivid mental 
 image growing out of an experience is much more lasting 
 than a fact memorized through the repetition of meaning- 
 less language. 
 
 COMPLEXITY OP RELATIONS 
 
 Sometimes a pupil cannot solve a given problem be- 
 cause the relations existing among the data are too com- 
 plex. That is, failure may be due to lack of sufficient 
 power to reason. However, pupils often seem to fail 
 from lack of reasoning power when the real trouble is 
 that they have made no serious attempt to analyze the
 
 ANALYSIS AND SOLUTION OF PROBLEMS 185 
 
 situation and thus to discover the essential relations that 
 exist. They depend too largely upon some word or words 
 in the description of the situation, and then, through 
 memory alone, associate the problem with some one en- 
 countered in the past experience for their "cue" as to 
 what to do, rather than make any serious analysis of the 
 conditions described. 
 
 When the answer to a problem is known, rather than 
 reason out the solution, the pupil is very apt to try to 
 juggle the figures in some way that will give the result. 
 Recently I saw a pupil attempt the following problem : 
 "If the interest of $500 for 2^ years is $50, what is the 
 rate?" She knew that the answer was 4%. Her solu- 
 tion was as follows : 
 
 2x$50=$125; $500 -J- $125 = 4% 
 
 In most schoolrooms, the teaching of a systematic 
 method of attack in the solution of a problem is greatly 
 neglected. The assignment of "the next ten problems" 
 with no thought given as to their fitness for the class is 
 entirely too common a practice. The recitation should 
 not be a place in which the teacher is to find out whether 
 assignments have been finished, but it should be a work- 
 shop in which the pupil is taught to translate the arithmet- 
 ical question into terms of his own experiences. 
 
 Home work should be given but sparingly and should 
 consist chiefly in performing the mechanical computa- 
 tions involved after the "what to do" has been discussed 
 in the classroom. When pupils are assigned home work 
 that they do not understand and have to depend upon
 
 186 THE TEACHING OF ARITHMETIC 
 
 help given by the parent, there is no uniformity of the 
 instruction of the various members of the class; and, in 
 general, the home instruction differs from that given by 
 the teacher. Hence, confusion arises. But worse than 
 that is the fact that much of the home work brought in 
 by the pupil is not his own but that of his parents, and 
 hence the teacher, knowing that the work is brought in, 
 does not realize the difficulties that the child has encoun- 
 tered or that he does not fully understand the work. 
 
 The classroom period should be given up to instruction 
 to real teaching and to rapid motivated drills. 
 The working out of mechanical details, as computing re- 
 sults, etc., should be the only part left for home work, in 
 most grades at least. 
 
 It is a common experience of teachers that pupils who 
 can solve problems when stated in the familiar terms of 
 the textbook in certain stereotyped forms of expression 
 fail to solve the same problems when stated in language 
 less common or when met in actual life. To overcome 
 this defect demands a more varied use of language in the 
 description of a situation as well as a wider range of ma- 
 terial used. 
 
 To develop a systematic method of attack in solving 
 a problem, the pupil must be led through simple types 
 of one-step, two-step, and three-step (or more) problems, 
 all stated in various ways and descriptive of a wide range 
 of material, all within the range of his own experiences 
 and interests. He must state the problem in his own 
 words, pointing out just what is wanted and what is given 
 in the problem that will help find this. Then, from a
 
 ANALYSIS AND SOLUTION OF PROBLEMS 187 
 
 clear mental image of the things involved, he will see 
 the essential relations that exist and will know what 
 to do. The numbers involved in any new problem should 
 be so small and the computation so easy as not to retard 
 thinking. 
 
 THE SOLUTION OF PROBLEMS 
 
 The solution of a problem requires an act of judgment 
 and cannot be made merely the application of a rule or 
 formula. Hence, there is as large a variety of solutions as 
 there are variations in the nature of problems. 
 
 After reasoning out what to do, the computation should 
 be made as simple as possible. When there are several 
 steps in the solution, it is best to reason entirely through 
 to the end before performing the computation. Some- 
 times one step will balance or cancel a later one and thus 
 work is saved. For example, to find the cost of 18,760 
 Ib. of hay at $18 per ton, it is seen that to divide 18,760 
 by 2000 gives the number of tons, which is 9.38. Then 
 9.38 X $18 gives the cost of all. But observing the 
 processes to be performed, we see that by multiplying 
 18,760 by 9 and pointing off three decimal places the wprk 
 can all be done without a pencil. If the work is to be 
 written down, it should be done in such a way that the 
 pupil does not think that he is multiplying pounds by 
 something else and getting dollars. The solution might 
 be written out logically as follows : 
 
 9 
 
 I|^X$Z8 = $170.840 
 2000 
 
 1000
 
 188 THE TEACHING OF ARITHMETIC 
 
 In finding 18,760 X $9, the logical multiplier is 18,760, 
 but the actual one is 9. The denomination of the answer 
 depends upon the denomination of the number multi- 
 plied, of course; but the real process is that of finding 
 the product of the two abstract factors, 18,760 and 9, 
 either of which may be used as the multiplier. 
 
 When several multiplications and divisions occur, 
 all work should be written down, like factors canceled, 
 and the multiplication performed before dividing. Most 
 problems of mensuration are of this type. Thus to find 
 the capacity of a bin 15 feet long, 8^ feet wide, and filled 
 to a depth of 6 feet, using .8 bu. per cubic foot, the 
 solution is : 
 
 2 
 
 After canceling like factors the multiplication can be 
 performed without a pencil. 
 
 To find the capacity of a tank 27 inches by 42 inches 
 by 22 inches, using 231 cubic inches per gallon, the solu- 
 tion is as follows : 
 
 Again all computation is done without a pencil. In- 
 stead of writing 231, its factors were written to facilitate 
 cancellation. 
 
 To discount a bill of $134.50 33% and 10% work is
 
 ANALYSIS AND SOLUTION OF PROBLEMS 189 
 
 saved by discounting by 10% first, for 134,50 will not 
 contain 3; but after taking off 10% of any 
 number the remainder will always contain 3. fq ^ 
 Hence, the solution might be as given in the ' 
 
 margin. 4Q 35 
 
 The same problem, however, can be solved 
 
 without a pencil by observing that, after 33^% 
 
 and 10% have been deducted, but 60% remains. And 
 
 60% of $134.50 = $80.70. 
 No fixed form of solution should be required. The 
 
 pupil should reason out what to do and look to see if any 
 
 required process will cancel any other. 
 
 ERRORS IN LABELING STEPS IN THE SOLUTION 
 
 It is a common practice to require the pupil to label 
 each step. This invariably leads to erroneous reasoning. 
 This is clearly shown in the usual solution to the follow- 
 ing type of problems: "Allowing 7 pk. of seed wheat 
 to the acre, how many bushels are needed for a field 35 rd. 
 by 42 rd.?" In many schoolrooms you will see the solu- 
 tion shown in the margin. It will 
 be seen that every principle of multi- 35 rd. 
 
 plication and division has been violated. 42 rd. 
 
 Rods cannot be multiplied by rods. 70 
 
 Square rods divided by 160 does not 140. 
 give acres. Acres multiplied by 7 160)1470 sq. rd. 
 does not give pecks ; and pecks divided 9 ^ A. 
 
 by 4 does not give bushels. None of 7 
 
 the steps should have been labeled in 4)64 ^ pk. 
 
 such a solution. The following solution 16 -fa bu.
 
 190 THE TEACHING OF ARITHMETIC 
 
 would not only have been correct, but it would have been 
 much more economical : 
 
 7 21 
 
 30X42^7, 1029 , 
 - X -. bu. = bu. =16^ bu. 
 Z00 4 64 
 
 32 2 
 
 It is not the type of solution, however, that is impor- 
 tant. But the important thing is that the solution of the 
 problem is an act of the pupil's own judgment and not 
 the application of some meaningless stereotyped rule or 
 formula.
 
 CHAPTER XV 
 PLANNING THE LESSON 
 
 NEED OF PREPARATION 
 
 BEFORE conducting a class exercise of any sort, a teacher 
 should make a careful preparation for the work she 
 expects to do. She should have definitely . in mind just 
 what she is going to present and why and how she is going 
 to present it. She should understand the whole subject 
 so well that she knows the logical steps to follow in order 
 to accomplish her aim; that is, she should know just 
 what knowledge the pupils have had in order to take up 
 the work which she has planned for the day. 
 
 THREE TYPES OF LESSONS 
 
 There are three general types of lessons in the teach- 
 ing of arithmetic. The work of the day may consist of 
 developing some new fact or process ; or it may be devoted 
 to drilling upon previously developed facts or processes 
 in order to give automatic control of them; or it may 
 be an application of principles and processes already 
 learned and made automatic to the solution of problems. 
 The nature of the teacher's part in the class exercise 
 depends upon which of these three phases is under con- 
 sideration. These phases will be considered under: 
 
 191
 
 192 THE TEACHING OF ARITHMETIC 
 
 (1) the drill lesson; (2) the inductive lesson; and (3) 
 .the deductive lesson. 
 
 THE DRILL LESSON 
 
 The pupil must have an automatic control of all of 
 the fundamental facts and processes of arithmetic. This 
 control can be accomplished only through well-chosen 
 motivated drill. In some schools there is too great an 
 emphasis placed upon drill. In others there is too little. 
 The teacher should consider carefully the three phases 
 of her work in order that they may be well balanced. 
 
 The teacher should realize, however, that accuracy 
 and a f air degree of rapidity in the four fundamental pro- 
 cesses are of first importance, and to secure this all other 
 desiderata must be subordinated. To secure and keep 
 this skill, constant practice and drill are indispensable, 
 not only in the lower grades, when the fundamental pro- 
 cesses are first taught, but throughout the entire course. 
 
 NEED OF MOTIVATION 
 
 While drill exercises may seem mechanical and depress- 
 ing, there is no substitute for them, and the ingenious 
 teacher will seek to make them attractive through com- 
 petitive tests, by working with a time limit to make the 
 pupil conscious of his growing power, or by other forms 
 of motivation suggested in the preceding chapters. 
 
 It is not sufficient for a child in the lower grades to be 
 told that his future success depends upon the habits 
 developed by this drill. He must have an immediate 
 motive; and the more immediate the motive the
 
 PLANNING THE LESSON 193 
 
 stronger the incentive the greater the attention that 
 will be given to the work, and the more rapid the progress 
 that he will make. 
 
 DRILL MUST INCLUDE EACH FACT OF A SERIES 
 
 Often the drill lessons are not systematically planned. 
 They are made up at random and lack system. Easy 
 combinations, for example, may be given more often 
 than more difficult ones. If the drills are made up at 
 random during the class period, many of the combinations 
 may be repeated uselessly while others may be omitted 
 entirely. The class period may have merely " killed time " 
 and served as little purpose as some so-called " busy work." 
 
 When the object of the drill is to secure an automatic 
 response to a series of facts, the teacher must see to it 
 that every fact of the series receives the required atten- 
 tion. If the drill is upon the forty-five primary facts 
 of addition, the teacher must make sure that every fact 
 is included in the drill. This does not imply that each 
 fact should recur exactly the same number of times. 
 Thus, when drilling from an addition chart like the one 
 given in Chapter IV, the latter part of the chart should 
 receive much more emphasis than the first part of it. 
 
 Unless great care is exercised, the problems given in 
 the lower grades for the purpose of motivating drill work 
 are likely to make too great use of certain combinations 
 to the exclusion of others. This is necessarily so, for the 
 problems should represent real conditions as to prices, 
 sizes, quantities, etc., and this limits the numbers to be 
 used. For this reason, problems for the purpose of fur-
 
 194 THE TEACHING OF ARITHMETIC 
 
 nishing drill work in the fundamental facts should be used 
 but sparingly, and the teacher should exercise great care 
 in order to get variety in the combinations involved. 
 
 As indicated above, the greater part of the time given 
 to a drill lesson should be spent upon those facts or pro- 
 cesses that present the greatest difficulty. Frequent 
 testing will show the teacher which facts have come under 
 the desired control and which ones need special drill, 
 and her drills should be planned accordingly. 
 
 THE REPETITION OF DRILLS 
 
 When a table is learned, it must be drilled upon daily 
 until it can be accurately and quickly given. But this 
 is not sufficient. A table may be given ever so glibly 
 by a pupil after a few days of drill, and yet the facts that 
 are not used for a week or so may be entirely forgotten 
 when needed again. When tables are first learned, the 
 periods between drills should be short. Thus, if a pupil 
 does not have occasion to use certain facts of addition for 
 several weeks after first learning them, he may entirely 
 forget them. Hence, as the tables of a new process are 
 learned, drill upon those already learned cannot be 
 neglected. However, the frequency in which they'Tteqes- 
 sarily recur gradually changes. As the child matures, 
 periods between drills may gradually be lengthened. 
 
 HOLDING THE ATTENTION 
 
 The results accomplished in any kind of drill work 
 depend upon the degree of attention given by the pupil. 
 How to secure a maximum of attention is the vital problem,
 
 PLANNING THE LESSON 195 
 
 then, that confronts the teacher. It is difficult for chil- 
 dren to fix then* attention for a considerable length of 
 time. A teacher could hardly expect that a maximum of 
 attention could be given a drill that extended throughout 
 the entire class period without variation. When the 
 pupils show signs of lagging in attention, the form of the 
 drill should be changed. 
 
 In concert work a large number of the class are not 
 really attending to the drills but are saying what the 
 leaders say and are thus getting no value from the exer- 
 cise. So, while some concert drill work may occasionally 
 be given, it should not consume an important part of the 
 class period. 
 
 On the other hand, individual drill has to be done very 
 skillfully, or the one reciting may be the only one paying 
 attention to the work that is being done. It frequently 
 becomes necessary, however, to have an individual 
 pupil recite an entire series of facts ; but there must be 
 some incentive to get each pupil of the class silently to 
 go through the same recitation. That was the purpose 
 for which many of the games given in Chapter VIII were 
 invented. 
 
 A method of securing the maximum attention of a 
 class cannot be outlined. Much depends upon the per- 
 sonality and the versatility of the teacher. She cannot 
 create interest and keep the attention of the class fixed 
 upon the work unless she too is interested in it. A wide- 
 awake, alert, resourceful teacher will readily devise means 
 of securing attention as occasion for it arises; while a 
 listless teacher, or one who lacks the power to sense a
 
 196 THE TEACHING OF ARITHMETIC 
 
 situation, could not hold the attention of a class even 
 though she knew all the laws upon which attention 
 depends. 
 
 While much depends upon the resourcefulness of the 
 teacher, among the various means of securing attention 
 that may be listed are : (1) variety in the forms of drill ; 
 (2) working with a time limit ; (3) desire to excel former 
 records; (4) the desire of a pupil to do as well as other 
 members of 'the class ; (5) the desire of the class as a whole 
 to do as well as or to excel some former class ; or (6) the 
 desire of the class to do as well as or excel a class of the 
 same grade in some other school. 
 
 THE INDUCTIVE LESSON 
 
 Although much of the time in the primary grades is 
 spent in drill work, still the teaching of arithmetic fur- 
 nishes many opportunities to stimulate and train the pupil 
 to analyze a situation and draw a conclusion. The tradi- 
 tional "development lesson," designed to stimulate think- 
 ing, is usually treated under five formal steps : (1) prep- 
 aration ; (2) presentation ; (3) comparison ; (4) generali- 
 zation; and (5) application. While there is no longer a 
 rigid adherence to these five steps followed in the order 
 given, and without any elimination or combination, still 
 a recognition of them will be of help whenever the induc- 
 tive development of a fact or process seems desirable. 
 These five formal steps, however, must be considered 
 as suggestive rather than as furnishing a fixed mode of 
 procedure.
 
 PLANNING THE LESSON 197 
 
 THE PEEPARATION 
 
 To prepare a pupil to think, a problem whose solution 
 is necessary must first be presented. Thought is stimu- 
 lated only when one realizes some aim that is to be satis- 
 fied by the process. This is formally called "develop- 
 ing an aim." 
 
 The solution of this new problem, however, depends 
 upon some old experience; hence, another factor in the 
 preparation is a review of the related old experience. 
 These two phases of the preparation : (1) the recognition 
 of a problem whose solution is necessary, and (2) a review 
 of the old facts upon which the solution depends, need 
 not follow each other in this order. The second phase 
 may even be eliminated when the teacher knows without 
 a review that all the old facts are fresh in the pupil's 
 mind. 
 
 Thus, before a pupil takes up the addition of two 
 fractions whose units are unlike as -g- + f , he has met and 
 solved the two problems in the addition of two fractions, 
 the one whose units are alike, and the other in which 
 one unit can be changed to the other, as for example : 
 i + f = f , and ? + i = l + T = -t. Now, it is not neces- 
 sary to spend time in reviewing these two problems if 
 they have immediately preceded this one, except to show 
 wherein the new one is different from the old ones, and thus 
 get the pupil to see the new problem before him, which is 
 the changing of fractions to like units. 
 
 To spend time reviewing well-known facts instead of 
 proceeding at once to the new will cause a loss of interest.
 
 198 THE TEACHING OF ARITHMETIC 
 
 THE PRESENTATION 
 
 After a full realization of the problem to be solved, 
 the next step is to relate the elements of the new to facts 
 already known. No rule can be laid down that will fully 
 cover the teacher's part in this. Sometimes, through 
 a series of questions, she gets the pupil to relate the new 
 with the old. At other times she finds it more economical 
 to illustrate or demonstrate the relations and ask fewer 
 questions. Thus, the presentation of the area of a circle 
 requires many more directions and demonstrations than 
 mere questions. After the pupil has realized that the 
 circle cannot be divided into a number of square inches, 
 as in the case of the rectangle, and, as it stands, it cannot 
 be divided up into triangles, parallelograms, or trapezoids, 
 as all plane rectilinear figures could, he has grasped the 
 problem. At least he sees that here is an area to be meas- 
 ured that differs in form from the others and seems to 
 demand a new "rule." Now in the presentation the 
 teacher can show that by dividing the circle into a number 
 of equal sectors and properly fitting them together, the 
 circle may be converted into what is practically a rectangle 
 or parallelogram with known dimensions. Now through 
 a series of questions she gets from the pupil that the area 
 must equal that of a rectangle whose base is half the 
 circumference and whose width is the radius, which is 
 really the next two of the formal steps the comparison 
 and the generalization. 
 
 Avoid questions, however, that are so suggestive that 
 they fail to stimulate thought.
 
 PLANNING THE LESSON 199 
 
 THE COMPARISON 
 
 With the problem before us and the necessary data 
 furnished, the pupil is led to compare the new with what 
 is already known. Thus in the illustration given above, 
 in finding the area of a circle, the pupil recognizes the 
 problem and is furnished with a circle cut into sixteen or 
 more equal sectors which are rearranged to somewhat 
 resemble a rectangle. The pupil observes several such 
 circles in their new arrangement and compares the forms 
 they now assume and draws the conclusion that they are 
 equivalent to rectangles with known dimensions. 
 
 THE GENERALIZATION 
 
 After making comparisons and forming conclusions 
 until satisfied of certain fixed facts that hold good in all 
 cases, the pupil summarizes these facts into a definition 
 or principle, or into a working rule for later use. Thus, 
 in the example given, the pupil generalizes his conclusions 
 and states that the area of a circle is equal to that of a 
 rectangle whose base is half the circumference and whose 
 width is the radius. Or, stated in the form of a rule, to 
 find the number of units hi the area of a circle, multiply 
 the number of units in half the circumference by the 
 number in the radius. 
 
 Through the aid of the teacher, the rule is then fitted 
 into the conventional form to be memorized, as :
 
 200 THE TEACHING OF ARITHMETIC 
 
 Besides aiding directly in forming habits of right 
 thinking, the discovery and generalization of facts and 
 processes by the pupil for himself give him a better 
 grasp of the subject so that it is used more intelligently 
 and remembered much more easily. The mere ability 
 to repeat parrot-like the rules or definitions of arithmetic 
 with no concrete background as a basis for them is prac- 
 tically valueless. 
 
 THE APPLICATION 
 
 When conclusions have been reached and generaliza- 
 tions made, opportunity for them to be usecl in some 
 practical problem must be furnished. In so far as possible, 
 the applications found in the textbook must be supple- 
 mented by problems that some one in the life about them 
 needs to solve. 
 
 SOME CAUTIONS IN USING THE INDUCTIVE METHOD 
 
 A teacher in planning the development of a rule or 
 principle must not feel that in order to do it properly 
 the five formal steps must all be followed as five distinct, 
 exclusive steps. Often there is no sharp differentiation 
 between them. In seeking to bring out the five steps, 
 she may make the work formal and uninteresting. Long, 
 tedious presentations should be avoided. If a truth has 
 been grasped and can be applied, that is the supreme 
 test. 
 
 Do not give too much help. Some teachers interpret 
 the inductive method to mean that, through a series of 
 carefully planned questions, the pupil is brought to state
 
 PLANNING THE LESSON 201 
 
 the generalization. Often these questions are so suggestive 
 as to require no thought upon the part of the pupil. After 
 the problem is fully presented to the pupil, the more he 
 does for himself, the better. It is the teacher's part to 
 stimulate the pupil to seek the solution, but to give as 
 few hints and to ask as few leading questions as possible. 
 
 THE FORMAL STEPS IN TEACHING THE AREA OP A 
 RECTANGLE 
 
 This is taken up here to illustrate the five formal steps 
 of an inductive lesson that have just been discussed. By 
 referring to the method of developing the various topics 
 given in former chapters the teacher should be able to 
 thus arrange the steps in the teaching of any topic. 
 
 Preparation. Recall the methods of measuring other 
 magnitudes, as lines, the contents of vessels, etc. See 
 that the pupil understands that to measure any magnitude 
 is to find how many of some standard unit it will contain. 
 Show some standard units used in measuring surfaces, 
 as a square inch, a square foot, etc. Show some rectangle 
 whose dimensions are integers and measure it by placing 
 enough square units on it to cover it. Raise the question 
 of how the number needed to cover it could have been 
 calculated so as to save the necessity of covering the area 
 with square units. The problem is thus fully presented ; 
 that is, the teacher has developed "the pupil's aim." 
 
 Presentation. Furnish pupils with rectangles of card- 
 board and with but one square unit each with which to 
 measure them. Let them find the area in any way they 
 can. Encourage them to thus lay off the rectangles into
 
 202 THE TEACHING OF ARITHMETIC 
 
 square units. Ask them to compare the number of square 
 units in one row along the length with the number of 
 linear units in the length. Then ask them to compare the 
 number of such rows with the number of linear units in 
 the width. 
 
 Comparison. Have the pupils make several compari- 
 sons as above and draw conclusions until they can tell 
 without drawing that a rectangle 3 in. by 4 in. is 
 composed of 3 rows of 4 sq. in. each, that one 5 in. by 
 6 in. is composed of 5 rows of 6 sq. in. each, etc. From 
 this have them see that in a rectangle 3 in. by 4 in. 
 there are 3X4 sq. in. In one 5 in. by 6 in. there are 
 5X6 sq. in., etc. 
 
 Generalization. From the conclusions formed above, 
 have the pupils formulate the fact that the number of 
 square units in the area of a rectangle is the product of 
 the number of linear units in the two dimensions. 
 
 Application. Test the pupils' understanding of the 
 principle just stated by applying it to problems of the 
 textbook, then to those found out of the text. In measur- 
 ing rectangles, select those whose measurement is neces- 
 sary. Thus, how much did the walk in front of the school 
 cost? Get the local price per unit and thus answer the 
 question. There is an abundance of real problems on 
 every hand that require the measurement of rectangles. 
 
 THE LESSON PLAN 
 
 In writing out a lesson plan embracing the five steps 
 as given above, it is customary to ask questions to show 
 the "teacher's part" and then answer them to show the
 
 PLANNING THE LESSON 203 
 
 "pupil's part." While a teacher should have clearly in 
 mind just the questions that she intends to ask, she must 
 realize that the pupils will not always answer as she has 
 expected them to do and hence that her questions will 
 have to be changed. Aside from the general questions, 
 then, she must fully realize what each step consists of 
 and how she expects to attain the ends sought and then 
 secure the greatest possible self-activity from the pupil. 
 Questions are often so leading that the pupil has to do 
 but little, if any, thinking. This should be carefully 
 avoided. The supreme test of teaching is the ability 
 of the teacher to arouse the self-activity of the pupil. 
 
 THE DEDUCTIVE LESSON f" 
 
 The complete thought process involves both the induc- 
 tive and deductive types of thinking. A lesson is rarely 
 ever purely inductive or purely deductive. All laws are 
 first arrived at through the process of induction; but, 
 in verifying them, the thought is deductive. The present 
 plan of developing the principles of arithmetic is more 
 inductive than deductive. The applications of these 
 principles are more deductive. However, in teaching a 
 process or rule, the plan sometimes becomes more deduc- 
 tive than inductive. Thus, in teaching the rule for 
 finding the area of a rectangle given under the inductive 
 plan, had the teacher equipped the class with rectangles 
 ruled in squares or with squared paper and then asked 
 them to verify the fact that the number of square units 
 was equal to the product of the number of linear units
 
 204 THE TEACHING OF ARITHMETIC 
 
 in the two dimensions, the process would have been de- 
 ductive. Whether the teaching process is to be inductive 
 or deductive, the first step is the preparation; that is, 
 needed knowledge is reviewed and the problem fully 
 presented. A generalization is then assumed, and, by 
 appealing to known facts, it is verified through a course 
 of reasoning. 
 
 In the solution of a problem, the type of reasoning 
 is largely deductive. Thus, there is first a clear recogni- 
 tion of the problem. Then follows an analysis of condi- 
 tions in seeking the principles to be applied. In general, 
 it is more difficult to select and classify the formal steps, 
 particularly in the solution of a problem, than it was in 
 the inductive type. Thus, if one is to find a selling price 
 of an article costing $2.10 that will give a profit of 30% 
 of itself, the problem is first studied, and it is clearly 
 seen that this problem is one in which the basis upon 
 which the per cent is reckoned is not given, but is wanted. 
 Hence, it cannot be the first type of problem. Then one 
 reasons that, since 30% of the selling price must be 
 profit, the remainder of it, or 70% of it, must represent 
 the cost. Hence, the relation 70% of the selling price 
 equals $2.10 is seen. Now, if this type of relation is a 
 seemingly new one, the reasoning again is somewhat as 
 follows: This means .70 X an unknown selling price = 
 $2.10. But this means that the product of two factors 
 and one of the factors is known and the problem is to 
 find the other. But from the meaning of division the 
 solution must be $2. 10 -f-. 70. Hence, the selling price 
 must be $3. To verify this, the gam must be 30% of
 
 PLANNING THE LESSON 205 
 
 $3 or 90 cents. And this leaves $2.10 for the cost as it 
 should. Hence, the solution is correct. Thus, it is seen 
 that there is a series of problems arising, each followed 
 by an analysis and a conclusion of which principles to 
 apply, and finally a verification of the conclusion.
 
 CHAPTER XVI 
 THE COURSE OF STUDY IN ARITHMETIC 
 
 ONE of the chief factors in efficient school work is the 
 course of study. And, while it must not be inflexible, 
 yet it must be based upon certain fixed fundamental 
 principles. Each year's work must be based as nearly 
 as possible upon the needs, interests, experiences, and 
 personality of the learner, and made to function with his 
 life both in and out of school. That is, the material used 
 must be based upon some social issue of present or near 
 future value and not given for the mere purpose of training. 
 
 There are three natural divisions of the course in arith- 
 metic, each- with its own aims, methods, and material 
 all rather distinctly marked. These three divisions fall 
 naturally under the so-called Primary, Intermediate, 
 and Grammar School grades. Textbook makers have 
 followed this division in their texts, and most modern 
 textbooks thus become a sort of course of study. But 
 the matter of any textbook must be adapted to the ends 
 to be attained by omitting and supplementing to meet the 
 needs of the child, for it must not be forgotten that it 
 is the child and not the subject that is being developed. 
 It is his temperament, his interests, his needs, and his 
 abilities that must be studied, and the instruction must 
 be shaped to fit them. 
 
 206
 
 THE COURSE OF STUDY IN ARITHMETIC 207 
 
 THE PRIMARY GRADES 
 
 The primary grades include the work of the first four 
 years. The aims of these grades are clearly defined and 
 definite. They are : (1) to teach the reading and writing 
 of numbers; (2) to give an automatic control of the 
 primary number facts (the so-called tables) ; and (3) to 
 develop some skill in the written processes with whole 
 numbers. While, of course, all these will find applica- 
 tions within the child's interests and needs, the problems 
 of the primary grades are of minor importance except in 
 so far as they make clear the meaning of the processes and 
 motivate the drill work. Neither is the rationalization 
 of the fundamental processes with whole numbers of 
 importance. It is proper habits of procedure that are of 
 chief importance. Thus the chief problem of the teacher 
 of these grades is how to bring about a control of the 
 primary facts and dependable habits of procedure in the 
 written processes with the greatest economy of time and 
 effort. The problem, then, is a study of methods of drill 
 and of the proper gradation of the written work a study 
 of the means of vitalizing the work so as to secure a maxi- 
 mum of attention. These have all been discussed in 
 former chapters of this book. 
 
 THE FIRST GRADE 
 
 Following the principle that the work of each year 
 shall as nearly as possible be based upon the child's needs 
 and interests, there is a rapidly growing tendency to defer 
 drill in formal number work until the second school year.
 
 208 THE TEACHING OF ARITHMETIC 
 
 However, there are needs of counting and of expression 
 during the first year that should be met. In most locali- 
 ties the child's needs during the first year are limited to 
 
 (1) reading numbers to 100, or possibly to reading num- 
 bers of three figures, in order to find a page in his book or 
 the number on a house, if he lives in a city ; (2) reading 
 the Roman numerals to XII on a dock face, in order to 
 tell time ; (3) counting by ones and tens to 100, in order to 
 count the words or lines in a lesson or the objects with 
 which he works in other school activities; (4) possibly 
 the recognition of one half, one third, and one fourth 
 of an object that has been divided ; and also (5) famil- 
 iarity with the common units of measure used by him in 
 or out of school. All this may be done in connection with 
 other school work, without setting aside a period in the 
 program especially for number work. 
 
 THE SECOND GRADE 
 
 The work of this grade should be entirely oral. The 
 most important aims of the year's work are : (1) to make 
 automatic the forty-five primary number facts of addi- 
 tion and the corresponding eighty-one subtraction facts ; 
 
 (2) to develop some skill in calling the sum of any two- 
 figured number added to a one-figured number; and (3) 
 to develop some ability in adding three or four one-figured 
 numbers. Compared with these three aims, the other 
 arithmetic work of the year is of minor importance. 
 At the end of the second year, then, there should have 
 been accomplished the following: 
 
 1. Reading and writing numbers to 1000.
 
 THE COURSE OF STUDY IN ARITHMETIC 209 
 
 2. Reading Roman numerals to XII. 
 
 3. Telling time to hours and half hours. 
 
 4. Reading and writing dollars and cents. 
 
 5. Interpreting the signs +, , =,$,. 
 
 6. The forty-five primary facts of addition and the 
 corresponding eighty-one subtraction facts. 
 
 7. Some skill in adding at sight any one-figured num- 
 ber to any two-figured number. 
 
 8. Some skill in adding single columns of three or 
 four one-figured numbers. 
 
 9. Recognition of halves, thirds, and fourths of single 
 objects. 
 
 10. Those measuring units used in any of the child's 
 activities either hi or out of school. 
 
 SUGGESTIONS. 1. Careful instruction and much practice 
 in making the figures and signs should be given, so that in later 
 written work the making of these will not retard thinking. 
 While all the work of this grade is oral, the pupil should spend 
 much time in copying figures and his "tables" just as he copies 
 letters and words. 
 
 2. Addition and subtraction should be treated as correlative 
 processes. Thus, when a child learns that 3 and 5 are 8, he 
 learns that either part taken from 8 leaves the other. 
 
 3. Drill with charts and flash cards until all the combinations 
 can be given instantly at sight. 
 
 4. Observe from Chapter IV that the forty-five primary num- 
 ber facts of addition are more economically treated by dividing 
 them into two groups: (1) those twenty-five combinations 
 whose sums do not exceed 10; and (2) those twenty whose 
 sums do exceed 10. The sums of the first group are found 
 through counting in order to fix the meaning of addition. The 
 sums of the second group are not found through counting. A 
 child is not able to image eight or nine things, much less their 
 sum. No aid, then, to^memorizing the second group is secured
 
 210 THE TEACHING OF ARITHMETIC 
 
 through finding the sums through counting, and the finding of 
 the first group is sufficient to fix the meaning of addition. (See 
 Ch. IV.) 
 
 5. The drills should be written in column form as the child 
 is to see them later in written work. Sight work should precede 
 dictation work, but both forms of drill should be used. 
 
 6. Make much use of games to vitalize the work and secure 
 the maximum of attention. 
 
 THE THIRD GRADE 
 
 The pupil should have a textbook in this grade and 
 written work begins here. The most important aims of 
 this year are : (1) to develop some skill in written addi- 
 tion and subtraction ; (2) to make automatic the primary 
 number facts of multiplication and the corresponding 
 division facts; and (3) to teach written multiplication 
 and division by one-figured multipliers and divisors, 
 the dividend being limited to an exact multiple of the 
 divisor. Compared with these, the other aims of the 
 year are of minor importance. In all, the work of the 
 year should include : 
 
 1. Reading and writing numbers to 10,000. 
 
 2. Reading and writing Roman numerals to XII. 
 
 3. Reading and writing unit fractions , , i, to . 
 
 4. Review, of the primary facts of addition and subtrac- 
 tion for accuracy and rapidity. 
 
 5. More practice in adding one-figured numbers to 
 two-figured numbers. 
 
 6. More practice in adding at sight single columns 
 of three or four one-figured numbers and extended to in- 
 clude five or six one-figured numbers. 
 
 7. Written addition limited to five or six addends-.
 
 THE COURSE OF STUDY IN ARITHMETIC 211 
 
 8. Written subtraction. 
 
 9. The primary facts of multiplication and the corre- 
 sponding division facts. 
 
 10. Written multiplication and division when the mul- 
 tipliers and divisors are one-figured numbers. 
 
 SUGGESTIONS. 1. Facility in reading and writing numbers 
 of three or four orders should come largely incidentally from the 
 ordinary use of numbers, not from teaching and drilling upon 
 "notation and numeration" as separate topics. 
 
 2. The fractions arise in the partition phase of division and 
 should be confined to unit fractions. 
 
 3. Drill upon the primary facts of the second grade should 
 be continued until all are recognized instantly. 
 
 4. Children should not be given much column addition until 
 they can add two-figured and one-figured numbers with con- 
 siderable skill. 
 
 5. Pupils need not understand the "why" of carrying or 
 borrowing in this grade. The "how" is the important thing. 
 
 6. Pupils should make their multiplication tables through 
 adding equal addends in order to see what multiplication means. 
 
 7. Include any unit of measure needed by the pupil to inter- 
 pret his other work, using the unit itself as it is being taught. 
 
 8. All applications should be kept within the range of the 
 pupil's experiences and interests. Dramatized activities as 
 playing store, etc., are much better for this grade than problems 
 from the world of adults. 
 
 THE FOUKTH GRADE 
 
 The greater part of the time of this year is devoted to 
 securing greater skill in the work begun in the third year. 
 The new work of the year consists chiefly in long multi- 
 plication and long division, using multipliers and divisors 
 of two or three figures. At the end of the fourth grade 
 the work finished should include :
 
 212 THE TEACHING OF ARITHMETIC 
 
 1. Reading and writing numbers to 100,000,000. 
 
 2. Reading and writing any of the Roman numerals. 
 
 3. Skill in written addition and subtraction and power 
 to check results so as to turn in work 100% accurate. 
 
 4. Complete mastery of the multiplication and division 
 tables. 
 
 5. Skill in written multiplication and division where 
 multipliers and divisors do not exceed three-figured num- 
 bers, and power to check work in order to turn in work 
 that is 100% accurate. 
 
 6. Power to apply the knowledge thus far acquired 
 to problems that come within the pupils' needs and 
 experience. 
 
 SUGGESTIONS. 1. Do not perplex pupils with drill in "no- 
 tation." The reading of numbers is of more importance. Facil- 
 ity in reading and writing numbers com.es incidentally through 
 the ordinary use of numbers rather than through special drill. 
 
 2. The reading and writing of the Roman numerals is of minor 
 importance and should not consume time needed for more impor- 
 tant parts of the year's work. 
 
 3. All drills of the preceding grades should be continued to 
 give greater skill. 
 
 4. Confine the work of denominate numbers to the needs of 
 the pupil. When a new unit is introduced it should be shown 
 the class. If square and cubic units are taken up, see that 
 pupils get a clear conception of them and their use. 
 
 < THE INTERMEDIATE GRADES 
 
 This includes the work of the fifth and sixth grades. 
 The teaching in these grades takes on new phases. Ra- 
 tionalization plays a much more important part. The 
 processes with fractions and decimals recur less frequently
 
 THE COURSE OF STUDY IN ARITHMETIC 213 
 
 than those with whole numbers, hence rationalization is 
 needed hi order to help the memory retain the "how" 
 of the processes. Not only on that account, but because 
 the rationalization of the subject of fractions and deci- 
 mals develops greater power to use the subjects properly 
 when applied to problems is rationalization urged. 
 
 The applications of arithmetic to problems becomes 
 much more important in these grades than in the pri- 
 mary grades. The problems are not given now merely to 
 furnish occasions for calculation, but they are connected 
 more with some social issue of interest to the pupil and 
 thus serve to develop the habit of looking upon the quanti- 
 tative side of life. He begins, too, in these grades to get 
 some insight into the use of arithmetic hi commercial 
 and industrial phases of the world's work. 
 
 THE FIFTH GRADE 
 
 Drill in the work of the preceding grades should continue 
 in order to secure greater skill. Greater use of the work 
 of whole numbers in problems should be made. But 
 the new work of this year is chiefly the development of 
 the fundamental processes with fractions and mixed 
 numbers and application of these processes to problems 
 that need solution. The complete work of the year 
 should include the following : 
 
 1. Review and extend the work in whole numbers. 
 
 2. Develop multiplication where the multiplier con- 
 tains one or more zeros. 
 
 3. Take up division in which zeros occur in the quotient. 
 
 4. Teach the meaning of abstract and concrete numbers.
 
 214 THE TEACHING OF ARITHMETIC 
 
 5. Teach all four fundamental processes with fractions 
 and mixed numbers. 
 
 6. Teach the meaning of ratio and use ratio within 
 the limits of familiar numbers. 
 
 7. Teach the measurement of areas of rectangles, 
 parallelograms, and triangles. 
 
 8. Teach the surfaces and volumes of right prisms. 
 
 SUGGESTIONS. 1. In the applications do not allow a pupil 
 to speak of the concrete factor as the multiplier. 
 
 2. Develop the notion of a fraction objectively. 
 
 3. Be careful in developing the notation of a fraction that 
 the pupil sees the use of each term. The development of the 
 processes depends upon the notation. 
 
 4. Show that as long as the meaning of a process does not 
 change, the method does not change. 
 
 5. Relate all processes in fractions with their corresponding 
 processes with integers. 
 
 6. Have pupils see clearly that the so-called multiplication 
 by a fraction is a new meaning of multiplication, fxf does 
 not mean "f times f " but "f of ." 
 
 7. Seek local applications for all processes as fast as learned. 
 
 THE SIXTH GKADE 
 
 Fractions, whole numbers, and mensuration are taken 
 up in this grade for review and extended in their applica- 
 tions. The new work of this grade consists of decimals 
 and percentage. These, then, demand the greatest care 
 in presentation. The outline of the year's work is as 
 follows : 
 
 1 . Teach the decimal place-value feature of our notation. 
 
 2. Continue drill in whole numbers, using larger 
 numbers.
 
 THE COURSE OF STUDY IN ARITHMETIC 215 
 
 3. Apply addition and subtraction to keeping accounts. 
 
 4. Teach short methods of multiplication and division, 
 using aliquot parts and special fractions. 
 
 5. Review and extend the meaning and use of com- 
 mon fractions. 
 
 6. Develop cancellation. 
 
 7. Develop the meaning of ratio and how to express 
 a ratio as a fraction. 
 
 8. Teach the fundamental processes with decimals. 
 
 9. Express any ratio in decimal form. 
 
 10. Develop per cent as a new name and notation for a 
 special fraction. 
 
 11. Study the application of percentage, confining the 
 work to the two direct applications of the subject : 
 (1) finding a per cent of a number; and (2) finding 
 what per cent one number is of another. 
 
 12. Review the mensuration given in the fifth grade and 
 extend its applications. 
 
 SUGGESTIONS. 1. In this final study of fractions seek to 
 develop the full meaning and use of a fraction : (1) one or more 
 of the several parts of some whole; (2) an expressed division; 
 and (3) the expression of the ratio of one magnitude to another. 
 
 2. Develop the decimal fraction as an extension of our deci- 
 mal place-value notation to the right of ones' place, not as a 
 special notation of a common fraction. 
 
 3. It is very important that a pupil be able to express the 
 ratio, as a common fraction, between any two magnitudes, then 
 change this ratio to a decimal, for this is the foundation of one 
 of the most important problems of percentage. 
 
 4. Present percentage as a problem of decimals, the notation 
 being the only new feature. 
 
 5. But two problems of percentage should be taught this 
 year to find a per cent of a number, and to find what per cent
 
 216 THE TEACHING OF ARITHMETIC 
 
 one number is of another. Make clear that both of these prob- 
 lems were encountered in decimals, multiplying by a decimal, 
 and expressing a ratio as a decimal. 
 
 6. Nothing is gained in clearness of thought by using the 
 terms " base," "rate," and " percentage," terms still used in some 
 textbooks. Hence it is better not to use them except when 
 "rate" is used in such expressions as "rate of gain," "rate of 
 interest," "rate of discount," etc. 
 
 7. While the applications now begin to take on more and 
 more of the adult point of view, keep them well within the pupil's 
 experiences. Encourage pupils to bring in advertisements, 
 sample business papers, bills, accounts, notes, etc. Make the 
 commercial applications more real and concrete by dramatiz- 
 ing the adult activities that are studied. Children like to play 
 " Going into business." 
 
 8. In local applications of mensuration have the pupils make 
 their own measurements in getting the needed data. 
 
 THE GRAMMAR SCHOOL GRADES 
 
 This includes the work of the seventh and eighth 
 grades. Educational opinion seems less settled upon 
 what should be included in these two grades than in the 
 earlier grades. In most schools the work still consists 
 almost entirely of arithmetic. In schools organized 
 under the Junior High School plan some algebra and 
 observational geometry are being introduced in these 
 grades and arithmetic continued into the ninth grade. 
 
 It seems clear, however, that in so far as the maturity 
 of the pupil will permit, the work of these two grades 
 should furnish the mathematics needed by the average 
 intelligent citizen outside of a specialized vocational 
 need. The needs of such a person are (1) power to see 
 and express, and to interpret the expressions of the 
 quantitative relations that come within one's needs and
 
 THE COURSE OF STUDY IN ARITHMETIC 217 
 
 interests ; (2) the habit of looking upon the quantitative 
 side of life and seeing these relations, particularly those 
 vital to one's welfare; and (3) a knowledge of commer- 
 cial and industrial practices through which one may 
 interpret references met in general reading and in one's 
 business and social intercourse with those with whom he 
 comes in contact. 
 
 With these aims, then, as the basis of the course, the 
 subject is organized around some social topic instead of 
 around some arithmetical topic, as borrowing and loan- 
 ing money, buying stocks, bond investments, taxes, insur- 
 ance, etc. Or, these are sometimes included as parts of 
 larger units, as problems of thrift, problems of investment, 
 problems of protection, problems of public expense, prob- 
 lems of transportation, problems of industrial life, etc. 
 The algebra taught in these grades is usually limited to 
 the interpretation of the formula and to the use of the 
 simple equation. The mensuration of the past is now 
 sometimes supplemented by a little constructive and 
 observational geometry and listed under the general 
 topic of "geometry." Regardless of the grouping, the 
 following topics are usually included in these two grades : 
 
 THE SEVENTH GRADE 
 
 1. A review of whole numbers, fractions, and decimals, 
 including short methods and emphasis upon checking 
 results and estimating results. 
 
 2. The two problems of percentage that were taken 
 up in the sixth grade as applied to profit and loss; dis- 
 count ; commission ; simple interest ; bank discount.
 
 218 THE TEACHING OF ARITHMETIC 
 
 3. The measurement of the areas of rectangles, tri- 
 angles, parallelograms, trapezoids, and circles; and the 
 measurement of the surfaces and volumes of prisms and 
 cylinders as applied to practical problems. 
 
 4. Drawing to a scale and sketching and interpreting 
 plans and diagrams. 
 
 SUGGESTIONS. 1. Attention this year to the applications 
 of arithmetic is the important thing. These applications in- 
 clude denominate numbers, mensuration, and percentage. 
 
 2. Confine the work of percentage to the two practical prob- 
 lems of the sixth-grade outline. Fix in memory all the impor- 
 tant fractional equivalents by constant use and drill. 
 
 3. Seek to make all work applied to commercial transactions 
 as realistic as possible. Play "Going into business" if it adds 
 interest and makes the work more realistic. All commercial 
 schools do this with students of much greater maturity. 
 
 4. Encourage pupils to bring in real problems, commercial 
 papers, and any material that will vitalize the work. 
 
 5. Distinguish between real problems met in real life, and 
 problems "for analysis" about real things. Remember that a 
 "real" problem is not necessarily concrete to the pupil. 
 
 6. In the application of mensuration, encourage pupils to 
 bring in real problems from measurements which they have 
 made for themselves. 
 
 7. Problems in mensuration are usually made clearer by dia- 
 grams or drawings made to a scale. 
 
 THE EIGHTH GRADE 
 
 The work of the eighth grade is an extension of the 
 work of the seventh to a wider range of applications and 
 it takes on a more adult point of view. If square root 
 is to be taught at all, it is taken up in this grade. If 
 the work of arithmetic is given in order that the pupil 
 may have the ability to interpret the quantitative phases
 
 THE COURSE OF STUDY IN ARITHMETIC 219 
 
 of his environment, it would seem that the time spent in 
 square root and its applications might better be spent 
 with other problems. 
 
 At the end of this year fundamental principles should 
 be thoroughly understood, habits of accuracy fixed, and 
 readiness and a fair degree of speed in ordinary computa- 
 tions attained. Power to state a problem clearly, to 
 analyze it logically, to choose a good method of solution, 
 and to do the work by the shortest method should have 
 been acquired. The work, then, should include : 
 
 1. A general application of the work of the preceding 
 grades to problems of human interest. 
 
 2. An extension of percentage to insurance; taxes; 
 national revenues; trade discount; successive trade 
 discounts; simple interest; bank discount; stock in- 
 vestments; bond investments. 
 
 3. An extension of mensuration to pyramids, cones, 
 and the Pythagorean Theorem. Also some simple geo- 
 metric constructions and observations; and the three 
 forms of graphs used in representing statistics. 
 
 4. The use and interpretation of formulae as "short- 
 hand expressions" of the principles and rules of mensura- 
 tion. 
 
 5. The use of ratio and proportion in expressing the 
 relation of similar figures and in the problems of simple 
 machines, as the lever, inclined plane, screw, wedge, etc. 
 
 SUGGESTIONS. 1. If time permits, the indirect problem 
 of percentage may be taken up. There are but few real appli- 
 cations of it. It is needed to find a selling price that will yield 
 a certain per cent of profit upon itself when the cost is known ;
 
 220 THE TEACHING OF ARITHMETIC 
 
 also to find a list price that may be discounted and yet yield a 
 certain profit when the cost is known. 
 
 2. The applications of percentage are difficult to teach only 
 because the transactions are outside of the experiences of many 
 of the pupils. Seek to make the work as realistic as possible. 
 Pupils of this age usually like to play at "make-believe" bill- 
 ing clerks, retail merchants, wholesalers, bank cashiers, etc. 
 This is done in all commercial schools. 
 
 3. Encourage pupils to bring in all kinds of commercial pa- 
 pers price lists, invoices of goods, receipts, checks, promissory 
 notes, insurance policies, tax bills, etc. As these are collected 
 by the pupils they should be carefully preserved and kept "in 
 stock" as part of the illustrative material of the school. 
 
 4. In "stock investments" there are but two real problems, 
 viz. : (1) Stock bought at one price and sold at another is a gain 
 or a loss to the buyer of how much ; (2) stock costing a certain 
 price and paying a certain dividend makes the investor what 
 per cent of the investment. These problems are no more diffi- 
 cult than many encountered much earlier in the course. The 
 pupil's difficulty with them lies in the fact that they are not 
 concrete to him. They lie entirely without his range of experi- 
 ences. Seek to make the topic realistic by a study of some local 
 corporation, by the issuing of stock certificates upon "make- 
 believe" corporations organized by the pupils, and by use of 
 problems made by the pupils from the market reports of the 
 daily papers. 
 
 5. The study of bonds may be made more real in connec- 
 tion with the study of civil government in considering the meth- 
 ods used by municipalities, states, or governments of raising 
 money. It is best to begin with a study of your own city. 
 
 6. It is well to explain the functions of savings banks, banks 
 of deposit, and other corporations. A visit to sjuch places is 
 well worth while.
 
 CHAPTER XVII 
 MEASURING RESULTS 
 
 UNTIL recently the rating of a pupil has depended upon 
 the personal opinion of the teacher and upon set examina- 
 tions. During the last few years considerable work has been 
 done along a method of measuring results, known as " stand- 
 ard tests," that promises a more scientific means of measur- 
 ing the products of certain lines of school training. 
 
 USES OF STANDARDS IN MEASURING EFFICIENCY 
 
 There are at least three general uses or purposes of 
 standards in measuring efficiency. There is a need of tests 
 and standards to be used in the administration of a school 
 system, also, for determining the rating of an individ- 
 ual for purposes of promotion ; and, finally, tests for the 
 purpose of analyzing the primary defects that lead to 
 backwardness in any subject. These may be classified 
 as administrative, grading, and diagnostic tests. A 
 standard test that will meet one of these three purposes 
 need not necessarily meet another. Thus, from the 
 standpoint of administration, the test must show merely 
 the general ranking of a large group, or of an entire school 
 system in a finished product ; as, for example, written addi- 
 tion. But the ability, to do written addition is a complex 
 
 221
 
 222 THE TEACHING OF ARITHMETIC 
 
 consisting of several primary abilities with which the ad- 
 ministration is not directly concerned. 
 
 From the standpoint of grading a pupil for entrance 
 or promotion, the test must measure the individual ability 
 instead of group ability, and, hence, must be so made as 
 to eliminate more nearly the element of chance error. 
 Thus the average score from a single example in addition 
 given to several thousand pupils might mark fairly well 
 the general ability of that group in that subject and thus 
 answer the purposes of an administrative test ; but such 
 a test would be entirely unreliable for a single individual, 
 owing to the element of chance. A grading test, moreover, 
 must be a test of the ability of the individual in a finished 
 product, not of abilities needed in that finished product. 
 
 The diagnostic tests are tests given to discover causes of 
 backwardness in performing any finished process. They 
 must locate the particular primary ability or abilities that 
 need special training. The weakness in any final process 
 may be due to a weakness in one or more of several prac- 
 tically unrelated primary abilities. Thus failure in the 
 finished product of written addition may be due to the 
 lack of any one of the following primary abilities : (1) the 
 automatic control of the primary number facts ; (2) abil- 
 ity to hold in mind a two- or a three-figured number and 
 add it to a one-figured number; (3) ability to carry in 
 mind the "tens" of each column- sum and add them to 
 the next column; and (4) ability to record accurately 
 the "ones" figure of each column-sum. Ability in any 
 one of these does not imply ability in any other, so diagnos- 
 tic tests to show the source of weakness must be made
 
 MEASURING RESULTS 223 
 
 before corrective drill work can be wisely applied. Thus, 
 if the weakness in final addition is due to (2), outlined 
 above, as is often the case, drill upon (1) is of almost no 
 help, yet this is often the only source of training prepara- 
 tory to written work. Thus it is seen that tests to be of 
 any real service in supervising, promoting, or correcting 
 weaknesses must be worked out carefully according to 
 the purposes that they are to serve. 
 
 ATTEMPTED SOLUTIONS OF THE PROBLEM 
 
 While many persons have worked at the problems out- 
 lined above, usually the work has been restricted to special 
 classes or school systems. Of those who have worked in a 
 wider field, the work of Rice, Stone, Courtis, and Woody 
 is best known. The real problem of Rice and Stone, 
 however, was not that of establishing a standard of meas- 
 urement, but since then* questions and results are used 
 by school systems in comparing their own schools with 
 those examined by these men, their questions are often 
 classed among the "standard tests." 
 
 THE WORK OF RICE 
 
 The tests given by Mr. J. M. Rice 1 in 1902 were not 
 intended to become standards. They consisted of five 
 sets of questions of eight each, for grades four to eight, 
 inclusive, and were given to the pupils of seven city sys- 
 tems, aggregating over 6000 pupils, in order to determine 
 some of the factors upon which successful school work 
 depends. They are of interest in showing the beginning 
 i The Forum, XXXIV (October-December), 1902.
 
 224 THE TEACHING OF ARITHMETIC 
 
 of the attempts that later led to more scientific attempts 
 at standardization. 
 
 THE QUESTIONS USED BY RICE 
 FOURTH YEAR 
 
 1. A man bought a lot of land for $1743, and built upon it a 
 house costing $5482. He sold them both for $10,000. How 
 much money did he make? 
 
 2. If a boy pays $2.83 for a hundred papers, and sells them at 
 4 cents apiece, how much money does he make? 
 
 3. If there were 4839 classrooms in New York City, and 47 
 children in each classroom, how many children would there be in 
 the New York schools ? 
 
 4. A man bought a farm for $16,575, paying $85 an acre. 
 How many acres were there in the farm? 
 
 5. What will 24 quarts of cream cost at $1.20 a gallon ? 
 
 6. A lady bought 4 pounds of coffee at 27 cents a pound, 16 
 pounds of flour at 4 cents a pound, 15 pounds of sugar at 6 cents a 
 pound, and a basket of peaches for 95 cents. She handed the 
 storekeeper a $10 note. How much change did she receive? 
 
 7. I have $9786. How much more must I have in order to be 
 able to pay for a farm worth $17,225 ? 
 
 8. If I buy 8 dozen pencils at 37 cents a dozen, and sell them 
 at 5 cents apiece, how much money do I make ? 
 
 FIFTH YEAR 
 
 1. A man bought a lot of land for $1743, and built upon it a 
 house costing $5482. He sold them both together for $10,000. 
 How much did he make ? 
 
 2. If a boy pays $2.83 for a hundred papers, and sells them at 
 4 cents apiece, how much does he make? 
 
 3. What will 24 quarts of cream cost at $1.20 a gallon? 
 
 4. If I buy 8 dozen pencils at 37 cents a dozen, and sell them 
 at 5 cents apiece, how much money do I make ? 
 
 5. A flour merchant bought 1437 barrels of flour at $7 a barrel. 
 He sold 900 of these barrels at $9 a barrel, and the remainder at 
 $6 a barrel. How much did he make?
 
 MEASURING RESULTS 225 
 
 6. How many feet long is a telegraph wire extending from New 
 York to New Haven, a distance of 74 miles? There are 5280 
 feet in a mile. 
 
 7. A merchant bought 15 pieces of cloth, each containing 62 
 yards. He sold 234 yards. How many dress patterns of 12 
 yards each did he have left? 
 
 8. Frank had $3.08. He spent J of it for a cap, \ of it for a 
 ball, and with the remainder bought a book. How much did 
 the book cost ? 
 
 SIXTH YEAR 
 
 1. If a boy pays $2.83 for a hundred papers, and sells them at 
 4 cents apiece, how much does he make? 
 
 2. What will 24 quarts of cream cost at $1.20 a gallon? 
 
 3. If I buy 8 dozen pencils at 37 cents a dozen, and sell them 
 at 5 cents apiece, how much do I make? 
 
 4. A flour merchant bought 1437 barrels of flour at $7 a barrel. 
 He sold 900 of these barrels at $9 a barrel, and the remainder at 
 $6 a barrel. How much did he make ? 
 
 5. If a train runs 31$ miles an hour, how long will it take the 
 train to run from Buffalo to Omaha, a distance of 1045 miles ? 
 
 6. If a map 10 inches wide and 16 inches long is made on a 
 scale of 50 miles to the inch, what is the area in square miles that 
 the map represents? 
 
 7. The salt water which was obtained from the bottom of a 
 mine of rock salt contained 0.08 of its weight of pure salt. What 
 weight of salt water was it necessary to evaporate in order to ob- 
 tain 3896 pounds of salt? 
 
 8. A gentleman gave away \ of the books in his library, lent 
 J of the remainder, and sold \ of what was left. He then had 420 
 books remaining. How many had he at first ? 
 
 SEVENTH YEAR 
 
 1. If a map 10 inches wide and 16 inches long is made on a 
 scale of 50 miles to the inch, what is the area in square miles that 
 the map represents? 
 
 2. The salt water which was obtained from the bottom of a 
 mine of rock salt contained 0.08 of its weight of pure salt. What
 
 226 THE TEACHING OF ARITHMETIC 
 
 weight of salt water was it necessary to evaporate in order to ob- 
 tain 3896 pounds of salt ? \3 1 f '<* * 
 
 3. A gentleman gave away | of the books in his library, lent 
 \ of the remainder, and sold \ of what was left. He then had 420 
 books remaining. How many had he at first ? 
 
 4. A farmer's wife bought 2.75 yards of table linen at $0.87 a 
 yard and 16 yards of flannel at $0.55 a yard. She paid in butter 
 at $0.27 a pound. How many pounds of butter was she obliged 
 to give? ^7^ 
 
 5. If coffee sold at 33 cents a pound gives a profit of 10 per 
 cent, what per cent of profit would there be if it were sold at 36 
 cents a pound ? 
 
 6. Sold steel at $27.60 a ton, with a profit of 15 per cent, and a 
 total profit of $184.50. What quantity was sold ? 
 
 7. If a woman can weave 1 inch of rag carpet a yard wide in 
 4 minutes, how many hours will she be obliged to work in order to 
 weave the carpet for a room 24 feet long and 24 feet wide ? No 
 deduction to be made for waste. ^3*f*7 ^" 
 
 8. A fruit dealer bought 300 apples at the rate of 5 for a cent, 
 and 300 at 4 for a cent. He sold them all at the rate of ,8 for 5 
 cents. What per cent did he gain on his investment ? 
 
 EIGHTH YEAB 
 
 1. If a map 10 inches wide and 16 inches long is made on a 
 scale of 50 miles to the inch, what is the area in square miles that 
 the map represents? 
 
 2. The salt water which was obtained from the bottom of a 
 mine of rock salt contained 0.08 of its weight of pure salt. What 
 weight of salt water was it necessary to evaporate in order to ob- 
 tain 3896 pounds of salt? 
 
 3. A gentleman gave away f of the books in his library, lent \ 
 of the remainder, and sold \ of what was left. He then had 420 
 books remaining. How many had he at first ? 
 
 /4. A man sold 50 horses at $126.00 each. On one half of them 
 he made 20 per cent, and on the other half he lost 10 per cent. 
 How much did he gain ? 
 
 5. Sold steel at $27.60 a ton, with a profit of 15 per cent, and a 
 total profit of $184.50. What quantity was sold? 
 
 6. A fruit dealer bought 300 apples at the rate of 5 for a cent,
 
 MEASURING RESULTS 
 
 227 
 
 and 300 at 4 for a cent. He sold them all at the rate of 8 for 5 
 cents. What per cent did he gain on his investment ? 
 
 7. The insurance on f of the value of a hotel and furniture cost 
 $420.00. The rate being 70 cents on $100.00, what was the value 
 of the property? 
 
 8. Gunpowder is composed of niter 15 parts, charcoal 3 parts, 
 and sulphur 2 parts. How much of each in 360 pounds of 
 powder ? , 
 
 RESULTS FOUND BY RICE 
 
 The results of the tests and many interesting conclusions 
 derived from them can be found in an article by Mr. Rice 
 in The Forum for October-December, 1902. The follow- 
 ing chart shows the averages for the schools in each of the 
 seven cities. 
 
 /"I, 
 
 GRADE 
 
 GRADE 
 
 GRADE 
 
 GRADE 
 
 GRADE 
 
 SCHOOL 
 
 Vj/ITT 
 
 IV 
 
 V 
 
 VI 
 
 VII ,. 
 
 VIII 
 
 AVERAGE 
 
 III 
 
 68.4 
 
 79.5 
 
 79.3 
 
 81.1 
 
 91.7 
 
 80.0 
 
 I 
 
 72.7 
 
 84.7 
 
 80.4 
 
 64.2 
 
 80.9 
 
 76.6 
 
 I 
 
 
 
 80.3 
 
 80.9 
 
 43.5 
 
 72.7 
 
 69.3 
 
 I 
 
 54.5 
 
 74.7 
 
 72.2 
 
 63.5 
 
 74.5 
 
 67.8 
 
 I 
 
 60.0 
 
 70.8 
 
 69.6 
 
 54.6 
 
 66.5 
 
 64.3 
 
 II 
 
 81.3 
 
 78.2 
 
 71.2 
 
 33.6 
 
 36.8 
 
 60.2 
 
 III 
 
 70.1 
 
 53.6 
 
 43.7 
 
 53.9 
 
 51.1 
 
 54.5 
 
 IV 
 
 70.5 
 
 73.2 
 
 58.9 
 
 31.2 
 
 41.6 
 
 55.1 
 
 IV 
 
 62.9 
 
 70.5 
 
 59.8 
 
 
 
 22.5 
 
 . 53.9 
 
 IV 
 
 53.5 
 
 53.5 
 
 42.3 
 
 16.1 
 
 48.7 
 
 42.8 
 
 IV 
 
 59.8 
 
 65.3 
 
 54.9 
 
 35.2 
 
 43.5 
 
 51.5 
 
 V 
 
 38.5 
 
 67.0 
 
 44.1 
 
 29.2 
 
 51.1 
 
 45.9 
 
 VI 
 
 28.1 
 
 38.1 
 
 68.3 
 
 33.5 
 
 26.9 
 
 39.0 
 
 VI 
 
 41.6 
 
 45.3 
 
 46.1 
 
 19.5 
 
 30.2 
 
 36.5 
 
 VI 
 
 36.8 
 
 55.0 
 
 34.5 
 
 30.5 
 
 23.3 
 
 36.0 
 
 VII 
 
 59.3 
 
 53.7 
 
 35.2 
 
 29.1 
 
 25.1 
 
 40.5 
 
 VII 
 
 47.4 
 
 65.4 
 
 35.2 
 
 15.0 
 
 19.6 
 
 36.5 
 
 VII 
 
 41.1 
 
 37.5 
 
 27.6 
 
 8.9 
 
 11.3 
 
 25.3 
 
 Gen. Av. 
 
 59.5 
 
 69.4 
 
 60.7 
 
 39.4 
 
 49.4 
 
 55.7
 
 228 THE TEACHING OF ARITHMETIC 
 
 THE WORK OF STONE 
 
 Dr. C. W. Stone, 1 in 1908, made a study of "Arithmetical 
 Abilities and Some of the Factors Determining Them." 
 This was a study of the nature of the product of the first 
 six years' work in arithmetic. The material gathered was 
 over 6000 test papers from 152 classrooms hi 26 different 
 school systems. The questions and results have been so 
 widely used that they are sometimes known as "Stone's 
 Standard Tests." A study of them will show that they do 
 not meet the needs of a standard test for any of the three 
 uses outlined in the first part of this chapter, nor were they 
 so intended by Dr. Stone. They have their value, how- 
 ever, and are reproduced here on account of the general 
 interest in them. Pupils were allowed 12 minutes for the 
 test on fundamental operations and 15 minutes for the 
 test on reasoning. 
 
 THE STONE TESTS 
 TEST IN FUNDAMENTAL OPERATIONS 
 
 Work as many of these problems as you have time for ; work 
 them in order as numbered. 
 
 1. Add 2375 
 
 4052 
 6354 
 260 
 6041 
 1543 
 
 1 C. W. Stone, Arithmetical Abilities and Some Factors 
 Determining Them, Published by Teachers College, Columbia 
 University, New York City, 1908.
 
 MEASURING RESULTS 229 
 
 2. Multiply 3265 by 20. 
 
 3. Divide 3328 by 64. 
 
 4. Add 596 
 
 428 
 
 94 
 
 75 
 302 
 645 
 984 
 897 
 
 5. Multiply 768 by 604. 
 
 6. Divide 1918962 by 543. 
 
 7. Add 4695 
 
 872 
 7948 
 6786 
 
 567 
 
 858 
 9447 
 7499 
 
 8. Multiply 976 by 87. 
 
 9. Divide 2782542 by 679. 
 
 10. Multiply 5489 by 9876. 
 
 11. Divide 5099941 by 749. 
 
 12. Multiply 876 by 79. 
 
 13. Divide 62693256 by 859. 
 
 14. Multiply 96879 by 896. 
 
 TEST IN REASONING 
 
 Solve as many of the following problems as you have time for ; 
 work them in order as numbered : 
 
 1. If you buy 2 tablets at 7 cents each and a book for 65 cents, 
 how much change should you receive from a two-dollar bill ?
 
 230 THE TEACHING OF ARITHMETIC 
 
 2. John sold 4 Saturday Evening Posts at 5 cents each. He 
 kept % the money and with the other he bought Sunday papers 
 at 2 cents each. How many did he buy ? 
 
 3. If James had 4 times as much money as George, he would 
 have $16. How much money has George? 
 
 4. How many pencils can you buy for 50 cents at the rate of 
 2 for 5 cents? 
 
 5. The uniforms for a baseball nine cost $2.50 each. The 
 shoes cost $2 a pair. What was the total cost of uniforms and 
 shoes for the nine ? 
 
 6. In the schools of a certain city there are 2,200 pupils ; | are 
 in the primary grades, J in the grammar grades, f in the High 
 School and the rest in the night school. How many pupils are 
 there in the night school? 
 
 7. If 85 tons of coal cost $21, what will 5 tons cost? 
 
 8. A news dealer bought some magazines for $1. He sold 
 them for $1.20, gaining 5 cents on each magazine. How many 
 magazines were there? 
 
 9. A girl spent | of her money for car fare, and three times as 
 much for clothes. Half of what she had left was 80 cents. How 
 much money did she have at first ? 
 
 10. Two girls receive $2.10 for making button-holes. One 
 makes 42, the other 28. How shall they divide the money ? 
 
 11. Mr. Brown paid one third of the cost of a building ; Mr. 
 Johnson paid the cost. Mr. Johnson received $500 more annual 
 rent than Mr. Brown. How much did each receive? 
 
 12. A freight train left Albany for New York at 6 o'clock. 
 An express left on the same track at 8 o'clock. It went at 
 the rate of 40 miles an hour. At what time of day will it 
 overtake the freight train if the freight train stops after it has 
 gone 56 miles ? 
 
 PERCENTAGE OF MISTAKES 
 
 Stone published 38 tables from the scores obtained. 
 Those of most interest to teachers and supervisors of arith- 
 metic are Tables VIII and IX, showing errors in reasoning 
 and in addition.
 
 MEASURING RESULTS 231 
 
 TABLE VIII TABLE IX 
 
 MISTAKES IN REASONING 
 
 MISTAKES IN ADDITION 
 
 ' - a> s|s 
 
 ill 
 
 oa?-* 3 
 
 QJT5 M 
 
 fg&l 
 
 
 
 li 
 
 - a 
 
 & 
 
 No. of 
 problems 
 incorrect 
 
 No. of 
 problems 
 attempted 
 
 Systems in 
 order of 
 per cent of 
 mistakes 
 
 Per cent 
 incorrect 
 
 1 
 
 
 
 "8 11 
 
 o'ffl-S 
 Z 
 
 No. of 
 steps 
 attempted 
 
 XVI 
 
 45.1 
 
 359 
 
 796 
 
 XXII 
 
 14.5 
 
 196 
 
 634 
 
 
 XVII 
 
 44.9 
 
 335 
 
 746 
 
 XX 
 
 13.5 
 
 139 
 
 595 
 
 XXIII 
 
 41.1 
 
 238 
 
 579 
 
 I 
 
 13.4 
 
 115 
 
 861 
 
 Ill 
 
 36.7 
 
 282 
 
 769 
 
 XVII 
 
 10.5 
 
 96 
 
 918 
 
 I. _ 
 
 34.7 
 
 269 
 
 776 
 
 XVIII 
 
 10.4 
 
 102 
 
 982 
 
 XV 
 
 33.7 
 
 249 
 
 739 
 
 VI 
 
 10.1 
 
 91 
 
 908 
 
 VI 
 
 31.8 
 
 233 
 
 733 
 
 X 
 
 9.9 
 
 81 
 
 819 
 
 XXII 
 
 30.9 
 
 196 
 
 634 
 
 IX 
 
 9.6 
 
 86 
 
 898 
 
 X... 
 
 29.7 
 
 232 
 
 781 
 
 V 
 
 9.2 
 
 89 
 
 967 
 
 XXIV. 
 
 28.9 
 
 167 
 
 577 
 
 XIII 
 
 8.9 
 
 75 
 
 847 
 
 XIII 
 
 28.8 
 
 230 
 
 799 
 
 XV 
 
 8.8 
 
 72 
 
 818 
 
 XXVI 
 
 28.6 
 
 276 
 
 964 
 
 VII 
 
 8.76 
 
 87 
 
 993 
 
 VIII 
 
 27.9 
 
 192 
 
 689 
 
 IV 
 
 8.5 
 
 81 
 
 951 
 
 XVIII 
 
 27 
 
 175 
 
 648 
 
 XXV 
 
 8.3 
 
 61 
 
 739 
 
 XIX 
 
 26.4 
 
 255 
 
 965 
 
 XXIII 
 
 8 
 
 56 
 
 703 
 
 XXV 
 
 25.3 
 
 150 
 
 592 
 
 VIII._ _ 
 
 7.5 
 
 60 
 
 803 
 
 IV 
 
 25.1 
 
 147 
 
 585 
 
 XVI. _ 
 
 7.04 
 
 67 
 
 952 
 
 XX 
 
 23.4 
 
 139 
 
 595 
 
 II 
 
 7 
 
 60 
 
 857 
 
 VII 
 
 23.1 
 
 189 
 
 819 
 
 Ill 
 
 6.5 
 
 55 
 
 843 
 
 XIV 
 
 22.9 
 
 175 
 
 765 
 
 XII 
 
 6.3 
 
 56 
 
 896 
 
 XII 
 
 22.3 
 
 185 
 
 831 
 
 XIV.__ 
 
 6.2 
 
 57 
 
 921 
 
 IX 
 
 20.3 
 
 161 
 
 794 
 
 XXIV 
 
 5.9 
 
 54 
 
 918 
 
 II 
 
 19.7 
 
 137 
 
 696 
 
 XXVI 
 
 5.8 
 
 54 
 
 930 
 
 V.. 
 
 18.6 
 
 171 
 
 919 
 
 XIX 
 
 5.78 
 
 57 
 
 987 
 
 XXI 
 
 15.7 
 
 106 
 
 674 
 
 XXI 
 
 5.1 
 
 44 
 
 860 
 
 XL. 
 
 14.4 
 
 112 
 
 776 
 
 XI 
 
 4.7 
 
 42 
 
 888 
 

 
 232 THE TEACHING OF ARITHMETIC 
 
 THE WORK or COURTIS 
 
 Mr. S. A. Courtis, 1 then head of the department of 
 mathematics in the Detroit Home and Day School, 
 through the inspiration received from the work of Stone, 
 began his investigations in 1909. From the first, his 
 purpose was to develop a series of standard tests to be 
 used in measuring the results of instruction in arithmetic, 
 and he is the first to take up this particular problem. The 
 first of the " Courtis tests" consists of eight tests known as 
 "Series A." The first published standards were derived 
 from the average results of testing a total of about 9000 
 pupils in all grades from the third to the eighth inclusive. 
 His later standards for the same series are made up from 
 the results obtained from 67,000 pupils distributed pretty 
 generally throughout the country. The standards are 
 not the average of conditions as they were found, but the 
 estimated scores that should exist at the end of the year. 
 " Series A," however, was full of defects. While more 
 diagnostic than administrative, it was neither, yet it 
 was evidently intended for the latter, for Courtis says, 
 "The general purpose of testing work is not to measure 
 the abilities of individuals to determine their fitness 
 for promotion, but to reveal the efficiency of school 
 procedure." 
 
 But Courtis was not blinded by enthusiasm to the de- 
 fects of his first work. In 1913 he published a new set of 
 tests and standards known as "Series B" which greatly 
 
 1 Courtis Standard Research Tests, published by Department of 
 Cooperative Research, 82 Eliot Street, Detroit, Michigan.
 
 MEASURING RESULTS 233 
 
 eliminated the defects of "Series A" from the standpoint 
 of its purpose as expressed in the above quotation. 
 
 This new series consists of but four tests, each consisting 
 of but a single written process, and the scores are median l 
 scores rather than average scores. With the exception 
 that Courtis has made but one test in each process, to be 
 used in all grades from the third to the eighth inclusive, 
 instead of a series for primary grades and another for 
 grammar grades with a known relation between them, 
 he has a series exceptionally well suited to measure the 
 abilities in the four fundamental processes. Test one is 
 an eight-minute test in written addition and consists of 
 twenty-four exercises of equal weight, each being made of 
 nine three-figured numbers. Such a test allows enough 
 time devoted to a single process to eliminate more nearly 
 the accidental errors and thus it becomes a much more 
 reliable measure than does a test of mixed exercises given 
 for a shorter period. 
 
 Test two is a four-minute test in written subtraction 
 consisting of twenty-four exercises with eight-figured 
 numbers. Test three is a six-minute test in multiplica- 
 tion, and test four an eight-minute test in division. 
 
 There are four forms of the same weighting in each test. 
 One form of each is given here. 
 
 1 The median score is found by arranging the scores in order 
 of magnitude from the best down to the poorest and taking the 
 middle case. Thus of 35 scores so arranged the 18th is the 
 median, for there are 17 poorer and 17 better. It differs slightly 
 from the average, for each score influences the median only as a 
 single case, whereas, in calculating an average, very small or very 
 large scores have a very marked effect upon the general average.
 
 234 THE TEACHING OF ARITHMETIC 
 
 COURTIS STANDARD RESEARCH TESTS 
 
 Arithmetic Test No. 1 Addition 
 Series B Form 1 
 
 SCORE 
 
 No. Attempted.. 
 No. Right... 
 
 You will be given eight minutes to find the answers to as many 
 of these addition examples as possible. Write the answers on this 
 paper directly underneath the examples. You are not expected 
 
 to be able to do them all. You will be marked for both speed and 
 
 accuracy, but it is more important to have your answers right 
 than to try a great many examples. 
 
 927 297 136 486 384 176 277 837 
 
 379 925 340 765 477 783 445 882 
 
 756 473 988 624 881 697 682 959 
 
 837 983 386 140 266 200 594 603 
 924 315 353 812 679 366 481 118 
 110 661 904 466 241 851 778 781 
 854 794 547 355 796 535 849 756 
 965 177 192 834 850 323 157 222 
 344 124 439 667 733 229 953 626 
 
 637 664 634 672 226 351 428 862 
 
 695 278 168 253 880 788 975 159 
 
 471 345 717 948 663 705 450 383 
 
 913 921 142 629 819 174 194 451 
 
 564 787 449 936 779 426 666 938 
 
 932 646 453 223 123 649 742 433 
 
 559 433 924 358 338 755 295 599 
 
 106 464 659 676 996 140 187 172 
 
 228 449 432 122 303 246 281 152 
 
 677 223 186 275 432 634 647 588 
 
 464 878 478 521 876 327 197 256 
 
 234 682 927 854 571 327 685 719 
 
 718 399 516 939 917 394 678 624 
 
 838 904 923 582 749 807 456 969 
 293 353 653 566 495 169 393 761 
 423 419 216 936 250 491 625 113 
 955 756 669 472 833 885 240 449 
 619 314 409 264 318 403 152 122
 
 MEASURING RESULTS 235 
 
 COURTIS STANDARD RESEARCH TESTS 
 
 Arithmetic Test No. 2 Subtraction 
 Series B Form 1 
 
 SCORE 
 
 No. Attempted. 
 No. Right.. 
 
 You will be given four minutes to find the answers to as many 
 of these subtraction examples as possible. Write the answers on 
 this paper directly underneath the examples. You are not ex- 
 pected to be able to do them all. You will be marked for both 
 speed and accuracy, but it is more important to have your answers 
 right than to try a great many examples. 
 
 107796491 
 77197029 
 
 75088824 
 57406394 
 
 91500053 
 19901563 
 
 87939983 
 72207316 
 
 160620971 
 80361837 
 
 51274387 
 25842708 
 
 117359208 
 36955523 
 
 47222970 
 17504943 
 
 115364741 
 80195261 
 
 67298125 
 29346861 
 
 92057352 
 42689037 
 
 113380936 
 42556840 
 
 64547329 
 48813139 
 
 121961783 
 90492726 
 
 109514632 
 81268615 
 
 125778972 
 30393060 
 
 92971900 
 62207032 
 
 104339409 
 74835938 
 
 60472960 
 50196521 
 
 119811864 
 34379846 
 
 137769153 
 70176835 
 
 144694835 
 74199225 
 
 123822790 
 40568814 
 
 80836465 
 49178036 
 
 Name.. 
 
 School. 
 City 
 
 , , Age last birthday. 
 
 BOT OB GIRL 
 
 Grade Room.... j ... 
 
 .State.... ....Date....
 
 236 THE TEACHING OF ARITHMETIC 
 
 COURTIS STANDARD RESEARCH TESTS 
 
 Arithmetic Test No. 3 Multiplication 
 Series B Form 1 
 
 SCORE 
 
 No. Attempted. 
 No. Right. _. 
 
 You will be given six minutes to work as many of these multi- 
 plication examples as possible. You are not expected to be able 
 to do them all. Do your work directly on this paper ; use no 
 other. You will be marked for both speed and accuracy, but it 
 is more important to have your answers right than to try a great 
 many examples. 
 
 8246 3597 5739 2648 9537 
 
 29 73 86 46 92 
 
 4268 7593 6428 8563 2947 
 
 37 640 68 207 63 
 
 5368 4792 7942 3586 9742 
 
 95 84 72 36 69 
 
 6385 8736 5942 6837 4952 
 
 48 602 39 680 47 
 
 3876 9245 7368 2594 6495 
 
 93 86 74 25 19 
 
 Name , , Age last birthday. 
 
 BOT OB GIRL 
 
 School.... Grade.... ....Room.... ;... 
 
 City State. _ Date.
 
 MEASURING RESULTS 237 
 
 COURTIS STANDARD RESEARCH TESTS 
 
 Arithmetic. Test No. 4. Division 
 Series B Form 1 
 
 SCORE 
 
 No. Attempted.. 
 No. Right. _. 
 
 You will be given eight minutes to work as many of these divi- 
 sion examples as possible. You are not expected to be able to 
 do them all. Do your work directly on this paper ; use no other. 
 You will be marked for both speed and accuracy, but it is more 
 important to have your answers right than to try a great many 
 examples. 
 
 25)6776 
 
 94)85352 
 
 37)9990 
 
 86)80066 
 
 73)68765 
 
 49)31409 
 
 68)43520 
 
 52)44252 
 
 37)14467 86)60372 94)67774 26)9750 
 
 68)39508 49)28420 52)21112 73)33653 
 
 45)33796 
 
 76)67000 
 
 93)28458 
 
 82)29602 
 
 Name , Age last birthday. 
 
 BOY OB GIRL 
 
 School Grade Room.... j 
 
 City.... State Date
 
 238 THE TEACHING OF ARITHMETIC 
 
 STANDARD SCORES l 
 
 In the table below will be found median speeds and accu- 
 racies based upon distribution of many thousands of individual 
 scores in tests given in May or June, 1915-1916. The distribu- 
 tion for each grade was made up of approximately equal numbers 
 of classes from large-city schools and from small-city and county 
 schools. One or two of the medians have been adjusted slightly 
 that the results as a whole might yield smooth curves. Half 
 year divisions have been combined to make whole grades. Com- 
 parison of these results with those in the tables that follow will 
 show the relation of the median scores, and of the values adopted 
 as standards, to the scores from tests in various cities. 
 
 TABLE 
 
 
 M. 
 
 s. 
 
 M. 
 
 8. 
 
 M. 
 
 S. 
 
 M. 
 
 S. 
 
 3 
 
 6.3 
 
 4 
 
 41 
 
 100 
 
 5.6 
 
 5 
 
 49 
 
 100 
 
 4 
 
 7.4 
 
 6 
 
 64 
 
 100 
 
 7.4 
 
 7 
 
 80 
 
 100 
 
 5 
 
 8.6 
 
 8 
 
 70 
 
 100 
 
 9.0 
 
 9 
 
 83 
 
 100 
 
 6 
 
 9.8 
 
 10 
 
 73 
 
 100 
 
 10.3 
 
 11 
 
 85 
 
 100 
 
 7 
 
 10.9 
 
 11 
 
 75 
 
 100 
 
 11.6 
 
 12 
 
 86 
 
 100 
 
 8 11.6 12 76 100 12.9 13 87 100 
 
 M. S. M. S. M. S. M. S. 
 
 3 .8 .6 
 
 4 6.2 6 67 100 4.6 4 57 100 
 
 5 7.5 8 75 100 6.1 6 77 100 
 
 6 9.1 9 78 100 8.2 8 87 100 
 
 7 10.2 10 80 100 9.6 10 90 100 
 
 8 11.5 11 81 100 10.7 11 91 100 
 
 M = Adjusted Median. 
 
 S = Scores adopted as standard. 
 
 1 From Bulletin Number Four, Courtis Standard Research 
 Tests.
 
 MEASURING RESULTS 239 
 
 FURTHER COMMENTS UPON SERIES B 
 
 The attempt is to give exercises of the same weight. 
 An examination of the exercises would seem to indicate 
 that this has been well done except in the case of division. 
 A glance at the answers and divisors shows a much greater 
 variation in weight than do the other three tests. For 
 in division the difficulty does not depend upon the number 
 of figures involved, but upon the character of the divisors 
 and quotients, since estimating the quotient figures con- 
 stitutes the greatest difficulty. 
 
 The results show that a 7th grade pupil can try 
 about 11 or 12 exercises in both addition and subtrac- 
 tion, and about 9 or 10 in both multiplication and division. 
 This eliminates more of the chance errors of the individual 
 and gives a more reliable result than those obtained in 
 Series A. For this reason they become not only a measure 
 of the efficiency of an entire school, for which they were 
 intended, but a rather reliable measure of the abilities of 
 the individual for promotional purposes. Yet a single 
 attempt will not furnish a very safe measure of the indi- 
 vidual, owing to the chance of accidental errors. 
 
 FURTHER COMMENTS ON SERIES A 
 
 When Series A first appeared, the novelty of measur- 
 ing any type of school efficiency by a fixed standard 
 awakened great enthusiasm among school men through- 
 out the country and no doubt led to a great many con- 
 clusions not warranted by the tests, and to undue emphasis 
 upon drills to develop special abilities rather than upon the
 
 240 THE TEACHING OF ARITHMETIC 
 
 completed process. Not only that, but in many cases no 
 doubt undue emphasis was placed upon drill in compu- 
 tation to the neglect of the applicative side of the subject 
 to the solution of problems. It is well, then, to examine 
 the first series, which are too well known to be repeated 
 here, to see what each of them really tests in order that 
 teachers may examine more critically other tests that 
 appear. 
 
 The first four tests are " speed tests " in recording the 
 primary number combinations of addition, subtraction, 
 multiplication, and division. It is a question whether 
 these tests test the knowledge of the fundamental number 
 facts as much as they do the motor dexterity of the pupil 
 in recording the fact. That is, a high score need not neces- 
 sarily show superior control of the facts, nor a low score 
 the lack of such a control, but the score may be condi- 
 tioned by the pupil's ability to make figures. But even if 
 they do measure the control of the primary number facts, 
 they are diagnostic tests needed in the analysis of the 
 cause of backwardness, and not administrative tests to 
 determine the efficiency of school instruction. The 
 administration is chiefly interested in the finished process. 
 
 The tests, however, are not suited for diagnostic tests 
 a thing for which they were not intended for the stand- 
 ards or scores are the work of but 1^ minutes upon each 
 test. Manifestly the work of an individual for 1^ minutes 
 cannot be taken as a reliable estimate of his ability. For 
 all who have worked with children know how largely the 
 element of chance enters into the work of so brief a 
 period.
 
 MEASURING RESULTS 241 
 
 The first four tests, then, as they now stand, are not 
 tests for measuring the abilities of the individual either to 
 determine his fitness for promotion or to discover weakness 
 that must be corrected in order to secure efficient work in 
 the written processes, and they should not be used for 
 such purposes by the teacher or supervisor. 
 
 Courtis in Bulletin Number Four says : " In June, 1912, 
 a set of tests of the four fundamental processes in arithme- 
 tic was issued, because investigations then completed had 
 proved that the tests of Series A were of little practical value 
 except as instruments of research." 
 
 Test five is a speed test in copying figures and thus tests 
 but motor control. It is thus more elementary than any 
 of the first four and belongs even more nearly to the diag- 
 nostic class of tests than they do. As an administrative 
 test it is of but little use, but it is a very necessary diag- 
 nostic test for determining the cause of slow work in any 
 written process. 
 
 Test number six is called "a speed test in reasoning." 
 In attempting to cover so wide a range as from the third 
 to the eighth grades inclusive, the results cannot be of 
 great value. One's ability to discover what to do in a 
 problem depends upon his ability to read understandingly, 
 upon his experiences through which he can interpret the 
 situation described, and, finally, upon his knowledge of the 
 meaning of the processes. So a set of problems that are 
 difficult enough to be a real test of ability for eighth grade 
 pupils would be too hard to be a test for third or fourth 
 graders. Likewise, a test fit for children in the lower 
 grades is too easy to test the efficiency of upper grades.
 
 242 THE TEACHING OF ARITHMETIC 
 
 Test number seven is, by its nature, the most satisfactory 
 test of the series from an administrative standpoint. 
 The test is a general one covering the written processes 
 in addition, subtraction, multiplication, and division. 
 But the examples are not of the same weight. In order 
 to give work for all grades from the third to the eighth 
 inclusive, the first examples are very easy. Since the 
 scores of the early grades are based upon the very easy 
 examples, and since all are reckoned as equal units in 
 making up the scores, the growth from grade to grade is 
 much greater than appears from the scores. Hence it is not 
 a test to be used in estimating growth from grade to grade. 
 
 Test number eight is a test in reasoning and computa- 
 tion combined. It consists of two-step problems. As to 
 length of statement, all are mechanically of the same 
 length. That is, there are about the same number of 
 printers' ems in the composition of each of them. But 
 that does not insure that they are all of equal units as to 
 difficulty. The degree of difficulty depends upon the 
 experiences of the pupil. But granting that the prob- 
 lems are equal in weight, manifestly, a test in reasoning 
 sufficiently difficult to measure such ability of a pupil in 
 the eighth grade is too difficult to measure the power of a 
 third or fourth grade pupil. 
 
 THE TESTS A REAL CONTRIBUTION TO EDUCATION 
 
 While the work of Courtis, being the pioneer work in 
 this new field of investigation, has its defects, and some of 
 his early tests were not well chosen for the purpose for 
 which they were intended, he has rendered the cause of
 
 MEASURING RESULTS 243 
 
 education a distinct service in making the beginning. 
 His standards, however, are based upon the present attain- 
 ments of the schools examined. They should not be taken 
 by school men as an ideal of attainment toward which their 
 schools should work. They may be too high or too low. 
 But surely a standard made as these were is much more 
 probably a safe one than one made from the snap judgment 
 of some individual, and such a standard goal of attain- 
 ment is going to lead to more uniform work throughout 
 the country. 
 
 But before we can have much more than a mere opinion 
 of what a reasonable standard should be, there must be a 
 much wider range of investigation. Standards based upon 
 conditions as they are in no way show what degree of 
 efficiency is desired; that is, they do not show a most 
 economical use of a pupil's time. If we can find by tests 
 how much skill attained in any grade is carried over into 
 future school work or out into the various adult activities, 
 and the relative loss of high or low attainments during long 
 or short periods of disuse, and many other such problems, 
 we shall then be in a much better position to state a more 
 scientific standard of efficiency. That is, we will be in a 
 much better position to decide when training in computa- 
 tion, or any other ability, in any grade, ceases to be an 
 economical use of a pupil's time. 
 
 There should be established, carefully and scientifically, 
 a measure known to all teachers, and one that can be 
 used by them at any time and without extra expense or 
 extra use of time. Thus, we should be able to say that 
 ability to add twelve exercises, each consisting of six
 
 244 THE TEACHING OF ARITHMETIC 
 
 three-figured numbers, in ten minutes, ten of the twelve 
 sums being correct, is ability of a certain grade. Such a 
 standard for each grade and each process would encourage 
 a teacher to work with a definite aim in mind. 
 
 THE WORK OF WOODY 
 
 In 1916 Dr. Clifford Woody 1 made an arithmetic 
 scale of measurement differing from those of Courtis in 
 several particulars. Instead of all exercises being of equal 
 weight, they ranged from the simplest primary number 
 facts through fractions, decimals, and compound numbers. 
 Thus they measure abilities too primary to be measured 
 by the Courtis tests and also measure a wider range of 
 abilities. There are two distinct series, known as "Series 
 A" and "Series B." Twenty minutes is allowed for each 
 scale in Series A, and ten minutes is allowed for each in 
 Series B. The author recommends that in Series B all 
 tests be taken in succession. 
 
 The tentative standards set up by the author were based 
 upon a total of about 20,000 test papers. There are two 
 methods of scoring used. The methods and scores are 
 not reproduced here, for before the standards of achieve- 
 ment would be of value to a teacher she would have to 
 know the conditions under which the tests were given and 
 the methods of computing the scores. For these the 
 reader is referred to Dr. Woody's monograph on the sub- 
 ject. To show the nature of the tests, however, Series A 
 is reproduced here. 
 
 1 Measurements of Some Achievements in Arithmetic, Teachers 
 College, Columbia University, New York City, 1916.
 
 MEASURING RESULTS 
 
 245 
 
 THE WOODY ARITHMETIC SCALE 
 
 SERIES A 
 ADDITION SCALE 
 
 When is your next birthday? How old will you be?.. 
 
 Are you a boy or girl? ....In what grade are you? 
 
 (1) (2) (3) 
 
 2 2 17 
 
 342 
 
 3 
 
 (10) (11) 
 
 21 32 
 
 33 59 
 
 35 17 
 
 (12) 
 
 43 
 
 1 
 
 2 
 
 13 
 
 45 
 
 (13) 
 23 
 25 
 16 
 
 (5) (6) 
 
 72 60 
 
 26 37 
 
 (14) 
 25+42 = 
 
 (7) 
 3+1 = 
 
 (15) (16) 
 
 100 9 
 
 33 24 
 
 45 12 
 201 15 
 
 46 19 
 
 (8) 
 
 2+5+1: 
 
 (9) 
 20 
 10 
 2 
 30 
 25 
 
 (17) (18) 
 
 199 2563 
 
 194 1387 
 
 295 4954 
 
 156 2065 
 
 (19) 
 $ .75 
 
 1.25 
 .49 
 
 (21) 
 
 $8.00 
 
 5.75 
 
 2.33 
 
 4.16 
 
 .94 
 
 6.32 
 
 (22) (23) 
 
 547 i+J 
 
 197 
 
 685 
 
 678 
 
 456 
 
 393 
 
 152 
 
 240 
 
 152 
 
 (24) 
 
 4.0125 
 
 1.5907 
 
 4.10 
 
 8.673 
 
 (25) 
 
 Sf 
 
 62| 
 12} 
 37J 
 
 (27) 
 
 (28) 
 
 (29) 
 
 5i 
 
 (30) 
 
 (31) 
 113.46 
 49.6097 
 19.9 
 9.87 
 
 .0086 
 18.253 
 6.04 
 
 (32) 
 
 i+i+i' 
 
 (33) 
 .49 
 .28 
 .63 
 .95 
 
 1.69 
 .22 
 .33 
 .36 
 
 1.01 
 .56 
 .88 
 .75 
 .58 
 
 1.10 
 .18 
 .56 
 
 (34) (35) 
 
 +i = 2ft. 6 in. 
 
 3 ft. 5 in. 
 
 4 ft. 9 in. 
 
 (36) 
 
 2 yr. 5 mo. 
 
 3 yr. 6 mo. 
 
 4 yr. 9 mo. 
 
 5 yr. 2 mo. 
 
 6 yr. 7 mo. 
 
 (38) 
 25.091 +100.4 +25 +98.28 +19.3614 =
 
 246 THE TEACHING OF ARITHMETIC 
 
 SERIES A 
 SUBTRACTION SCALE 
 
 Name 
 
 
 
 
 
 
 When is your nert b 
 Are you a boy or gir 
 
 (1) (2) (3) 
 862 
 501 
 
 irthday? 
 
 How old will 
 
 you be?. 
 
 1 ? . . . In wl 
 
 lat grade are you? 
 
 (7) (8) 1 
 13 59 
 8 12 
 
 (4) (5) (6) 
 9 4 11 
 347 
 
 :) do) (ID 
 
 78 7 -4 = 76 
 37 60 
 
 (12) (13) (14) (15) (16) 
 27 16 50 21 270 
 3 9 25 9 190 
 
 (17) (18) 
 393 1000 
 178 537 
 
 (19) (20) 
 567482 2J-1 = 
 106493 
 
 (21) (22) 
 10.00 3i-} = 
 3.49 
 
 (23) 
 80836465 
 49178036 
 
 (24) (25) 
 8} 27 
 5} 12f 
 
 4yd. 
 2yd. 
 
 (26) 
 1 ft. 6 in. 
 2 ft. 3 in. 
 
 (27) 
 5 yd. 1 ft. 4 in. 
 2 yd. 2 ft. 8 in. 
 
 (28) 
 10-6.25 = 
 
 (32) 
 1912 6 mo. 8 da. 
 1910 7 mo. 15 da. 
 
 (29) 
 75f 
 52} 
 
 (30) 
 9.8063-9.019 = 
 
 (34) (35) 
 6} 81-11- 
 
 (31) 
 7.3-3.00081 = 
 
 (33) 
 
 A-&- 
 
 SERIES A 
 MULTIPLICATION SCALE 
 
 Name 
 When is 
 Are you 
 
 (1) 
 3X7- 
 
 (8) 
 50 
 3 
 
 
 
 
 
 
 
 your next birthdi 
 
 iy? He 
 
 w old will you be? 
 
 i ftr von ? 
 
 In what grade 
 
 (2) 
 5X1 = 
 
 (9) (10) 
 254 623 
 6 7 
 
 (3) (4) (5) 
 2X3= 4X8= 23 
 Jt 
 
 (11) (12) (13) 
 1036 5096 8754 
 868 
 
 (6) 
 310 
 
 _4 
 
 (14) 
 165 
 40 
 
 (7) 
 7X9 = 
 
 (15) 
 235 
 23 
 
 (16) 
 7898 
 9 
 
 (17) 
 145 
 206 
 
 (18) (19) 
 24 9.6 
 234 4 
 
 (20) 
 287 
 .05 
 
 (21) 
 24 
 2J 
 
 (22) 
 8X5| = 
 
 (23) 
 UX8- 
 
 (30) 
 2.49 
 36 
 
 (35) 
 987} 
 25 
 
 (24) 
 16 
 
 (31) 
 
 iiXJ| = 
 
 (36) 
 3 ft. 5 in. 
 5 
 
 (25) (26) 
 1X1" 9742 
 59 
 
 (32) 
 6 dollars 49 cents 
 8 
 
 (27) 
 6.25 
 3.2 
 
 (33) 
 2|X3i 
 
 (38) 
 .0963 i 
 .084 
 
 (28) 
 .0123 
 9.8 
 
 (29) 
 JX2- 
 
 (34) 
 
 i+i- 
 
 (39) 
 8 ft. 9} in. 
 9 
 
 (37) 
 2iX4|XH =
 
 MEASURING RESULTS 
 
 247 
 
 Name 
 
 SERIES A 
 DIVISION SCALE 
 
 When is your next bi 
 Are you a boy or girl 
 
 (1) (2) 
 375 972T 
 
 rthday? 
 
 B 
 
 .ovr old will you be? 
 
 !e are you ? 
 
 y 
 
 ..In what grad 
 
 (3) 
 
 &T 
 
 (4) 
 ITS 
 
 (5) 
 9735 
 
 (6) 
 373T 
 
 (7) (8) 
 4 -5-2 = 970 
 
 (9) 
 
 17T 
 
 (10) (11) 
 6X =30 2713" 
 
 (12) 
 2-1-2 = 
 
 (13) 
 
 (14) 
 8)5856 
 
 (19) 
 
 248 -f-7 = 
 
 (15) 
 1 of 128 
 
 (20) 
 2.17253 
 
 (16) 
 68)2108 
 
 (21) 
 25)9750 
 
 (17) 
 50-J-7- 
 
 (22) 
 2)13.50 
 
 4)24 Ib. 8 oz. 
 
 (18) 
 13755055 
 
 (23) 
 23)159" 
 
 (24) 
 75)2250c 
 
 BB 
 
 (25) 
 2400)504000 
 
 (26) 
 1272^76 
 
 (27) 
 J of 624 = 
 
 (28) 
 .003) .0936 
 
 (29) 
 3i-S-9 = 
 
 (30) 
 J-i-5- 
 
 (31) 
 8-1 = 
 
 
 (32) 
 9f-i-3f = 
 
 (33) 
 52)3756 
 
 (34) 
 62.50 -J-li- 
 
 
 (35) 
 531)37722 
 
 
 (36) 
 
 9)69 Ib. 9 os. 
 
 THE CLEVELAND SURVEY TESTS 
 
 A series of fifteen tests used in the Cleveland, Ohio, 
 Survey * of 1915 is given here. As will be seen, they 
 range from the primary number facts to simple operations 
 with fractions. It was the intention of the authors of the 
 tests to make tests that would show both the complexity 
 of the processes which a given grade can master and also 
 the number of examples of a given type that can be per- 
 formed in a given time. 
 
 1 Cleveland Education Survey, by C. H. Judd. Published by 
 The Survey Committee of the Cleveland Foundation, Cleveland, 
 Ohio, 1916.
 
 248 
 
 THE TEACHING OF ARITHMETIC 
 
 The Cleveland Survey 
 Arithmetic Tests 
 
 June 1915 
 INDIVIDUAL SCORE SHEET 
 
 Name Age 
 
 Grade Date .... 
 
 School.... ....Teacher ... 
 
 SCORE IN NUMBER OF EXAMPLES RIGHT 
 
 Set A 
 
 SetB 
 
 SetC 
 
 SetD 
 
 SetE 
 
 Set F 
 
 Set G 
 
 Set H 
 
 Set I 
 
 Set J 
 
 Set K 
 
 Set L 
 
 Set M 
 
 Set N 
 
 Set O 
 
 Total Score in Points for Whole Test.. 
 
 INSTRUCTIONS FOR CHILDREN 
 
 1. Obey promptly all signals from the examiner, who will tell 
 
 you when to begin working and when to stop. 
 
 2. Do all your work directly on this paper. Work steadily and 
 
 rapidly, but do not hurry. Only the answers that are 
 right will be counted.
 
 MEASURING RESULTS 
 
 249 
 
 SET A Addition 
 
 1 
 
 2 
 
 6 
 6 
 
 9 
 5 1 
 
 4 
 2 
 
 1 
 3 
 
 7 
 7 
 
 9 3 
 6 
 
 2 
 
 4 
 
 1 
 5 
 
 3 6 
 8 9 
 
 
 
 3 
 
 8 9 
 
 7 
 
 8 
 
 2 
 
 1 4 
 
 8 
 
 
 
 2 3 
 
 4 
 
 7 
 
 3 
 
 1 
 
 2 
 
 5 
 
 6 7 
 
 5 
 
 8 
 
 6 9 
 
 4 
 3 
 
 2 
 2 
 
 9 7 
 3 8 
 
 4 
 
 
 5 
 2 
 
 7 
 1 
 
 4 8 
 9 6 
 
 
 
 
 3 
 
 4 
 
 9 2 
 
 1 8 
 
 5 
 
 7 
 
 
 4 
 
 6 2 
 3 1 
 
 4 
 8 
 
 5 
 9 
 
 1 
 
 
 6 3 
 2 3 
 
 7 
 4 
 
 9 
 8 
 
 4 
 6 5 
 
 
 
 
 
 
 
 Total 
 
 Number 
 
 Right 
 
 SET 
 
 B 
 
 Subtraction 
 
 9 
 9 
 
 7 
 3 
 
 11 
 6 
 
 8 
 1 
 
 12 
 3 
 
 1 
 
 
 9 
 
 7 
 
 13 
 8 
 
 4 
 3 
 
 12 
 6 
 
 8 
 
 
 11 
 9 
 
 12 
 7 
 
 5 
 1 
 
 10 
 2 
 
 6 
 
 
 11 
 7 
 
 15 
 8 
 
 10 
 _9 
 
 12 
 4 
 
 2 
 1^ 
 
 7 
 5 
 
 13 
 
 7 
 
 3 
 2 
 
 10 
 _5 
 
 1 
 1 
 
 6 
 3 
 
 15 
 _9 
 
 4 
 2 
 
 8 
 3 
 
 4 
 4 
 
 10 
 7 
 
 13 
 5 
 
 10 
 1 
 
 9 
 4 
 
 5 
 6 
 
 8 
 6 
 
 17 
 9 
 
 6 
 4 
 
 11 
 JJ 
 
 6 
 
 
 12 
 
 9 
 
 15 
 6 
 
 5 
 3 
 
 16 
 8 
 
 7 
 
 
 8 
 5 
 
 16 
 
 7 
 
 9 
 1 
 
 11 
 4 
 
 Total Number Right
 
 250 
 
 THE TEACHING OF ARITHMETIC 
 
 SET C Multiplication 
 
 3 
 
 4 
 
 9 
 
 
 
 5 
 
 4 
 
 2 
 
 7 
 
 4 
 
 9 
 
 2 
 
 7 
 
 8 
 
 2 
 
 6 
 
 1 
 
 9 
 
 6 
 
 
 
 5 
 
 9 
 
 6 
 
 4 
 
 7 
 
 6 
 
 2 
 
 3 
 
 9 
 
 
 
 7 
 
 1 
 
 2 
 
 8 
 
 
 
 5 
 
 1 
 
 3 
 
 6 
 
 5 
 
 4 
 
 1 
 
 2 
 
 7 
 
 
 
 8 
 
 7 
 
 3 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 7 
 
 6 
 
 3 
 
 1 
 
 8 
 
 9 
 
 
 
 3^ 
 
 1 
 
 4 
 
 8 
 
 
 
 4 
 
 1 
 
 6 
 
 8 
 
 
 
 9 
 
 6 
 
 4 
 
 9 
 
 3 
 
 5 
 
 4 
 
 2 
 
 8 
 
 7 
 
 3 
 
 1 
 
 3 
 
 6 
 
 
 
 3 
 
 2 
 
 6 
 
 7 
 
 6 
 
 4 
 
 7 
 
 4 
 
 8 
 
 
 
 9 
 
 2 
 
 3 
 
 9 
 
 5 
 
 6 
 
 Total Number Right 
 
 SET D Division 
 
 3)9 4)32 6)36 2)0 7)28 9)9 3)21 
 
 6)48 1)1 6)10 2)6 4)24 7)63 6)0 
 
 8)32 1)8 5)30 8)72 1)0 9)36 1)7 
 
 2)10 7H2 1)1 6)18 3)6 4)20 7)49 
 
 1)3 2)8 6)6 3)27 8)64 1)2 4)16 
 
 6)0 3)24 9)36 2)4 8)24 7)7 2)18 
 
 6)42 3)0 7)21 4)4 3)16 9)81 7)0 
 
 Total Number Right
 
 MEASURING RESULTS 
 
 251 
 
 SET E Addition 
 
 5 
 
 2 
 
 9 
 
 2 
 
 6 
 
 1 
 
 4 
 
 9 
 
 2 
 
 8 
 
 8 
 
 8 
 
 3 
 
 4 
 
 6 
 
 7 
 
 2 
 
 8 
 
 
 
 6 
 
 4 
 
 2 
 
 5 
 
 1 
 
 
 
 6 
 
 7 
 
 
 
 8 
 
 5 
 
 3 
 
 5 
 
 4_ 
 
 1^ 
 
 6 
 
 6 
 
 8 
 
 
 
 4 
 
 3 
 
 6 
 
 2 
 
 6 
 
 8 
 
 6 
 
 4 
 
 1 
 
 3 
 
 7 
 
 7 
 
 2 
 
 5 
 
 9 
 
 
 
 4 
 
 7 
 
 8 
 
 3 
 
 3 
 
 1 
 
 6 
 
 8 
 
 1 
 
 2 
 
 5 
 
 4 
 
 9 
 
 3 
 
 3 
 
 5 
 
 8 
 
 9 
 
 5 
 
 1 
 
 3 
 
 8 
 
 8 
 
 5 
 
 4 
 
 6 
 
 SET F Subtraction 
 
 Total Number Right 
 
 1009 
 269 
 
 1248 
 709 
 
 1335 
 419 
 
 908 
 258 
 
 768 
 295 
 
 1365 
 618 
 
 707 
 277 
 
 519 
 324 
 
 1269 
 772 
 
 1092 
 472 
 
 816 
 335 
 
 1236 
 908 
 
 615 
 527 
 
 716 
 344 
 
 1157 
 908 
 
 1344 
 818 
 
 854 
 286 
 
 SET G Multiplication 
 
 Total Number Right 
 
 2345 
 2 
 
 9735 
 5 
 
 8642 
 9 
 
 6789 
 2 
 
 2345 
 6 
 
 9735 
 9 
 
 2468 
 3 
 
 6789 
 6 
 
 3579 
 3 
 
 2468 
 7 
 
 5432 
 4 
 
 5432 
 8 
 
 9876 
 8 
 
 8642 
 5 
 
 3579 
 7 
 
 9876 
 4 
 
 3689 
 5 
 
 2457 
 6 
 
 9863 
 4 
 
 7542 
 7 
 
 Total Number Right
 
 252 
 
 THE TEACHING OF ARITHMETIC 
 
 SET H Fractions 
 
 3 1 = ?_1 = 
 
 5 + 5~ 9 9~ 
 
 M- 
 
 9^9 
 
 9 9 
 13 
 
 9 + 9 : 
 
 7 1 
 
 5 5 
 
 6_2 
 7 1 
 
 8 8 
 
 4_3 
 
 8 8 
 
 SET I Division 
 
 4)55424 7)65982 
 
 -4-1= 8 _ 7 
 
 9 9~ 9 9 = 
 
 1 4_ 6_2 
 
 6 1_ 6_6 
 
 9*9 99 
 
 2 1 = ^_* = 
 
 7 + 7~ 9 9~ 
 
 4 2 = Z_5 = 
 
 it 99 
 
 Total Number Right 
 
 2)58748 6)41780 
 
 9)98604 6)57432 3)82689 6)83184 
 
 8)51496 9)75933 8)87856 4)38968 
 
 Total Number Right 
 
 10
 
 MEASURING RESULTS 
 
 253 
 
 SET J Addition 
 
 7 9 
 
 4 
 
 7 
 
 2 
 
 9 
 
 6 
 
 7 
 
 7 
 
 8943 
 
 2 
 
 .0 
 
 1 
 
 5 2 
 
 5 
 
 1 
 
 9 
 
 6 
 
 9 
 
 1 
 
 8 
 
 0631 
 
 1 
 
 fc 
 
 > 
 
 4 4 
 
 8 
 
 9 
 
 4 
 
 2 
 
 6 
 
 5 
 
 5 
 
 7377 
 
 6 
 
 
 
 
 
 2 8 
 
 1 
 
 4 
 
 8 
 
 4 
 
 7 
 
 1 
 
 4 
 
 1476 
 
 6 
 
 
 
 6 2 
 
 4 
 
 3 
 
 5 
 
 7 
 
 
 
 4 
 
 1 
 
 8609 
 
 1 
 
 
 
 7 
 
 8 
 
 2 
 
 1 
 
 1 
 
 4 
 
 6 
 
 8 
 
 6226 
 
 8 
 
 
 
 5 5 
 
 5 
 
 8 
 
 5 
 
 3 
 
 3 
 
 5 
 
 2 
 
 1393 
 
 6 
 
 
 
 1 3 
 
 1 
 
 5 
 
 2 
 
 9 
 
 7 
 
 3 
 
 1 
 
 3964 
 
 9 
 
 
 
 8 6 
 
 3 
 
 2 
 
 4 
 
 2 
 
 1 
 
 3 
 
 3 
 
 7266 
 
 7 
 
 
 
 3 1 
 
 9 
 
 7 
 
 3 
 
 3 
 
 6 
 
 7 
 
 9 
 
 4234 
 
 6 
 
 
 
 2 4 
 
 6 
 
 7 
 
 6 
 
 8 
 
 
 
 6 
 
 8 
 
 9842 
 
 2 
 
 
 
 9 8 
 
 3 
 
 1 
 
 7 
 
 6 
 
 6 
 
 1 
 
 4 
 
 4689 
 
 2 
 
 
 
 9 8 
 
 6 
 
 9 
 
 6 
 
 5 
 
 6 
 
 7 
 
 6 
 
 4689 
 
 4 
 
 
 
 Total Number Right 
 
 SET K Division 
 
 21)441 
 
 32)672 
 
 23)483 
 
 61)1173 
 
 71)1662 42)882 32)992 61)1342 
 
 63)1166 22)462 21)1071 62)1092 
 
 61)1122 41)861 31)961 41)1681 
 
 61)1281 22)484 31)661 
 
 33)693 
 
 Total Number Right 
 
 16 
 
 16
 
 254 
 
 THE TEACHING OF ARITHMETIC 
 
 SET L Multiplication 
 
 8246 3597 
 29 73 
 
 5739 2648 9537 
 86 46 92 
 
 | Number 
 
 
 ? 
 
 I 
 
 CO 
 
 4268 
 37 
 
 7593 
 64 
 
 6428 8563 2947 
 68 207 63 
 
 
 
 
 Total 
 
 Number Right 
 
 
 30 
 
 
 SET M Addition 
 
 
 
 
 7493 
 9016 
 6487 
 7691 
 6166 
 
 8937 8625 
 6345 4091 
 2783 3844 
 4883 8697 
 1341 7314 
 
 2123 
 1679 
 6555 
 6331 
 
 6808 
 
 5142 3691 
 0376 4526 
 4955 7479 
 9314 2087 
 5507 8165 
 
 
 30 
 
 
 5226 
 2883 
 2584 
 0058 
 2398 
 
 9149 6268 
 8467 7726 
 0251 8331 
 7536 5493 
 5223 3918 
 
 9397 
 6168 
 3732 
 4641 
 7919 
 
 7337 8243 
 2674 6429 
 9669 9298 
 6114 7404 
 8154 2575 
 
 
 
 Total 
 
 Number Right 
 
 SET N- 
 
 Division 
 
 
 
 
 
 
 67)32763 
 
 48)28464 
 
 97)36084 
 
 69)29382 
 
 78)69888 
 
 88)34496 
 
 69)40296 
 
 38)26562 
 
 
 
 Total 
 
 Number Right 
 
 
 30 

 
 MEASURING RESULTS 
 SET O Fractions 
 
 3m 
 1~ 
 
 4^18 
 
 255 
 
 14 4 
 
 35 20 1 
 
 4 X 6 = 2l"6 = 
 
 12^8 6 21 6 20 12 ' 
 
 G.^m~ A ' IK 
 o 10 o 15 
 
 Total Number Right 
 
 30 
 
 Instructions for Examiners 
 
 Have the children fill out the blanks at the top of the first page. 
 Have them start and stop work together. Let there be an inter- 
 val of half a minute between each set of examples. The time al- 
 lowances given below must be followed exactly. 
 
 Set A 30 seconds Set F 1 minute Set K 2 minutes 
 
 Set B 30 seconds Set G 1 minute Set L 3 minutes 
 
 Set C 30 seconds Set H 30 seconds Set M 3 minutes 
 
 SetD 30 seconds Set 1 1 minute Set N 3 minutes 
 
 Set E 30 seconds Set J 2 minutes Set 3 minutes 
 
 Have the children exchange papers. Read the answers aloud 
 and let the children mark each example that is correct, "C." 
 For each set let them count the number of C's and write the 
 number at the end of the set in the first column ; also on the 
 first page. 
 
 The teacher should multiply the number of examples right by 
 the number in the second column, writing the result in the third 
 column. Then find the sum of the scores in the third column for 
 a total score.
 
 256 
 
 THE TEACHING OF ARITHMETIC 
 
 STANDARD SCORES 
 
 MEDIANS IN EACH ARITHMETIC TEST FOR ALL 
 GRADES 
 
 TEST 
 
 GRADE 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 A 
 
 13.4 
 
 17.8 
 
 22.2 
 
 24.8 
 
 26.7 
 
 27.5 
 
 B 
 
 9.3 
 
 13.4 
 
 17.2 
 
 19.8 
 
 21.5 
 
 26.0 
 
 C 
 
 6.5 
 
 12.0 
 
 15.5 
 
 16.6 
 
 17.7 
 
 19.0 
 
 D 
 
 6.3 
 
 12.4 
 
 15.7 
 
 18.5 
 
 20.8 
 
 22.5 
 
 E 
 
 4.3 
 
 5.3 
 
 6.3 
 
 6.8 
 
 7.5 
 
 7.8 
 
 F 
 
 2.0 
 
 4.9 
 
 6.7 
 
 7.5 
 
 8.6 
 
 10.1 
 
 G 
 
 2.0 
 
 3.9 
 
 5.2 
 
 5.5 
 
 5.9 
 
 6.6 
 
 H 
 
 0.0 
 
 0.0 
 
 5.0 
 
 5.5 
 
 7.7 
 
 8.5 
 
 I 
 
 0.6 
 
 1.1 
 
 2.0 
 
 3.1 
 
 4.0 
 
 4.7 
 
 J 
 
 1.9 
 
 3.2 
 
 4.0 
 
 4.4 
 
 4.9 
 
 5.7 
 
 K 
 
 0.0 
 
 4.0 
 
 6.8 
 
 8.5 
 
 10.1 
 
 12.5 
 
 L 
 
 0.0 
 
 1.7 
 
 2.5 
 
 2.8 
 
 3.2 
 
 3.9 
 
 M 
 
 1.4 
 
 2.5 
 
 3.2 
 
 3.8 
 
 4.4 
 
 5.1 
 
 N 
 
 0.0 
 
 0.8 
 
 1.3 
 
 1.7 
 
 2.0 
 
 2.6 
 
 
 
 0.0 
 
 0.0 
 
 0.0 
 
 3.1 
 
 4.1 
 
 5.5 
 
 SOME CAUTIONS TO OBSERVE 
 
 It must be observed by teachers and supervisors that 
 the "standard tests" which we have discussed are but 
 measures of the mechanical aspects of the subject of 
 arithmetic. There is but little if any relation between the 
 abilities to compute and the power to interpret a problem 
 and reason out what processes to apply. There is a dan- 
 ger that this may be overlooked by some and that a
 
 MEASURING RESULTS 257 
 
 teacher may overemphasize the mechanical side of the 
 subject, which is so easily measured, to the neglect of the 
 more important part of it, for which there can be no such 
 definite tests. 
 
 For just as abilities to spell and to write are of no value 
 unless we have thoughts to express through such abilities, 
 so abilities to compute are of no value unless we can inter- 
 pret a problem and know what processes to apply to its 
 solution.
 
 Accuracy, 
 
 standards of, 6, 221. 
 Adding zeros, 36. 
 Addition, 
 
 carrying in, 40. 
 
 checks in, 43. 
 
 decimals, 118. 
 
 drills in, 38, 43. 
 
 importance of drills, 33. 
 
 order of drills, 32. 
 
 primary facts, 30, 33. 
 
 single columns, 39. 
 
 use of objects in, 32. 
 
 written work in, 39. 
 Addition facts, 
 
 chart of, 37. 
 Aims, importance of, 8. 
 Analysis of problems, ch. XIV. 
 Applications, 
 
 business, 130. 
 
 of percentage, 129. 
 Approximating results, 181. 
 Areas, 152. 
 
 of a circle, 157. 
 
 of a parallelogram, 154. 
 
 of a rectangle, 153. 
 
 of a trapezoid, 156. 
 
 of a triangle, 155. 
 Arithmetic, 
 
 aims of, 8. 
 
 essentials of, 22. 
 
 practical values, 12. 
 
 traditional values, 9. 
 
 Banking, 143. 
 
 Business applications, 130. 
 
 Carrying in addition, 40. 
 Checking work, 43. 
 Cleveland survey tests, 247. 
 Colburn, Warren, 1. 
 Commercial discount, 132. 
 Commission, 130. 
 Complex fractions, 18, 115.] 
 Compound interest, 138. 
 Counting, ch. III. 
 
 rational, 25. 
 
 rote, 25. 
 
 to 20; to 100,28. 
 Course of study, ch. XVI. 
 
 grammar grades, 216. 
 
 intermediate grades, 212. 
 
 primary grades, 207. 
 Courtis, S. A., 232. 
 Courtis tests, the, 232. 
 
 Decimal fractions, ch. X. 
 
 addition of, 118. 
 
 applications of, 122. 
 
 division of, 120. 
 
 multiplication of, 119. 
 
 subtraction of, 118. 
 Decimal notation, 117. 
 Denominate numbers, ch. XII. 
 
 fundamental processes in, 151. 
 
 reduction of, 150. 
 Discipline, formal, 11. 
 259
 
 260 
 
 INDEX 
 
 Discount, 131. 
 
 bank, 144. 
 
 commercial, 132. 
 Division, ch. VII. 
 
 charts for, 69. 
 
 decimals, 120. 
 
 fractions, 112. 
 
 long, 66. 
 
 measurement, 62. 
 
 partition, 62. 
 
 short, 65. 
 
 tables of, 61. 
 
 written, 65. 
 
 Doctrine of formal discipline, 11. 
 Drill work, 3. 
 
 importance of, 33. 
 
 motivation of, 165. 
 
 repetition of, 194. 
 
 Economy in teaching, 1. 
 Educational principles, 3. 
 Efficiency in computation, 13. 
 Elimination of topics, 17. 
 Estimating results, 182. 
 Exchange, 21. 
 
 Figures, 27. 
 
 Forecasting results, 182. 
 Formal discipline, 11. 
 Fractions, common, ch. IX. 
 
 adding, 103. 
 
 complex, 115. 
 
 division of, 111. 
 
 early work in, 95. 
 
 expressing a ratio, 102. 
 
 fundamental processes, 97. 
 
 ideas involved in, 99. 
 
 improper, 106. 
 
 indicated division, 101. 
 
 Fractions Continued 
 
 multiplication of, 110. 
 
 notation of, 96. 
 
 objective presentation of, 98. 
 
 part of a whole, 100. 
 
 reduction of, 106. 
 
 reduction to decimal, 121. 
 
 subtraction, 108. 
 
 unit of, 63. 
 Fractions, decimal, 
 
 (See Decimals). 
 
 Games, ch. VIII. 
 
 "make believe," 74. 
 
 motor activity, 83. 
 
 purpose of, 70. 
 
 school room, 71. 
 
 scoring, 71. 
 Grades, work of, 
 
 eighth, 218. , 
 
 fifth, 213. 
 
 first, 207. 
 
 fourth, 211. 
 
 second, 208. 
 
 seventh, 217. 
 
 sixth, 214. 
 
 third, 210. 
 Grammar grades, 
 
 work of, 216. 
 Graphs, 159. 
 Greatest Common Divisor, 17. 
 
 Habit of seeing relations, 170. 
 Habituation, 5. 
 
 Insurance, 139. 
 
 fire, 139. 
 
 life, 141. 
 Interest, 137. 
 
 compound, 138.
 
 INDEX 
 
 261 
 
 Interest Continued 
 
 methods of computing, 137. 
 
 simple, 134. 
 Intermediate grades, 
 
 work of, 212. 
 Inverse problems, 20. 
 Inverting divisor, 113. 
 
 Labeling steps, 189. 
 
 Least Common Denominator, 105. 
 
 Least Common Multiple, 17. 
 
 Lessons, 
 
 deductive, 201. 
 
 drill, 192. 
 
 inductive, 196. 
 
 three types of, 191. 
 Lesson, planning a, ch. XV. 
 Lesson plan, 202. 
 Loss and profit, 133. 
 
 Measurement, division, 62. 
 Mensuration, 152. 
 
 circles, 157. 
 
 parallelograms, 154. 
 
 rectangles, 163. 
 
 solids, 158. 
 
 trapezoids, 156. 
 
 triangles, 155. 
 
 Measuring results, ch. XVII. 
 Metric system, 22. 
 Mixed numbers, 
 
 adding, 107. 
 
 dividing, 114. 
 
 multiplying, 111. 
 
 subtracting, 109. 
 Motivation, 45, 165. 
 Multiplication, 
 
 decimals, 119. 
 
 development of tables, 52, 56. 
 
 Multiplication Continued 
 drills in, 57. 
 fractions, 110. fc 
 mixed numbers, 111. 
 written, 58. 
 
 Number, 
 
 begun too early, 1. 
 child's need of, 2. 
 
 Objective teaching, 32. 
 Oral arithmetic, 23. 
 
 Parallelograms, 154. 
 Partial payments, 20. 
 Partition, division, 62. 
 Partnership, 21. 
 Per Cent, 
 
 ability to express, 129. 
 
 finding a, 124. 
 
 notation of, 123. 
 
 notion of, 123. 
 Percentage, ch. XI. 
 
 applications of, 129. 
 
 three problems of, 124. 
 Pestalozzi, 1. 
 
 Planning a lesson, ch. XV. 
 Power to see relations, 168. 
 Primary grades, 
 
 problems of, 163. 
 
 work of, 207. 
 
 Principles of teaching, 3, 175. 
 Problems, 
 
 analysis and solution of, ch. 
 XIV. 
 
 book vs. local, 178. 
 
 clarifying a process, 45, 164. 
 
 concrete, 176. 
 
 contribute to social insight, 172.
 
 262 
 
 INDEX 
 
 Problems Continued 
 
 failure to solve, 179. 
 
 grammar grade, 167. 
 
 motivating drills, 165. 
 
 nature and use of, 45. 
 
 primary grades, 163. 
 
 purpose of, 163. 
 
 solution of, 187. 
 
 wording of, 166. 
 Profit and loss, 133. 
 Pythagorean Theorem, 21. 
 
 Quantitative relations, 16. 
 
 Ratio, 102. 
 
 Rationalization, 5. 
 
 Reading and writing numbers, 29. 
 
 Rectangle, 153. 
 
 Reduction, 
 
 of denominate numbers, 150. 
 
 of fractions, 106. 
 Rice, J.M., 223. 
 Root, square, 21. 
 
 Social insight, 15, 172. 
 Speed and accuracy, 6. 
 Square root, 21. 
 Stock investments, 145. 
 Stone, C. W., 228. 
 
 Stone's tests, 228. 
 Subject matter, 
 
 elimination of, 17. 
 
 essentials, 22. 
 
 grades, 207. 
 Subtraction, 
 
 addition method, 47. 
 
 decimals, 118. 
 
 fractions, 108. 
 
 meaning of, 47. 
 
 taking away method, 50. 
 Standard tests, 221. 
 Standards of accuracy, 6. 
 Statistics represented, 
 
 by graphs, 159. 
 
 Taxes, 142. 
 
 Unit fractions, 63. 
 
 Values of arithmetic, 
 
 practical, 12. 
 
 traditional, 9. 
 Volumes, 158. 
 
 Weights and measures, 149. 
 Woody, C., 244. 
 work of, 244. 
 Woody's tests, 245. 
 Work by grades, 207.
 
 UNIVERSITY OF CALIFORNIA AT LOS ANGELES 
 
 THE UNIVERSITY LIBRARY 
 This book is DUE on the last date stamped below 
 
 AW 18 193f 
 JUL 12 1938 
 
 otf * 
 
 
 NOV8 
 
 MS 13, 949 
 
 AUG 1 3 1949 
 HAR 
 
 MAR 3 
 
 AUG 1 2 1950 
 
 V* 
 
 Form L-9-15m-2,'36 
 
 iDOVSOttff 
 OMUD 
 
 REC'DLO 
 
 E MAR 2 9 i$b? 
 
 MAR 20 Era 

 
 D Q o -7 ' /Bl