^tatc QJolbgp of K^timltntt At (Cornell Imuecaitg 3tlfata, N. ^. Slibtatg Cornell University Library QA 135.J43 The supervision of arithmetic, 3 1924 002 963 704 The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002963704 THE SUPERVISION OF ARITHMETIC THE MACMILLAN COMPANY NEW YORK • BOSTON ■ CHICAGO • DALLAS ATLANTA • SAN FRANCISCO MACMILLAN & CO., Limited LONDON • BOMBAY - CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO THE SUPERVISION OF ARITHMETIC BY W. A. JESSUP UNIVERSITY OF IOWA AND L. D. COFFMAN UNIVERSITY OF MINNESOTA THE MACMILLAN COMPANY 1916 AU rights reserved Copyright, 1916, ' By the MACMILLAN COMPANY. Set up and electrotyped. Published September, 1916. Worfajooli ^ttss J. S. CuBhing Co. — Berwick & Smitb Co. Norwood, Mass., U.S.A. PREFACE This is not a teacher's handbook for the teaching of arithmetic ; it does not contain a detailed description of the methods and devices teachers should employ in teaching arithmetic, nor is it intended to be in any sense a critical analysis of the many scholarly articles that have appeared within recent years bearing upon the subject. It is frankly the result of a number of surveys and investigations conducted by the authors themselves or their students of certain problems relat- ing to the supervision of arithmetic. It is not main- tained that every problem relating to the supervision of arithmetic has been solved. This book gives the inquiring and progressive super- visor certain criteria for judging his course of study in arithmetic and in addition it gives him certain tests for measuring the attainments of his pupils. Certainly its greatest value lies in the fact that it raises to con- sciousness and outlines more clearly than heretofore a number of problems connected with the supervision of arithmetic. THE AUTHORS. CONTENTS CHAPTER PAGB I. Subject Matter of Arithmetic . . . . i II. The Grade Distribution of the Different Arithmetical Topics .... 21 III. Time Allotment for Arithmetic . . 39 IV. Dominance of Methods in the Teaching of Arithmetic . . . -65 V. The Sequence of the Multiplication Tables . 79 VI. Oral Work in Arithmetic . . -87 VII. Drill in Arithmetic .... 92 VIII. Grade for Introduction of Text in Arithmetic 97 IX. Judging Textbooks . . . . 106 X. Algebra and Geometry in Grades . 136 XI. Problems Related to Current Business Life . 145 XII. Tests and Results as Shown by Special Inves- tigation 154 Appendix A 218 Appendix B 220 THE SUPERVISION OF ARITHMETIC CHAPTER I SUBJECT MATTER OF ARITHMETIC Current Criticisms Most critics of the public schools are inclined to main- tain the thesis that the work ia arithmetic of to-day is less satisfactory than the work which was done in this subject at an earlier period. They say that the old- fashioned teacher spent more time in drill, and less time in rationalization or explanation. The critic of to-day is disposed to call attention to the fact that certain fac- tors and processes which are being taught in arithmetic are of questionable value as far as the practical demands of the day are concerned. The usual illustrations cited are : tables — furlongs, apothecaries' weights, foreign money, folding paper, longitude and time, and the like. Much criticism is also heard agaiast the teaching of true discount, cube root, partnership, unreal fractions, and alligation. B I I 2 THE SUPERVISION OF ARITHMETIC Again, it is not infrequently stated that more atten- tion should be given both in the fundamentals and in the applications of arithmetic to social conditions con- nected with the problems of saving, banking, borrowing, bviilding and loans, investments, bonds and stocks, taxes, levies, public expenditures, and insurance. Both of these criticisms contain certain elements of validity. Not only is it difficult for the school to dis- criminate between the essential elements of the experi- ence of the past, but it has no well-defined standards for discriminating values of the present. The immatur- ity, lack of experience, and inadequacy of training of teachers have resulted in a more or less slavish depend- ence upon the textbook. Indeed, in many instances the textbook serves as the entire course of study, and furnishes the only criterion of method. The expense of changing textbooks and the difficulties attending their adoption have contributed to a certain amount of conservatism and inflexibility in their use. Again, the teacher's immaturity and lack of experience have tended toward the exclusive use of the problems and examples found in the textbook. In other words, there has been too little disposition on the part of teachers to supplement intelligently and to make con- crete the prescribed material of the textbook. The result of these tendencies is that the material has been more or less formal and artificial. subject matter of arithmetic 3 Changing Conceptions of Arithmetical Values Despite the fact that men and women are to-day investing in stocks, bonds, insurance, i buildings, and loans, the school tends to give but sHght attention to these forms of apphed arithmetic, but clings to those appUcations of arithmetic which relate to older forms of investment, such as surveyiag, the buying and selEng of land and other tangible property such as houses and stores. Although it is not infrequently stated by school teachers that stocks, bonds, and insurance should not be taught, there is, on the contrary, much evidence to indicate that we should spend much more time in pre- senting these phases of arithmetic than we have hereto- fore expended. However, if we expect pupils to profit by the school's attempts to teach these new fields, it is necessary that they be taught by teachers who thoroughly understand these forms of quantitative experience. Other forms of mathematical experience with which pupils should be familiarized are pubUc expenditures, tax levies, and assessments. It is imperative that the good citizen of the future understand fully the arith- metic involved in the assessment and levying of taxes, the issuance of public bonds, and pubHc expenditures. Many of the so-called "practical" courses which have been introduced in recent years have failed to emphasize 4 THE SUPERVISION OF ARITHMETIC the quantitative relationships of our social and economic life. Consequently children have been fed stereotyped problems. The child who studies arithmetic at the present time has a right to be familiarized with the quantitative problems of the day. The failure of many teachers to understand clearly the arithmetical apphca- tions in the various fields of social activity has been responsible for much of the criticism directed at the product of our schools by business men. Teachers of arithmetic have been satisfied to have the children "work the problems" in the book. The children have been satisfied to find the "answers." Neither teachers nor pupils have been forced to go to the bottom of the subject in such a way as to understand the issues in- volved. Recently an editorial writer of one of the leading American periodicals made the statement that arith- metic properly taught would make it impossible to exploit wild-cat mining schemes promising to pay fabu- lous returns. The writer said further that proper arithmetic teaching should include a thorough discussion and interpretation of sound business principles. Two Phases of Arithmetic Experimental researches in modern educational psy- chology have tended to divide arithmetic into two well- defined fields : one, that of drill wherein a child is SUBJECT MATTER OF ARITHMETIC 5 taught to add, subtract, multiply, and divide skillfully whole numbers and fractions; the other, wherein a child is taught to appreciate arithmetical situations, to realize differences in value, and to understand the prin- ciples operative, in business to-day. A clearer recog- nition of this differentiation would mean an increased emphasis on the drill operations of arithmetic, as well as a clear-cut descriptive treatment of the broader ap- plications of these habits of arithmetical procedure in concrete social and economic situations. That there is a demand for greater skill on the part of children in abihty to add, subtract, multiply, and divide is indubitably true. There is a demand for greater accuracy and speed. The supervisor will find it worth while to devise plans for an increased expendi- ture of energy on the part of the teacher in this direc- tion. We may, no doubt, expect to see an increased amount of carefully directed drill work in connection with this phase of arithmetic. On the other hand much of that part of the subject matter of arithmetic which is taught for the sake of giving the child an opportunity to understand arith- metical relationships in pubUc and private Ufe should be treated by detailed discussions setting forth the factors involved. There should be whole recitation periods during which attention should be focused en- tirely upon a discussion of material of -an explanatory 6 THE SUPERVISION OF ARITHMETIC character, and upon a critical analysis of those social relationships which involve arithmetic. For example, a teacher might spend several days in teaching the issues involved in life insurance, in taxation, or in pub- lic school expenditures. The Supervisory Problem If the results indicated in the foregoing discussion are to be attained, it will be necessary for the supervisor to assure himself that his teachers are actually familiar with the issues involved in the formation of arithmetical habits on the one hand, and with the underlying mathe- matical principles involved in the interpretation of busi- ness situations on the other. While it is true that it is a more or less difficult thing to do, yet it is by no means impossible for the super- .visor to develop an inquiring type of mind on the part of teachers, which will result in an understanding of the principles involved in social and economic arithmetical situations. Many commercial teachers visit commer- cial institutions and, indeed, not a few seek employ- ment in such institutions in order to gain a first-hand knowledge of the commercial practices involved. While it is not recommended that classroom teachers resign for the purpose of famiharizing themselves with the commercial phase of arithmetic, it is urged that they take the steps necessary to secure this knowledge. SUBJECT MATTER OF ARITHMETIC 7 Teachers might profitably interview bankers, insurance solicitors, pubUc accountants, and brokers. Now and then supervisors might profitably assemble the teachers of arithmetic for a frank consideration of such im- portant aspects of the subject. Textbook writers might wisely devote more atten- tion to these matters. At the present time many of the textbooks afford very meager descriptive material deal- ing with such activities and processes. It should be said, however, that there is already a tendency on the part of textbook writers to give more attention to these phases of the subject, and it is to be hoped that the present tendencies will be accentuated. It should be borne in mind, however, that this will not relieve the supervisor of the responsibility of utilizing every rational agency for making his teachers conscious of the funda- mental character of this problem. Nor will the text- book relieve the teacher of the same necessity of draw- ing as much of this material as possible from outside activities. The studies in elimination and retardation which have been carried on in recent years reveal the fact that a large proportion of pupils do not complete the elementary school course. In view of this condition it is of great importance that they be taught the arith- metical applications to social situations in an elemen- tary way as early as possible in their school career. 8 THE SUPERVISION OF ARITHMETIC It is certainly unwise to postpone all such instruction until the eighth grade. Attitxtoe of Superintendents In order to ascertain, something of the attitude of city superintendents toward the elimination of certain questionable subject matter and the introduction of certain other new subject matter, a questionnaire^ was sent by the authors to seventeen hundred superin- tendents. Replies returned from more than eight hundred city superintendents indicate that, at the present time, there is a well-defined disposition to- ward the elimination of certain more or less obsolete material. This attitude is expressed quite clearly in Table I. The meaning of this table becomes clear when read thus : — 72 per cent of the superintendents in cities of 100,000 and over favor the elimination of apothecaries' weight ; 75 per cent favor the elimination of troy weight ; S3 per cent of the superintendents in all cities favor the elimination of apothecaries' weight ; and 42 per ' cent favor the elimination of troy weight. 1 This investigation was made in connection with a report for the Committee of Economy of Time of the National Education Association. Parts of this material were published in the Proceedings of the N. E. A., the Elementary School Teacher, and the Fourteenth Year Book of the National Society for the Study of Education. SUBp:CT MATTER OF ARITHMETIC TABLE I Per Cent . of Superintendents Who Favor Elimination of the Topics, Distributed so as to Reveal Differences in Attitude m Large Cities, Cities and Counties, and for All Cities Topic ClTffiS OF 100,000 AND Over Cities and Counties All Cities Per Cent Per Cent Per Cent Apothecaries' weight 72 51 S3 Troy weight ... 75 40 42 Furlong \ 85 70 72 Rood in square measure 13 10 20 Dram ... 78 19 60 Quarter in avoirdupois . 81 66 68 Surveyors' tables . . . 72 43 47 Foreign money . . . 25 27 28 Folding paper 63 34 35 Reduction of more than two steps 25 21 ■ 22 Long measure of G. C. D. 66 33 35 Least common multiple . . 25 20 22 True discount 66 44 47 Cube root .... 56 41 46 Partnership . . 50 23 25 Compound proportion . . . 72 49 52 Compound and complex fractions 28 24 26 Cases in percentage 28 18 20 Annual interest . . 63 40 41 Longitude and time 13 7 8 Unreal fractions . . . 72 70 74 Alligation 88 82 85 Metric system . . . 41 19 20 Progression .... 88 63 67 Aliquot parts .... 9 19 21 Chart I shows the per cent of superintendents who favored ]the "elimination" of certain topics, or the giving of "less attention" to certain topics, or were lO THE SUPERVISION OF ARITHMETIC "satisfied" with the emphasis given to the topics. The cross-hatched portion indicates the per cent of super- intendents who favor the elimina- tion of each of these subjects. The shaded portion in- dicates the per cent of superintend- ents who favor giving less atten- tion to these sub- jects; and the re- maining portion indicates the per cent of superin- tendents who are satisfied with the present condition. It should be borne in mind, however, in interpreting this chart, that it is possible that those superintendents who are satisfied with the pres- ent degree of emphasis may have already decreased the amount of attention given to the various topics in their own schools ; Kkewise, it is clear that the use of the word "less" in this particular is a relative matter; Long, and Time Cases in fo Aliquot parts Metric System Rood L.C.1VI. Reduction Partnership Comp. Fractions Foreign Money G. C. D. Paper Folding Annual Interest Troy weight Cube Root Surveyor's Tables Discount Comp. Proportion Apothecaries wt. Dram Progression Quarter Furlong Unreal Fraction Alligation 10 20 30 40 60 CO 70 80 90 100 PER CENT Chart I. SUBJECT MATTER OF ARITHMETIC II we do not know what "less than" means. The re- sponse, however, is such as to convince a student that the superintendents of this country are not in favor of paying much attention to these topics. Graphic Representation of Eliminations Charts II and III present new distributions of the facts found in the preceding table. The two charts show thatgreatervari- ^^^^^^ ^^ ^^ eliminated ability of opinion exists among su- perintendents in cities of 100,000 and above than among the su- perintendents of the smaller cities or cities in gen- eral. They can be read easily. For example, Chart II shows that 17 per cent of the superintendents in cities of 100,000 or more favor the elimination of longitude and time, 40 per cent favor the eHmination of cases in per- centage, 10 per cent aliquot parts, 45 per cent the Long, and Time r"^ ' Cases in ^0 ^ Metric -— : >>-. L Rod ► ^ L. C. Multiple . Reduction •,A PartnershiD •■/ Complex Fractions i Foreien Monev — ' -U- Paper — — A — ~~''f — — Annual Interest :^ f Trov Weie-ht — ^~ Cube Root ^ *~^ -t Survey Tables — t •\ Discount >— -' Como. ProD'n. — ? -7 Apothecary — Class iH Proeressions ■^ / 1 — Class II- Quarter ~-i- -h Furlone > V ■ T Unreal Fractions /^ AUieration — ' — — 1 !-■■ 1 1 1 1 100 90 80 70 60 60 40 30 20 10 PER CENT Chart II. 12 THE SUPERVISION OF ARITHMETIC metric system, etc. Chart III may be read in the same way. Recommendations Because of the actual support given by the superin- tendents to the theoretical reasons for emphasizing material of dis- tinct social value, it seems safe to recorn- mend that the following sub- jects be given little or no attention in elementary courses of study in arith- metic, alliga- tion, unreal fractions, fur- longs, progression, dram, apothecaries' weight, compound proportion, true discount, surveyors' tables, cube root, troy weight, annual interest, paper folding, long method in greatest common divisor, foreign money, compound frac- tions, partnership, reduction of more than two steps, least common multiple, rod in square measure, metric system aliquot parts, cases in percentage, longitude, and time. TOPICS TO BE ELIMINATED Lone, and Time — ■^'^' Cases in % ~TT— r Aliauot ' — iiT Metric — ^=^ -— liL Rod 'r L. C. Multiple ill Reduction "T Partnership *=^ /! Complex Fractions t ;( Foreign Money -r, G. C. Divisor r rr Paper 2: Annual Interest V —h Troy Weight /,' Cube Root ; / Survey Tables s '.' Discount > / Comp. Prop'n. -t Yi Apothecary / Dram _.'^, Progressions * - — —^ ill Citie Quarter ~i 7" Furlonff • 1st Class Cities | | Unreal Fractions V 1 1 AUipration 1 1 100 90 80 70 60 50 40 30 20 10 PER CENT Chart III. SUBJECT MATTER OF ARITHMETIC 13 No doubt there are other subjects which might be added to this list. This does not necessarily mean that these topics should be left out absolutely. It does mean, however, that instruction in these fields might properly be minimized in the arithmetic of the elementary grades. Longitude and time, for ordinary purposes, might be very properly taught in connection with the general topic of railroading, wherein the different kinds of time might be explained in a purely descriptive way, so as to enable the children to understand the facts in connection with standard time throughout the different divisions of the United States. Partnership might be taught more profitably under the general head of Cor- porations in connection with the issuance of stocks and bonds. For the most part, however, little time should be given to the topics hsted above. The fact that superintendents throughout the country are dissatisfied with conditions as they exist to-day should encourage us to believe that we are about to change our practice with regard to these topics. Since a large proportion of the pupils do not com- plete the elementary school course, it is necessary that the school take cognizance of this so as to guarantee that the normal child who leaves school at an early age be proficient in the manipulation of the fundamentals of arithmetical procedure. To do this, serious attention should be given to the problem of teaching the funda- mental processes effectively. 14 THE SUPERVISION OF ARITHMETIC Topics Demanding Increased Emphasis The same superintendents who were asked to express their opinions with regard to their attitude toward the subject matter indicated above, expressed themselves as overwhelmingly in favor of giving more attention to the fundamentals of addition, subtraction, multiphca- tion, division, and fractions. The replies of the superintendents favoring increased emphasis are presented in tabulated form in Table II. TABLE II Percentage or Superintendents Who Favor the Plan of Giving More Attention to the Topics Listed, Distributed so as to Reveal Differences in Attitude in Large Cities, Cities and Counties, and for all Cities Topic Cities of 100,000 AND Over Cities and Counties All Cities Addition Subtraction Multiplication Division .... Fractions Percentage Interest Saving and loaning money . Banking Borrowing Building and loan association , Investments Bonds and stocks . Taxes Levies . Public expenditure Insurance Profits . . , Public utilities 59 SO S9 S6 S6 31 25 SO 38 22 \ 13 16 9 2S 6 28 31 28 34 75 68 72 69 66' SI 41 61 40 37 46 44 20 S3 36 54 S4 47 57 7S 69 72 70 65 SO 39 61 39 37 48 44 20 53 35 55 5S 46 57 Chart IV presents these data in graphic form. SUBJECT MATTER OF ARITHMETIC 15 In view of the theoretical advantage of increasing the emphasis upon these subjects and because it corre- sponds with the experience of the superintendents of this country, it is recommended that additional em- phasis be placed upon the following subject matter in arithmetic courses of study : addi- tion, multiplica- tion, subtraction, division, frac- tions, saving money, public utilities, public expenditures, in- surance, taxes, percentage, prof- its, building and loans, invest- ments, interest, banking, borrow- ing, levies, stocks and bonds. Additional subjects might be added to this Kst, and no doubt will be added from time to time. This does not mean that the school should give necessarily the same amount of time to all these subjects. It is moreover conceded that certain of the subjects might be properly united. For instance, pubHc expenditures, pubUc utilities, stocks and Bonds Levies Borrowing Banking Interest Investments Building and Loan Profit Percentage Taxes Insurance Public Expenditure Public Utilities Saving Money Fractions Subtraction Division Multiplication Addition 10 20 30 40 50 60 70 I PER CENT Chart IV. l6 THE SUPERVISION OF ARITHMETIC taxes and levies might be handled all under one head, such as public finances. The important point, however, is that children should be taught to understand the issues in- volved in the expenditure of pubhc money, the purchase of pubhc utilities, and the' levying of taxes. Since the school is to train pubhc officials and the tax payers of the future, it is important that an increased amount of attention be given to such topics as these. Saving money might be handled in connection with banking, building and loans, insurance, or in connection with problems of economical purchasing or expenditures. The point of vital importance is that children under- stand the conditions involved in these various activities. It should be noted that the conditions under which children are taught addition, multiplication, subtrac- tion, and di\dsion are different from those under which they are taught the arithmetical applications. The method in the one is that of intelligent drill ; the method in the other is that of intelhgent appreciation, detailed description, definite clearing up of points. Van Houten's Investigation As a check upon the returns secured by the question- naire and also for the purpose of presenting a more de- tailed analysis of the topics whose retention is regarded as questionable, we include the results of an investiga- tion conducted by Mr. L. H. Van Houten. SUBJECT MATTER OF ARITHMETIC 17 TABLE III Elimination Specifically Mentioned in 148 Coitrses op Study Subject Annuities .... Average of accounts . Average of payments Denominate numbers : Compound numbers Apothecaries' weight Denominate fractions Paper measure . . Surveyors' measure . Troy weight . . . Terms : Bale . Bundle Chain Dram Echo Furlong Hand Link Quarter, avoirdupois Quintal Rood .... Sign Decimals, circulating . ' Duties and customs Equation of payments Exchange : BiUsof Foreign . . Domestic . . ... Foreign money . . ... Fractions : Comparison Complex except in simple form Compound ... . . Government land ... i8 THE SUPERVISION OF ARITHMETIC TABLE 111 — Continued Sdbjeci Interest : Annual All but 6 % method .... Compound Exact . . . . . . Interest commission . . . . Present worth .... . . . Bank discount . . . . . True discount Partial payments ... . . . . Except U. S. rule .... Except two indorsements Insurance : Life Fire Involution and evolution . . . . Cube root ... ........ L. C. M. and G. C. D. except by factor Longitude and time Standard time Measurements : Convex surface of and volume of, cone, sphere, pyramid. Circular measure Painting . . Papering . . Plastering . . . Roofing ... .... Sphere Rhomboid and rhombus Trapezoid and trapezium . . . Round log . . . Surface measures Markings . . . Metric system . ... Notation, scales of Partnership . With time . Progressions . . . ... SUBJECT MATTER OF ARITHMETIC 19 TABLE III — Continued Subject Cases Proportion : Compound . .... Partitive . .... Percentage : Commission and brokerage . . . Finding B when R and P are given Trade discount Stocks and bonds . ... Taxes PoU Temperature Specific gravity Reinvestment and net proceeds Casting out nines . Savings bank accounts . Optional subjects : Investments Ratio .... Square measure . . Tariff . . ... 13 3 Note. — Mr. L. H. Van Houten, Superintendent o£ Schools, Toledo, Iowa, made an intensive study of one hundred forty-eight courses of study in arithmetic and then checked his inferences by a questionnaire investi- gation. His results were accepted as a Master's thesis at the University of Iowa, and are on file in the library of that institution. Mr. Van Houten has granted the authors the privilege of drawing heavily upon his material. A comparison of the results secured by the two in- vestigations warrants the conclusion that the course of study makers are slightly more progressive when they consider elimination impersonally than when they are pubhshing courses of study and reports for their con- 20 THE SUPERVISION OF ARITHMETIC stituencies. While this Gomparison of topics has no particular scientific merit, it does reveal a situation full of human interest. It also confirms the impression secured from the former study that a simplification of texts and courses of study in arithmetic is demanded. Summary Summarizing, it should be said that the present tendency among superintendents is in favor of the ehm- ination of questionable matter in the field of arith- metic. While as yet there is not perfect agreement as to just what all of this questionable subject matter is, there is rather clear agreement in regard to certain subjects, such as apothecaries' weight, troy weight, fur- long, rood in square measure, dram, quarter in avoir- dupois, surveyors' tables, foreign money, folding paper, reduction of more than two steps, long measure, greatest common divisor, least common multiple, true discount, cube root, partnership, compound proportion, compound and complex fractions, annual interest, longitude and time, unreal fractions, alhgation, metric system, progression, ahquot parts. Again, the attitude of the superintendents indicates a tendency to give increased attention to the following topics : addition, subtraction, multiplication, division, fractions, percentage, interest, saving and loan- ing money, banking, borrowing, building and loan asso- ciations, investments, bonds and stocks, levying of taxes, pubhc expenditures, insurance, profit, public utihties. CHAPTER II THE GRADE DISTRIBUTION OF THE DIFFERENT ARITHMETICAL TOPICS Factors Influencing Choice and Sequence of Topics A STUDENT of the courses of study in arithmetic is immediately impressed with the fact that wide varia- tion exists in grade distribution of the topics. The choice and sequence of topics may be determined by the mental maturity of the child. Or the choice and se- quence may be determined by the logic of the case; that is to say, there may be certain topics in arith- metic which should precede others, because they are preparatory and basic to these later topics. A third factor in the determination of the sequence of the topics in arithmetic may be that of the social and economic conditions affecting the school. For example, while it may be desirable, so far as the maturity of the child is concerned, for us to postpone certain business applica- tions of arithmetic until the later years of the child's school hfe, we are compelled, nevertheless, to take into consideration the fact that large numbers of children leave school without completing the elementary course; _ 2 2 THE SUPERVISION OP ARITHMETIC and consequently it becomes necessary for us to choose between having the child study certain business applica- tions at an early date, or not get them at all. The supervisor is interested in knowing the adjust- ment the American schoolmaster has made to these different factors. To what extent has he recognized the Hmitations of immaturity? To what extent has he accepted the logical demands of the subject matter? To what extent has he recognized the necessity for an early introduction of the social and business applica- tions of arithmetic? Investigations Three important investigations have been made in this field with a view toward finding out the adjustment which the school has made in actual practice. Dr. Bruce R. Payne in his study of the elementary school curriculum in 1906 reported on the grade distribution of twenty-nine topics in ten American cities. The Balti- more School Commission in 1911 compared the poHcy of thirteen American cities with reference to seven selected phases of arithmetic work. Mr. L. H. Van Houten's investigation covers one hundred forty-eight courses of study distributed throughout the United States. The following table based on the study of Dr. Payne shows the grade distribution in ten American cities.^ ' Payne, Elementary School Curriculum, Silver, Burdett & Co. DISTRIBUTION OF THE ARITHMETICAL TOPICS 23 TABLE IV Table Showing Twenty-nine Topics in Arithmetic and Their Distribution by Grades in the Public Elementary Schools OF Ten American Cities (Payne) Grade I II III IV V VI VII VIII Numeration . . . . 10 10 10 8 s 3 I Notation . 10 10 10 8 5 3 I Relation of numbers ... 7 4 3 I I Addition . . . . 8 9 10 8 3 I Subtraction ... . . S 9 10 8 3 I Multiplication . . . . 2 7 7 8 4 I Division . . . . . . 2 S 6 8 6 3 Fractions 3 4 6 8 10 9 3 4 Denominate numbers C ,5 4 9 7 10 6 6 Involution and evolution . I 3 2 2 9 Decimal fractions 4 8 7 3 Mensuration . . ... I 2 2 3 7 Multiplication tables 4 5 4 I I Commission and brokerage S Insurance . . . . 5 I Percentage . . . I 2 5 7 5 Ratio and proportion . ... I 3 S Partnership ... 2 4 Partial pajonents 2 4 G. C. M. and L. C. M. 2 5 Longitude and time . . . . 2 2 Profit and loss 4 I Taxes . . . . 5 Duties . . . . I Banking ... 7 I Exchange ... 2 2 Simple interest I 3 7 5 Stocks and bonds 3 I Business forms I 3 ' I It will be observed that wide overlapping existed. For instancef, numeration and notation were taught in seven grades. Fractions and denominate numbers were taught in every grade. Addition, multiplication, division. 24 THE SUPERVISION OF ARIXflMETIC and subtraction were taught in six grades. All ten of the cities agree that numeration and notation should be taught in the first three grades ; addition and subtraction in the third grade ; fractions in the fifth grade ; and de- nominate numbers in the sixth grade. The figures reveal a great disparity of opinion in regard to the other topics. The following table shows the information gathered by the Baltimore Report.^ TABLE V The Year op the Cottrse in Which Specified Topics in Arithmetic Are Treated in the Certain Cities (Baltimore Commission) < , ,g g H tl s 1 1 Cities s^ Stag i ii r i-i r 6g^ h Nfew York . 2 3 3 4 5 S 6 Chicago .... 2 4 4 S S 6 6 PhUadelphia . 2 2 3 S S 6 6 St. Louis . . 2 3 4 3 4 4 S Boston . . 2 4 4 S 6 5 6 Cleveland . . 2 4 4 S 6 S 6 Baltimore . . 2 3 4 3 4 4 6 Pittsburgh . 2 3 4 4 S S 6 Detroit . . 2 3 4 4 S S 6 Buffalo . . 2 • 3 4 S 5 S 6 San Francisco 2 3 4 4 S 4 6 Milwaukee . 4 4 4 S S 6 7 Cincinnati 2 3 3 4 5 4 6 ^ Report of the Commission Appointed to Study the System of Educa- tion in the Public Schools of Baltimore. Bureau of Education Bulletin, igii, No. 4, p. 76. DISTRIBUTION OF THE ARITHMETICAL TOPICS 25 The Commission drew the following conclusions from the foregoing table: "From the above table it appears that the forty-five combinations in arithmetic are learned in all but two of the cities in the second grade. The most common grade in which the multiplication tables are learned is the third. Long division is taught most conimonly in the fourth grade ; addition and subtrac- tion, multiplication and division of fractions, and decimals are taught most commonly in the fifth grade ; and percentage in the sixth." Addition and subtraction of fractions, however, are taught in these cities in grades as low as the third and as high as the fifth ; multiplication and division of frac- tions appear in grades as low as the fourth and as high as the sixth ; decimals are taught in grades as low as the fourth and as high as the sixth. The investigation of Dr. Payne and the Baltimore Commission related to the grade distribution of topics in a small group of large cities only. The most thor- oughgoing investigation of the problem was made by Mr. L. H. Van Houten in 1913. The Van Houten Investigation Mr. Van Houten based his report upon one hundred and forty-eight courses. Cities ranging in population from III 8 to that of New York City are included in his report. These cities are located in thirty-seven 26 THE SXJPERVISION OF ARITHMETIC states and in the District of Columbia. State courses of study were obtained from nearly every state and were used when referred to by the city courses. Some of the topics selected were chosen on account of their commonly accepted importance and othters because of the discussion which has arisen concerning their eHmination. Care was taken to verify all the material so far as possible by consulting the texts in use when the course was not sufficiently specific. In many cases the text was the sole guide, since only text assignments were given. TABLE VI Frequency Table Showing Grade Occurrence of Arithmetic Topics in 148 Cities ( Van Houten) Topic Notation Niuneration Addition Subtraction . . . . Multiplication . . . . Division Factoring Cancellation Fractions . . Denominate numbers . . Involution and evolution Decimal fractions . . . Mensuration . . . Multiplication tables . . Grade 12 3 4 lOI 108 98 38 28 83 126 I2S 148 143 108 97 no 94 139 139 148 146 147 146 3 17 124 102 90 132 129 132 145 I4S 145 145 30 62 138 97 42 93 92 96 97 117 125 117 117 65 43 147 103 2 121 92 13 6 7 8 9 S8 S8 90 88 90 91 35 3 104 121 2 122 121 4 17 18 39 10 41 41 2 43 34 123 27 96 3 Modal Grade 3 3 3>4 3 3 3 S 4 5 6 DISTRIBUTION OF THE ARITHMETICAL TOPICS 27 TABLE VI — Continued Grade Topic MODAI. Grade 1 2 3 4 5 I 6 43 7 86 8 36 9 Commission and brokerage . 7 Insurance .... I 6 84 52 7 Percentage . . . . I 12 42 105 108 62 7 Ratio and proportion . . . 2 7 21 53 91 3 8 Partnership . 10 44 3 8 Partial payments . . I 33 60 3 8 G. C. D. and L. C. M. I I 6 8 Longitude and time . . I 15 ■34 51 I 8 Profit and loss . . . . I 49 90 20 7 Taxes I 7 83 55 I 7 Duties I 67 52 I 7 Banking ... 23 58 2 8 Exchange . . . 28 72 2 8 Simple interest . ... 6 9 72 117 61 I 7 Stocks and bonds . . 2 28 73 2 8 Business forms ... 2 39 62 66 44 30 6 Simple accounts . . I IS 31 39 37 30 I 6 U. S. money ... 48 54 99 85 71 45 16 5 3 Approximations 4 II 2 6 Commercial discounts 7 10 46 82 46 I 7 Bank discount . . . I 12 SI 67 3 8 Analysis . . ... 9 IS 59 '58 7 Metric system . . . I 2 2 22 80 8 Note. — This table should be read as follows : Notation is taught in the first grade in 98 cities, in the second grade in 1 26 cities, etc. From the foregoing it may be seen that wide varia- tion exists in practice — in fact addition is the only topic reported in all of the cities in either the second or third grade. Subtraction, multiplication, and division are almost uniformly taught in the third and fourth grades. Only eleven topics are taught in the first and 28 THE SXIPERVISION OF ARITHMETIC second grades in any of the cities,, while almost one third of .the cities did not teach any of the topics in these grades. Although .five additional topics are taught in a few schools in the third grade, the greatest agreement is on teaching notation, numeration, addition, subtraction, multiplication, diAdsion, fractions, denominate numbers, multiplication tables, and United States money. In the fourth and fifth grades there is a decided in- crease in the number of cities teaching factoring, can- cellation, decimal fractions, simple accounts, and busi- ness forms. The upper grades show a marked tendency to select fewer of the formal topics and more of the topics which have to do with the apphcation to social and economic situations. Perhaps the most striking fact of the table is that there seems to be so Httle agreement in grade distribu- tion of topics. Such variation suggests the desirability of our knowing more about the relative success of the different schemes of distribution. Summary The following is a summary showing the modal grade in which the various topics are taught : Grade I. Variation is such that no mode appears. Grade II. Variation is such that no mode appears. DISTRIBUTION OF THE ARITHMETICAL TOPICS 29 Grade III. Grade IV. Grade V. Grade VI. Grade VII. Notation Numeration Addition Subtraction Multiplication Division Multiplication tables U. S. money Addition Cancellation Factoring Fractions Decimal fractions Denominate numbers Mensuration Business forms Simple accounts Commission and brokerage Insurance Percentage Profit and loss Taxes Duties Simple Interest Commercial discounts Analysis 3° THE SUPERVISION OF ARITHMETIC Grade VIII. Involution and evolution Ratio and proportion Partnership Partial payments G. C. D and L. C. M. Longitude and time Banking Exchange Stocks and bonds Bank discounts Metric system It is thus seen that the four fundamental processes are most frequently taught in the first four grades. Factoring is taught most frequently in the fifth grade. Cancellation is taught most frequently in the fourth grade. Fractions are taught most generally in the fifth grade. Decimal fractions are taught most frequently in the fifth and sixth grades. Denominate numbers and mensuration are taught quite generally throughout the whole course of eight grades, although there is a tendency for this material to be taught in the sixth grade or lower. Involution and evolution are taught predominately in the eighth grade. DISTRIBUTION OE THE ARITHMETICAL TOPICS 3 1 Percentage and its applications appear most frequently in the sixth and seventh grades. The metric system is taught most generally in the eighth grade. Grade Occurrence of Seven Specified Topics Table VII was made by taking the seven specified topics used by the Baltimore Commission's report. It supplements the information given in the preceding table, by telUng definitely where emphasis is placed on a subject. In this table no grades are mentioned unless the topics receive their principal treatment there. In cities where two grades seemed to be of equal impor- tance, both are given. As found by the Baltimore Commission in a study of the courses of ten cities, the forty-five combinations are completed by a majority of schools in the second grade. About ten per cent complete these in the third grade and only four per cent in the first grade. The mul- tiphcation tables are generally completed in the third grade, though there is greater variation than in the com- binations. The fourth grade is the grade of long divi- sion. About twenty per cent of the schools teach the topic in the third grade, and three per cent in the fifth. Formal fractions are taught in the fifth grade by the majority of schools. Twenty-eight schools teach addi- tion and subtraction of fractions and multipHcation and 32 THE SUPERVISION OF ARITHMETIC •division in different grades. In nineteen of .these cases the first two processes are taught iij the fourth grade and the latter in the fifth. Decimal fractions likewise have the fifth as the modal grade, though over forty per cent of the schools offer the subject in the sixth grade. In seventeen cities they are continued through two grades. Grade six is the time when the principles of percentage are studied. The following table shows the practice in the cities studied : TABLE VII The Year op the Coxjuse in Which Seven Specified. Topics in Arithmetic are Treated (Van Houten) Cities Aberdeen, S. Dak Akron, Ohio . Albia, Iowa . Altoona, Pa. . Ansonia, Conn. Appleton, Wis. Astoria, Ore. Athens, Ga. . Atlanta, Ga. Atlantic, Iowa Baraboo, Wis. Belleville, 111. Berkeley, Cal. Birmingham, Ala. Boise, Idaho . Boone, Iowa Boston, Mass. Boulder, Colo. Topics S 6 6 6 S S S S S 7 S&6 S s s 6 s 5 5 DISTRIBUTION OF THE ARITHMETICAL TOPICS 33 TABLE VLl — Continued Topics Cities 1 2 3 4 6 6 7 Bowling Green, Ky. 2 3 4 4 S&6 6 7 Bradford, Pa. . 2 3 4 S 5 s 7 Brockton, Mass. ? 3 4 S 6 6 7 Burlington, Iowa 2 3 4 S S 5 7 Cambridge, Mass. 2 3 4 4 4 5 6 Canton, Ohio . 2 4 S S 6 6 7 CenterviUe, Iowa 2 3 3 s s s 7 Chester, Pa. 2 4 4 4 S S 6 Chicago, 111. 2 3 5 5 s 6 7 Cheyenne, Wyo. 2 3 4 S s S 6 Cincinnati, Ohio 2 3 4 S 6 6 7 Cleveland, Ohio . . 2 4 4 5 s 6 7 Columbus, Ga. . 2 3 4 S s s 6 Connersville, Ind. 2 . 4 4 5 5 s 2 Cortland, N. Y. 2 3 4 5 5 s 6 Covington, Ky. 2 3 4 5 S 7 7 Crawfordsville, Ind 2 4 4 S 5 s 6 Danbury, Conn. 2 3 4 5 5 5 6 Davenport, Iowa 2 3 3 S 5 S 6 Detroit, Mich. . 2 3 4 5 S S 6 Dover, N. H. . 2 4 4 S 5 5 6 Dubuque, Iowa . I 3 4 4 S 5 6 Dunkirk, N. Y. . I 3 3 4 S 5 6 Eau Claire, Wis. . 2 4 4 S s S 6 Elgin, lU. ... 2 4 4 S 5 5 7 Englewood, N. J. . 2 ' 3 3 4 4 S 6 Enid, Okla. . . . 2 3 4 5 5 S 6 Erie, Pa, .... 2 3 4 4 4 5 7 Everett, Wash. 2 4 4 5 S 6 6 Fort Smith, Ark. . 2 3 4 5 5 S 6 Fort Wayne, Ind. . 3 ' 4 4 5 S 6 6 Frankfort, Ky. . I 2 3 4 5 5 7 Freeport, 111. 3 3 4 S S 5&6 6 Fresno, Cal. . . . 3 4 5 5 S S&6 7 Fulton, N.Y 2 3 4 S 5 S 6 Galesburg, HI. 2 3 4 5 5 S S 34 THE SXJPERVISION OF ARITHMETIC TABLE Wll — Continued Cities Topics 1 2 3 4 5 6 7 GloversviUe, N. Y. . . . 2 3 4 6 6 6 7 Grand Junction, Colo. 2 4 4 S S S 6 Guthrie, Okla 2 4 4 5 S S 6 Harrisburg, Pa 3 3 4 4 4,S&6 5 8 Hartford, Conn . . 2 3 4 S S 6 6 Indianapolis, Ind. . . . 3 4 4 S S 6 6 Ironwood, Mich. . . . 3 4 4 S 5 6 5&6 Jamestown, N. Y. . 2 3 4 5 6 6 7 Jefferson, Iowa . . . 3 4 4 S S 6 6 Jersey City, N. J. . . . 2 3 3 5 S,6&7 S 6 Johnstown, Pa 3 4 4 5 5 S 6 Lancaster, Ohio . . 2 4 4 S 5 6 6 Lansing, Mich. . . . 2 3 4 4 4,S&6 4&S 4&S Laramie, Wyo 2 4 4 S S S 6 Lincoln, Neb. . . 2 4 4 S 5 6 6&7 Long Beach, Cal. . . . 2 4 4 S 6 S 7 Los Angeles, Cal. . 3 4 4 S 6 S 7 Lynn, Mass. . . 2 3 S S S 6 6 Madison, Wis 2 4 4 S S 6 6 Manchester, N. H. . . 2 3 4 4 S 6 6 Manistee, Mich, . . . 2 4 4 S S 6 7 Marengo, Iowa .... 2 3 4 S S S 6 Marion, Ind 2 4 4 6 6 6 7 Mason City, Iowa . . . 2 3 4 S S 5 6 Memphis, Tenn. . . 2 3 4 4 S 6 Menominee, Mich. . . . 2 3 3 4 4 4&6 6 Milwaukee, Wis. . . . 2 3 4 S S 6 7 Minneapohs, Minn. 2 4 4 S 5 6 6&7 Monessen, Mass. . . 2 3 4 S S S 6 Muscatine, Iowa . . . 2 4 4 s S 6 6 Muncie, Ind. . . . 2 4 4 5 S S 7 Muskogee, Okla. . . . 2 4 4 6 S&6 6 6&7 Norfolk, Va. . . 2 3 4 6 6 6 7 Nashville, Tenn. . . . 2 3 4 4 4 S S New Hampton, Iowa 2 4 3 S S S 6 New Haven, Conn. . . 2 4 4 S S S 6 DISTRIBUTION OF THE ARITHMETICAL TOPICS 35 TABLE Vll — Continued Cities Topics 1 2 3 4 5 6 7 Newton, Mass. . 2 3 4 4&S 6 S,6&7 7 New York, N. Y. . . . 2 4 4 S 5 S 6 Niles, Ohio .... 2 4 4 s S 6 6 Oakland, Cal. . . 3 4 s S 5 6 Oklahoma City, Okla. . . 2 4 4 5 5 5 6 Glean, N. Y 2 3 3 5 S S 6 Ownesboro, Ky 2 4 5 5 S 5 Paterson, N. J. . . . 2 3 4 S 5 6 7 Pensacola, Fla. . . 2 4 4 5 5 S 7 Philadelphia, Pa. . . 2 3 3 5 5 6 6' Phoenix, Ariz. 2 4 5 6 6 6 7 Piqua, Ohio . . , 2 3 3 4 5 S 6 Plainfield, N. J 2 3 4 4 5 s 5 Pomona, Cal 3 3 4 5 5 s 6&7 Portland, Me. ' . . 2 3 4 4 5 5 6 Providence, R. I. . . 2 3 4 5 5 ■ 6 7 Raleigh, N. C. 2 3 3 4 4 5 6 Reading, Pa. . . . . 2 3 3 S S 6 6 Reno, Nev. . . . 2 3 3 4 5 6 6 Richmond, Va. . 2 3 4 5 5 6 6 Riverside, Cal. 3 4 4 5 S 6 7 Roanoke, Va. . . 2 3 3 4 4 3&4 6 Saginaw, Mich. . 2 3 4 5 5 5&6 7 Salem, Ore 2 4 4 5 6 5&6 7 Salt Lake City, Utah. . 2 3 4 S S 6 6 San Francisco, Cal. . . 2 3 3 4&5 5 6 6 San Jos6, Cal. . . 2 3 4 5 S 6 7 Sault Ste. Marie, Mich. . 2 4 4 S 5 5 6 Schenectady, N. Y. . . 2 3 4 5 S S 6 Scranton, Pa. 2 , 3 4 S 6 6 7 Sedalia, Mo. 2 3 4 S 5 S 6&7 Shamokin, Pa. . . 2 2 3 4 4 5 6 Sheboygan, Wis. 2 3 3 4&S 5 S 6 Spokane, Wash. . . 2 3 4 4 5 S 6 Springfield, Mass. . 2 4 4 4 S 5 5&6 Springfield, Mo. I 2&3 3 4 4 S 6 36 THE SUPERVISION OF ARITHMETIC TABLE Vll — Continued Cities Springfield, Ohio St. Joseph, Mo. . St. Paul, Minn. . Superior, Wis. -Syracuse, N. Y. . Tacoma, Wash. . Terra Haute, Ind. Tulsa, Okla. . . Vincennes, Ind. . Washington, D. C. Watertown, N. Y. Wausau, Wis. Webb City, Mo. West Chester, Pa. Winston, Pa. . . Worcester, Mass. Utica, N. Y. . . York, Pa. . . . Topics S&6 6 6 S&6 S 6 6 S 5&6 6 S 6 S&6 S 6 S s 6 5&6 6 6 6 6 6 7 6 6 7 6 7 6 6 7 6 6 6 The foregoing table should be read thus : In Aberdeen, South Dakota, the forty-five combinations are completed in the second grade ; the multiplication tables are learned in the third grade. Long division is taught in the fourth grade. Addition and subtraction of fractions are taught in the fifth grade. Multiplication and division of fractions are taught in the fifth grade. Decimals are taught in the fifth grade; percentage is taught in the sixth grade. Column i represents the forty-five combinations learned; 2, Multiplication tables learned; 3, Long division taught ; 4, Addition and subtraction of fractions taught ; 5, Multiplication and division of fractions taught ; 6, Decimals taught; 7, Percentage taught. DISTRIBUTION OF THE ARITHMETICAL TOPICS 37 The table below is a summary of the foregoing and shows the prevailing practice in these particulars. TABLE VIII Feequency Table Showing Grade Occukrence of Seven " Specified Topics Cities Grade 1 2 3 4 5 6 13 18 66 93 7 8 3 46 8 Forty-five combinations completed Multiplication tables com- pleted Long division taught . . Addition and subtraction of fractions taught . . Multiplication and division of fractions . . Decimals taught . . . Percentage taught . . . S 122 3 14 26 8 53 107 29 S 108 120 9 2 Summary In summarizing this, Mr. Van Houten says : "The forty-five combinations are completed by a majority of schools in the second grade. About ten per cent complete these in the third grade and only four per cent in the first grade. The multiplication tables are generally completed in the third grade, though there is greater variation in this particular than in the completion of the forty-five combinations. The fourth 38 THE SUPERVISION OF ARITHMETIC grade is the grade of long division. About twenty per cent of the schools teach the topic in the third grade, and three per cent in the fifth. Formal fractions are taught in the fifth grade by the majority of schools. Twenty-eight schools teach addition and subtraction of fractions and multipUcation and division of fractions in different grades. In nineteen of these cases the first two processes are taught in the fourth grade and the latter in the fifth. Decimal fractions likewise are taught in grades five, six, and seven, yet the fifth grade is pre- dominantly the grade in which decimal fractions are taught. Although percentage appears in the sixth, seventh, and eighth grades, and in some cases as low as the fifth, yet the sixth grade is the most frequent grade in which percentage is taught." CHAPTER III TIME ALLOTMENT FOR ARITHMETIC Varying the Emphasis upon Arithmetic Arithmetic has been held in varying degrees of esteem in the past. At certain stages of civilization little at- tention was given to the quantitative interpretation of the environment of the individual. The early Massa- chusetts enactment of 1647 in regard to education ordered that reading and writing schools be established in every community of fifty householders. No doubt a certain amount of arithmetic was taught in these early reading and writing schools, yet the facts indicate that arith- metic when taught at aU was taught incidentally. No formal recognition of the subject was given in connec- tion with the time allotment in the daily program. Even though arithmetic was included in the "Seven Liberal Arts, " it should be borne in mind that, as a sub- ject for elementary education for the masses, it was not considered of importance until relatively late. The demand for a knowledge of arithmetic came with the development of commerce in New England in the latter 39 40 THE SUPERVISION OF ARITHMETIC part of the eighteenth century and with the expansion of commercial activity in the nineteenth century. The early grammar school, both in New England and elsewhere in this country, gave arithmetic a place. The arithmetic of the grammar school period was of an elementary character, and was not infrequently differ- entiated into several parts. This tendency to differ- entiate the subject matter of arithmetic into various parts was even more strikingly emphasized during the "academy period" (1780-1835), when courses were offered in navigation, accounts, bookkeeping, and the like. The academy gave the subject a definite allot- ment of time. With the establishment of high schools in Boston, New York, and elsewhere, arithmetic received definite recognition. It was included in the list of subjects for which provision was made by the Massachusetts law of 1827 for all communities of fifty or more famihes or households, and from this time on it was almost invariably included in the curriculum. Not only was arithmetic taught in what we would call to-day the elementary schools, but it was included in the curricu- lum of the Boston Latin School, the Boston English High School, the Boston Girls' High School, and the Leicester Academy. Arithmetic was required in all elementary schools in the state of JNIassachusetts by the law of 1789. The TIME ALLOTMENT FOE ARITHMETIC 41 fact is that with the rise of the academy and the high school it received definite recognition in the daily pro- gram of almost every school. With the establishment of the "Common Schools" throughout the United States, which accompanied the development of the West, arith- metic was included as an essential part of every pubhc school curriculum. While legislation and public opinion both responded to the social demands that arithmetic should be taught, it remained for the schoolmaster to determine the amount of time it should receive. As the general pub- hc from time to time placed new emphasis upon the arithmetical attainments of the pupils, the school- master tended to devote more and more time to the subject, with the belief that increased proficiency de- pended upon an increased amount of time being allotted to the subject. Thus the popular criticism to the effect that the students were inefficient in arithmetic resulted in many cases in an increase in the number of school years in which arithmetic was studied. This adjustment has been questioned recently. ,,A few years ago Mr. J. M. Rice undertook to analyze the conditions under which arithmetic was being taught. He found that wide variation existed in the amount of time which cities were devoting to the subject. After analyzing the results of a somewhat elaborate series of tests which he gave to elementary school children in a 42 THE SXIPERVISION ^OF ARITHMETIC number of cities, he arrived at the following conclusions : "The amount of time devoted to arithmetic in the school that attained the lowest average, twenty-five per cent, was practically the same as in the one where the highest average, eighty per cent, was obtained. In the former the regular time for arithmetic in all grades was forty-five minutes a day, but some additional time was given. In the latter the time varied in the differ- ent grades, but averaged fifty-three minutes daily. The schools showing the most favorable results cannot be accused of making a fetish of arithmetic. These state- ments are further justified by the fact that the four schools which on the whole stood highest gave practically the same amount of time to arithmetic as the schools which stood lowest." ^ Dr. C. W. Stone ^ later conducted an investigation on "Arithmetical abilities and some of the factors deter- mining them," partly for the purpose of determining whether the amount of time given to the arithmetic recitation was a factor in arithmetical efficiency. He found among the thirteen schools that devoted less than the median time to the arithmetic recitation that the work was slightly better than in the thirteen schools which devoted more than the median time. Dr. Stone says : "What is claimed is that as the present practice ' Rice, Scientific Management in Education, p. 119. ' Stone, Arithmetical Abilities, p. 62. TIME ALLOTMENT FOR ARITHMETIC 43 goes, a large amount of time spent on arithmetic is no guarantee of a high degree of efficiency in arithmetic. If one were to choose at random among the schools with more than the median time given to arithmetic, the chances are about equal that he would get a school with inferior products : and conversely, if one were to choose among the schools with less than the median time given, the chances are about equal that he would get a school with a superior product. in arithmetic." While the foregoing statements do not dispose of the debatable points as to the amount of time which may be given profitably to arithmetic in the public schools, they do tend to shake our faith in the belief that ia- creased arithmetical efficiency is a function of time ex- penditure. Experimental education will no doubt have additional contributions to make to this problem within a comparatively short time. In the meantime, how- ever, the school superintendent who wishes to profit by the experience of other superintendents is interested in the policy of schools throughout the country with refer- ence to this matter. One of the important contributions to this problem has been made by Dr. Bruce R. Payne. Dr. Pa3aie undertook to ' discover the relative amount of time which was being given to arithmetic recitations in the leading cities in this country in 1904. He collected and collated data with regard to the percentage of recitation 44 THE SUPERVISION OF ARITHMETIC time given to each of the subjects in the elementary school curriculum in the following cities : New York, Boston, Chicago, Cleveland, San Francisco, Columbus, Ga., Louisville, Jersey City, New Orleans, and Kansas City, Kan. The following table shows the average per cent of recitation time given to arithmetic in each grade in these cities : TABLE IX I II III IV V VI VII VIII Average Arithmetic . 13.6 iS-4 18.2 18.0 17.6 18.3 17.7 17.0 16.97s He found that arithmetic was second only to reading and Kterature in the amount of time expenditure in the daily programs of these cities. In the investigation of this same topic in 191 1 by the committee appointed to study the system of education in the pubKc schools of Baltimore, of which Dr. E. E. Brown, then commissioner of education of the United States, was chairman, it was discovered that there was a wide variation in the total amount of time given to the study of arithmetic and algebra in New York, Chicago, Philadelphia, St. Louis, Boston, Cleveland, Baltimore, Pittsburgh, Detroit, San Francisco, Mil- waukee, and Cincinnati. The range of variation in recitation time per week was from 11 25 minutes in St. TIME ALLOTMENT FOR ARITHMETIC 45 Louis to 2080 minutes in Cincinnati. The percentage of school time devoted exclusively to the study of arithmetic and algebra in these cities varied from 10 per cent in Chicago to almost 19 per cent in Cincinnati. This variation, in view of the findings of Mr. Rice and Mr. Stone, suggests the probabiUty that the time given to arithmetic is not being used to an equal advantage in all these cities. It is possible that Chicago, which is giving only 10 per cent of its time to arithmetic, is get- ting just as good results as Cincinnati, which is giving almost twice as much time to arithmetic. The material which is presented in the following tables indicates that these cities have been doing more or less experimenting with this important problem. TABLE X Per Cent of Recitation Time Given to (Payne) Arithmetic 1888 1904 1888 1904 New York . . . Boston .... Chicago . . . 26.2 16.6 9-3 12.0 16.2 18.6 St. Louis .... Louisville . . . 19-3 16.7 15.2 17.2 It should be noted in the above that New York and St. Louis in the period from 1888 to 1904 made definite attempts to decrease the relative time given to arith- metic recitations, while Chicago increased the percent- 46 THE SUPERVISION OF ARITHMETIC age of time. Boston and Louisville remained practically the same. The Baltimore Commission furnished the following table, which shows conditions in 1890 and 1910 and 1911 : TABLE XI Per Cent of Recitation Time Given to Abithmbtic (Baltimore Survey) 1890 1910-11 1890 1910-11 New York . . . 26.2 134 Detroit . . . 17.2 16.0 Boston .... 16.6 iS-S San Francisco . 14.0 16.6 Chicago . . . 9-3 lO.O Milwaukee . . iS-5 14.7 St. Louis . . 193 15.0 Cincinnati . . 134 18.8 Cleveland . . 14.1 iS'S Average . . 16.51 15-38 Baltimore . . . 19s 18.3 It is interesting to note in the above that more than one half of the cities reduced the per cent of time expenditure during the twenty years included in this comparison. New York, which in 1890 was giving 26.2 per cent of her total time to arithmetic, dropped back to 13.4 per cent in 1910. The average time given by the ten cities in 1890 was 16.51 per cent, and in 1910-11 it was 15.38 per cent. The situation in foreign countries does not differ greatly from the situation m America. Dr. Payne col- lected data from ten leading cities in England, ten in Germany, and ten in France, showing an average per cent of time given to the recitation of arithmetic. TIME ALLOTMENT FOR ARITHMETIC 47 TABLE XII Grades I II III IV V VI VII VIII Averages England . Germany . France 19.8 21.2 19.4 22.3 12.S 19.4 18.7 12.5 22.7 17.6 16.6 21.3 iS-6 16.6 20.5 15.2 16.6 18.9 15.2 16.8 16.5 15-3 19.87 17.64 15.26 Dr. Payne found that "in arithmetic the EngUsh courses show 3 per cent more time than American, with about 5 per cent more in the earher grades." He says that as in America the relative time assigned to arith- metic makes it second in importance to language. The variation between ten German cities is about the same as between ten American cities. The relative time rangers from 14 per cent to 22.8 per cent in the German cities, and between 12 per cent and 19.5 per cent in American cities. The above table shows that the French schools devote a smaller proportion of time to arithmetic than do the schools of the other countries. In 1 9 13 a study was made by the authors of the amount of time devoted to recitations in arithmetic in six hundred and thirty cities in the United States having a population of four thousand and over. It was found that the variation in the number of minutes per week given to arithmetic ranged from no time at all up to 450 minutes. The time distribution for the difierent grades is given in the table below : 48 THE SUPERVISION OF AKITHMETIC TABLE XIII Grades No. OF Minutes PER Week 1 2 3 4 5 6 7 8 9 o 136 31 4 24 439 IS 7 2 20 2S 18 I I I I 30 10 7 5 I [ I I 35 I 40 6 4 2 5 2 I I 2 4S 50 99 36 3 I 4 4 2 2 55 60 27 22 10 5 3 4 5 5 4 6S I 70 3 3 I I I I 2 I I 75 103 127 59 12 4 2 80 6 5 10 II 2 3 I I I 8S 90 5 6 12 3 4 2 3 3 2 95 100 108 83 tS6 129 71 31 14 10 3 OS 10 I I 6 5 3 - 2 I 15 20 6 10 13 13 12 19 22 18 4 1 25 17 40 90 "3 122 109 59 36 5 30 3 4 S 6 9 3 2 35 40 2 4 6 I 9 8 3 45 150 53 66 98 129 158 196 210 192 30 55 60 I 2 4 4 6 2 7 8 5 TIME ALLOTMENT FOR ARITHMETIC 49 TABLE Xni — Continued No. Grades Minutes PER Week 1 2 3 i 6 6 7 8 9 65 70 2 3 3 I 2 I7S 6 8 9 20 31 28 35 37 6 80 2 5 6 3 5 9 16 IS 4 8S 90 I I 95 200 II 33 54 56 52 72 96 121 60 05 10 3 2 2 5 7 IS 20 1 I 2 I I 2 2 225 4 10 25 19 22 28 31 38 27 30 I 2 2 2 35 40 4 8 8 4 4 7 I 45 250 I - 9 25 39 SI SI 45 34 13 55 * 60 I I 2 2 2 65 70 I I 3 2 I I 275 2 2 8 9 8 S I To 300 4 14 24 30 30 38 33 5 To 350 3 ■> 3 7 8 2 To 450 I I I 2 3 Appendix A shows interesting variations in the time schedules of forty selected cities. so THE SUPERVISION OF ARITHMETIC From left to right this table means that there are> 136 cities that give no time to recitations in arithmetic in the first grade, 31 cities that give no time to it in the second grade, and 4 that ignore it in the third grade. From the top down the table reads : 136 cities give no time to arithmetic recitations in the first grade ; 7 give fifteen minutes per week in the first grade ; 18, twenty-five minutes ; 10, thirty minutes, etc. The range for the first grade is from no time to 450 minutes per week, and it is practically the same for the other grades. The median number of niinutes per week devoted to recitations in arithmetic in the different grades is shown, in the following table : TABLE XIV Grades I II III IV v VI VII vni Median number of minutes per week 7S 100 125 15° iS° iS° iS° 170 Chart V shows graphically the difference in variation and the central tendency in practice. Attention is directed to the upper and lower quartiles. It will be noted that the cases included within the middle 50 per cent are close to the median. The lower 25 percentile shows that one fourth of the cities spend 25 minutes or less in the first grade ; 75 minutes or less in the second grade; 100 minutes or less in the third TIME ALLOTMENT FOR ARITHMETIC 51 and fourth grades; 125 minutes or less in the fifth and sixth grades; and 150 minutes or less in the seventh and eighth grades. Again, the upper quartile brings out the fact that another fourth of the cities spend 100 minutes or more in the first grade ; 125 minutes or more MEDIAN MINUTES PER WEEK GIVEN TO RECITATIONS IN ARITHMETIC 150 a 125 / \ / ,\ / • y 1 / • • • / \ / \ / \ / Median \ IV V VI GRADE Chart V. VII VIII IX in the second grade ; 1 50 minutes or more in the third grade ; 200 minutes or more in the fourth, fifth, sixth, seventh, and eighth grades. From the foregoing analy- sis it may be seen that some cities spend relatively far more recitation time than others on arithmetic. 52 THE SUPERVISION OF ARITHMETIC Comparison of the 150 cities spending the greatest amount of time with the 150 cities spending the' least amount of time indicates that already many of these cities are making headway in the economy of time. If one fourth of the cities can get satisfactory results with an expenditure of from 5 to 20 minutes per day or less in each- of the first four grades, there is reason for inquiry as to the accomplishment of cities which spend from 20 to 40 minutes or more per day in these grades. Again, if one fourth of the cities are able to get satisfactory results in from 20 to 30 minutes per day or less in the fifth to eighth grades, certainly we have cause to question the reason why another one fourth of the cities spend from 40 to 60 minutes or more per day in those grades. On the whole, it seems safe to say that the wide variation of recitation time in the various cities of the United States suggests the possi- bility of attempting to effect an economy of time by means of standardizing the number of miniites in the recitation period. It is true that we need a compari- son of the actual results obtained in all cities giving a large amount of time to the recitation of arith- metic, with all cities giving a small amount of time. In the absence of scientific data touching all cities, however, it is important that we take cognizance of the fact that the investigations which have thus far been made by Rice, Stone, and others, all indicate that there TIME ALLOTMENT FOR ARITHMETIC 53 is no assurance of doubling the efSciency in arithmetic by doubling the recitation time. Therefore, it seems safe to recommend that the supervisor modify practice in a particular city on the basis of the experience of supervisors elsewhere, and such experimental evidence as has been obtained. If those cities spending more than the median amount of recitation time would reduce their schedule to the median time, a decided economy might be effected. In doing this no one would be adopting an untried policy, as the 157 superintendents who are included within the first quarter have already adopted this policy. It is difficult to believe that the results attained in the 315 schools reporting the median time expendi- ture are distinctly inferior to the results attained in the other schools. Further analysis of these data indicates that the North Atlantic and South Atlantic cities spend from 25 per cent to 50 per cent more time in arithmetic than do the cities in the other parts of the country. Large cities seem to give more time to arithmetic recitations than do small cities, although the difference is not great. The county superintendents report a very much smaller amount of time given to the arithmetic recitations in the rural schools. This is, no doubt, due to a differ- ence in the organization of the daily program in the rural schools. However, it should be noted that no 54 THE SUPERVISiqN OF ARITHMETIC section or city differs so much from the central tendency as to make it difhcult to work toward some part of the lower half of the curve of time distribution for recita- tions in each grade. That is to say, each city might wisely try the following time expenditure : TABLE XV Number of Minutes or Less per Week in Each Grade Grades I II III IV V VI VII VIII Median minutes . 7S TOO I2S iS° ISO iS° iS° 170 This table becomes clear when read as follows : The median time expenditure per week in the first grade is 75 minutes, or 15 minutes -per day; in the second grade 100 minutes, or 20 minutes per day ; in the third, 125 minutes, or 25 minutes per day; in the fourth, fifth, sixth, and seventh grades 150 minutes, or 30 minutes per day; in the eighth grade 170 minutes per week, or a little less than 35 minutes per day. Relative Time Expenditure The supervisor who adopts the standard proposed above may do so with the assurance which comes with the knowledge that this is the generalized experience of other teachers, principals, and supervisors. This TIME ALLOTMENT FOR ARITHMETIC 55 standard more nearly represents the judgment of the American schoolmaster, based on generations of ex- perience in teaching arithmetic, than does any other standard. The amount of time arithmetic should receive in relation to the other subjects of study is a factor of importance in making school programs. A critical study of the time distribution in one hundred forty-eight cities was made by Mr. Van Houten for the purpose of determining this relationship. The material was first distributed to show the total number of minutes per week given to arithmetic. This distribution is shown in the next table. The meaning of this table becomes clear when read thus : Akron, Ohio, spent 250 minutes per week on arithmetic in the first grade, 300 minutes in the second grade, 300 minutes in the third grade, 300 minutes in the fourth grade, 275 minutes in the fifth grade, 275 minutes in the sixth grade, 300 minutes in the seventh grade, 300 minutes in the eighth grade. In all 2300 minutes were given over to arithmetic. 11,100 minutes were given over to recitation in all subjects each week. Arithmetic received 20.7 per cent of the total time. Table XVI is a summary of Table XV. 56 THE SUPERVISION OF ARITHMETIC s s Q, H n 3 H f^ fl> X H 01 o M W) 'K ^ H CAl (1) -a Ak hn M 1 ^ ^ c3 pq tfi ^ (/) O =3 1 g B sl - jS c o ?i,H u t« u •a a-1 < ■g o s :^| 4-) ^.9 U2 i« H II K fa t>oo CO q >^t^>ntot^'*'*co"* t^oo ■o dvod »n »i d d >oo6 ciNooNwOMtAiA |lll|iP»lli|liP|lf!illl||lll|||l PPPPIs|£tH|§^pipg|p|p|||§HH s s a l&B HS&s^s&H H^HH&Hs&^IHsSi&HH sHHS&H5&HI&8s&Hns§HsEHsE58&H&s ?8|Hl2HSsH&SI°2^S?l&|B?2|aHFa8gRS^8 sSB8H3nE§HS?ll2^§S§S^8RSRas^^8^R8„a8 IHs|sS^3S°sBH&2l§S§§&iisS&§sH3&sB|8 S^»-|K§&2^s8&8a§RaaKS8^|8a^a|8R88a8a|8 S§sK8 = 88aSaH"2Hl^i-&2S8&§SF&^&8§§H SaSRS ^Bft- ^S-!S8R 8!= s8:=S8Ra8 §5 8 8 R ^R I^^^^^IJ=IJSni^aJlidKsiii^rfliS ijiiliiil iMiililliiliil 58 THE SUPERVISION OF ARITHMETIC 1^ t^M POO NOIOIH MM W«0« tO^l^l^Wt^H WW '°"?'?'*'t»^. *0"**0 ds'o N did woo tHodooo dici d»od d* ^ i>- ei w rood o d nvo d »^i^»on M M MM M H H HMHM HMW MMMIHC) g 2 III liHPlMlilllffilllM fffilHll |l |Sli|P|^llliill|HHif||iHsHi| 1 A 8 & 00 la^siS&Sssa&l&lsgaai-si^s&ISassis t- |a^2S&&8s2-&SsSSg8|§||slH2lilg| tD ISiHjssiHES&IH&ss&HHsHalSHI lO Hsss&SHssHH&ssHHsIHglllH ^ S§§s5&S&ss§H8H* JoUet . . Joplin . . . Kalamazoo Kansas City Keokuk . . Lancaster . Lansing . Lewistoni . Lincoln . . . Long Beach . Los Angeles Louisville Manchester . Manhattan Mankato Mason City Memphis Meriden . . Milwaukee Minneapolis . Muncie Muscatine . Muskogee . Nashvme . New Castle New Haven New Orleans Newport . Newton New York NUes TIME ALLOTMENT FOR ARITHMETIC 59 0>o0g0r00i-*0i00« ro coo W rO« WOO « TfM lONVO ^ f-vO fO t^OO -O NMVOvOO>OOO^^tO £> 0~ O" Cfi m" o' o" o" a o" cS o" O" « o" O" O" O" 0~ O" O" O'oo" O" O" O' (n" m" h' o' o'oo" OOOiO»«ioOOO»niO»rtOOioQOO"OirtO OQOOiOi/lOOQOinOOQ O 10»0 l~-(SW»nO»0«Nt^iO»Ot--000 0'OOCTi OOMirj t^vO r)-N O O r^cOu^O \0 moo 00 t^ 'to IN \0 t-1 u^tr^t~. 0>0 -S- W CO W 0\ t^OO -^t^O t-fOMOOCO t^Oit--* 8 8 t* - ^^a ^ ^ s 8 §H Eg lO ^■s-s S, o o B s K8 8 O m 1 8? as O JOO to V 8 8 ro o m m IN si o O lO o o 8 s O to P« 8 SS? 8 ro lO O as 588!? 8 »/l 8 1 lO r^ M 8 m o O (N r-. o o S" 8 8 1 ISS 1 ss R88;r o 8 S? 8 8 O 8;r8 tfJH M 8 O O !? si y o 8 8 8 asgg,^ 8 22 as S888 o 8 ;? 8 8 o ^8 8 8 O 8 O S 88 s o 8-8 M M 8 as 'i- to 1 S8 O 2~ I- sHs a 8 8 8<8 o 88 8 s O 8 8 to S S8 8 o 8 8 o t- I& O 22 i "2 t- 1— 88S,!S 8 8 8 8 8 o N t^ »oO t^ ■5 S 8 8 i? o r- o lOO >rt as 8 »0 M Vi 8s.aaa t~- O H 5 as i t- t-. 8 8 o O 8^ to lo o 8 .as 6o THE SUPERVISION OF ARITHMETIC Hi 1° O.00 o o >n -t 'C°° "i "t c^oo P o w >^ lO « ti 4 0» coo H d( H (O ti H t^ li! 00_ t*. lO -^ O O MO t^ '^O oo_ W OO '^ C d" w m" n o" m" o" ph" n c o" o" o" lO III. OOOOOOQiowioOgOirtOQ A lO 00 OOOOioOi'iWWtnOOOOQO CO >o»o^N O w « r-t^ioo »o»oO »o MWNHoioioioO QioO OO MM«HN(OWMHM«COW«CO>H (0 0»oOOviO»oiotoOOOOOOO mi^O ^ON « ^»lolOloO »oO ^n ta O^OOt/iOOOOOOOOOOO lot^O ■*« 0n»«00 Owi lHM«l-(C«CO«l-ll-lHMMMWCOH ^ s^gssls^&s'^ssHs' CO OioQiowiO irtioOviQ 0»oOQ>o HOCIMMC^H HMHH «»OM M ogo»oQvooioo«oooiooio MIHM «WH HMMH MCOH iH Qio loOQio 0»oOO OOO t^M t^ONt^ cow lOO O >0 O « M MM M M IH u Tacoma . . . Tulsa . . Vincennes . . Waterbury . . Watertown . . Waterville . . Waukegan . . Wausau . Webb City . . West Chester . Wilkesbarre . Winona . Worcester Yorki . . Youngstown TIME ALLOTMENT FOR ARITHMETIC 6l TABLE XVI All Grades. Distribution of Time Spent on Arithmetic in Minutes per Week Time Cases Time Cases Time Cases 640 I 1150 2 1750 3 650 I "55 I 1765 66s I "75 2 1770 700 I 1180 2 1775 720 I 1200 2 1780 725 2 1205 1790 75° 2 1225 1800 770 I 1250 1820 775 4 125s 1825 800 3 1270 i860 810 127s 187s 815 1325 4 1930 82s 1350 2 1940 830 1360 2 1950 840 1375 I 1975 842 1400 4 1980 87s 3 1410 2010 900 4 1425 2020 910 I 1450 2025 92s 3 1460 2066 930 2 1500 2080 95° 4 1525 2100 955 1555 2150 2175 970 1575 975 1600 990 1625 2225 1000 1035 163s 1650 2250 2280 1050 1660 2255 1 100 1675 2300 1 1 20 i68s 2320 1125 1700 2340 1129 1725 2410 "35 1740 3075 The most noticeable feature of this table is the ex- treme variability that prevails among cities as to the 62 THE SUPERVISION OF ARITHMETIC total amount of time per week given to arithmetic. The range is from 640 to 3075 minutes. No mode or central tendency is discoverable in the table. The same situation prevails when the facts presented in the foregoing table are brought together in closer formation by using a larger unit. TABLE XVII Closer Distribution of Time According to Minutes per Week Spent on Arithmetic TiitE Cities 601- 700 .......... 4 701- 800 13 801- 900 13 901-1000 .......... 15 looi-iioo .......... 6 1101-1200 .......... 14 1201-1300 .......... 6 1301-1400 .......... 13 1401-1500 .......... 17 1501-1600 .......... 14 1601-1700 .......... 12 1701-1800 ..'........ IS 1801-1900 .......... 4 1901-2000 .......... 6 2001-2100 .......... 6 2101-2200 .......... 2 2201-2300 6 2301-2400 .......... 2 2401-2500 .......... I 3001-3100 .......... I Total 148 Median — 1338 minutes per week Av. Dev. — 407-24 minutes ist quartile — 950 3d quartile — 1746 These tables, showing a range of 2500 minutes and an average deviation of 407 minutes, clearly indicate the absence of a time standard. Note. — The seventy-sixth case was taken as the median. The de- viation for each group was found by finding the variation of the average of the group from the median and multiplying by the frequency. TIME ALLOTMENT FOR ARITHMETIC 63 The next two tables give the percentile distribution of the total school time devoted to arithmetic : TABLE XVIII Table Showing Distribution of Cities According to Per Cent OF Total School Time, Exclusive of Recesses and Opening Exercises, Devoted to Arithmetic Per Cent Cases Per Cent Cases Per Cent Cases 6.1 3 II.O 3 16.0 2 6.4 2 II. I 5 16.1 3 6.7 I II-3 I 16.4 I 6.8 I 11.4 I 16.S 2 11.6 I 16.6 2 7.0 3 11.9 2 7-1 I 17.1 I 7.2 3 12.2 I 17.2 2 7-3 2 12.3 3 17-3 I 7-4 2 12-5 I 17.6 2 7-5 I 12.6 I 17.7 t 7.8 3 12.7 2 17.8 2 8.0 2 12.9 I 17.9 [ 8.1 2 18.0 3 8.2 I 13.0 I 18.2 I 8.4 I I3-I I 18.8 2 8-5 2 13.^ I 13-3 I 19.0 I g.o I 13-4 4 19.1 I 9-3 3 13-5 I 19.2 I 9.4 I 13-7 2 19-3 I 9-5 I 13.8 I 19.4 I 9.6 I 19-5 t 9-7 2 14-5 3 19.6 I 9.8 I 14-7 I 19.7 I lO.O I 15.0 I 20.4 I 10.2 I I5-I 2 20.S 2 10.3 I 15-2 3 20.7 I 10.4 I 15-4 2 20 9 I lo.s 3 15-7 2 22.3 2 10.7 I 15-8 2 10.8 3 iS-9 I 28.8 I 64 THE SUPERVISION OF ARITHMETIC TABLE XIX Closer Distribution of Time According to Per Cent op the Total Recitation Time Devoted to Arithmetic Percentages ' Cases 6.1- 7 lo 7-1- 8 14 8.1- 9 .......... lo 9.1-10 .......... 10 lO.I-II .......... 13 11.1-12 .......... II 12.1-13 9 13-1-14 II 14-1-15 S 15-1-16 14 16.1-17 8 17-1-18 13 18.1-19 .......... 4 19.1-20 .......... 7 20.1-21 .......... 5 21. 1-22 .......... o 22.1-23 .......... 2 28.1-29 .......... I Total 147 Median — 12.7% Av. Dev. — 3.69% ist quartile — 9.3 % 3d quartile — 16.5% These tables show that about 13 per cent of the total school time is devoted to arithmetic. They do not teU us whether this is too much or too little. But if the cities in the lower quartile get satisfactory results with only 9 per cent of their total time given to arithmetic, then it is reasonable to suppose that all cities might reduce the time at least as far as the median. Note. — The seventy-fourth case is tak^n as the median. The deviation for each group was found by finding the variation of the aver- age of the group from the median and multiplying by the frequency. Note. — There are only 147 cases in this table, since it was impossible to ascertain the total time spent in recitations other than arithmetic in one case. CHAPTER IV DOMINANCE OF METHODS IN THE TEACH- ING OF ARITHMETIC The Topical vs. the Spiral Method Methods in the teaching of arithmetic have been stressed differently at different times. Naturally the modes of instruction have been colored by the various stages through which the subject matter has evolved. Arithmetic grew by piecemeal ; new topics, sections, or divisions were added as they were seen to bear a logical relation to old ones. Under the influence of scholars, the subject gradually assumed a highly unified and logical character. Later, however, it was observed that such an organization was not necessarily adapted to the most economical learning of the facts and processes of arithmetic. Consequently a reaction set in against the formal, adult, scientific attempts at the organiza- tion of the subject, and a readjustment of the materials tb harmonize with the maturity levels of children began to receive increased attention. The older of these at- tempts at the organization of subject matter resulted in what is currently known as the topical method of r 65 66 THE SXJPERVISION OF ARITHMETIC instruction; the younger, in what is currently known as the spiral method of instruction. By topical method is meant the presentation of topics sequentially related without any reference to the facility with which they may be learned, each topic being completed before the next one is presented. By spiral method is meant the presentation of recurring topics in widening concentric circles in harmony with the age and ability of the chil- dren being instructed. The topical method was pre- sumed to give a notion of unity and continuity to the subject, while the spiral method was presumed to cor- respond to the psychological conditions of learning and to insure the fixation of the processes and skills of the subject. Like every method, each of these, when it was not modified by the restraining and clarifying influence of the other, tended to swing to an unnatural extreme; this was particularly true when either was taught by inade- quately trained or unsupervised teachers. We are not yet entirely rid of the baneful influence of these over- emphases. Although each in its time was an epoch- making method, reaction and readjustment .were in- evitable. As new topics were forced into the textbooks in arithmetic, the spirals were shortened, and the pupils were bewildered by the more frequent recurrence of the same topics. Textbook makers, catching the drift of this criticism as it came with ever increasing volume METHODS IN THE TEACHING OF ARITHMETIC 67 from every section of the country, began to reduce the number of the spirals and to lengthen them. The SpmAL and Topical Methods in Dieeerent Geographical Regions The present trend of educational sentiment with reference to the dominance of these two methods is shown in the tables that follow. These tables should be of service to textbook writers, to supervisors who are selecting textbooks and are plarining courses of study, and to professionally inchned teachers who are interested in the drift of educational theory and practice. TABLE XX Spikal Method Topical Method Combination OF THE Two Total Num- ber OF Cities North Central states North Atlantic states . Western states South Central states South Atlantic states . Counties 7 4 I 3 I 60 52 13 16 6 26 20s 177 37 50 22 68 272 233 SI 69 68 95 Total 16 173 559 748 Table XX should be read as follows : 7 superintend- ents out of 272 in the North Central states favor the exclusive use of the spiral method, 60 favor the ex- clusive use of the topical method, while 205 favor a combination of the two. 68 THE SUPERVISION OF ARITHMETIC TABLE XXI (Table XX reduced to per cents) Spiral Topical Combined North Central states .... North Atlantic states .... Western states South Central states . . . South Atlantic states .... Counties 2.9 1.9 2.0 4.0 O.I I.O 22.0 22.3 2S-S 23.2 21.4 27.4 75-1 75-8 72-S 72.4 78.6' 71.6 2.2 23.1 74-7 Table XXI is Table XX reduced to per cents. Table XXI should be read as follows: 2.9% of the superintendents in the North Central states favor the spiral method; 22%, the topical method; and 75%, the combination method. 2.2% of all the superin- tendents irrespective of lo- cation favor the spiral method ; 23 %, the topical method ; and 75 %, the combination method. The most striking feature of these tables is their uniformity. Clearly, opinion among school superintendents as to the de- sirability of either or both of these methods is well stand- ardized. The number of superintendents advocating the spiral plan is almost negligible. Three fourths of the superintendents are of ^ Spiral ^ Topical I i Combined Chart VI. METHODS IN THE TEACHING OF ARITHMETIC 69 the opinion that a combination of the two methods results in the most successful practice. The relative strength of the three groups is shown by Chart VI. From the foregoing facts it would seem that which method shall prevail is no longer a mooted question. Sanity in combining the two methods is almost univer- sally demanded by practitioners. For large geographical areas there is almost no perceptible variation in practice. With reference to this matter one section expects and demands what every' other section expects and demands. The Spiral and Topical Methods in Cities or Different Size Although experience has been generalized for the different sections of the United States and for the United States as a whole, it remains to be seen whether there is any marked variation when these methods are dis- tributed for different-sized cities. Not all of the super- intendents who replied indicated the size of the city whose schools they superintend. The replies that were clear are distributed in the next two tables. The data were distributed so as to show differences, if any, in practice in cities of ten different population classes. Cities in Rank I had a population of 1,000,000 or over ; Rank II, 200,000 to 999,999 ; Rank III, 100,000 to 199,999; Rank IV, 50,000 to 99,999; Rank V, 30,000 to 49,999, etc. 7° THE SXIPERVISION OF ARITHMETIC TABLE XXII Kank City Spiral Topical Combined Total I 1 ,000,000 and over 2 2 II 200,000 to 999,999 2 14 16 III 100,000 to 199,999 I 8 9 IV 50,000 to 99,999 I 7 23 31 V 30,000 to 49,999 8 32 40 VI 20,000 to 29,999 3 14 26 43 vn 15,000 to 19,999 I 12 31 44 VIII 10,000 to 14,999 I 16 68 8S rx 8,000 to 9,999 2 16 62 80 X 4,000 to 7,999 Total 7 71 225 303 IS 147 491 6S3 TABLE XXni (Table XXII reduced to per cents) Rank City Spiral Topical Combined I 1,000,000 and over lOO.O II 200,000 to 999,999 12.5 87.5 III 100,000 to 199,999 II. I 88.9 IV 50,000 to 99,999 3-2 22.6 74.2 V 30,000 to 49,999 ' 20.0 80.0 VI 20,000 to 29,999 6.9 32.6 60.5 VII 15,000 to 19,999 2-3 27-3 70.4 VIII 10,000 to 14,999 0.2 ig.8 80.0 IX 8,000 to 9,999 2-S 20.0 77.2 X 4,000 to 7,999 AU cities ... 2.7 23.1 74.2 2-3 22.5 7S-2 Table XXII shows that 2 superintendents in cities of 1,000,000 inhabitants or over favor a combination of the two methods; that 2 in cities between 200,000 METHODS IN THE TEACHING OF ARITHMETIC 7 1 and 1,000,000 prefer the topical, and 14 the combined methods; that i in a city between 125,000 and 250,000 prefers the spiral, 7 the topical method, and 23 the combined methods. Table XXIII presents the same facts reduced to per cents for each of the different-sized cities. It is clear that the number contending for the spiral plan is too small to be worthy of serious consideration. It is true that there is a variation of opinion in nearly every sized city, except the very largest, but no signifi- cant conclusions may be drawn from this variation. The fact of most import shown by these tables is that the great majority of superintendents, regardless of the size of the cities, are agreed that a combination of the spiral and topical plans is better than either alone. The Sanction of Usage We are not here concerned directly with achievement or results, but with the opinions of school men as to the desirability of certain methods. Naturally the efficacy of the two methods is involved, but, so far as we know, no trustworthy tests have been made to determine this. It is doubtful if any are needed. The laborious processes of trial and error, of success and failure, under countless varying conditions, have sanctioned in no uncertain manner the discontinuance and elimination of the spiral 72 THE SUPERVISION OF ARITHMETIC and topical plans as such, and have warranted the as- sumption that a combination of the two produces the best results. In the long run experience gained in this way is likely to be right. Until some one actually dis- proves that it is wrong or produces data that question its soundness, superintendents both young and old wiU be warranted in subscribing to those conditions and standards which their educational forbears have spent years in evolving. The weight of the testimony is so preponderantly in favor of a combination of the spiral and topical methods as to leave little room for doubt. It is true that many questions relating to these two methods have been left unanswered. Our data do not show the exact manner in which this combination should be effected nor do they show how the material should be distributed grade by grade. Matters of such paramount importance as these have already attracted the attention of textbook makers, and there is scarcely a text of the last half dozen years that does not rep- resent an attempt to make the proper adjustments. These are still matters of opinion that should be sub- mitted to scientific scrutiny and investigation. But the men who are in the best position to do this are the super- intendents themselves, and they are not hkely to have the time or the inchnation to do work of this character. Unless the experimental educationist comes to their rescue — and he is always likely to be handicapped by METHODS IN THE TEACHING OF ARITHMETIC 73 remoteness from or unfamiliarity with actual school- room situations — we shall again "cut and try" through years of painful and more or less bUnd experimenting until by some happy turn of the wheel of fortune we shall arrive at a solution of our problem. Methods of Subtraction There are other phases of method concerning which superintendents have rather decided opinions. One of these relates to the manner in which subtraction shall be taught. Two methods are in vogue ; one, the method of "taking from" ; and the other, the so-called Austrian method of addition. Those who advocate teaching subtraction by "taking from" insist that there is no subtraction in the Austrian method, and those who advocate the Austrian method insist that it is far more psychological than the method of "taking from " because it does not involve a new mode of learning. The advocates of the Austrian method also contend that the skill acquired by its use will be more serviceable since it corresponds to the making change method employed by the business world. We are not now concerned with a presentation of the arguments of the two contending groups. Our prob- lem is to determine the extent to which each of the devices has the sanction of usage. No doubt there are those who will maintain that usage is no measure of 74 THE STIPERVISION OF ARITHMETIC the value of a tool. Such a criticism is entitled to consideration when the tool is used by an unintelligent or unskilled class of people; but when it is employed day after day and year after year with children of varying ages and circumstances by teachers and super- intendents of reputed training and skill, such a criticism seems groundless. Surely the testimony of superin- tendents and teachers with reference to the intellectual instruments they daily use should be regarded as expert testimony. Variation of Subtraction Methods in Different Geographical Areas The following tables show that the Austrian method is universally regarded as second in importance, and that only a relatively small number and per cent of superin- tendents favor both methods. TABLE XXIV Addition Taking FROM Both Aix Cities North Central North Atlantic Western ... ... South Central . . South Atlantic . . . . Counties 96 24 16 ir 12 134 107 21 44 14 47 15 8 4 3 I 3 24s 220 1 49 63 26 82 Total 264- 367 34 68s METHODS IN THE TEACHING OF ARITHMETIC 75 TABLE XXV (The above table reduced to per cents) Addition Taking from Both North Central North Atlantic . . . Western ... South Central . . South Atlantic . ... Counties 39-2 47-7 48.9 25-4 42-3 39-0 54-8 48.6 42.8 70.0 54.0 57-3 6.0 3-7 8.3 4.6 3-7 3-7 Average . . 41-5 53-6 4.9 According to Table XXIV, 96 of 245 superintendents in the North Central states would teach subtraction by the Austrian method, 134 would teach subtraction by the "taking from" method, and 15 would use both methods. The same facts are presented in per cents in Table XXV. Neither of the arrays in- dicates that there is a pro- nounced tendency in either direction. The surprising fea- ture about the tables, although the range for both methods is from 3.7 per cent for county superintendents to 8.3 per cent for superintendents in the Western I I Taking from W^ Addition ^Both Chart VII. 76 THE SUPERVISION OF ARITHMETIC states, is the agreement in the different geographical areas in regard to the undesirabiUty of using both methods. For the country at large less than 5 per cent of the total number of superintendents is convinced of the value of using both methods. Here, as elsewhere in this material, we have fairly conclusive evidence that superintendents favor a specific way of doing things. s. An examination of the percentile array shows that there is wide divergence of opinion among the different sections of the country. Seventy per cent of the super- intendents in South Central states favors "taking from," while only 42.8 per cent of the superintendents in the Western states favors this method; this is a difference of 28.2 per cent. Such a difference cannot be accounted for by a difference in the mathematical necessities of the two sections of the country, for the practice of the South Atlantic states, as shown by the percentages, corresponds almost exactly to the central tendency of the country as a whole. It should be noted that the division of emphasis found in country schools closely corresponds with the division for the United States in general. From the data thus far presented one cannot main- tain that there is a marked tendency in either direction, nor can one safely predict what the practice of to-morrow will be. METHODS IN THE TEACHING OF ARITHMETIC 77 Variation or Subtraction Methods in Cities The data presented in the two preceding tables were redistributed to determine the variability of the use pf these devices in different-sized cities. TABLE XXVI Cities Addition Taking FROM Both All Cities 1,000,000 and over 2 I 3 200,000 to 999,999 7 6 I 14 100,000 to 199,999 7 4 I 12 50,000 to 99,999 8 19 I 28 30,000 to 49,999 • 13 21 I 3S 20,000 to 29,999 22 17 39 15,000 to 19,999 9 26 3 38 10,000 to 14,999 32 43 4 79 8,000 to 9,999 28 45 4 77 4,000 to 7,999 124 138 16 278 Total 252 320 31 603 TABLE XXVII' (Table above reduced to per cents) Cities Addition Taking from Both 1,000,000 aiid over 66.7 32-3 0.0 200,000 to 999,999 50.0 42.8 7.2 100,000 to 199,999 S8.3 33-3 8.4 50,000 to 99,999 28.5 67.8 3-7 30,000 to 49,999 37-1 60.0 2.0 20,000 to 29,999 56-4 43-6 0.0 : 15,000 to 19,999 23.6 68.4 8.0 10,000 to 14,999 44-4 54-4 1.2 8,000 to 9,999 36.3 58.4 5-3 4,000 to 7,999 44-5 49.6 5-9 Averages 41.8 53-1 5-1 78 THE SUPERVISION OF ARITHMETIC These tables show that the large cities favor the Austrian method, while the smaller cities favor the "taking from" method. It is admitted that educa- tional progress usually takes root first in the large cities, and that tradition and conservatism are clung to more tenaciously in the rural and semi-rural districts. If these generalizations apply in this case, then we have here a positive tendency. It is true that a superintendent in a given sized city can by inspection determine whether he is to be counted with the majority or the minority, but he learns nothing from these tables as to what the tendency is, unless it be true that the practice of the large cities represents educational progress. He will know the extent to which his practice varies from current practice, and the remodification of emphasis that must be made in order to make his practice correspond more closely to the central tendency. Certainly super- intendents in cities of Class I and those in Class VII, whose schools represent the extreme variations from the central tendency, need to justify their practice. It is to be regretted that there are no conclusive results to subiriit regarding this phase of instruction. We merely have here another problem awaiting solution. Intelligent observation and experimentation are needed to determine which of the two methods is the more economical and which will produce the better results. CHAPTER V THE SEQUENCE OF THE MULTIPLICATION TABLES The Influence of Tradition Custom and convention have heretofore determined the order in which the multiplication tables have been taught. The custom has prevailed of teaching them in the order of the digits. This custom arose because it was presumed that the digits had originated in a i, 2, 3 order. There is not the slightest evidence that this is true. So far as we have any evidence it tends to show that the digits were not invented in any regular order. It has long been a question with which school super- visors everywhere have been concerned, as to whether or not the present logical order of presentation is after all the most pedagogical. It is barely possible that the most difficult order of mastering the multiplication tables is the' I, 2, 3 order. Numerous experiments at varying this order have been tried, but, so far as we know, none of them bear the stamp of careful scientific work under controlled conditions. As yet we have no reliable in- formation as to the best sequential arrangement of the tables for teaching purposes. 79 8o THE SUPERVISION OF ARITHMETIC The Undermining of Tradition Over 500 superintendents furnished us with testimony as to the order that they think is conducive to best re- sults. Their repUes are presented in the following tables : TABLE XXVm Oeder or Teaching Multiplication Tables Regular DSDER 2, 4, S, 10, ETC. No Tables BUT Com- binations Not Im- portant Misc. North Central North Atlantic . Western . . . South Central South Atlantic . Counties . . . Total . . S8 S8 14 17 9 39 57 49 S 7 S 9 132 S 18 2 s 3 33 IS 9 I 4 I 8 38 SI 29 6 8 2 16 112 TABLE XXrx (The above table reduced to per cents) Regular Order 2, 4., S, 10, ETC. No Tables but Com- binations Not Im-- portant Misc. North Central . . . 31.2 30.7 2.7 8.1 27-3 North Atlantic .... 33-5 28.3 10.4 S-2 22.6 Western . . . . . 50.0 17.8 7.2 3-6 21.S South Central 41.S I7.I 12.3 9-7 19.9 South Atlantic 53° 29.7 0.0 S-8 ii-S Counties S2.0 12.0 4.0 10.7 21.3 Average .... 37-S 25-4 6.4 7-3 23-4 THE SEQUENCE OF THE MULTIPLICATION TABLES 8 1 According to Table XXVIII, out of i86 superintendents in the North Central states 58 maintain that the multi- plication tables should be taught in the regular order, 57 that they should be taught in a 2, 4, 5, 10 order, 5 that the combinations should be presented without any reference to order, 15 that an order is of no importance whatever, and 51 that any miscellaneous order will be satisfactory. The distribution of the replies from the other geographical divisions may be read in the same way. Table XXIX is Table XXVIII reduced to per cents. The second column in each of these tables — the one headed 2, 4, 5, 10 — is unsatisfactory, for the reason that the 132 superintendents who insisted upon this order did not agree as to the order that should prevail for the other tables. Some said 3, 6, 7, 8, 9 ; others, 8, 3, 6, 7, 9 ; still others, 8, 3, 9, 7, 6 ; still others, 8, 3, 6, 9, 7 ; every possible combination, in fact, was presented. It will be noted from these tables that a majority of the superintendents are of the opinion that the regular order is not the best order. Thirty-three or 6.4 per cent of the superintendents insist that no tables at all should be given, — that the combinations should be presented without reference to any systematic arrange- ment. Thirty-eight or 7.3 per cent insist that the order of the tables is not a matter of importance, and 23.4 per cent beheve that a miscellaneous presentation of them 82 THE SUPERVISION OF ARITHMETIC insures the most satisfactory results. Only 37.5 per cent cling to the traditional order. Chart VIII shows roughly that four superintendents in every ten are satisfied with the regular order and that a little over one half are not in agreement as to the order that should be used. An inspection of the vari- ation of the different geo- graphical regions as indicated by the first column of Table XXIX suggests the lack of agreement. The range is from 31.2 per cent in the Central states to 53 per cent in the South Atlantic states. The tables indicate that the North Central states are least dis- posed to cHng to tradition and that the South Atlantic states are most conservative in this matter. The difference between the extremes in the miscel- laneous column — the North Central and the South Atlantic states — is 16.8 per cent. The divergences in the other cases are so marked as to indicate that much experi- menting needs to be done before we shall arrive at a final solution of this important problem. Undoubtedly we are warranted in concluding that we have evidence here of a decided tendency to reconstruct a mode of instruction. I I Regular Order ^^2=,4',5',10',etc. ^ ' J Miscellaneous i^W Combinations ^^ Not iniportant Chart VIII. THE SEQUENCE OF THE MULTIPLICATION TABLES 83 If a number of cooperative movements could be es- tablished and different orders were tried simultaneously in a number of places and the results carefully checked by some uniform system, we might hope to determine at an early date upon the best order of presentation. We have distributed this material on the basis of the different-sized cities from which the data were collected. TABLE XXX Oedee or Teaching Multiplication Tables Population Regular 2, 4, 5, 10, No Not Im- All (By size of city) Order ETC. Tables portant Cities 1,000,000 I I 2 200,000-999,999 4 3 2 2 13 100,000-199,999 . 2 I 2 I 6 50,000- 99,999 . . 6 S 4 5 22 30,000 49,999 II 8 3 4 26 20,000— 29,999 10 5 3 6 26 15,000- 19,999 • 10 10 2 5 30 10,000— 14,999 20 22 2 " 58 8,000- 9,999 . 29 17 9 57 4,000- 7,999 . . 63 51 12 53 195 Table XXX gives the number of superintendents replying to each question for each of the different- sized cities, and Table XXXI gives the same facts reduced to per cents. It should be noted that there are ten population classes, later referred to as Class I to Class X, from largest to smallest. 84 THE SUPERVISION OF ARITHMETIC TABLE XXXI (Above table reduced to per cents) Regular 2, 4, 5, 10, Combina- Not Im- All Order ETC. tions portant Cities 1,000,000 . . 50.0 50.0 0.0 0.0 0.0 200,000-999,999 ■ • 30.8 ■ 23.0 lS-4 iS-4 15-4 100,000-199,999 . . 33-3 16.7 0.0 33-3 16.7 50,000- 99,999 . . . 27-3 22.7 9.1 18.2 22.7 30,000- 49.999 • ■ • 42.3 30.7 0.0 II.6 23.1 20,000- 29,999 38-4 19.2 7-7 11.6 23.1 15,000- 19,999 • • • 33-3 33-3 lO.O 6.7 23.1 10,000- 14,999 34.S 37-9 S-2 3-S 18.9 8,000- 9,999 . . . S0.9 30.0 S-3 0.0 15-8 4,000- 7,999 . . . 32.3 26.2 8.2 6.1 27.2 Average . . . 35-9 28.3 6.9 22 22 These tables are more instructive than the preced- ing tables. The two superintendents from cities of over a million inhabitants are evenly divided upon the ad- visability of clinging to the regular order, but, with the exception of these and those of Class IX, nearly two thirds of the superintendents in the other cities are agreed that we need a reorganization and readjustment of the customary order employed in teaching the multi- plication tables. The distinguishing characteristic of any single column is its variabiKty. For example, only 16.7 per cent of the superintendents in cities ranging from 100,000 to 199,999 beHeve in the 2, 4, 5, 10 order, while 37.9 per cent of those in cities between 10,000 and 14,999 ^.nd 50 per cent of THE SEQUENCE OF THE MULTIPLICATION TABLES 85 those in cities of 1,000,000 or over advocate this order. None of the superintendents in cities of Class I, III, or V refer to teaching multiphcatiori wholly by combinations, while 10 per cent of those in Class VII and 15.4 per cent of those in Class II urge this plan. Combining the re- plies of those listed in columns four and five, those who would insist upon some' order but who are not con- cerned about any exact order, we have a variation ranging from 15.8 per cent in cities of 8000 to 9999 to 50 per cent in cities of 100,000 to 199,999. No decided tendency can be detected in any of the columns nor in any of the dififerent-sized cities, unless it be the general dissatisfaction that seems to prevail with the existing mode of procedure. Apparently superin- tendents are still at sea with reference to this phase of technique. The Effect of Educational Dissatisfaction An examination of such tables as these tends to inten- sify the general feeling of unrest that these tables reveal. The prevalence of a widespread dissatisfaction with the regular order of teaching the multiplication tables is, we believe, healthy and will in the long run result in more economical instruction. A superintendent may utilize the tables to discover the particular group of reactionaries or conservatives to which he belongs. For it will be observed that super- 86 THE STJPERVISION OF ARITHMETIC intendents in Class IV apparently are not so conserva- tive as those in Class IX, while those in Class IV are more radical than those in Class I. After all, the par- ticular group to which one belongs is a relative matter. School supervisors of every degree of experience will find many questions in this part of this investigation. Those who have reached a final conclusion regarding this whole matter, if there be any such, should give their results to the world so that the uninitiated may not stumble into the pitfalls of tradition. On the basis of our returns, it seems that sound advice cannot be given to a young superintendent. The only conclusion to which we can come is that the old order is still the prevailing order, and that the prevailing tendency is to try to find some other. CHAPTER VI ORAL WORK IN ARITHMETIC While it is true from one point of view to state that all teaching in the elementary grades is of an oral charac- ter, yet in recent years there has come to be a more or less sharp differentiation between the oral and other types of work. It is stated in many courses of study that work of a certain grade is to be treated orally, or that a certain bit of subject matter is to be treated orally. Conse- quently it is very dif&cult to make a true interpretation of any statements relating to the amount of oral work done in any subject. The authors asked the superintendents in the cities throughout the country in towns of four thousand and over,' to state " the per cent of recitation time in each grade which should be given to oral work in each grade." Grade I 2 3 4 5 6 7 8 9 Per cent . . — — — — — — — — — In view of the whole schedule, the authors have deemed it safe to interpret these results according to their face 87 88 THE SUPERVISION OF ARITHMETIC value. The replies from one half of the superintendents throughout the country are tabulated in Table XXXII, so as to show the median per cent of recitation time pro- posed for oral work in arithmetic in the different geo- graphical sections. TABLE XXXII Median Per Cent of Recitation Time Proposed for Oeai Work in Arithmetic Geographical Divisions I II III ^ IV V VI VII VIII North Central . North Atlantic . Western . . . South Central . South Atlantic . 66.0 47-S SS-° 54-0 54-0 68.0 SCO 6o.o 44.0 42-S 49-0 42.0 52.5 3S-0 32.0 39-0 31-3 3S-0 24.6 28.2 32.0 29.6 3°-S 19.0 17-3 29.0 25.0 32.0 16.0 IS-2 18.0 19.0 27.6 13.0 10.2 17.0 15.2 20.0 9.0 10.2 55-3 52-9 42.1 31-6. 25-7 23-4 17.6 143 The table should be read as follows : In the North Central territory the median per cent of recitation time devoted to the oral work in arithmetic is 66 per cent in the first grade; in the North Atlantic territory, 47.5 per cent; in the Western territory, 55 per cent; in the South Atlantic territory, 54 per cent. The aver- age of the medians for the country as a whole for the first grade is 55.3 per cent. It will be observed from an examination of this table that there is a regular and fairly gradual reduction in the median amount of time given to oral work with each succeeding grade. This no doubt is what we should expect. As students in- ORAL WORK-IN ARITHMETIC 89 crease in maturity and in facility they should grow more and more able to work independently of the teacher. One of the measures of the effectiveness of instruction is the extent to which pupils can work independently of the teacher in those subjects in which they have been instructed for a long time. Attention is directed to the fact that the median time given -to the oral recitation work in arithmetic in the first, second, fourth, and fifth grades is high in the North Central states. The Western states show the high per cent of recitation time given to oral work in arith- metic in the third, sixth, seventh, and eighth grades. However, these variations are not great. In order to ascertain whether differences in regard to the median per cent of time given to the oral work in arithmetic exist in different-sized cities, the following table was prepared : TABLE XXXIII Median Pee Cent of Recitation Time Pkoposed for Oral Work IN Arithmetic Size of Citv I II III IV V VI VII vni 100,000 and over . 30,000 to 100,000 15,000 to 30,000 10,000 to 15,000 8,000 to 10,000 4,000 to 8,000 57- 42. 49. 34- 26. 23- 49- 46. 64. 44. SI- SC- 43- 40. 47- 34- 33- 36. 32- 28. 33- 27. 29. 3i- 27. 25- 25- 24. 26. 27- 23- 21. 24- 23- 20. 27- 18. 13- 21. 16. 17- 25- 14. 12. 19- 12. 12. 29. Average 39- SI- 39- 3°- 26. 23- 18. 16. QO THE SUPERVISION OF ARITHMETIC This table should be read as follows : In cities of 100,000 or bver the median per cent of recitation time devoted to oral work in arithmetic in the first grade is 57 per cent; in cities between 30,000 and 100,000, 42 per cent; between 15,000 and 30,000, 49 per cent; between 10,000 and 15,000, 34 per cent; between 8000 and 10,000, 26 per cent; between 4000 and 8000, 23 per cent ; and the average of the country as a whole being 39 per cent. It will be noted that the median per cent of recitation time given to oral work in arithmetic is highest in the first grade. The highest median per cent of recitation time for the second and third grades is found in cities of from 15,000 to 30,000 population. No sharp Hnes of distinction can be drawn for grades four, five, and six. The median per cent of recitation time given to oral work in the seventh and eighth grades is greater in the smaller cities. It seems that the larger the city the greater the amount of time given in the lower grades to oral work and the more constant is the reduction of time grade by grade, while the smaller the -city the more uniform is the amount of time given to oral work grade by grade. While it is true that these data may not be absolutely accurate, yet in consideration of the fact that these replies were received from cities of varying sizes, located in differ- ent parts of the country, we believe that they are fairly reliable, and of importance to the supervisor of the teaching of arithmetic. ORAL WORK IN ARITHMETIC 9I The opportunities for waste in connection with the oral treatment of a subject like arithmetic are great. It is desirable for the supervisor to know what is going on during this period of oral work. In the first three grades about one half of the recitation time is assigned to this type of activity,; in the intermediate grade about one third ; in the grammar grades from one sixth to one seventh. Hence, there are numerous opportunities for wastefulness in the recitation throughout these different grades. CHAPTER VII DRILL IN ARITHMETIC Much interest attaches to the general problem of the proportion of time which should be given over to drill in teaching arithmetic. Many of the theories of modern edu- cation have been such as to discourage teachers in the matter of drill, and to encourage them to emphasize ration- alization of the arithmetical work. This has led many teachers to teach the technique of arithmetic inadequately. While it is true that no conclusive answers have been arrived at by the educational experimentalist, yet it is nevertheless important to note that the experiments which have been made with a view toward determining the value of drill have tended toward the conclusion that drill within 'certain limits is of distinct educational value. Such studies as those of Mr. J. C. Brown and Dr. T. J. Kirby (referred to in a later chapter) indicate clearly that short rapid drill is of importance. Dr. Kirby experimented with a group of children in the upper grades, wherein an oppor- tunity was given for evaluating the effectiveness of drill periods of different lengths. His conclusions were that much might be gained through the introduction of a short drill period in every arithmetic recitation. In a recent investigation conducted by the authors 92 DRILL IN ARITHMETIC 93 data concerning the per cent of recitation time which should be given to drill were received from 564 cities. These data were distributed for the different sections of the United States and also for the cities of different size for the purpose of discovering the variations in practice in each grade. The table below shows this variation. TABLE XXXIV , The Median Per Cent of Recitation Time Favored for Strictly Drill Work by 564 Superintendents Distributed through- out the Doterent Sections of the Country Grade I II III IV V VI VII VIII North Atlantic . . . . 49 60 6q 45 42 32 24 18 South Atlantic . . . . ,S7 44 40 31 .^2 IQ 17 12 North Central .... 2Q 53 52 46 ,S8 28 21 17 South Central . . 45 46 45 42 ,S5 28 23 12 Western 26 48 52 52 42 45 28 21 United States as a whole . 43 S° 52 45 39 31 22 17 The table should be read thus : The median per cent of recitation time favored for strictly drill work in the North Atlantic section is 49 for the first grade, 60 for the second grade, 69 for the third grade, etc. It may be seen that there is a disposition to give a much smaller proportion of the time to drill in the upper grades than in the lower grades. Though there is a considerable variation in the different sections of the country, the general emphasis is quite similar. The table below shows the data distributed for the cities of different size. 94 THE SUPERVISION OF ARITHMETIC TABLE XXXV Median Per Cent of Recitation Time Favored for Drill by Superintendents in Cities of Different Size CiTIKS Grade I II III IV V VI VII vm 100,000 and over . . . 30,000 to 100,000 . . . 15,000 to 30,000 .... 10,000 to 15,000 .... 8,000 to 10,000 .... 4,000 to '8,000 .... 33 25 38 31 42 44 46 . 29 SI S2 53 53 57 SO 58 57 52 SO 51 44 45 48 42 45 41 35 37 39 36 36 31 32 29 31 27 21 29 21 19 22 21 22 18 15 17 16 16 17 This table should be read thus : The median per cent of recitation time favored for strictly drill work in cities of 100,000 and over for the first grade is 33, for the second grade 46, for the third grade 57, etc. It is interesting to note that there seems to be no striking differences in the attitude of the superintendents in large cities or in small cities. Chart IX shows the attitude of superintendents toward drill work in the recitations in each grade in the 564 cities. The upper line shows that 75 percentile, that is to say, one fourth of the superintendents favor the percentage of time indicated or more for each grade. For example, one fourth of the superintendents favor the giving of 79 per cent or more of the time to strictly drill work in the first grade ; 81 per cent or more to strictly drill work in the second grade ; 74 per cent or more to strictly drill work in the third grade ; 64 per cent or more to strictly drill work in the fifth grade ; and so on. The middle line represents the median, which DRILL IN ARITHMETIC 95 90 Mill 80 CITY SCHOOLS " \ Jpper Quartile — 70 s^ Median S 1 60 \ \, 5,60 -~ \ • ^. \ g40 \ s \ s 30 -•f-. \ \, , *-.^ \ s 20 \ , '• ~- 10 ■■■-. a I 11 in IV V VI VII VIII GRADE means that one half of the superintendents favor giving 43 per cent or more of time in the first grade to drill work ; 50 per cent or more in the second grade ; 2 2 per cent or more in the third grade ; 26 per cent or more in the eighth grade. The lower line is the lower quartile, and shows that one fourth of the superintendents are in favor of giving 24 per cent or less of the time to drill work in the first grade ; 27 per cent or less in the second grade ; 34 per cent or less in the third grade ; 31 per cent or Chart IX. Per cent of reci- , • . 1 r ii • 1 1 ^ tation time to drill, less in the fourth grade ; and 6 per cent or less in the eighth grade. Chart IX shows the median per cent of recitation proposed for strictly drill work in cities of different size throughout the country. The close agreement is striking. GeneraUzations based on such wide experience are of im- portance in arriving at any satisfactory solution of the problem. The fact that the curves descend after the third grade is of special interest. In the absence of experimental knowledge as to the exact per cent of time which should be given to drill, the supervisor is interested in knowing what the attitude of other supervisors may be in regard to this policy. While we have no adequate means of evaluating the comparative 96 THE SUPERVISION OF ARITHMETIC arithmetical efl&ciency obtained in the schools differing in the amount of time given over to drill, it is important that the amateur know the prevalent practice. It would seem safe to say that if one fourth of the cities are devoting 80 per cent or more of the recitation time in the third grade to drill ; 50 per cent or more of the recitation time in the fifth grade ; and 25 per cent or more of the recitation time in the eighth grade, we should inquire into the effectiveness of the work in the 140 cities devoting 35 per cent or less in the fifth grade, decreasing to 5 per cent or less in the eighth grade. In other words, such wide variation suggests the probabiHty of waste in the method at this point. A thoroughgoing comparison of results attained in cities with widely different standards for the time given to drill would be of great value if constructive recommendations are to be made. However, in view of the results of the detailed investigations on the value of drill in arithmetic which have thus far been made, we are certain that short drill periods produce the best results. In the absence of more satisfactory data for the evalu- ation of the different degrees of emphasis on drill, it is of importance to know the central tendency of practice, based on the experience in hundreds of cities. No doubt many supervisors will wish to adopt the median for each grade proposed above by the superintendents as the most satisfactory time limit. (Chart IX shows this clearly.) Note. For more detailed information concerning the amount of time given to drill work see Appendix B . CHAPTER VIII GRADE FOR INTRODUCTION OF TEXT IN ARITHMETIC The supervisor who has been interested in the prac- tice of his neighbor with reference to the introduction of a textbook in arithmetic has no doubt been impressed with the fact that wide variations exist in this particular. Any investigation covering a small number of cities presents such wide variations as to make it impossible to form an intelhgent opinion as to the practice in this connection. With data from hundreds of superintend- ents distributed throughout the country, however, it is possible to make certain generahzations with regard to the prevailing practice. The writers, in connection with the report of the N. E. A. Committee on Economy of Time in Arithmetic, received replies from 754 cities bearing upon this problem. General surveys of this sort are both interesting and valuable. They show the spread of practice and they reveal the standardization of opinion. The data collected from the 754 cities are presented in the following tables : H 97 98 THE SUPERVISION OF ARITHMETIC TABLE XXXVI Showing Grade in Which an Arithmetic Text is Introduced By Geographical Divisions I II III IV V VI Total North Central North Atlantic Western . . I 2 O O O 2 i8 9 6 13 4 i6 i6o 122 44 ig S3 64 78 17 17 5 28 22 IS 2 I 2 4 2 3 267 226 SO 7S 30 106 South Central South Atlantic Counties S 66 423 209 46 S 7S4 The meaning of this table becomes clear when read as follows : Of the 267 cities reporting from thS North Central territory, i introduced a text in the first grade ; 18, in the second grade ; 160, in the third grade ; 64, in the fourth grade; 22, in the fifth grade; and 2 in the sixth grade. Again, of the five cities introducing a textbook in the first grade, i is in the North Central territory; 2, in the North Atlantic territory; and 2, in the country schools reported by the county su- perintendents. Of the 66 schools introducing a text- book in the second grade 18 are in the North Central territory ; 9, in the North Atlantic territory ; 6, in the Western territory; 13, in the South Central territory; 4, in the South Atlantic territory; and 16, in the counties reported by the county superintendents. At- tention is directed to the variation represented by isolated cases ; for example, 5 superintendents introduce GRADE FOR INTRODUCTION OF TEXT 99 a textbook in the first grade while 5 other superin- tendents do not introduce a textbook until the sixth grade. Again, the 66 superintendents who introduce a textbook in the second grade disagree in policy and practice with the 46 superintendents who intro- duce a textbook in the fifth grade. However, despite this variation, it is of significance to note that the prevailing tendency is to introduce a textbook in the third grade or the fourth grade; thus experience seems to point to these as the standard grades for the introduction of a textbook. (It should be noted that the distribution resembles the distribution to be expected by chance.) The following table shows the same facts reduced to per cents. The third and fourth grades are even more clearly shown to be the dominant grades for the intro- duction of a textbook. Almost 85 per cent of the cities introduce a textbook in one or the other of these grades. There seems to be no striking differences due to geographical location, the third grade being the modal grade in each section of the country, and the fourth grade standing second in each section of the country. About the same percentage of superintendents wait until the fifth grade to introduce the text as introduce it in the second grade, and exactly the same percentage wait until the sixth as introduce it in the first grade. lOO THE SUPERVISION OF ARITHMETIC TABLE XXXVII (Preceding table reduced to per cents) I II III IV V VI Total North Central . . North Atlantic . . Western . . . South Central . . South Atlantic . . Counties .... 0-3 o.g o.o o.o o.o i.g 6.8 4.2 12.0 17.3 13-3 15.2 60.0 53-8 50.0 58.7 63.3 SCO 24.0 34-4 34-0 22.7 16.7 26.4 8.2 6.7 4.0 1-3 6.7 3-7 0.7 0.0 0.0 0.0 0.0 2.8 100 100 100 100 100 ^00 Average 0.7 8.7 S6.i 27.7 6.1 0.7 100 The meaning of this table becomes clear when read as follows : In the North Central territory .3 per cent of the schools introduce a textbook in the first grade; 6.8 per cent, in the second grade ; 60 per cent, in the third grade; 24 per cent, in the fourth grade; 8.2 per cent, in the fifth grade ; and .7 per cent, in the sixth grade. In the absence of striking sectional differences, the ques- tion arises as to whether or not differences in the year in which a textbook is introduced may not be correlated with the size of the city; that is, are textbooks in arithmetic introduced earlier or later in large cities than in small cities? If one were to hazard an opinion on the basis of the foregoing facts, he would infer that they are introduced earlier in the smaller places. The following table shows how nearly current such an opinion is. GRADE FOR INTRODUCTION OF TEXT lOI TABLE XXXVIII Showing Grades in which Arithmetic Text is Introduced Population (By size of city) I II III IV V VI Total 1,000,000 and over I I 200,000 to 999,999 I 9 5 IS 100,000 to 199,999 8 4 I 13 50,000 to 99,999 2 20 8 I 31 30,000 to 49,999 2 22 12 3 39 20,000 to 29,999 I I 26 II 3 42 15,000 to 19,999 I 30 9 I 41 10,000 to 14,999 10 45 26 7 0. 88 8,000 to 9,999 4 41 25 7 2 79 4,000 to 7,999 2 29 168 81 19 299 3 50 370 181 42 2 648 The meaning of this table becomes clear when read as follows : In the one city of 1,000,000 and over re- porting, textbooks are introduced in the third grade. Of the fifteen cities of 200,000 to 999,999 reporting, I introduces a text in arithmetic in the second grade; 9, in the third grade ; 5, in the fourth grade. It is interesting to note that the greater variations appear in the smaller cities. All the cities introducing arithmetic in the first grade are in towns with a popula- tion of 30,000 or less. Four fifths of the cities introducing a textbook in arithmetic in the second grade are in cities of 15,000 or less. Three fourths of the cities introducing a textbook in arithmetic in the fifth grade are in towns of 15,000 or less. The variation is revealed even more clearly in the table of percentages below. I02 THE SUPERVISION OF ARITHMETIC TABLE XXXIX (Preceding table reduced to per cents) Population I II III IV V VI Total 1,000,000 and over 0.0 0.0 lOO.O 0.0 0.0 0.0 100 200,000 to 999,999 0.0 6.7 60.0 33-3 0.0 0.0 100 100,000 to 199,999 0.0 0.0 6I.S 30.7 7.6 0.0 100 50,000 to 99,999 0.0 c-s 64-5 25.8 3-2 0.0 100 30,000 to 49,999 0.0 S-2 ' 564 3°-7 7-7 0.0 100 20,000 to 29,999 2.4 2.4 61.8 26.2 7.2 0.0 100 15,000 to 19,999 0.0 2-S 73-2 21.8 2-S 0.0 100 10,000 to 14,999 0.0 "•3 Si-i 30-° 7.6 0.0 100 8,000 to 9,999 0.0 S-2 Si-9 31-7 «.7 2-S 100 4,000 to 7,999 0.8 g.6 56.2 2.7.1 6.3 0.0 100 0.6 7-7 57-1 28.0 6.2 0.4 100 The meaning of this table becomes clear when read as follows : In cities of 1,000,000 population or over, 100 per cent introduce a textbook in the third grade; in cities of 200,000 to 999,999 population, 6.7 per cent introduce a textbook in the second grade ; 60 per cent in the third grade; 33.3 per oent in the fourth grade, etc. Here again it is clear that experience has been standardized in cities of every size; the third grade is the modal grade for the introduction of a text- book. Chart X presents this conclusion graphically. From the foregoing presentation of replies from super- intendents distributed throughout the United States, and in cities of different size, we are justified in the following conclusions : A superintendent who introduces GRADE FOR INTRODUCTION OF TEXT 103 a textbook as early as the first or second grade, or who postpones the introduction of such text as late as the fifth or sixth grade, will do so in the face of generaUzed practice. While we do not know as the result of careful investigation the best time to introduce a textbook in arithmetic, we do know that in the experience of the thousands of grade where text is introduced teachers and of the hundreds of superintend- ents repre- sented in this study the third grade is the best grade for the introduc- tion of this subject, with the fourth grade standing second. It would be of great administrative impor- tance for us to know about the results obtained in arithmetic work secured in a school which postpones the introduction of a textbook until the fifth or sixth grade. From an investigation of isolated instances, where the textbooks have been introduced very late, we have reason to beheve that much of the arith- Chart X. I04 THE SXIPERVISION OF ARITHMETIC metic work which is commonly associated with the textbook is really taught in the earlier grades. In other words, the extreme postponement indicated in the foregoing table in all- probability represents an attempt to get away from the use of the textbook, rather than an attempt to get away from the actual teaching of arithmetic. Again, students of this problem are concerned with the question as to which is the better grade for the introduction of a text, the third or the fourth grade. This can only be determined by careful tests, but the amount of time to be saved is of sufficient importance to justify the attempt to determine which practice is the better. This experimentation is going on, as is shown in the foregoing tables. What is needed now is a thoroughgoing cooperative investigation of results ob- tained under the different systems. The advocates of the policy of concentration of the energy of the school toward the mastery of reading in the first three grades may find much to encourage them in this report. If a third of the schools are already post- poning the introduction of -a textbook in arithmetic until the fourth grade, there need be little difficulty in getting more time for reading during the first three grades. Students of educational administration who take cognizance of the wide variation in age and maturity GRADE FOR INTRODUCTION OF TEXT 10$ of children in a particular grade may be led to question the advisabihty of postponing formal instruction in arithmetic to the upper grades. Again, the student who is conscious of the enormous amount of elimination which goes on in the early grades may question the policy of allowing children to postpone the introduction of a textbook in arithmetic until so late in their scho- lastic career. CHAPTER IX JUDGING TEXTBOOKS Society through legislation fixes the curriculum of the school. Legal enactments do not go so far, however, as to prescribe the topics within the field of a particular subject. That task is left to the school oflicers. The schoolmaster usually is assigned the responsibihty of selecting the particular topics within the various fields of subject matter, and of determining the necessary time and sequence to be given them. The Textbook as a Basis eor the Course OF Study An analysis of courses of study pubhshed throughout the country by different school boards indicates that the particular topics taught, with their sequence and time allotments, are determined largely by the nature of the textbook in use. The most common statement found in a course of study is that the fifth grade is to take the material found in the adopted text from pages 1 20 to 160 in a given period. Thus, the school oQicials leave the choice and arrangement of particxilar topics 106 JUDGING TEXTBOOKS 107 to the textbook writers. Indeed, it may be said that textbook writers in America are the ones who determine not only the choice and sequence of topics but also their gradation. In view of this dependence upon the textbook, it is important for us to know that textbooks vary in all three of the foregoing features. Some textbooks ignore completely certain topics which are heavily stressed in others. The sequence of topics varies widely. The gradation likewise presents astounding differences. If the school officials desire to maintain that a text with 300 pages, which is to be used in the fifth and sixth grade, is to be divided so that 150 pages are to be taught in the fifth grade, 150 pages in the sixth grade, and that in turn 75 pages are to be taught in the first semester, 75 pages in the second semester, 25 pages in the first third of the first semester, and so on, they should be assured that the textbook is so graded that the children wiU find the units of 25 pages for a six weeks' period to be of equal difficulty. But we do not have such assurance. If we are to adapt textbooks to this scheme of supervision, it will be necessary for textbook writers to grade their material more scientifically. No doubt reUance upon the author's selection of topics with their sequence and gradation has been responsi- ble for a large share of the unsatisfactory results in teaching. I08 THE SUPERVISION OF ARITHMETIC Principles Underlying Course or Study Making In arithmetic it is of especial importance that the supervisor or teacher select each topic with the greatest care, with a clear knowledge of the social, economic, and psychological features involved. It is likewise necessary that the sequence with which these topics are taught should be determined by the psychology involved rather than by the logic contained in the subject matter. As was indicated in an earher chapter,- there is a clear tendency toward breaking away from the supposed sequences in arithmetic. The teaching of the multi- plication tables in the one, two, three order was based on the supposed logic of the case. Many teachers have found it economical, however, to present the material in a two, four, five, ten order. Again, certain theoretical considerations have suggested the desirabiHty of post- poning decimal fractions until after the completion of common fractions. However, it is a fact, that children who have completed a study of United States money are able to make a study of decimal fractions without the knowledge of common fractions. Other instances of a similar character abound. The supervisor who is intent upon securing maximum results is no longer satisfied to trust the sequence of topics within a given textbook. Rather wiU he be interested in establishing a sequence of topics on the basis of the child's diificulty JUDGING TEXTBOOKS IO9 in mastering them and in light of the social need of the topics. The theory of the graded courses of study is that the work will be adapted to the different stages of mental maturity. To do this is a difficult task and requires keen analysis, experimentation, and constant attention. Every supervisor and teacher must recognize the varia- tion in ability that exists between classes within the same grade and adjust the materials accordingly. If the teacher rests content after presenting the material according to the plan of the textbook, there need be no surprise if the results are unsatisfactory. Every wide-awake teacher has recognized the fact that equal page allotments of material are of unequal difficulty. Reliance upon Textbooks German and French teachers do not rely upon the textbook as heavily as American teachers. Foreign teachers present their material fresh to the children from day to day, grading it as it is given to the needs of the particular class under instruction. Whenever we have teachers who know the quantitative demands of social and industrial life, who know arithmetic as a science, and who know children and their individual difficulties, we shall have arithmetic taught more effectively, and with far less effort than is being expended at the present time. no THE SUPERVISION OF ARITHMETIC Changes in text are not always made because the new book is better than the old book, but because the old book has been in use for a considerable number of years. It is the fashion to change books periodically, and adult human beings — even school teachers and superintendents — have not lost J:heir interest in new things. The failure to examine new books critically is often due to the absence of rational standards. Changing Character of Textbooks A consideration of the adaptability of textbooks is one of the duties that falls naturally within the province of the supervisor. He looks to them as representing a registration of educational progress. Textbook makers in the main stand for sanity in practice ; they are inter- mediators between tradition and radicalism ; they seek to make progress slowly. An investigation of a number of textbooks in arithmetic, ranging over several decades, shows that eUminations and additions have followed fairly closely upon the changes in business life. The table on the opposite page presents a comparison of the changes that have occurred in a period of sixty- five years in percentage and its applications, as shown by a study of ten textbooks. It is believed that these texts are fairly typical of the periods in which they were used, although it is not known that they were used more extensively than others. JUDGING TEXTBOOKS III i-l xsTaaxNi aNnodPioo ^. f^ ^* ^- t^ fo oi^o iNnoosiQ •iviOHajmo3 t^ ro r- oo O O >rt w >n isaaaiNi aTdrers CO O -3- « ^ Ol IrtOO M M saixaa ONV sHoisa^ WHHt^O'O fOCO saxvx ■*'0 M O t^-^iOUiOO t- aoNVHnsNi IflC-lVOOO Tj-O>»OM00 IN soNoa ONV saooxs H M 4 M fO M CO aovaaxoHa ONV noissikmoo (St-OlVOOOOO M t- rOTj-MNMM CO M NIVO ONV SS01 lO \o t^oo ^\o 00 -fl- lo n- ■«i-vo lo ■* a si xovHxsay Oc^^o0^oOOwOO axanoNoo « inmcoo M MOO -d-t^ xovaxsay 00> Ph aaNisKOO X Xl>< iv^ijg X avDidOx XX«X X w m a a B S OOVO M <0>Ot^Ch^O N ^lOOOCOiOQ M 1-1 OOOOOOOOOOOOCO SiOOi 112 THE SUPERVISION OF ARITHMETIC It is a matter of regret that when this study was being made we were unable to secure a book that was in current use between 1856 and 1892. It would probably not have revealed any very striking changes, as the 1856 text was the most widely used book during this entire period. The Failure op the Spiral Plan A significant modification in the general plan of text- book making is indicated by this table as taking place near the close of the last century when Hall's arithmetics, No. 9, afterwards known as the Werner arithmetics, came into vogue. The distinguishing feature of these books was the spiral plan by which each topic was re- peated and expanded with mechanical regularity upon a later page. This concept was almost hterally epoch making in the field of school method ; it practically revolutionized a number of old texts and called forth a flood of new ones. But in their ardor to attain a very desirable end, many authors overworked the device, and, as a consequence, a reaction set in almost immedi- ately. The great virtue of the spiral plan, however, was that it made us more self-conscious with reference to the possibiUties of teaching technique. This in- creased self-consciousness of the value of method has resulted in numerous recent attempts to find a happy combination of the two methods, the spiral and the topical ; for it is generally recognized that either alone JUDGING TEXTBOOKS II3 is unjustifiable, — a fact which we- have discussed in another connection. The Integration of the Separate Units There is another very interesting feature of these books, — a feature not readily reducible to tabular form, — that distinguishes the old from the new books. Formerly the different topics or divisions of the subject were in no way connected ; each new division of the subject was begun and treated as if it had no connection with any- thing that preceded or followed. Now these topics or divisions are as closely connected as human ingenuity can make them. Where there were once pages of ex- planations and definitions relating to the new subject, there are now problems and questions which are in- tended to bridge the gap. After a half page of explana- tion, including the "General Rule" and an example, Ray's first problem in Profit and Loss is "If my rate of gain is 25 %, how much should I mark goods for sale that cost me $ 8 ; $ 7.50 ; $ 6.25 ; $ 4.75 ; $ 3.87I ; $ 2.62^ ; I1.93I; .62^^; 15 (ii a yard?" In dealing with this topic the first words of the eighth book listed in Table XL are "Whatis|2.so with 10% added? with 20 % added ? " Seven such questions as these, becoming progressively more difficult, are presented before the suggestion is offered. " The per cent of profit or loss is always reckoned on the cost of the goods." Then an example (an analyzed problem) is presented. 114 the supervision of arithmetic Concrete Problems op Different Periods : Illus- trations There is still another feature not presented in the fore- going table, that is entitled to some consideration. We have reference to the changing character of the problems, included in the texts. Beginning with Ray's arithmetic, we have selected three of the very first supposedly con- crete problems from each of the texts. No. 2 : — A man owing f of a ship, sold 40 per cent of his share. What part of the ship did he sell and what part did he stiU own ? Out of a cask containing 47 gal. 2 qt. 1 pt. leaked 6f per cent. How much was that ? A found f s which was 13J per cent of what he had before. How much had he then ? No. 4 : — A cistern with a capacity of 84 gal. is 41I per cent full. How many gallons does it contain ? A owed B a certain sum of money. After paying him 20 per cent of the debt, 25 per cent of the remainder, 50 per cent of what then remained, and 83 i per cent of the third remainder, what part of the debt was still unpaid? In a school of 42 pupils, 7 were in one class, 14 in another, 6 in a third, 12 in a fourth, and the remainder in a fifth. Give the per cent of the school in each class. No. 5 : — A farmer having 400 sheep loses 20 per cent of them and sells 25 per cent of the remainder. How many does he seU? Corn shrinks 20 per cent from the time it is first husked. How many bushels will 6800 lb. of corn measure after shrinking, allowing 56 lb. to the bushel? If a certain cloth shrinks 4J per cent of its length, what is the shrinkage of a piece containing 38 yd. before shrinkage? JUDGING TEXTBOOKS IIS No. 8 : — If a man sells 12 per cent of his hens and has 66 left, how many had he at first? If you have read all but 40 per cent of a book, and have read 234 pages, how many pages has the book ? A stenographer can write 95 words a minute. Three months ago she wrote 20 per cent less than now. What was her rate then ? No. 10 : — The weight of a live chicken is 41 lb. When dressed it weighs only 3 lb. What per cent of the live weight is waste ? In his examination in arithmetic a boy had 10 problems out of 1 2 right. His grade was what per cent ? Emery paper costs $4.63 per ream, emery cloth costs $13.25 per ream. What is the difference in the cost? What is the per cent of difference? Constants and Variables in Percentage A cursory examination of these problems shows that the character of problem making has been undergoing changes. One feels that when some of the older authors wrote their problems down, they must have said, " Now see if you can get that." Problem makers now give less attention to the hj^othetical discipUnary value of problems and more to their social utiHty. Although there has been no tendency whatever to increase the absolute space given to this phase of arith- metic, the ninth column shows that the actual per cent of total book space given to percentage and its applica- tions has actually increased from fifteen to thirty-eight. As the years have gone by there has been a shght decrease in the number of applications treated. The Il6 THE SUPERVISION OF ARITHMETIC number of rules stated dropped to the jninimum about fifty years ago. This may be a startling revelation to some of those "modern" schoolmasters who have re- cently discovered the futihty of teaching arithmetic by rules. Practically no rules are presented in any of the new books, and practically none have been presented in half a century. The number of problems, however, has shown a tendency to increase ; the number of examples has varied irregularly, but the necessity of using them seems to be about as pronounced as ever. Oral and con- crete problems seem to be increasing, while abstract problems (if they may be so named) have practically been eliminated. Taxes, insurance, and simple interest are the only topics that have a place in each of the ten books. Gain and loss, commission and brokerage, stocks and bonds, duties, bank discount, and' compound interest and partial payments each appear in eight of the books. Commercial discount appears in seven books, and is clearly increasing in importance. Of these eleven topics, partial pa}rments and compound interest are losing in importance ; the others may be said to represent the constants of this part of the curriculum. Apparently there is a persistent demand to retain percentage and its applications because of its interpretative rather than because of its utilitarian value, for most of these topics are not taught for the purpose of making children adept JUDGING TEXTBOOKS II7 in certain skills. A utilitarian claim may be established for interest, but such a claim would hardly be considered sufficient to justify the presence of the other subjects. Perhaps the query may be made as to how such material as this helps a superintendent. It must be admitted that this particular study does not supply him with definite quantitative standards for the judging of textbooks. Perhaps no such standards are possible, and, certainly with reference to some aspects of arith- metic, they are not desirable. But this study does furnish a number of suggestions that should receive consideration when textbooks in arithmetic are being examined. In the first place it indicates what the variables and constants are in percentage. In the second place it suppHes a rough basis, as indicated by the num- ber of problems, for estimating the relative emphasis each topic should receive. The Shifting Content of Mensuration^ For the purpose of studying some topic intensively a second investigation was made, — this time with mensuration. Only two things were considered : whether there had been any change (i) in content or (2) in the relative emphasis of the topics as indicated by the amount of space devoted to them. Thirty-one 1 The authors are indebted to Mr. L. O. Bright for most of the ma- terial in the investigation of this topic. ii8 THE SUPERVISION OF ARITHMETIC PflSES Devotco to MEnSURJITIOn no-Paecd MeruL XIS .--TTl ^- ■JHH-I-T 1 1 1 ,^ /• , / /A -> y 1 1 _ 1 1 % Mena Pagea 1* no, Prob.Men». Fig. I. no. OF MiscELLAHEOus McnsuRflTion Problems »5 |— 1 i» ?''^ '*"=~, Is Z ^^^ ^---^ ^-2 = '" ! / ^^ ^:> 6 T'^ \ JU- ' ^^^^ _ _ 9i Mcfl5. Prob. all, til. **, 1% I83S 1650 1865 Fig. II. Q.UADR1LATCRALS riO.OF PtiOt. — - p« lU y\.t LO an IM ..- ^.■ ■' ,.. ~ ■~ rxi ~ T.O P ■' _^ -- ^ •• ... -" / — — ro- AU Q, flD ^* y ,' ^ - - — .^ ~; — r«, PS. .11/ 1 ' ,' ^ / ' _ ^ ^ _^ > - ' ^ ■^, ^ -■ — ~^ , -- ^ #^ ^'' ,.- c: .- .^ ^ > ^ ^' -- - - ._ . ■I ^ '' / .< ^ ■A •' _ ., — ' _, .. !!' r- Ta ^ J. d 1> ^ s£ '•- — ~ J IB50 IS65 Fig. III. IS90 1900 JUDGING TEXTBOOKS rfo.of Paa«» 119 Via. of Prob. ■ 1 — z Tn S^J l» „. ,„., ,.»..H ...=° ^ ^ 'Z '" 7ir ;k ==»' J! ig "^ „."° J.- ,-.'•' "-•"■ ^ •».«, ttnJ - ■". .* ■-- ■^ .-^" - " .-Vr ??f t" ^ rf ^ — J 1796 l&ZO IfidS. (650 1665 Fig. IV. no. of P^C5 laeo I690 (900 1915' ■ ~1 ~ _ Pr sm , — -~ ^ ^ ~" Cjl nJ »f:s / __ ■ '- / -^ " ' ~- -- -^ — - / ,^ ^ "~ — ' " ~ ^ ^ — - .^ ^ ^ - "~ -- ■- ._ __ — -- / _ "^ ■ ^ :: — " 1796 l«20 lejS laSO 1665 1860 1690 1900 1915 Fig. V. Ptrhmid flriD Come rio.of Pigcs ■ ^ ^ — " /^ ■" .-^ ^ ' /■ i. — . --' IT96 18^0 IBS5 Mo.of Po-gea leso IS&5 teso 1690 Fig. VI. 1900 19IS iSpHETiagL "" " - .... ■ ~ ■' . ^ ,• ^ *■ ,/ / *" ^ ^ __ __ -^ ^ _ - / ' ~^ — ■ " ^ ' ' f" _ 1796 102.0 1659 IS50 Xt&S IdOO (S90 1900 mS Fig. VII. 120 flo.of PAsgea THE SUPERVISION OF ARITHMETIC rio.ofProb. =: = E. II. ise z: - 3 T7il if fi 1' ^ "' ^ - ■— • — ~ =^ ^ A- "" V -' ^^. ^ '- _ C' ^ _ *^^, I82A 1835 I&50 1665 I880 1590 I900 Fig. VIII. rSo. of *'4ges — Ll ml >cr y / 7j ~ B ■(C ; f Tid S fad e / / ^ ^ ; ^ -J — -^ ^^ .. ,^ r. -J L< ^ ' / r_ y' ^, ^ -. ^ -- ;< V -" CN _j ^ - ' ^ ^ _ _ -^ J 1796 (&£0 1835 1650 1665 IBSO ld90 I900 (915 Fig. IX. QmemB of Cask^ Jio.of PAgci Ho. of Pro b. " " ^ - ^, ,- ,-" ^ _ ;^- ■'' ^^ ^ s f =^ — ,s > •^ ^ ir96 1620 16^^ lefO 1065 (060 I&90 I900 1915 Fig. X. grammar school texts, ranging according to their dates of pubHcation, from 1796 to 1912 were examined. The amount of space and the number of problems appearing under each of the following topics were checked : Quad- rilaterals, including trapezoids, trapeziums, and parallelo- JUDGING TEXTBOOKS 121 grams; circles; triangles; prisms; cylinders; pyra- mids and cones; spheres, ellipses; similar figures; lumber measure; brick and stone measure, and gaug- ing of casks. The results can be shown more clearly by the graphs here given than by the original table. In each instance the arithmetic mean of the period is taken as the ordi- nate of the period. In Figure I the black line shows the numbet of pages devoted to mensuration, indicated by the scale at the left. The red line shows the percentage of the total number of pages devoted to mensuration. From this figure one can see that mensuration came into use in textbooks shortly after 1820 and that it has gradually increased in importance until in 191 2 about twenty-one pages or 8 per cent of the whole book was given to it. Figure II shows the rather irregular increase in the number of miscellaneous mensuration problems appear- ing in textbooks. In Figures III to X the black lines denote the number of pages, read from the scale to the left, and the red lines show the number of problems given to each topic. Quadrilaterals have gradually risen in importance from about one third of a page with two problems to three pages with eighteen problems. The trapezium, which appeared in the texts about 1845, about ten years after the trapezoid, is receiving less attention than the 122 THE SUPERVISION OF ARITHMETIC trapezoid. The parallelogram, which appears in the texts about 1825, now receives almost twice the space of the trapezium or the trapezoid. From the earhest introduction of mensuration, we have had circles, tri- angles, cyUnders, prisms, pyramids and cones, and spheres, and each of these has gradually increased in importance. The elHpse, which came in about 1880, disappeared about 1900. The study of similar figures has slowly increased since 1850. Brick and stone measure have disappeared, but lumber measure has increased since 1870. Gauging of casks lasted from 1840 to 1900. A Comparison of Five Elementary School Arithmetics A critical study of five elementary arithmetics now in use was made for the purpose of comparing them as to the similarity of their structure. It was presumed that uniformities would be readily discoverable and that they would furnish the uninitiated with standards for estimating the structural character of any elemen- tary text in arithmetic. A sunmiary of this analysis is presented in Table XLI, the meaning of which becomes clear when read thus : Textbook number I contained 273 pages, 3 chapters, 11 topics, 37 lines of footnotes, 228 explanations, 183 pictures, 1949 examples, 1526 written exercises, 905 review problems, 54 definitions, etc. JUDGING TEXTBOOKS 123 TABLE XLI Table Showing Totals tor Vaeious Items in Five Elementary Aeithmetics I n III IV V Number of pages . . 273 256 243 264 135 Number of chapters 3 3 4 5 Number of topics . ... II 12 12 12 8 Number of notes (lines) 37 20s 18 153 Number of explanations . . 228 73 163 108 37 Number of pictures 183 162 29 62 51 Number of examples 1949 1 201 335 1214 605 Number of written exercises 1526 849 830 1026 1021 Number of review problems 905 148 395 730 124 Number of definitions .... 54 21 40 48 18 Number of rules . . 15 S 52 15 Number of drills . ... 32 46 30 49 39 Number of home problen^s 84 84 143 124 200 Number of school problems . 39 29 26 95 68 Number of farm problems . . . lOI 176 193 123 37 Number of business problems . . 220 213 338 336 232 Number of arithmetical problems . 481 494 282 263 444 It will be seen from the above lists that dissimilarity rather than similarity is the custom. Note, for ex- ample, that the number of notes varies from none to 205, the number of lines of explanations ranges from 37 to 228, the number of pictures from 29 to 183, the num- ber of examples from 335 to 1949, the number of written exercises from 526 to 1026, the review problems from 124 to 905, the number of rules from none to 52, and so on. That part of the table which d^^ls with the more strictly arithmetical relationships shows that writers of 124 THE SXIPERVISION OF ARITHMETIC arithmetics are not agreed as to the standards that should prevail in the construction of textbooks. Less diversity is manifest in those topics that deal with the applications of the arithmetical concepts and principles. Greater similarity is found when we examine the order in which the topics are treated in these books. The next table presents this information. TABLE XLII Order of Topics in Five Elementary Arithmetics I n Ill IV - V 1. CouQting I Counting I. Counting I. Lines, Angles - Counting ^. Addition ^. Addition ^. Addition 2. Counting 2. Size and position Combination 3. Subtraction 3- Subtraction 3- Subtraction 3- Addition 3- Addition and Sub- traction 4. Multiplica- 4- Multiplica- 4- Multiplica- 4- Subtraction 4- Multiplica- tion tion tion tion 5. Division 5- Division 5- Measuring S- Multiplica- tion 5- Division 6. Measures 6. Roman Numerals 6. Division 6. Division 6. Measure- ments 7. Roman 7- Fractions 7. Roman 7. U. S. Money 7- U. S. Money Notation Numerals 8. U. S. Money 8. Measure- ments 8. Fractions 8. Measure- ments 8. Roman Numerals 9. Fractions 9- Analysis 9- Decimals 9. Fractions 9- Mail 10. Decimals 10. Decimals ID. Bills 10. Ratio 10. Bills II. BiUs II. Bills and Statements II. Factors and Multiples II. BiUs 11. Fractions 12. 12. Farm Prob- lems 12. Percentage I.. Percentage 12. 13. LongDivision Markets Relative Emphasis of Given Topics It will be seen that about the same topics are treated and that they are treated in about the same order in all of the books. Clearly counting, addition, subtrac- JUDGING TEXTBOOKS 1 25 tion, multipKcation, division, measurement, and frac- tions are the constants; such topics as ratio, Roman numerals, analysis, and percentage are variables. The next table gives some notion of the relative emphasis these topics are receiving in each of Jhe five arithmetics. It seems to us that the differences are more interest- ing than any central tendencies that might be computed, for these, after all, would really not be descriptive of practice. Considering the fact that the first two books on the list are enjoying a wide sale and the others are having almost no sale at all, we were disposed to examine the first two more closely in the hope that similarities may be noticeable. But such an examination was futile ; it shed almost no Hght upon the relative values of these books. A Dictionary or Teems and Names As a final resort, therefore, we had every word in the first fifty pages of these books counted and its frequency noted. This may appear to be a foolish task. But we were really interested in determining whether these books are arithmetics or something else. It seems logical to assume that an arithmetic should fix certain arithmetical terms, principles, or concepts. Recurrence or repetition is necessary to insure this fixation. If this repetition recurs only very 126 THE SUPERVISION OF ARITHMETIC 1 > lo COCO r-. •«*• w ^ N M <0 H H M & Or*CTi« ^-^WO^0v> W CO W t— 1 N M « 0";'*OU,N H fOOs^l-^^coOOO l-f OcOiHNMiO':i-- NCOt^HVOlO'^OM > )-H O QO »o ■* T^ CM-t -ftO M lO^O ro w t— 1 CM^ O H (OO »OtOO\roO « H ON ■*« rH H M B CO O cooo CO n CO O lOH lOcO H ^ O t^oO H O CN Coo '"' MM )-H tow «r-0 cO'-i-O O H H W CO M M M 1 > <0 W 0\ coco CO M O lO CN CO W M M H >| OvO ^r-0 ■* w M w OvO M M 3' t-OOcO'^rowcoM H OilOM H O Tf ■* coo to M OO l-H O ■* t>.ao tUn o w MM.OOO,H Lines of notes to teachers . Number of explanations Number of pictures . Number of examples . . . Number of mental exercises Number of review problems Number of definitions Number of rules , . . Number of drills . . a n > ooooooooo > ooooooooo B 0iJ<0O0WHt-.0 ^ co« 1— 1 OOOoOooOHt^O *-^ O O PO coco W CM N i 1 > OioO'^hOnOO fe O 0\ O coo Om^ O O io« )-H iOO\H O-OioO Osw a O-ovOO-inioO H H IOOn CO M l-H O Ov 0>0 CO M t- Ov H I^ Ov Ov w > HM-tOWlOOOCN H H M VO N > 1-* N O ■^vo ■* O \0 MM M M 1— ( OvO »rtO 'Nvoovo H O ^OO »^^ OVO - movovo HCO w o CO M -.t »OQO 00 ■* moo CO M O > t— 1 OOvNiot^OoowO H 1— 1 1— ( »— « Ovf00Hio0«00 » ■*ioO Lines of notes to teachers . Number of explanations Number of pictures . . . Number of examples . Number of exercises . . . Number of review problems Number of definitions . . Number of rules .... Number of drills .... JUDGING TEXTBOOKS 1 27 infrequently, the student, not being familiar with the laws of habit formation, will drift aimlessly through the subject, spending two or three years acquiring a few fundamental skills that should be assured him at the outset. It was such fundamental propositions as these that induced the authors to prepare Table XLIII. An examinati9n qf the Table gives one certain common sense notions as to why some of the books are necessary and others failing. It will be observed, for example, that Book V gives little attention to explanations, to review problems, or to common and decimal fractions. While Book IV emphasizes review problems in the fundamentals, it does not do so consistently in the other operations, and it neglects to give any consider- able attention to fractions. Book I is the most consist- ent in the emphasis it gives to the various topics. In the first fifty pages of these books 94 proper names are used 342 times. The names appearing most frequently are Ella and Kate, which appear 6 times each, Helen 8, Henry 9, Carl 12, Fred and James 16 each, Frank 24, Mary 27, and John 40. There are in these same pages 224 words beginning with c, and these words are used a total of 1403 times. As a sample of the variety of things that may appear under a, single letter of the alphabet we print two of these tables, — the tables Usting the words beginning with s and w. 128 THE SUPERVISION OF ARITHMETIC TABLE XLIV Words Beginning with " S " with Their Frequencies in Five Elementary Arithmetics WOKBS I II III IV V Words I II III IV V sacks . . . s shad . I saddle . . . 2 shaU. . . . I s sale . . I share . . 3 Sam . . . . I 7 she .... 7 II I 31 28 same . . . . I lO 2 2 sheep . 6 lO II I 3 Sara . . . . I 3 sheet .... 3 satisfactory . I shelf .... I I 28 Saturday . . I 2 shock . . . I I 8 sat . . . I I short I 9 3 2 sauces . . . I should . . 4 2% II save .... I 3 2 3 show .... 7 24 3 6 savings . . . I shot . . . I saw . . . . I 2 shrubs . . . I say .... S I 3 side .... I 2 4 9 4 school . . . 3 2 2 6 21 sidewalk . . I scissors . . 4 sight . . lO I 2 score . . . 2 8 sign . . . 6 13 lO scored . . . 9 signal . . . 2 sealed . . . I silk I I search . . . I sill . . I season . . . I simple . 2 I seat . . . I S 5 sing . . . I seated . . . I I I single 2 2 I second . . I I 2 13 7 sister . . . 8 I 3 sections 6 2 I sitting . . . I see .... 2 3 lO 8 situation I seeds . . . 3 six . . 22 I 9 5 4 seek . . . I sixteen . . . I 2 I I seen 2 I sixths . . 6 I 4 select . . . I I sixty . 4 2 4 sell .... 4 5 2 6 size . . . 4 2 selling . T , I I I skates . . . I send . . . I sketches . . 2 sent .... I 3 sleeps 2 separate 3 I slightly I session . . . I slip 3 set .... 2 7 slowly . I seven 20 I 2 3 3 small 2 I s seventeen 2 I I I Smith . . . 6 seventy . . . S 5 so .... 3 I 10 4 several . . . 3 soao . . . T T I JUDGING TEXTBOOKS 129 TABLE 'KLVJ — Continued Words Beginning with " S " with Their Frequencies in Five Elementary Arithmetics Words I II III IV V Words I II III IV V soda . . I stimulate I sold . 7 5 10 5 16 stock I soldiers . . . I 3 stopped . . . I solve .... 10 store .... 4 4 6 some . . . 28 2 8 •stormy . I somethings 2 I story . . 2 6 I I sometimes 5 I straight I 7 soon .... I I I strawberries . I 4 sounds . . . I I street . . . I 5 soup . 2 strengthen . . 2 source I strings . . . I 2 south . . . 3 strip . . I I 2 souvenirs . I stripes . . . 2 spaces . 2 4 2 stroke 7 spade 2 struck I sparrows 2 I study 4 5 speak I stumble . . 4 speckled . . 4 subdivisions . I spelling . . . 6 subscribers I spend . . . 2 3 6 2 subtract 12 10 12 7 2 spider 2 subtraction 3 2 17 I 7 splints . 3 16 subtrahend 3 I spoke 2 succeeding . . I spools . . I 2 such . . 2 3 sponge . . I sugar 4 2 sprinkling 2 suggest 4 I spruce . . . 2 sums . . 19 13 13 I 3 square- . . . 34 28 4 16 II Sunday . . 3 2 stable . I supper I stacks 3 supply 4 7 I stage I supposing I stairway 6 surely I stamp . . . 4 27 surface . 2 stand . . 21 I 6 Susan . 2 standard I 4 swallows I I start . 8 2 3 swing I state . . 2 4 I syrup 2 stays . . . 3 I Total . 279 182 304 271 381 sticks . . . 65 I 8 6 No. different stiff . I words 62 25 74 67 113 stm . . . . 3 I30 THE SUPERVISION OF ARITHMETIC TABLE XLIV — Cmtinued WoEDS Beginning with "W" with Their Frequencies in Five Elementary Arithmetics Words I II III IV V Words I II III IV V wagon 4 s 3 I whom . . . 2 walk .... I 7 why . . I wall . . I wide . . 4 I 2 2 walnut . I width . I wanted . I I Wilbur . I warm I wiU . . 3 6 II 4 20 was_ . . 2 I I 17 Wmiam . 2 2 I 2 watch . 2 willow . I water 7 I I 3 win . I I wave . . 3 window . I I 2 2 way . . .. 5 14 I 2 winning we . . 27 37 wise . . I wear . . 3 wished . S weaving I wishes . I I 2 Wednesday- I I 3 with . . 6 10 14 23 week . . 14 I 2 19 without • 4 3 weigh . 2 13 women . 2 well . . 3 won . . I went . . I I 2 9 wood . . I I were . . 2 2 7 19 28 woodchucks . 2 what . . 20 6 27 25 40 words . 2 I 2 I wheel . 2 7 3 work . . . 2 31 I 4 when 2 7 I 12 worms . . . 4 where . 2 worth . 4 2 12 whether I would 3 7 which . T I 28 6 21 wren . . 2 whichever I write .... 43 13 SI 31 22 while 3 written . 25 8 27 • 8 I whistle . I I Totals 140 92 298 141 239 white 3 9 5 2 No. different who . . 3 I 3 words . . 23 19 35 29 40 whole . . . 7 6 I The Recurrence of Arithmetical Terms The next table, which presents the distribution of the arithmetical terms in each of the arithmetics, shows JUDGING TEXTBOOKS 131 TABLE XLV Table Showing Arithmetical Teems of First Fifty Pages Frequencies OF Arithmetical Terms ARITHMETICAL TeEMS I II III IV V add . . . 20 2 20 17 23 addition ... 2 2 16 I 2 as many as as much as . I count . . . . 32 6 12 8 20 counting . 10 difference 10 I 2 2 divide . 32 11 6 7 divided by . . 28 dividend 7 division . 13 I 3 divisor S equal 9 9 14 8 fraction 3 how many 272 328 54 112 262 how much 15 4 24 35 45 less, less than . . 81 24 2 minuend 2 I minus . . . . 2 more than II 9 14 multiplication u 12 I 2 multiply . I I 7 2 3 plus .... 2 quotient . . . . 4 remainder . . I 12 solve ... u I subtract ... 9 12 13 5 I subtraction I 16 I 8 subtrahend 3 I sum 17 12 I 2 take away ... I 19 times . . . . . 3 43 24 21 28 units ... . . 5 2 what part of . . 4 4 Totals . 481 494 282 263 444 132 THE SUPERVISION OF ARITHMETIC clearly that there has been no conscious attempt on the part of any of these authors to organize their books so as to insure the fixation of certain fundamental arith- metical expressions. We believe this to be a defect common to many arithmetics. An Analysis of Grammar-grade Texts Four current grammar-grade textbooks in arithmetic were selected for careful analysis. First of all, each text was inspected with a view to finding the particular topics included. Second, the number of problems on each topic was determined. Third, the number of Hues of introductory description was counted. Fourth, the teaching devices were counted. The texts were then compared in order to form an estimate of the de- gree of agreement which existed among them. It was immediately apparent that there is agreement on the part of the writers as to certain topics which are to be taught. All of these authors present addition, multi- plication, division, subtraction, denominate numbers, fractions, percentage, measureijient, interest, and the like. Two of the authors, however, would use no topics under the head of banking and exchange. Three had selected no topics under the head of discount. On the other hand, one text had selected topics in the field of insurance, taxes, public expenditures, transportation, and the like. The children in a school system using JUDGING TEXTBOOKS 133 one of these textbooks would have access to material touching these lines, while children in schools using the other texts would be denied the opportunity of coming into contact with such topics, unless the teachers or supervisor consciously introduced them. It would be worth while for each supervisor to make some such analysis as this of the textbook that is now being used, in order to find whether or not topics of questionable value are being presented, and whether important topics are being omitted. A further analysis was made of these books in order to determine the relative emphasis given to the different topics. For example, it was found that the book placing the most emphasis upon addition presented five times as many problems as the book giving the least emphasis to addition. The supervisor should ask himself the question : Do the children under instruction in this particular instance need the material in the proportion indicated by this analysis ? Are there too many prob- lems, or are there not enough problems? Again, the same variation was present in the case in fractions. The book placing the most emphasis upon fractions had almost five times as many problems as the book giving the least emphasis to fractions. What is the cause of this variation? Who is right? Do we need hundreds of problems in fractions, or do we need a few problems only? The supervisor or teacher should know whether 134 THE SUPERVISION OF ARITHMETIC or not the textbook in use provides enough problems for her particular purpose. Certainly it is an unsafe proposition to rely merely upon the textbook makers in this particular. Again, certain texts make a great point of review exercises. One text has sixteen times as many review problems as another. To what is this difference due? The supervisor or teacher should know the extent to which review exercises are profitable for the class under instruction. With this variation in emphasis on the text- books, it is clearly unwise to rely solely upon a textbook. The Only Adequate Standards are Common-sense Standards The supervisor who is intent upon results in arith- metic needs a more secure basis than the foregoing for the judging of textbooks. Adequate scientific standards for judging textbooks have not been determined. It is clear that textbooks should be adopted on the basis of what is in them, on the basis of the selection of topics, the sequence of topics, and the gradation of the material, but fixed and unalterable definitions cannot be deter- mined for these items. They will continue to be modi- fied in light of shifting social needs. It is probable that greater agreement would have been found had more books been studied and compared. The books studied were chosen because they were be- JUDGING TEXTBOOKS I35 lieved to be representative. They at least warrant the assumption that books must continue to be selected upon. the basis of common^sense standards, and these standards usually relate to whether the books are closely organized and teachable. Naturally the application of such standards may permit the incorporation of much obsolete material in the books. This material can be eliminated only when other criteria are used for judging it. Reference must be made to social conditions to determine whether the material in the book is still serv- iceable and reference must be made to the personal needs of children to determine whether the material may be learned most economically. Definite criteria of this sort are not available. CHAPTER X ALGEBRA AND GEOMETRY IN THE GRADES The Recommendations of the Committee op Ten The Committee of Ten, which reported in 1893, rec- ommended instruction in algebraic symbols and in simple equations. It is also stated "that a child's geometrical education should begin as early as possible. — At the age of ten years for the average child systematic in- struction in concrete or experimental geometry should begin and should occupy one hour per week for at least three years." It was not presumed in this report that algebra and geometry should appear as separate sub- jects, but that they should be taught in connection with arithmetic. The recommendations followed closely the prevaiHng practice in Europe. Although consider- able opposition has been voiced from time to time to this part of the report of the Committee of Ten, many mathematicians and superintendents now indorse the early introduction and use of algebraic formulae and certain fundamental concepts of geometry. 136 ALGEBRA AND GEOMETRY IN THE GRADES 137 The Attitude of Superintendents with Refer- ence TO THE Use of Algebraic Symbols The authors, in a recent investigation, sought informa- tion from a large number of superintendents in regard to this point. A little less than half of the superintendents in cities of 4000 and over, and more than one hundred county superintendents, replied to the queries as to whether algebraic symbols should be taught in the grades, and if so, in what grades. Their replies appear in the tables. Opinions were expressed by seven hundred ninety- five superintendents ; five hundred thirty-nine of whom expressed themselves definitely in favor of teaching Algebra symbols in the grade. Two hundred forty-one expressed themselves definitely as being opposed to the teaching of algebraic symbols. Fifteen expressed them- selves doubtfully in regard to the issue. TABLE XLVI Geographical Section Ves No Ques- tion- able All : II III IV V VI VII VIII North Central 182 97 II 290 : 6 7 12 17 32 103 182 North Atlantic 167 68 3 238 ; 4 6 9 15 37 83 167 Western . . 33 20 53 ; I I 7 16 S3 South Central 57 17 74 ; 4 5 6 8 21 so S7 South Atlantic 27 3 30 : I I I 5 10 17 22 Counties 73 36 I no ; s 6 8 10 20 44 66 539 241 IS 795 20 25 37 56 127 318 527 138 THE STJPERVISION OF ARITHMETIC TABLE XLVII (Reduced to per cents) Ques- Yes No tion- able All 11 ill IV V VI Vll vni North Central . . 62.7 33-4 3-9 100 3-.S 3-8 6.6 9-3 17.6 S6.6 100.0 North AUantic . . 70.2 28.5 1-3 100 2-3 3-6 .';-4 8.9 22.1 52.7 lOO.O Western .... 62.3 37-7 0.0 100 0.0 0.0 3-0 3-0 21.2 48.S 1 00.0 South Central . 77.0 23.0 0.0 100 7.0 6.3 lo.s 14.0 36.8 87.7 loo.o South Atlantic . . 90.0 lO.O 0.0 100 .•?-7 3-7 3-7 18.S 37-0 62.9 8I.S Counties .... 66.4 32-7 •9 100 6.9 8.2 10.9 13-7 27.4 60.3 90.4 67.8 30.3 1.9 100 3-7 4.6 7.0 10.4 23-S 59° 97.8 To the left of the dotted line, Table XL VI reads 182 superintendents out of 290 in the North Central states favor instruction in algebra in the grades, 97 are op- posed to it, 1 1 consider it questionable. The figures to the right of the dotted line show in what grades these superintendents think algebra should be taught. It will be observed that the totals to the right of the dotted line do not agree with the totals to the left of the dotted hne. This is due to the fact that a single superintend- ent may have indicated that the subject should ap- pear in more than one grade, and his reply is conse- quently recorded , for each of the grades indicated. It would have been possible for each grade in the North Central states to have received 182 votes, and so on for each of the geographical sections. Two thirds of the superintendents advocate the teach- ALGEBRA AND GEOMETRY IN THE GRADES 139 ing of algebra in some of the grades. Almost one third are opposed to instruction in the subject. These superintendents testified only with reference to the use of algebraic symbols. These tables supply no information whatever as to the extent to which definite operations like simple equations, simultaneous equations, quadratic equations, and the like, are advised. Although a small but insignificant number of superintendents ad- vocate the use of the symbols in the second, third, fourth, fifth, and sixth grades, it is not until the seventh and eighth grades are reached that any considerable number urge such instruction. From Table XL VII, which presents the facts of Table XL VI distributed in per cents, it will be seen that more than one half of those favoring the introduction of the symbols of algebra in the grades maintain that they should appear in the seventh grade, and the vote of this group is almost unanimous in favor of their use in the eighth grade. Attention is directed to the fact that the South Central and South Atlantic states show a slightly larger per- centage of superintendents who favor the use of algebraic symbols than do the other sections of the country. This is of especial interest, as these two groups of adminis- trators urge more strongly than the other groups the use of the symbols in the lower grades. A large percentage of the South Central superintendents mention the sec- I4C? THE SXIPKEVlSlON GF ARITHMETIC ond, third, and fourth grades, and a large percentage of the South Atlantic superintendents mention the fifth and sixth grades as" the grades in which the symbols should appear. Variation in Attitude of Superintendents in Cities oe Different Sizes The query arises as to whether or not the attitude of the superintendents in the large cities is the same as that of the superintendents of the small cities. These data consequently were distributed again in order to reveal differences, if any, in the attitude of the superintendents of schools in cities of different size. TABLE XLVIII, Use of Algebraic Symbols Yes No All In What Grade? n m IV V VI vn VIII 1,000,000 and over 2 2 I I I 200,000 to 999,999 II s 16 2 7 10 100,000 to 199,999 8 3 II I I 2 2 4 8 50,000 to 99,999 18 12 30 I 2 7 18 30,000 to 49,999 29 13 42 I I 3 IS 29 20,000 to 29,999 33 12 45 2 5 5 5 13 22 32 15,000 to 19,999 29 14 3 4b I I 2 2 9 20 29 10,000 to 14,999. S6 36 3 95 2 3 II 36 S6 8,000 to 9,999 SS 2b 2 «3 3 4 8 9 21 35 55 4,000 to 7,999 225 84 6 315 8 8 II 22 44 127 223 466 205 14 685 15 19 29 46 107 274 461 ALGEBRA AND GEOMETRY IN THE GRADES 141 TABLE XLDC (Above table reduced to per cents) Size of City Yes No < ] In What Grade? 1 II III IV V VI VII VIII 1,000,000 and over 200,000 to 909,999 100,000 to 199,999 50,000 to 99.999 30,000 to 49,999 20,000 to 29,999 15,000 to 19,999 10,000 to 14,999 8,000 to 9,999 4,000 to 7,999 100 68.8 72.7 60 69 73-3 63 58.9 66.22 71-4 0.0 31.2 27-3 40.0 31.0 26.7 304 37.9 31.6 26.0 0.0 0.0 0.0 0.0 0.0 0.0 6.6 3.2 2-S 2.0 100 1 0.0 100 1 0.0 100I12.S 100] 0.0 100] 0.0 100] 6.1 100 1 3.4 TOO' 0.0 1001 5.5 100, 3.5 0.0 0.0 I2-S 0.0 0.0 iS-i 3-4 0.0 7-3 3-5 0,0 0.0 0.0 o.c 3-S IS-I 6.8 3-6 I4-S 4-9 50.0 0.0 25.0 ss 3-5 151 6.8 5-4 16.4 9.8 0.0 18.2 25.0 II.O IO-3 39-4 31.0 19.8 38.2 19.6 50.0 63.6 So.o 38.8 SI.6 66.7 69.0 643 63.6 56.4 50 91 100 100 100 100 100 100 100 100 68. 30 2.0 100 1 3.2 4.1 6.3 9.8 22.8 58.8 99 No new facts are revealed for cities of any size by dis- tributing the replies on the basis of the grades in which the symbols are taught. In the form, however, in which the material is presented in this part of the table it will be easy for the reader to be misled. The totals for the grades in Table XL VIII are swelled because some su- per,intendents specified more than one grade. Again the per cents in Table XLIX are based upon the grade dis- tribution of Table XL VIII and not upon the 685 su- perintendents who repUed to the questionnaires. While very Httle of scientific value is known about the actual advantage to be gained from the use of alge- braic symbols in arithmetic, the working supervisor will be inclined to attach significance to the opinion of the 142 THE SUPERVISION OF ARITHMETIC majority of the superintendents throughout the country. In the absence of better data, there is some reason for this sort of response. Mr. Van Houten discovered that only three school superintendents out of 148 required the omission of al- gebra from the course of study, and that a considerable number of them offer one full term of it above the sixth grade. Algebraic Topics Taught in the Grades Superintendents in a number of typical American cities were asked to specify what topics should appear in a graded course in algebra. Their replies are listed in the following tables : TABLE L Topics Taught in Elementary Grade Algebra Per Cent of Cities Teaching Algebra Equation with one unknown Equation with two unknowns Addition and Subtraction Multiplication and Division Factoring Fractions Quadratics . . Schools using a separate text algebra 98.9 37-5 S5-2 46.0 36.0 Si.o 7-3 16.6 This table means that 95 of the cities replying give instruction in equations with one unknown and 36 in ALGEBRA AND GEOMETRY IN THE GRADES 143 equations with two unknowns, 53 in addition and sub- traction, 49 in fractions, 7 in quadratics. It is also shown that 98.9 % of all the cities teaching algebra pro- vide instruction in equations with one unknown, 37.5 % in equations with two unknown, 55.2 % in addition and subtraction, and so on. Sixteen of the schools repre- sented, or almost 17%, use separate texts in algebra. Views of Present-day Writers Recent writers on the subject of teaching of arith- metic, such as Young, Smith, Brown and Coffman, are inchned to favor the use of algebraic symbols. The latter writers say: "In the latter part of the sixth and in the seventh and eighth grades the teacher should not hesitate to introduce letters to represent numbers. It is no more unnatural, after a short time, for the pupil to use 'C to represent the cost, 'I' to represent interest, 'V to represent volume, and 'N' to represent a number than for him to use N. Y. to represent New York or a.m. to represent the time before noon. The pupil should recognize the symbols of algebra as a short- hand method of indicating magnitudes. Arithmetic does not become algebra by the use of letters instead of numbers. The solution of many of the problems of common and decimal fractions, or ratio and proportion, percentage and mensuration, is greatly simplified and abridged by the use of letters for the magnitudes which they represent." 144 THE SUPERVISION OF ARITHMETIC TABLE LI Grade Occotieences op Geometry Grade Cities Pee Cent or 149 7B I 0.6 7A . ... 2 1-3 8B . 4 2.7 8A 19 12.0 Schools offering one semester only i8 12.0 Total schools offering Geometry . . 22 14.8 Schools offering Geometry and not Algebra ... . . 2 TABLE LII Topics in Geometry Taught Topics Cities Percentage of Schools Teaching Geometry Constructive Geometry ... Similar figures Schools using text in Geometry . . 21 II 2 95-4 50.0 lO.O An examination of the second of these tables leads one to suspect that the geometry used is very elementary in character. Only two schools use a textbook in this field. While it can be safely asserted that there is a marked tendency to give instruction in the simpler forms of algebra in the two upper grades, no such statement can be made with reference to geometry. CHAPTER XI PROBLEMS RELATED TO CURRENT BUSINESS LIFE The Articulation of Subjects with Life Reduces Wast?; In every field of human activity efforts are being made to economize time. Usually these efforts result in a standardization of the processes employed in carrying on the work. Frequently economy of effort is secured by the introduction of time and labor saving devices. School officers everywhere are sympathetic with the movement to secure more satisfactory results in less time, but they are not always conscious of the agencies by which this may be accomplished. Certainly one means at the disposal of every superintendent for improving the efficiency of his schools is that of reconstructing the internal organization of the schools. Heretofore superin- tendents have devoted much of their time and energy to this problem, because it was close at hand and they could readily see the beneficial ^results that they were securing by the reorganizations they adopted. Im- 146 THE SUPERVISION OF ARITHMETIC portant as details of management and of organization are, they are entitled to no more consideration and are of no more importance in removing unnecessary waste and friction than are the adjustments of the school to shifting social and industrial conditions. Just to the extent that a proper articulation can be made between the materials of the school and those social situations which they represent, waste may be avoided. The Evolution of Subjects of Study It is a fact that every subject of study had a long period of preconscious evolution before it was consciously formulated. During this time the race was discovering that some of its experiences had a similarity and that these experiences were of service in securing adjustments of a Hke character. Obviously some of these experiences were transient in character and were soon forgotten. Others occurred so frequently and with such intensity and touched the hves of so many people that they were considered as vital and necessary to the Kves of all the people. These experiences were saved, and many of them were eventually incorporated in the various sub- jects of study. Each subject of study, therefore, rep- resents a special attempt at environmental adjust- ment. Each subject of study should give one control over some special phase of his environment. Now the unfortunate thing is that many teachers forget the con- PROBLEMS RELATED TO CURRENT BUSINESS LIFE 147 Crete situations that gave rise to the experiences that are incorporated in any particular study and teach the sub- ject as if it were an end in itself instead of a means to ' an end. Since each subject is merely a series of related problems, it should be taught not only to show its in- tegrity, but so that its facts and processes may function in situations outside the school that are similar to those that gave rise to the materials included within the subject. There is no subject that has suffered more in this par- ticular than arithmetic. Teachers have been chnging to obsolete phases of the subject, i.e., to aspects that are no longer used in the business world, with the hope that they might still be socially serviceable. Some teachers, however, have justified the retention of certain divisions of the subject on disciplinary grounds ; for example, the Euclidean method of finding the greatest common divisor. It is doubtful whether any one should argue for the retention of any subject in the curriculum purely on the ground of its mind-training value, and yet every one desires that the mind be trained. Surely the mind will get as much training from exercise upon materials that are useful as from exercise upon materials that are no longer useful. It is for this very reason that textbook makers in the field of arithmetic and superin- tendents are ehminating much of the material that the business world has ceased to use. Such topics as the 148 THE SUPERVISION OF ARITHMETIC greatest common divisor and the least common multiple (when numbers are not readily factorable), obsolete tables in denominate numbers, troy weight, apothecary weight, circulating decimals, cube root, progressions, compound proportion, and the like have disappeared or are disappearing from our textbooks and courses in arithmetic. Dangers Inherent in the Localization oe Subject Matter Believing that modern social theory had affected the opinions of school superintendents everjrwhere with reference to whether time could be saved by relating the problems in arithmetic more closely, to industries, occu- pations, and the like, a series of questions were sub- mitted to them bearing upon this point. We realize that there are certain dangers in attempting to make such adjustments. No school curriculum should be completely localized, for the reason that many of, those trained in any particular community wiU Hve else- where. Moreover, localized school curriculums would accentuate differences between communities. It seems to us that the primary purpose of the elementary school is to increase the homogeneity of our general population. On the other hand, the children should see that the material included in arithmetic is of service in carrying on the work of the community. problems related to current business lipe 149 The Value of the Socialization of Arithmetic There are some phases of arithmetic that are needed in practically all industries and occupations in every part of the country. These are the things which should receive the greatest emphasis. Seven hundred and sixty-two superintendents furnished testimony as to whether or not time can be saved by securing a closer correlation between the subject matter of arithmetic and the situations in which it is supposed to function. The first two tables show how these answers were dis- tributed. It will be noted that 551 or 72.3% of those repljdng were of the opinion that waste could be reduced by such' an arrangement. As a matter of fact this total should be increased by 9.3 %, as 71 additional indi- viduals, although a Httle uncertain, really belong in the affirmative column. It will be noted that about 13 % were of the opinion that nothing could be gained by such a procedure. The greatest conservatism was shown by the superintendents of the North Central and the North Atlantic states, — ■ where less than 70 % main- tained that time may be saved in this way; but the difference between these geographical areas and the other sections of the United States is not so great as it seems, for it will be noted that 10.4% in the North Central and 9.7% in the North Atlantic states thought that time might be saved by increasing the reality and concreteness of the problems. ISO THE SUPERVISION OF ARITHMETIC TABLE LIU Can Time be Saved by Relating Problems to Industries, Occupations, etc., of the Community? Geographical Divisions Yes No Think so Not Question- able All Cities North Central .... North Atlantic . . Western . . South Central . . South Atlantic Counties . i8o 157 40 61 26 87 31 20 7 3 3 9 28 22 3 10 2 6 II 9 2 I 4 17 17 2 I I 2 267 22s 54 76 32 108 551 73 71 27 40 762 TABLE LIV (Reduced to per cents) Geographical Divisions Yes . No Think so Not Question- able All Crrms North Central .... North Atlantic . . Western . . South Central .... 67.4 67.7 74.1 .80 2 II. 7 8.9 13.0 4.0 9-3 8-3 10.4 9-7 5-5 13.2 6-3 5-6 4.1 4.0 3-7 1-3 0.0 3-7 6.4 7-7 3-7 1-3 3-1 1.9 100 100 100 100 South Atlantic Counties .... 81.3 80.S 100 100 72-3 9.6 9-3 3-5 S-3 100 Important variations in practice are not discoverable until the replies of superintendents are distributed for the different-sized cities, irrespective of their location. But when this is done, it is not clear that one group of cities is more progressive or conservative than another group of cities. It is true that variations occur, but it seems that superintendents of small cities are no more likely to maintain that the proper way of increasing PROBLEMS RELATED TO CURRENT BUSINESS LIFE 151 efficiency in arithmetic is by increasing the social char- acter of the material than are superintendents of large cities. At any rate these tables are descriptive of a tendency that is characteristic of every section of the United States and of superintendents of every sjzed city. TABLE LV Can Time be Saved by Relating Problems to Industries, Occupations, etc., of the Community? Size of City Yes No Think So Not Ques- tionable All Cities 1,000,000 and over 3 3 200,000 to 999,999 9 4 2 I 16 100,000 to 199,999 6 2 I I 10 50,000 to 99,999 24 3 2 I 2 32 30,000 to 49,999 24 2 6 2 3 37 20,000 to 29,999 34 5 2 2 3 4b 15,000 to 19,999 29 7 3 2 I 42 10,000 to 14,999 69 9 5 3 4 90 8,000 to 9,999 57 6 10 3 8 84 4,000 to 7,999 209 26 35 9 IS 294 464 64 65 23 38 554 TABLE LVI (Reduced to per cents) Size of City Yes No Think So Not Qdes- tionXble 1,000,000 and over lOO.O 0.0 0.0 0.0 0.0 200,000 to 999,999 56.2 25.0 12.5 0.0 6.3 100,000 to 199,999 60.0 20.0 0.0 lo.o 1 0.0 50,000 to 99,999 75-0 9.4 6.2 3-2 6.2 30,000 to 49,999 65.0 5-4 16.2 5-4 8.0 20,000 to 29,999 74.0 10.8 4-3 4-3 6.6 15,000 to 19,999 69-5 16.6 7-1 4-7 2-3 10,000 to 14,999 76.7 lO.O 5-5 3-4 4.4 8,000 to 9,999 67,8 7.2 11.8 3-6 9.6 4,000 to 7,999 71. 1 8.8 11.9 3-1 5-1 70.9 9-7 9.9 3-5 6.0 152 the supervision of arithmetic The Social Character of the Material in the Indianapolis Course of Study Many courses of study show a tendency on the part of supervisors to reconstruct the course in arithmetic in accordance with social and business demands. One of the most noteworthy of these is the IndianapoHs course of study in which there is a special attempt at supplement- ing the arithmetic of the seventh and eighth grades with community problems. The problems outHned for the 7 A grade deal with the grocery, meat market, the de- partment store, lumber dealers, the cost of heating and Kghting the home, the cost of furnishings for the home, saving money, banking, interest, real estate, and the loaning of money. In addition to these the eighth-grade list includes problems referring to the dairy and milk department, fire department, city market, city hospital, library, street construction, transportation, insurance, stocks and bonds. In every instance these problems are directly related to local conditions. It could not, how- ever, be maintained that the training that the student is getting in the solution of these problems would be of no service to him elsewhere, as Indianapolis is a tj^ical American city. The cost of food, clothing, and furnish- ings, and the agencies that may be used for the saving of money are very much the same everywhere. IndianapoHs authorities have merely taken advantage PROBLEMS RELATED TO CURRENT BUSINESS LIFE 153 of local conditions for the purpose of teaching arithmetic facts and problems that people universally need. Such a device is calculated not only to increase the interest of students in arithmetic, but to insure the fixation of cer- tain fundamental phases of the subject. The students are encouraged to make original problems and to observe that arithmetic plays an important part in their daily life and in that of the community in general. We are of the opinion that such a device as this economizes time and energy, because it rationalizes the processes and, because the students are immediately conscious of the recurrence of their arithmetical experiences in the world outside. CHAPTER XII TESTS AND RESULTS AS SHOWN BY SPECIAL INVESTIGATIONS A Comparison of the Arithmetical Efficiency of To-day with that of the Past the springfield tests Within recent years there has been much discussion in regard to the! results attained under present-day con- ditions in the teaching of arithmetic as compared with the work of earlier days. For the most part the discus- sion has confined itself to statements of opinion or proof by isolated examples. However, if a supervisor wishes to compare the work of a particular school of to-day with that of a particular school of the past, means are available. A few years ago a set of examination questions with the papers and their markings, which were given in the Springfield, Massachusetts, schools in 1846, were dis- covered. The examination was given to eighty-five pupils in the ninth grade. The average marking of the class of 1846 was 29.4 per cent. In 1905 the eighth- grade class in Springfield, Massachusetts, averaged 65.5 1 54 TESTS AND RESITLTS 155 per cent. The Springfield questions in arithmetic have been given in many cities since they were brought to light. In practically all instances the children of to-day have attained a higher rank than the children in 1846. In Frankfort, Indiana, the class averaged 62.2 per cent; in Albia, Iowa, the class averaged 74.5 per cent. The questions are as follows: 1. Add together the following numbers : Three thousand and nine, twenty-nine, one, three hundred and one, sixty-one, sixteen, seven hundred and two, nine thousand, nineteen and a half, one and a half. 2. Miiltiply 10,008 by 8009. 3. In a town five miles wide and six miles long, how many acres ? 4. How many steps of two and, a half feet each will a person take in walking one mile? 6. What is one third of 1755? 6. A boy bought three dozen of oranges for 375 cents, and sold them for 15 cents apiece. What would he have gained if he had sold them for 25 cents apiece? 7. There is a certain number, one third of which exceeds one fourth of it by two. What is the number? 8. What is the simple interest of $ 1 200 for 1 2 years, 1 1 months, and 2g days, at six per cent ? THE PIONEER INVESTIGATION In 1902, Dr. J. M. Rice,^ who had earHer interested himself in testing the results secured in other subjects, ' J. M. Rice, Scientific Management in Education. Hinds, Noble, & Eldredge. New York, 1903. 156 THE SXIPERVISION OF ARITHMETIC gave a test to 6000 grammar-grade children distributed through eighteen school buildings in seven cities. He examined each of the pupils in the fourth, fifth, sixth, seventh, and eighth grades by means of eight examples. Concerning these examples. Dr. Rice says, "In prepar- ing my questions, I endeavored to arrange them in a way that would suit the individual grades of all schools, regardless of the methods or systems employed. From this standpoint I was successful, excepting that in a very few instances two of the examples were beyond the scope of the pupils in the first half of the fourth year, because they had not yet learned to multiply or divide with figures above twelve, and in the first half of the seventh year, where the classes had not yet had much practice in percentage. These points were carefully noted; but when the papers were marked, it was found that the effect upon the entire school average would not in any case exceed 2 per cent. I wish to add, furthermore, that for the purpose of studying the growth of mental power from year to year, some of the problems were carried through several grades. Thus, of the eight questions for the fourth grade, five were repeated in the fifth and three in the sixth, etc. Moreover, this repetition will enable us to see not only, for instance, how the results in the fifth and sixth grades, in regard to certain prob- lems, compare with those of the fourth in the same school, but also how the results in the fourth grade of some TESTS AND RESULTS 157 schools compare in these examples with those of the later grades of others, etc. " The problems for each were as follows : 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 seU them at 5 cents apiece, how much money do I make ? FrFTH 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 f 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 ? IS8 THE SUPERVISION OF ARITHMETIC 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 s 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 nauch did he make ? 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 1 2 yards each did he have left ? 8. Frank had $3.08. He spent J of it for a cap, f 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 win 24 quarts of cream cost at f 1.20 a gallon? 3. If I buy 8 dozen pencils at 37 cents a dozen, and sell them at s 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 3 if 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 TESTS AND RESULTS ^ 159 weight of salt water was it necessary to evaporate in order to obtain 3896 pounds of salt? 8. A gentleman gave away ^ of the books in his library, lent 5 of the remainder, and sold 5 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 weight of salt water was it necessary to evaporate in order to ob- tain 3896 pounds of salt? 3. A gentleman gave away j of the books in his library, lent g of the remainder, and sold J 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 but- ter at $0.27 a pound. How many pounds of butter was she ob- ligated to give? 6. 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 i inch of rag carpet a yard wide in 4 minutes, how many hours wiU she be obligated to work in order to weave the carpet for a room 24 feet long and 24 feet wide ? No deduction is to be made for waste. l6o THE SXIPERVISION OF ARITHMETIC 8. A fruit dealer bought 300 apples at the rate of s for a cent, and 300 at 4 for a cent. He sold them at the rate of 8 for 5 cents. What per cent did he gain on the investment ? EIGHTH YEAR 1. If a map 10 inches wide and 16 inches long is made on a scale of so miles to the inch, what is the area in square mUes 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 } of the books in his Ubrary, lent 5 of the remainder, and sold 5 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 ? 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? 6. A fruit dealer bought 300 apples at the rate of 5 for a cent, and 300 at 4 for a cent. He sold them at the rate of 8 for 5" cents. What per cent did he gain on his investment ? 7. The insurance of | 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 nitre 15 parts, charcoal 3 parts, and sulphur 2 parts. How much of each in 360 pounds of gun- powder ? I The average for each city is presented in the following distribution : TESTS AND RESULTS i6i Grades School Aver- age IV V VI VII VIII Results Results Results Results Results Results Minutes Daily City III . . City I . . City I . . City I City I . . City II . . City III City IV City IV . . City IV . City IV City V City VI . City VI . City VI City VII City VII City VII . . 68.4 72.7 S4-S 60.0 81.3 70.1 70.S 62.9 59-8 53-S 38.S 28.1 41.6 36-8 59-3 47-4 41. 1 79-S 84.7 80,3 74-7 70.8 78.2 53-6 73-2 70.S 65.3 53-5 67.0 38.1 45-3 55-0 53-7 654 37-5 79-3 80.4 80.9 72.2,- 69.6 71.2 43-7 58.9 S9-8 54-9 42.3 44.1 68.3 46.1" 34.5 35-2 35-2 27.6 81. 1 64.2 43-S 63-5 54-6 33-6 53-9 31-2 3S-2 16.1 29.2 33-5 I9-S 30-5 29.1 15-0 8.9 91.7 80.9 72.7 74-5 66.5 36-8 5I-I 41.6 22.5 43-S 48.7 Si-i 26.9 30.2 23-3 25-1 19.6 II-3 80.0 76.6 69-3 67.8 64-3 60.2 54-5 5S-I 53-9 Si-S 42.8 45-9 39-0 36-5 36.0 40.5 36.5 25-3 53 60 45 45 45 60 60 60 60 40 33 30 48 42 45 45 General Average . 59-5 69.4 60.7 39-4 49.4 55-7 — Number of pupils examined Total, 5963 Mr. Rice found great variation in the reaction of pupils to these questions. The variability was even greater in the advanced grades than in the earlier grades. In the seventh grade the class averages ranged from as low as 8.9 per cent to as high as 81. i per cent, and in the eighth grade the range was from 11. 3 per cent to 91.7 per cent. Not only were the extremes l62 THE SUPERVISION OF ARITHMETIC / widely separated, but "the averages for schools t^ken as a whole varied between 25 and 80 per cent." While it is true that wide variation was found in" each school, yet Mr. Rice found that the performance of the different schools in the same city were highly similar; that is to say, if one school in a system ranked high, all the other schools in the system showed a similar ranking, and if one school ranked low, the system probably ranked low. Concerning the causes of the differences in per- formance, Mr. Rice proposes that "two factors must be taken into cionsideration : first, the influence of the teaching; and, secondly, the resistance against that influence due to circurqstances over which the teacher has no direct control. . . . Analysis of the problem will show that the essential elements of which it is com- posed (resistance) do not exceed three in number : (i) the home environment of the pupils; (2) the size of the classes; and (3) the average age of the children." After Mr. Rice's detailed study of the specific effect of each of these factors, he says : " A study of the figures . . . will show conclusively that the influence of aU these fac- tors has been very much exaggerated, and, therefore, that the cause of unfavorable results must be sought, largely at least, on the pedagogical side. . . . That is to say, the school laboring under the poorest conditions in respect to home environment obtained a better standing than any one of. the so-called aristocratic schools. . . . TESTS AND RESULTS 163 Equally surprising, if indeed not more incredible, may appear the statement that no allowance whatever is to he made for the size of the class in judging the results of my tests. . . . The number of pupils per class was larger in the highest six schools than it was in the schools of City VI, and the classes were exceptionally small in the school that stands at the lower end. ... A glance at the ages will show that the average age of the pupils of the schools that showed the best results was about JBive months higher than that of the pupils of the schools that did the poorest. . . . But the factor of age may be completely eliminated by compar- ing the results of a given grade of the successful schools with those of a higher grade of the unsuccessful ones. . . . These facts certainly constitute a striking blow at the theory of those who believe that arithmetic is a matter of natural evolution." Mr. Rice, in the further analysis of the factors affect- ing the efficiency of the arithmetical performances of the children on the basis of the time expenditure, stated the fpUowing conclusions : "A glance at the figures will tell us at once that there is no direct relation between time and results; that special pressure does not neces- sarily lead to success, and conversely, that lack of pres- sure does not necessarily mean failure. " In the first place, it is interesting to note that the amount of time devoted to arithmetic in the school that 164 THE SUPERVISION OF ARITHMETIC obtained the lowest average, 25 per cent, was practi- cally the same as it was in the one where the highest average, 80 per cent, was obtained. From these few facts two important deductions may be made : first, that the unsatisfactory results cannot be accounted for on the ground of insufficient instruction ; and, secondly, that the school showing the favorable results cannot be accused of having made a fetich of arithmetic." The supervisor is especially interested in the analysis which Mr. Rice makes of the pedagogical aspects of the problem. He discusses this from two points of view: first, the part played by the teacher; and secondly, the part played by the supervisor. Concerning the former he says: "The elements brought into play by the teacher, though numerous, may be, for practical purposes, resolved into three primary factors : " I. The time devoted to arithmetic; "2. The methods of instruction; and "3. Teaching abihty, as represented by a combina- tion of education, training, and the personality of the teacher." Concerning the time element, Mr. Rice, in addition to the results presented earlier in this chapter, says: "In view of the results and of my interview with prin- cipals and teachers, I feel confident that home-work in arithmetic means a tax upon the time and energy of the pupil, for which he receives very meager, if any, TESTS AND RESULTS 165 compensation. Consequently, I wish to add to my suggestions, as to the amount of time to be appor- tioned to arithmetic, that the forty-five minutes daily should stand for the preparation and recitation com- bined. "Secondly, methods of teaching can certainly not be looked upon as the controlling element. ... In the schools that passed my test satisfactorily no special methods had been in use. . . . Thoroughness is, undoubtedly, one of the secrets of success. "Variations in the results cannot be accounted for by the dif- ferences in the general qualities of the teacher. Few will take exception to the statement that marked individual variations will be found among the members of every corps of teachers. Therefore, if general ability were the controlling factor, the ex- treme variations in results should be found in the different class- rooms of the same locality. But this condition does not appear in the table, where it is shown that in certain localities practically all the results were good, while in certain other cities practically 'all the results were poor." Thus Mr. Rice disposes of the factors of time, methods of instruction, and special teaching abihty; his con- clusion being that the variability in results does not parallel variability in these factors. He next attempts to analyze the results of the test with a -view of deter- mining the effect of their supervisory factors. Con- cerning this he says : "The facts have led me to beheve that it is here that the controlUng factor lies. My conviction is based on the circumstance that, in every instance, a variation in the results appears to accord 1 66 THE SUPERVISION OF ARITHMETIC with a variation in a special phase of the supervision. If my interpretation of the facts is correct, we are forced to conclude that the results secured in the average classroom do not represent the powers of the average teacher, but the response to what is expected of her; so that, ultimately, the problem of results becomes a question of demand and supply. And my deduction is this, that the teachers will supply what their super- visors demand, provided the demand be placed within reasonable bounds." Mr. Rice differentiated the functions of the super- intendent under five heads : 1. The preparation of the course of study; 2. The apportionment of time to the individual subjects; 3. Offering suggestions to teachers, during meetings and visits, as to methods of teaching and the treatment of children ; 4. The establishment of demands in regard to results ; and 5. The testing for results to see whether the teachers are living up to these demands/ After analyzing the varying factors, Mr. Rice made the following generalization: "The controlling factor in the accomplishment of results is to be found in the system of examinations employed, some systems lead- ing to better results than others. . . . The controlling element Hes, therefore, in that form of examination which is intended as a test of the teacher's progress." These tests were classified as follows : TESTS AND RESULTS 1 67 1. Tests made from time to time by the teachers themselves. Each teacher formulates her own questions, marks the papers of her own class, and submits the results to the superintendent ; but no tests are made by principal or superintendent. 2. Tests made in the same way by the teachers; but the teachep' tests are supplemented from time to time by those of the superintendent. 3. Tests made from time to time by the principals, each principal formulating the questions for his own school. The results are reported to the superintendent, but the latter does not make any tests of his own. 4. The same system of testing by the principals; but the principals' tests are supplemented from time to time by those of the superintendent. Of those foregoing classes of tests, Mr. Rice selects the fourth as being the system best calculated to attain a high standard of efficiency. The supervisor of the teaching of arithmetic finds much in the investigation of Mr. Rice to encourage closer supervision in the teaching of arithmetia His conclusions to the effect that results are more directly traceable to the supervisor's activity than to any other one cause, are of great significance not only to the superintendent but also to the teacher. Supervisors will do well to experiment definitely with the different types of tests which Mr. Rice proposes. It may be readily seen that one of the most difficult problems is that of determining the standard of results which are to be demanded of children in the different grades. 1 68 THE SUPERVISION OF ARITHMETIC Abilities of Children in the 6 A Grade in the Fundamentals and also in Reasoning Dr. C. W. Stone ^ in 1908 gave the same test under as nearly uniform conditions as possible to children in the 6 A grade distributed throughout twenty-six different cities in the United States. Some of the exercises were so planned as to test the ability of children in the fundamental operations. The other part of the ex- amination tested the reasoning abihty of the children. A time limit of twelve minutes was given for the solution of the fundamental problems, and fifteen minutes for the reasoning problems. The Ust of prob- lems in each case was so long as to make it impossible for any student to solve all of them in the time given. Each child had a chance to do all he could in the time allotted. Ordinarily, examinations are arranged in such a way as to enable the bright pupil to finish his work before the time is up, and as a consequence it is never known just how much more that pupil could have done in the time allotted had he worked at his maxi- mum speed. On the other hand, the slow pupil finds it impossible to finish. In other words, the ordinary examination tends to conceal the extreme range of va- riation which actually exists in the ordinary class. It is as though we were to announce that every one of a 1 C. W. Stone. " Arithmetical Abilities and Some Factors Determin- ing Them." Teachers College Record. New York City. igio. TESTS AND RESULTS 1 69 given group should run a hundred yards in fourteen seconds. Swift runners would cover the distance in less than fourteen seconds, but slow runners would re- quire a longer time. Dr.- Stone scored all the papers aUke. The rehability of system of scoring used was checked up by a prehmi- nary test. Interest at once attaches to the results attained by the 6 A children in the different cities. The fact that children in the same grade, distributed throughout the different cities, were tested by the same person, using the same questions, with the same time limit, and the same system of scoring, enables Dr. Stone to compare the arithmetical ability of the children in the different cities. It also enabled him to compare the results in one city with those in another city. Comparisons of this sort are of very great value for supervisory purposes. While they do not supply the supervisor with a remedy when he discovers that his schools are far above or far below the achievement of other schools, the comparisons at least reveal the place that requires a more careful study and diagnosis. As a matter of fact these comparisons did turn the attention of Dr. Stone to the study of the curriculum, methods of instruction, and supervisory helps for an explanation of the variation. The following table shows the ranking of the different systems, the range of variability of the systems as a whole. 170 THE SUPERVISION OF ARITHMETIC ACHIEVEMENTS OF THE SYSTEMS AS SYSTEMS MEASURED BY SCORES MADE TABLE I Scores of the Twenty-six Systems m Reasoning with Deviations FROM the Median. Scores from All Problems. M ='SSi. Systems in Order OF Achievement Scores Made Deviations from THE Median Deviations in Per Cent op the Median xxm .... 3S6 -19s -35 XXIV . 429 — 122 —22 XVII 444 -107 -19 rv. . . 464 -87 -16 XXV . 464 -87 -16 XXII. 468 -83 -IS XVI . > . 469 -82 -IS XX . . 491 -60 — II XVIII . 509 -42 -8 XV . . 532 -19 -3 Ill . . S33 -18 -3 VIII . . S38 -13 —2 VI. . . 55° — I — 2 I . SS2 I 2 X . . . 601 5° 9 II . . . 61S 64 12 XXI 627 76 14 XIII 636 8S IS XIV . . 661 no 19 DC. . . 691 140 20 VII . . 734 183 33 XII . . 736 i8s 34 XI. . . 7S9 208 38 XXVI 791 240 44 XIX . . 848 297 S4 V 914 363 66 TESTS AND RESULTS 171 TABLE II SCOKES OF THE TWENTY-SDC SYSTEMS IN FXJNDAMENTALS WITH DEVIA- TIONS FROM THE Median. Scores from all Problems. M = 3111. Systems in Order OF Achievement Scores Made Deviations prom THE Median Deviations in Per Cent of the Median XXIII .... 1841 — 1270 -41 XXV 2167 -944 -30 XX . 2168 -943 -30 XXII 2311 -800 -26 VIII 2747 -364 — 12 X 2749 -362 — 12 XV 2779 -332 — II m 2845 -266 -8 I 2935 -176 -6 XXI .... 2951 -160 -s II . 2958 -153 -s XVII .... 3042 -69 — 2 XIII 3049 -62 — 2 VI 3173 62 2 XI. . 3261 150 S IX 3404 293 9 XII 3410 299 10 XXIV . . 3513 402 13 XIV 3561 450 14 rv 3563 452 14 V 3S69 ' 458 15 XXVI . . . 3682 S7I 18 XVI . 3707 596 19 XVIII . . 3758 647 21 VII . . 3782 671 22 XDC . . 4099 988 31 It is important to note the wide range of ability dis- played. The supervisor is at once concerned as to the possible cause for the wide range of performance by children in the same grade. The supervisor asks, Is 172 THE SXJPERVISION OF ARITHMETIC this due to difference in time allotment, to difference in method, to difference in material, to difference in children, or to difference in supervision? Dr. Stone distributed his data so as jto show the rela- tionship which existed between achievement in arith- metic and time allotment. His conclusion follows: For these systems it is evident that there is practically no relation between time expenditure and arithmetical abilities; and, in view of the representative nature of these twenty-six systems, it is probable that this lack of relationship is the rule the coimtry over. This is not to say that a certain amount of time is not essential to the production of arithmetical abilities; nor that, given the same other factors, operating equally well, the product will not increase somewhat with an increased time expenditure. What is claimed is that, as present practice goes, a large amount of time spent on arithmetic is no guarantee of a high degree of efficiency. If one were to choose at random among the schools with more than the median time given to arithmetic, the chances are about equal that he would get a school with an inferior product, and conversely, if one were to choose among the schools with less than the median time cost, the chances are about equal that he would get a school with a superior product in arith- metic. Dr. Stone also attempted to compare the rankings of the performance of the 6 A children with the rankings of the courses of study. So far as he was able to deter- mine, there seemed to be little if any relation between these. This investigation corroborates that of Mr. Rice to the effect that the most important factor in TESTS AND RESULTS 173 attaining eflEiciency in arithmetic is that of close super- vision of the teacher. With reference to the relationships existing between the different abihties measured, Dr. Stone says: "It seems safe to say tentatively of the fundamentals that the possession of abihty in addition is the least guarantee of the possession of abihty in others ; that the posses- sion of abihty in multiphcation is the best guarantee of the possession in others ; and that this probably means that multiphcation is hke addition on its mechanical side and hke division on its thinking side. Hence, if it is desired to measure abihties in fundamentals by a single test, one in multiphcation would be best; and a test in division would probably be the best single measure of arithmetical abihty." All of which tends to confirm the thesis proposed by Thorndike and others ; namely, that there is no such thing as general ability in arithmetic. The fact that a child is able to do one thing in arithmetic is no guarantee that he can do other things that are apparently very similar. In other words, a supervisor of the teaching of arithmetic cannot be satisfied until he knows the degree of proficiency which each child possesses in each of the desired arith- metical abihties. Many supervisors have been interested in applying the same tests to their own 6 A grade children. For the benefit of others, who may wish to try this experiment, 174 THE Stn-ERVISION OF ARITHMETIC we print the questions used in this investigation. For the close student of the problem nothing short of a detailed study of this experiment will suffice. THE TESTS AS GIVEN Work as many of these problems as you have time for ; work n i in order as numbered : 1. Add 237s 4052 6354 260 S041 1543 2. Multiply 3265 by 20. 3. Divide 3328 by 64. 4. Add 596 428 94 75 302 64s 984 6. Multiply 768 by 604. 897 6. Divide 1918962 by 543. 7. Add 469s 872 7948 6786 567 858 9447 7499 TESTS AND RESULTS 175 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. 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-doUar bill ? 2. John sold 4 Saturday Evening Posts at 5 cents each. He kept J the money and with the other 5 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 imiforms 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 2200 pupils; J are in the primary grades, 3 in the grammar grades, | in the High School and the rest in the night school. How many pupils are there in the night school ? 7. If 35 tons of coal cost $ 21, what wfll 5I tons cost? 8. A newsdealer bought some magazines for $1. He sold them for $ 1.20, gaining 5 cents on each magazine. How many magazines were there? 176 THE SUPERVISION OF ARITHMETIC 9. A girl spent 5 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 buttonholes. One makes 42, and the other 28. How shall they divide the money? 11. Mr. Brown paid one third of the cost of -a biulding; Mr. Johnson paid 5 the cost. Mr. Johnson received $ 500 more annual rent than Mr. Brown. How much did each receive? 12. A freight train left i^lba-Qy ioi New York at 6 o'clock. An express left on the same track at 8 o'clock. It went at the rate of 40 rmles an hour. At what time of day wiU it overtake the freight train if the freight train stops after it has gone 56 mUes ? Additional Tests for Fundamentals and Reasoning Superintendent Giles ^ of Richmond, Indiana, pre- pared a series of simple tests or formulae by which any teacher might at any time measure, within reasonable Hmits, the ability of any school or pupil in the funda- mentals of arithmetic and in reasoning power with abstract numbers. The object of the tests or formulas, according to Superintendent Giles, is to determine the average percentage of accuracy, together with a meas- ure of the variation in the various fundamental opera- tions and in reasoning. The conditions imder which the proposed tests are to be given are as follows : ' J. T. Giles, " The Scientific Study of Arithmetic Work in School. " N. E. A. igi2,-pp. 488-492. TESTS AND RESULTS 177 1. Time for each test, from four to five minutes. 2. The tests are to be previously written on the blackboard, plainly, in a good light, and covered by a screen or curtain until the tests begin. 3. The tests are to be arranged in rows and columns, each row containing ten problems and a sufficient number of rows given to supply more problems than can be worked by any pupil in the time allowed. 4. Pupils write on papers previously prepared in cross-sections ten wide to correspond with the arrangement of problems on the board. 6. AU pupils to begin writing, giving answers only, as soon as the curtain is drawn and continue until curtain falls. 6. The numbers composing the test to be drawn by the teacher by lot in the following manner: Fifty strips of paper are pre- pared of uniform size, on each of which is written at regular intervals the natural series of digits omitting o and i. These strips are then cut into pieces of uniform size, each containing a digit of the series 2-9. These digits are then placed in a bag and drawn one by one, as needed in formulating the test. This arrangement insures an equal number of each of the digits to draw from. Zero and i are omitted to avoid the possibOity of chance combinations of exceptional ease of solution. Tests four minutes long and over, formed in this way, woidd vary but slightly in the degree of difficulty. 7. Where a chance drawing would result in an absurdity, as in the subtraction of a larger from a smaller number, the order of the last two digits drawn should be reversed. 8. The operation to be tested in each exercise is to be explained to the class by the teacher before drawing the curtain. It should also be written above the test as well as indicated by placing the digits in the position usually adopted for performing the various N 178 THE SUPERVISION OF ARITHMETIC operations, i.e., one above the other, with a line below in addition, subtraction, and multiplication and the usual arrangement for short division, omitting all the operation signs +, — , X, -^. 9. Pupils may grade and mark their own papers, which should be checked later by the teacher. From these marks the average or median accomplishment of the school can be quickly obtained and a measure of the variation easily derived. The subject matter of the proposed tests is as follows : (i) Addition in single combinations, also double column with two figures in column'. (2) The same arrangement for subtraction. (3) Multiplication in which both factors are single digits, also when the multiplicand is composed of two digits. (4) Short division in which the divisor is a single digit and the dividend two, also with three digits in the dividend. The reasoning tests include (i) three-quantity one-step abstract problems and (2) four-quantity two-step problems. The formulae may be expressed by letters which are to be re- placed with the digits drawn from the bag. Two letters written together mean a number of two digits, not multipUcation as in algebra. For addition we have : Find the sum : a c e g b d f h etc. Also ab ej ij cd gh kl etc. For subtraction Find the difference : a c e I g i b d / ft i etc. TESTS AND RESULTS 179 ab ef ij cd g_k kl Multiplication : Write products ; a c e S b d I h etc. ab de gh c f i etc. Division : a)bc d)ef g)hi, etc. a)bcd e)fgh i)jkl, etc. One-step reasoning formulae : a + b = c axb = c a -^ b = c from which are derived the following problems : Indicate by the abbreviation add., sub., mul., or div. the opera- tion to be performed in the two given numbers to get the required one. 1. The sum of two numbers is a. One is b, what is the other? 2. The difference of two numbers is a. One is b, what is the other? 3. The product of two numbers is a. One is 6, what is the other ? 4. The quotient of two numbers is a. One is b, what is the other? 5. What number added to b gives c ? 6. What number subtracted from 6 gives c ? 7. What number multiplied by b gives c ? 8. What number divided by b gives c ? 9. a added to what number gives c ? l8o THE SUPERVISION OF ARITHMETIC 10. a subtracted from what number gives c ? 11. a multiplied by what number gives c ? 12. a divided by what number gives c ? 13. o is 6 more than what number ? 14. a is J less than what number? 15. o is 6 times what number? 16. a is 6 divided by what number? The problems in the two-step reasoning test are derived from the formulas : a + b = c — d ab = cd (a + b) = cd 1. What number added to a is equal to the sum of c and d ? 2. What number subtracted from a is equal to the sum of c and df 3 . What number added to a is equal to the difference of c and d ? 4. What number subtracted from a is equal to the difference of c and d ? 5. What number multiplied by a is equal to the product of c and d ? 6. What number multiplied by a is equal to the sum of c and d ? 7. What number multiplied by a is equal to the difference of c and d ? 8. What number added to a is equal to the product of c and d ? 9. What number subtracted from a is equal to the product of c and d ? 10. What number multiplied by the sum of a and 6 equals c ? 11. What number multiplied by the difference of a and b equals c ? 12. What number divided into the sum of a and b equals c ? 13. What number divided into the difference of a and b equals c ? TESTS AND RESULTS l8l The order in which the reasoning problems are given is also to be determined by lot, two or more sets being used if necessary to provide sufficient problems for the time allowed. It will be observed that the tests are inexpensive, easily manipulated, and readily interpreted. As yet, however, they are of little value for comparing one school or one grade with another, for they have not been adequately standardized. Their ultimate standardiza- tion will require that a number of superintendents in different parts of the country cooperate in giving them and in sending the results to some common clearing house for tabulation. The standards derived may then be utilized in determining the relative station of any individual, class, or school. It will be better, and in the long run far more economical, for tests of this sort to receive a universal trial than for a multitude of new ones to be constructed. We can hasten the day when definite standards will supplant opinion by duplicating the work of others in enough places to make the results universally vahd. Experiment on Way of Gaining Facility in the Use of the Multiplication Tables Supervisors have been more or less seriously interested in a more economical way of teaching children the multiphcation tables. Many ways have been proposed, 152 THE SUPERVISION OF ARITHMETIC varying from that of having the children "stay in" after school in order to write the tables a given number of times, to that of teaching the tables purely inciden- tally. There have been very few experiments performed calculated to throw any hght on this problem. Mr. E. A. Kirkpatrick of the State Normal School at Fitchburg, Massachusetts, published ^ in 1914 an interesting experiment bearing upon this problem. In his study he recognized three characteristic ways of learning the tables : first, having the children memorize the multiplication tables in the traditional way ; second, having the children placed in possession of a multiplica- tion sheet which they could use as they needed, with the hope that after using it awhile, they would remember the combinations well enough so that they would be able to "work" the problems without the use of the sheet. He proposed as a third method that each child be taught to derive multiphcation combinations from a knowledge of the combinations in addition, — "figure out each combination as he needed it." As Mr. Kirkpatrick dealt with students in the Normal School who already knew the multiplication tables, he experimented upon the different ways of learning a new set of products, "of seven by the prime numbers from 17 to 53, inclusive, and the experiment was conducted 1 E. A. Kirkpatrick, "An Experiment on Memorizing vs. Incidental Learning." Journal of Educational Psychology, Vol. V., pp. 405-413. TESTS AND RESULTS 183 to determine the relative advantage of three methods of learning these combinations. A practice sheet or test sheet was prepared containing the prime numbers above named with a smaller figure seven beneath them arranged in lo hnes of lo such groups of figures each. Another sheet, known as the key sheet, indicated the products that could be substituted for each group of numbers. In two methods of testing the pupils were not informed that the numbers on the key sheet were products, and the majority of them did not discover the fact, and very few of those who did made use of their knowledge." Mr. Elirkpatrick then experimented upon two classes of young men in the normal school. After a preliminary test, one group practiced while the other group memo- rized for four or five days and then began practicing, using the key sheet if they needed it. . On the tenth day of the experiment, the fifth or sixth day of practice for the memorizers, their record of 17.2 seconds was slower than the fifth or sixth grade of practice for the practicers by 3.1 seconds. After an interval of about three weeks, during which nothing was done with the experiment, a final test was given. In this test they wrote as many products as possible without the key sheet in two minutes. The memory group wrote on an average of 40.9 answers, and the practicing group 46.2. It appears, there- fore, that from every point of view the results averaged better for the practice group than for the memory group. The next test was tried with two classes of freshmen normal 184 THE SUPERVISION OF ARITHMETIC school Students of about 25 each, all of them young ladies except four. One class, practiced, using the key as previously described, for eight days, while the other had no key, but multiphed the numbers and wrote the answers as rapidly as possible the same number of days. In the final test the average number of answers written in two minutes by the group practicing with the key was 25.4, whUe the computing group wrote 44.3 answers. The best of those practicing with the key were nearly, but not quite, as good as the best in the computing group, but only one wrote fifty or more answers and six not over twenty, while in the group that computed seven wrote fifty or more and only one less than twenty. Those who had memorized the key were helpless in the final efficiency test. Only two or three knew they could get the answers by multiplying. Those in the computing group were perfectly at home and only a little slower than those who incidentally memo- rized the final products. Mr. Kirkpatrick experimented in a similar way with children, but owing to an epidemic, the conditions were not quite under control. However, Mr. Kirkpatrick made the following conclusions : The result in the final efficiency test of two minutes was that the number of products written by the computers was 27.7 ; by the practicers, 19.1; by the fifth-grade memorizers, lo.i, and by the sixth-grade memorizers and practicers, 27.4. While Mr. Kirkpatrick does not maintain that his experiments are conclusive, yet he does feel that they are highly suggestive. He says that "it seems that memorizing apart from use is the poorest method of aU, drill in using somewhat better, while the method of using previous knowledge as a guide in practice is the TESTS AND RESULTS l8S best of the three. The results indicate that in many lines of teaching there has been a tremendous waste of time, energy, and interest in first memorizing, then later practicing, the use of what has been learned. Also that pupils do better when practice is guided by their own knowledge than when it is directed by author- ity of book or teacher. It is probable that at least a year in numbers is wasted in special drill on the various combinations beyond what would be necessary if em- phasis were first placed on learning how to compute tables, than upon working examples and problems with larger numbers, computing products till they were learned incidentally. This last statement has, of course, not been proved experimentally, but is merely an opinion based on inference from general principles and on a good deal of personal observation. Just how much time could be saved by the use of computation and incidental methods in number work as compared with memorizing and special drill methods is not known, but there can be no doubt that there is no need to have children memorize any tables." Practice in the Case oe School Children Two experiments, one in addition and the other in division, were made by Dr. T. J. Kirby^ with 1350 ' T. J. Kirby. " Practice in the Case of School Children." Teachers College Record, Columbia University. 1 86 THE SUPERVISION OF ARITHMETIC children in the schools of the Children's Aid Society, New York City. The work with addition was given to classes in the fourth year of the elementary schools ; the division, to classes in. the last half of the third year and the first half of the fourth year. Thorndike's "Addition Sheets" and the "Remainder Division Tables" were used as a basis for the praptice. These addition sheets, seven in number, each contain 48 columns of one-place numbers so arranged that any successive five columns are of nearly equal difficulty. The division combinations used include the entire series from "20 = — j's and — remainder" up to "89 = — 9's and — remainder," thus involving not only the combinations which are the inverse of the multiphcation tables through 9 times 9, but also the division of the intervening series of numbers which give a remainder in the answer. After an initial practice period of fifteen minutes, each class practiced the addition columns for forty-five minutes and then received a final test of fifteen minutes in length. The first and last practice or tests served as bases for determining the change in ability for each individual measured. The intervening practice of forty- five minutes between initial practice period and the final practice period was broken up for different groups of classes in four different ways. For Group I it was divided into two practice periods of 22 J minutes each ; for Group TESTS AND RESULTS 187 II, into three practice periods of fifteen minutes each ; and for Group III, into eight practice periods, seven of six minutes each and one of three minutes; for Group IV, into twenty-two practice periods, twenty-one of two minutes each and one of three minutes. The following table makes the plan clear: Gistoups Initial Intervening 45 Minutes Final Period I . . . . II ... . III. IV. IS min. IS min. IS min. IS min. 2 22J min. 3 IS min. 7 6 min. and i of 3 min. 21 2 min. and i of 3 min IS min. IS min. IS min. IS min. A similar plan was used in the division experiment. Each class had an initial practice period of ten minutes and a final period of ten minutes ; and the intervening practice of 40 minutes was divided in three different ways. In the division, as in the addition, the initial and final practice periods were identical in character with the intervening practice periods; but besides serving as practice periods, they served as measures of ability at the beginning and end of practice from which any change in abihty was measured. The first group had four intervening practice periods of twenty minutes each ; a second group had four intervening practice periods of ten minutes each ; a third group had twenty intervening practice periods of two minutes each. The following tabular statement presents the plan for each group ; THE SUPERVISION OF ARITHMETIC Groups Initiai, Period Intervening 40 Minutes FiNAi. Periods I II Ill lo rain, lo min. lo min. 2 20 min. 4 10 min. 20 2 min. 10 min. 10 min. 10 min. The pupils were encouraged to do their best, to work as fast as they could without making mistakes. Their papers were collected at the close of each practice and scored in a uniform manner, each column in addition and each quotient and remainder in division counting one. The conditions within the schoolroom and the time practice were kept as nearly uniform as possible from day to day. Dr. Kirby's results show that the 732 fourth-grade children, who received practice in addition, had a median ability at the end of the first fifteen minutes of practice of 23.3 columns added correctly and a median accuracy of 79 per cent. In other words fifty per cent of the pupils added 23.3 per cent or more of the columns while fifty per cent added 23.3 per cent or less of them correctly. A median of 79 per cent of accuracy shows that there were as many children who added correctly four fifths or more of their problems as there were who added cor- rectly four fifths or less. It will be observed that we have referred only to the problems worked correctly; the median number of problems actually worked was 29.5, the difference between the two medians being 6.2. In the final fifteen minutes of practice in addition the TESTS AND RESULTS l8g group added correctly a median of 10.7 more columns than in the initial fifteen minutes period. This meant a median percentile gain of 48 per cent for the practice. That speed and .accuracy are not directly related is shown by the fact that this gain of 10.7 problems added correctly was accompanied by a median loss in accuracy of .4 per cent. This negative relation is more apparent than real, for the pupils actually tried more problems and solved more correctly during the last fifteen minutes of practice. They gained in speed, but while they were doing so their accuracy remained almost at a standstill. The 606 children who took the practice in division showed an initial median ability of 34.5 combinations with a median per cent of accuracy of 93 per cent, and the final practice showed a median ability of 62 com- binations with a median percentile gain in accuracy of 2.6 per cent. We cannot be certain that Dr. Kirby's figures rep- resent norms that may be applied to all third and fourth 'grade children, but they should be accepted until further experimentation refines them. A fact of very great interest to every supervisor is the relative value of drill periods of different lengths. This was the particular problem with which Dr. Kirby was concerned. It will be remembered that he gave the practice in addition to four groups. His results in addition are summarized in the following tables : I go THE SUPERVISION OF ARITHMETIC Medians of the Groups or iNDiyiDUALS Median Initial Average Medlan Median Median Ability. Gross Gross Gain Gain in Examples Gain Gain Per Cent Accuracy CORKECT Group I . . . 22.9 II.O 95 45 35 Group II . . . 2S-4 13-6 II.O 43 1.6 Group m . 21. 1 10.7 9.6 42 i-S Group IV . . . 25.1 16.1 12.6 S6 2.7 In the above tables the individual in the group was considered as the unit; i.e., these averages or medians represent central tendencies secured by tabulating the scores of all the pupils, irrespective of the classes in which they were registered. The table shows that the greatest gain in speed was made by the two-minute drill group, while the only group that made any gain in accuracy was the 225-minute drill group. The central tendency for each class was also computed and the average of these central tendencies was found. These averages corroborate the evidence presented in the preceding table. Average op Class Median Group I Group II Group III Group IV Average Initial Median Ability OF Classes. Examples Correct 23-7 2S-7 21.3 2S-S Average Gross Median Gain ^ of Classes 10.2 9.6 9.4 13-9 Average of Median Percentile Gains 42 41 42 S8 TESTS AND RESULTS 191 Using now the best measure of efi&ciency — the num- ber of examples correct, which includes credit for both speed and accuracy — and the three methods of com- puting the gain, we have : Average Gross Gain of Individuals Median Gross Gain of Individuals Average of Median Gross Gain of Classes Group I Group II Group in Group IV II. o 13.6 10.7 16.1 9-5 II.O 9.6 12.6 10.2 9.6 94 13-9 It therefore appears that, taking the results at their face value, the 2-minute practice produces the best results. The summary in division is : Medians or the Groups of Individuals Group I Group II . Group III . Median Initial Ability 38-3 32.0 40-3 Average Gross Gain Median Gross Gain 25.1 25-S ■ 42.6 22.6 23-S 40.4 Median Gain Per Cent 60 73 94 Median Gain in Accuracy 2.1 3-5 2.3 It will be observed that there was a marked difference in the initial ability of the three groups, with reference to the number of divisions made, although there seems to be little difference in their initial ability. 192 THE SUPERVISION OF ARITHMETIC Group III gained almost twice as many combinations done correctly in the course of tlie practice as did the other two groups, whose gains were practically the same. The above measures were computed from the scores of the individuals comprising these groups. The fol- lowing measures are computed from the scores of the classes in the groups. The median for each class was found ; then the average of these medians was computed. Average oe Class Medians Average Initial Median Ability OF Classes Average Gross Median of Classes Average oe Median Percentile Gains Group I Group 11 Group III 38-4 33-4 41.4 20.6 2S-I 44-7 S8 77 114 Using the three methods of computing gross gain, we have: Group I Group II Group III Average Gross Gain of Individuals 25-1 2S-S 42.6 Median Gross Gain of Individuals 22.6 23-5 40.4 Average of Median Gross Gains OF Classes 20.6 2S-I 44-7 Considering the facts for both addition and division, it ap- pears that, subject to discounts for the inequalities of the groups in initial ability, there is considerable advantage in the short TESTS AND RESULTS 1 93 period lengths, when the length is two minutes. The advantage there is noteworthy, since in addition the gain is greater than in the longer-period groups, even when their ability was greater in the longer period (Group II) ; and since in division the gain is so very much greater than in the 20 or lo-minute group. The facts can be freed from the influence of inequalities in ability at the beginning of practice by comparing only those of equal initial ability. For example, we find in the case of division that those of initial abUity 15 averaged in gain 18.0, 26.3, and 23 according as they had practiced in 20-, 10-, or 2-minute periods. Making such calculations for those of each initial ability in division from 5 to 64 and allowing equal weight to each suc- cessive set of five successive groups, it appears that on the aver- age the 20-minute, lo-minute and 2-minute period varieties of practice brought to those of equal initial ability gains in the relation of 100, no J, and 177. In the case of addition the same procedure, carried out with those on initial abilities 5, 6, 7, and so on up through 49, gives the following results: according as the practice was in 225, 15, 6, or 2 minute divisions, it brought to those of equal initial abUity gains in the relation of 100, 121, loi, and 146I. It appears,. then, that the superiority of the shortest practice period length remains when inequalities of initial ability are eliminated. It appears further that the periods of intermediate length have really a greater superiority over the longest period than the results irrespective of difference in initial ability showed. There is a positive relation between initial ability and gross gain. Consequently Group III in addition and Group II in division, which happened to be groups of low initial ability, suffered in the comparison. Dr. Kirby cautions the reader against the conclusion that the short practice periods are wholly responsible for the difference in gain. He points out that those 194 THE SUPERVISION OF ARITHMETIC having the shortest practice periods had more days in which to profit from the practice and in which to profit from the regular school work, that they had a better opportunity of catching the spirit of the experiment, and that consequently they had a more powerful incentive to practice outside. Regardless of the causes, his material shows indisputably that the short periods pay the best dividends. We are therefore unable to sub- scribe to Dr. Kirby's conclusion that perhaps for ad- ministrative reasons it would be better to use the long periods in school. We see no especial administrative difficulties, certainly none that are insuperable, — in having daily drills of two minutes. This is a case, so it seems to us, where an administrative adjustment should be subordinated to a teaching adjustment. Tests given by Dr. Kirby several months after the final test of the experiment was given, showed a rela- tively high degree of permanence of the practice effect. Brown's Experiment on the Value of Daily Drill ^ how the investigation was conducted Tests were given in the sixth grades of three different public school systems and in the sixth grade of a large private school. The total number of cases recorded in . ' Used with the permission of Brown and CofEman, authors of How to Teach Arithmetic, Ross Peterson & Co. TESTS AND RESULTS 195 this study was 222; of these, no were boys and 112 were girls. The three public schools examined are in the Central West. City C has a population of seven thousand; City M, of twelve thousand; and City D, of thirty thousand. The private school is in New York City. The effects of the drill in fundamentals were shown by a comparison of sections subjected to the drill with sections of equal size and approximately equal abihty not subjected to the drill but otherwise undergoing the same arithmetical instruction. The object was to determine the improvement made by the drill class upon its previous record and the improvement made by the non-drill class upon its previous record. In a given class the tests were conducted at the same hour of the school day, in order to eliminate the time factor as far as possible. Immediately after the first test was given in each school, half of the classes examined in each city were given five minutes' driU each day upon the fundamental operations in arithmetic, — addition, subtraction, multipli- cation, and division. The first five minutes of the recita- tion period in arithmetic were devoted to the drill work. The drill was partly oral and partly written, and the time was about evenly distributed among the four operations. No special instructions were given to the teachers in charge of the drill sections, except that they were to em- 196 THE SUPERVISION OF ARITHMETIC phasize both speed and accuracy in the four operations, and were to cover the same daily assignments in the text- books as the class that had no driU. The teachers of the non-drill classes were asked to give no formal drill upon any of> the four fundamental operations during the time that this investigation was in progress. These instructions were carefully observed by the teachers. The drill classes in each city were able to cover the same subject-matter of the text as the non-driU classes of that city. No special tests were given to determine the comparative excellence of the textbook work, but in every case the teacher in charge of a drill class reported that five minutes devoted to drill at the beginning of each recitation seemed to act as a mental tonic. It seemed to energize the pupils and to make them keen for the textbook work that was to follow. All teachers of drill classes reported an improvement in textbook work. Formal drill work on the four fundamental operations had not been given prior to this investigation in any of the sixth grades examined. Whatever marked changes occurred in all of the drill sections that did not occur in the non-drill sections may reasonably be attributed to the results of the special drill. RESULTS OF THE DRILL If the number of problems worked in each test may be taken as a measure of the speed of the pupils, the drill TESTS AND RESULTS 197 class increased its speed by 16.9 per cent and the non- drill class by 6.4 per cent. Since practically all of the pupils finished at least the first six problems in each test, a comparison of the rec- ords made on these six problems will give a basis for determining the relative accuracy. Measured by this standard, the drill class made a gain of 11. 7 per cent in accuracy, whereas the non-drill class actually lost in accuracy ( — 1.8 per cent). The largest gain made by the drill class was in division, 34.2 per cent, which was more than twice the gain made in division by the non-drill class, 15.4 per cent. If we compare the gain made by the drill class upon its own record with the gain made by the non-drill class upon its own record, we find that the following results were attained : Drill class gained 2.64 times as much as non-drill class on problems worked. Drill class gained 2.72 times as much as non-drill class in addi- tion. Drill class gained 2.68 times as much as non-driU class in sub- traction. Drill class gained 2.21 times as much as non-drill class in multiplication. Drill class gained 3.13 times as much as non-drill class in division. Drill class gained 2.57 times as much as non-drill class in total number of points. ig8 THE SUPERVISION OF ARITHMETIC The drill classes made from two and one fifth to three and one tenth times as much improvement as the non-drill classes. It is worthy of note that the average age in the drill classes was exactly the same as the average age in the non-drill classes, being twelve and two tenths years in each case. In the following table the first test was given before the drill was begun, the second test was given immediately after the thirty- days' drill, and the third test was given on the first day of the fall term, after a vacation of twelve weeks : Comparison of the Results of the Third Test with the First AND Second ( " I " indicates combined drill sections; " II," the non-drill sections.) I did 26.4 per cent better than on first test and 4.1 per cent better than on second test in number of problems worked. II did 9.8 per cent better than on first test and same as on second test in number of problems worked. I did 25.4 per cent better than on first test and 6 per cent poorer than on second test in addition. II did 7.7 per cent better than on first test and 3.7 per cent poorer than on second test in addition. I did 46.2 per cent better than on first test and 6.7 per cent better than on second test in subtraction. II did 20.4 per cent better than on first test and 6.4 per cent better than on second test in subtraction. I did 31.3 per cent better than on first test and 1.5 per cent better than on second test in multipUcation. II did 1 1. 1 per cent better than on first test and 2.2 per cent poorer than on second test in multiphcation. TESTS AND RESULTS IQQ I did 36.7 per cent better than on first test and 7.3 per cent better than on second test in division. II did 1 1. 1 per cent better than on first test and 2.8 per cent poorer than on second test in division. I did 31.7 per cent better than on first test and 0.2 per cent poorer than on second test in total points. II did 12.16 per cent better than on first test and 2.29 per cent poorer than on second test in total points. I did 5.2 per cent better than on first test and 0.6 per cent poorer than on second test in first six problems. II did 3.7 per cent poorer than on first test and 1.3 per cent poorer than on second test in first six problems. The results of the third test indicated that the supe- riority of the drill class was maintained over the vacation period. The "period of hibernation" served to increase the speed, while those who had not had the advantage of the drill worked no faster than on the second test. The non-drill section either made no improvement or did worse than on the second test in everything except subtraction. No investigation has yet been made to determine the relative efhciency of drill periods from one to ten or fifteen minutes, or whether the same length of period is best for each of the fundamental operations. The Ability of Children to Express Mathe- matical Judgments Dr. Frederick G. Bonser ^ of the Teachers College of Columbia University made an investigation of the ' Frederick G. Bonser, ' ' The Reasoning Ability of Children." Teachers College Publications. No. 37. 1910. 200 THE SUPERVISION OF ARITHMETIC reasoning abilities of children. He tested 757 children in the fourth, fifth, and sixth grades in Passaic, New Jersey. He tested the mathematical judgment by giving the children twenty questions in arithmetic involving three steps: "First, the analysis of the situation by which the essential features of the problems are conceived and abstracted; second, the recall of an appropriate prin- ciple to be appHed to the abstracted problem, a search among various principles which may suggest themselves for the right one; and third, involving the second, the inference, the recognition of identity between the known principle and the new situation. Clearly these are examples of deductive reasoning of the usual scien- tific, involving data, principles, and inferences." The tests are as follows : Tests I and II I. A. Get the answers to these problems as quickly as you can. 1. If J of a gallon of oil costs 9 cents, what will 7 gallons cost? 2. John sold 4 sheep for $ 5 each. He kept 2 of the money and with the other ^ he bought lambs at $ 2 each. How many did he buy ? 3. A pint of water weighs a pound. What does a gallon weigh ? 4. At 1 25 cents each, how many more will 6 tablets cost than 10 pens at 5 cents each ? 5. At IS cents a yard, how much wiU 7 feet of cloth cost? TESTS AND RESULTS 20I B. 1. A man whose salary is $20 a week spends $ 14 a week. In how many weeks can he save $ 300 ? 2. How many pencils can you buy for 50 cents at the rate of 2 for five cents? 3. A man bought land for $ 100. He sold it for $ 120, gaining $ 5 an acre. How many acres were there ? 4. A man spent | of his money and had $ 8 left. How much had he at first ? 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? n. A. 1. 32 plus what number equals 36? 2. If John had 15 cents more than he spent to-day, he would have 40 cents. How much did he spend to-day? 3. What number minus 7 equals 23 ? 4. If James had 4 times as much money as George, he would have $16. How much money has George? 5. What number added to 16 gives a number 4 less than 27? B. 1. What number subtracted 12 times from 30 will leave a remainder of 6? 2. If a train travels half a mUe in a minute, what is its rate per hour ? 3. What number minus 16 equals 20? 4. What number doubled equals 2 times three ? 3 ? 6. If 7 multiplied by some number equals 63, what is the number ? 202 THE SUPERVISION OF ARITHMETIC "The results of tests I and II were combined, a single quantity thus representing the summarized valuation of each child's mathe- matical judgment. . . . Below are the tables and summaries of results for grade, age, and sex differences." Frequency of Abilities by Grades ABiLirv Grade 4A sB SA 6B 6A B G B G B G B G B G 2 I I 2 .3 I 3 I 4 4 6 2 2 I S I 3 2 I I 6 6 4 I 4 2 I I I 7 S I 2 I 8 8 4 I 3 I I 9 I 6 I I lO 7 S I S 2 3 I I 2 II 4 3 3 2 I I I 12 7 II 4 9 2 4 3 I I I 13 2 3 I 2 I 2 I I 14 4 4 6 3 3 s 2 3 3 IS .1 S I 3 3 2 I i6 4 7 7 6 2 2 2 6 2 17 2 I I 3 2 2 I 2 i8 8 2 I 2 2 3 I 3 ' 2 19 3 S I S I I 20 6 4 8 9 4 2 4 S 2 3 21 2 I 2 2 4 I 22 2 2 9 S 5 3 6 8 I S 23 6 I I 2 3 2 3 2 I 24 6 3 II 3 3 2 6 6 4 I 25 I 3 2 I 4 I 26 3 I 8 4 S 3 II 4 7 7 27 2 I I I I 3 28 I I I 6 3 II 6 6 29 I I 3 2 T 2 6 30 4 I 2 2 4 8 2 I 31 I 2 I I 3 32 I 4 I 4 2 10 S 6 8 TESTS AND RESULTS 203 The meaning of the foregoing table becomes clear when read thus : In grade 4a, one boy had ability 2, one boy had abiUty 3, four boys had ability 4, and so on down to the case of one boy with ability 32. In order to see more clearly the differences in results attained by boys and girls in the different grades, the following summary has been made : Median Ability and Variability poe Each Grade Grades M's Q's Boys Girls Boys Girls 4A SB SA 6B 6A 14.50 21.39 22.83 25-63 28.00 11.36 15-66 ig.oo 24.08 25.92 S-39 4-75 S-S8 4-95 3-96 4.21 4-71 6.46 6.42 5-95 The table above should be read thus : The median ability of 4a boys was 14.50, 4a girls 11.36. The ability of half of the boys was within 5.39 of the median for the boys. The ability of half of the girls was within 4.21 of the median for the girls. Q is used here as a measure of variability. "It is gotten by counting in from the low end of the distribution until 25 per cent of the cases are covered ; and likewise from the high end of the distribu- tion until the point marking 75 per cent of the cases is reached. These two values give the limit within which exactly 50 per cent of the cases lie. Subtracting the 204 THE SUPERVISION OF ARITHMETIC lower value from the higher and dividing the result by 2 gives the variability in steps of the unit of measure used." It should be noted that the median performance of boys and girls in response to this test of mathematical judgment steadily increases from the fourth grade to the sixth grade. The supervisor here has evidence sufficient to justify a clear differentiation in regard to the standard of mathematical judgment which should be set up for the fourth grade, the fifth grade, or the sixth grade. The difference in ability of boys and girls in each grade is also worthy of note. While it is commonly stated that boys exceed girls in the abihty to express a mathematical judgment, there is little scientific evidence to support this contention. The following summary by Dr. Bonser states the case very clearly: Other than the evident differences in the foregoing, the simplest summary of sex differences is in a statement of the per cents of the boys who reach or exceed the ability reached by 50 per cent of the girls as given below : Grade 4A sB SA 6B 6A Percent . . . 61.8 71-3 70.7 60.8 65.1 Differences in abihty at different ages for boys and girls are also evident, as is indicated by the following table from Dr. Bonser : TESTS AND RESULTS 205 Frequency of Abilities by Ages Abujiy Age 8 B 10 G 10 B IT G II B 12 G 12 B 13 G 13 B 14 G 14 B 16 G 2 I I I 1 3 2 I I I 4 I 4 I I I I 4 I I s 3 I I I 2 6 2 3 3 3 2 I 2 I I I I 7 I 2 3 2 I 8 3 I 3 I I 2 3 3 I 9 3 2 I I I I 10 2 2 S 5 S I 3 I I 2 II I I 2 2 3 2 I I 2 12 S S s 5 S s S 2 4 2 13 3 2 I 2 2 I I 14 4 4 3 3 6 3 3 5 2 IS 16 3 I 5 4 7 3 6 4 8 I 2 2 4 2 I I I 17 2 I 2 S I 2 18 2 4 I 2 7 4 2 I I I 19 I 3 I 4 I I I I 2 I 20 2 I 8 6 4 2 7 9 3 5 I 21 I I 2 2 2 2 I I 22 5 S 6 6 10 8 I 3 I 23 I 2 4 6 2 2 I 3 24 2 2 7 3 8 3 6 3 2 3 S I 2S 2 2 I I I 3 I 26 2 4 7 3 7 3 8 7 4 2 4 27 I I 3 I I I 28 I 5 3 3 3 II I 4 2 2 29 I 2 6 3 I I I 30 3 I 2 7 2 I 5 2 31 I I I 2 2 2 32 2 4 5 4 2 8 8 S 2 2 33 I 2 2 I I I I 34 2 I 2 3 I 2 2 I 3S 36 2 I 2 I I 5 I 2 I 4 I 2 I 37 38 I I I 2 I 2 2 I I I I I I 39 40 I I Cases 27 43 "§7 "87 lOI 93 96 93 S3 46 24 16 206 THE SUPERVISION OF ARITHMETIC After calculating the median and the ages for each sex, Mr. Bonser combined them into the following table : Median Ability and Variability for Each Sex Age M's Q's Boys GlKLS Boys GlBlS 8 to II .... 13 to i6 .... 21.50 23-50 15.20 19.00 6.05 6.98 6.21 7-33 In conclusion Dr. Bonser says : "A study of the median abilities of the respective grades shows progress through these from grade to grade for both boys and girls. . . ; The greatest gain in per cent from one grade to the next for both is from 4 A to 5B. The smallest gain is, for boys, from grade 5B to sA; for girls, from 6B to 6A. ... In these tests, variability diminishes from grade 4A upward." In view of the experiments which other investigators have made, it is likely that the reason that variability diminishes throughout the upper grades is that the children in the upper grades are more highly selected. The upper-grade students are the students who have survived the rigors of the elementary school curriculum. The upper-grade work is also probably more nearly standardized. Concerning age differenced, Dr. Bonser says : "From the array of median abihties of half years, the regularity of progression found on the basis of school grades is not at all in evidence. TESTS AND RESULTS 207 "A rhythm in ability is fairly apparent for the boys with its first crest at about 9 years 6 months, the second at about 12 years, and the third at about 14 years 6 months, each crest a little higher than the preceding. . . . Retardation seems evident in the pupils of each respective grade who are from two to four years older than the median age for that grade. There also seems evident another type of retardation in these special abilities, perhaps quite as important, in those pupils who are from two to three years younger than the median age for their respective grades. ... In tests in which progress from grade to grade and year to year is so very evident, in large groups as here shown, these wide divergencies in ability of lowest and highest quarters of these respective groups indicate that native abUity is measured by the tests quite as much as school training." In regard to sex differences, Dr. Bonser says : "The one marked sex difference is that of the superiority of the, boys in these tests. By every distribution, in every one of its respective divisions, the boys are shown to be more able than the girls excepting in two cases of selected groups, one from grade 4A, and the other from grade 6A. In the youngest 25 per cent of the former grade, the girls just equal the boys in median ability, while for the corresponding group for the latter grade the girls slightly excel. . . . The sex difference diminishes as we proceed up the grades from 4 to 6. Should we proceed far enough (test chil- dren in higher grades), we might reach the condition found by Fox and Thorndike in a study of 28 boys and 49 girls of high school age where ' girls do about 5 per cent better on the whole than boys.' " The percentage of all of the boys reaching or exceed- ing the ability reached by 50 per cent of all of the boys taken together is 71.43. The median ability of all of the boys taken together is 22.60 with a coefficient of varia- bility of .26 ; of the girls, is 17.75 with a coefficient of .39. 208 the supervision of arithmetic An Investigation in Regard to the Development OE Standard Tests Following the report of Dr. Stone, Mr. S. A. Courtis, now Director of Research in the Detroit public school, became interested in the development of a series of standard tests for the different grades and different abilities in arithmetic. As a result of his activities, the same arithmetic tests have now been given under as nearly uniform conditions as possible to thousands and thousands of children distributed throughout the United States, sooo children were tested in Detroit; 33,000 in New York; 20,000 in Boston. The tests have been given quite generally throughout smaller cities of the United States, as Mr. Courtis has succeeded in securing the cooperation of a large number of school supervisors throughout the country. By this means he was enabled to arrive at certain tentative standards of excellence to be attained by children in the different grades in certain abiHties in arithmetic. For example, despite the variabihty in performance of children in the fifth grade in addition, fifth-grade children tended to attain a certain standard of excellence, which was somewhat less than the standard of excellence attained in the same test by children in the sixth grade; which was, in a similar way, somewhat lower than that at- tained in the same test by children in the seventh grade, and so on. Thus Mr. Courtis has attempted to deter- TESTS AND RESULTS 209 mine the standards of attainment which we have a right to expect from normal children in the different grades in addition, subtraction, multipUcation, division, copying figures, one-step problems in reasoning, ab- stract examples in the four fundamentals, and two-step problems in reasoning.' For those who believe in the teaching of the formal side of arithmetic, these tests give a view of the com- pleteness and balance of training afforded by any particular school as compared with the average achieve- ments of schools in other places, and they furnish the teacher with accurate measures of the peculiarities and weaknesses of individuals which enable him to adjust his work accordingly. Three editions of this series, equal in value but differing in every figure, have been pubUshed, so that repeated tests may be made either during one year or in successive years. When this is done, the re- sults obtained in one grade may be passed along with the child for the guidance of teachers in the higher grades, the curves for different years being drawn upon the same graph sheet. It should be said that results secured by tests given under conditions unlike those described by Mr. Courtis ^ 1 The series is known as series A. A later series of Courtis tests is known as series B . 2 These tests have been prepared upon regular form sheets and can be secured together with the instructions for giving them from Mr. S. A. Courtis, 82 Eliot Street, Detroit, Michigan. p 2IO THE SUPERVISION OF ARITHMETIC or using the combinations in a different order cannot be compared with those obtained by Mr. Courtis. The conditions must be dupHcated in every respect if the results are to be compared. These (series A) tests have been given by Mr. Courtis himself, or by teachers under his immediate supervision, to more than 60,000 children in grades ranging from the third to the eighth. As a result of' this, standard scores have been computed for each of the grades above the second. The following table presents these scores : REVISED STANDARD SCORES FOR SERIES A June, 1913 Test No. No. I No. 2 No. 3 No. 4 No. s No. 6 No. 7 No. 8 Ats. Rts. Ats. Rts. Ats. Rts. Grade 3 . Grade 4 . Grade s . Grade 6 . Grade 7 . Grade 8 . 26 34 42 5°' S8 63 19 2S 31 38 44 49 16 23 3° 37 41 45 16 23 30 37 44 49 63 7S 84 92 100 108 2-5 3-S 4.2 4.9 5-6 6.4 '■■5 1.8 2.6 3-5 4-5 S-7 S-o 7.0 9.0 II.O 12.S 14.0 1-7 3-S S-2 6.7 8.2 9.4 2.S 2.9 3-1 3-4 3-7 4.0 °-S 0.7 I.O 1-4 1.9 2-S , This table should be interpreted as follows : The average third-grade child should do 26 additions in one minute; 19 subtractions; 16 multipHcations ; 16 divisions; should copy 63 figures; should attempt 2.5 of the speed reasoning problems and get 1.5 right, etc. The attainment of each grade is also indicated by the table. For example: An average fourth-grade child TESTS AND RESULTS 211 should do 8 more additions than an average third-grade child. A fifth-grade child should do 8 more than a fourth- grade child. Similar standards are shown for each of the grades. By the use of such standard scores as these, teachers can determine whether each pupil has attained the standard degree of ability' for his grade and he can also determine what pupils may be excused from taking work in those phases of the subject in which they have al- ready advanced beyond the standard set for their grade. Mr. Courtis has prepared a graph sheet by means of which one may graphically represent his attainment in each of these abilities or by which the attainment of any given class or room or school may be represented. Later Mr. Courtis, feeling the need of a simpler series of tests which would be less expensive in time and money, proposed his series B. In announcing this series Mr. Courtis says, "The experimental work of the past few years proves that if school work is to be made more efhcient, there must be : " I. Definition of Aim. Illustration : Eighth-grade teachers should practice their children in addition until they can add correctly in eight minutes, twelve examples, each example three colunms wide, each column nine figures long. "2. Limitation of Training. Illustration: As soon as an eighth-grade child's scores reach the standard, he should be excused from further work in addition, whether he makes a year's gain in one month or ten. 212 THE SUPERVISION OF ARITHMETIC " 3 . Specialization of Training. Illustration : The child that is up to standard in addition, but below in subtraction, should be given increased drill in subtrac- tion, the extra time coming from the limitation of training in addition. " 4. Diagnosis and Rem'^H r of Individual Defects. Illustration : If a child's scores do not rise with class practice, he should be studied individually, his symptoms observed, his difl&culties discovered, and the proper adjustment of his work made." The following median scores for series B are submitted. It should be noted that these are tentative only, and subject to such modifications as may be suggested by additional data : MEDIAN SCORES: SERIES B February, 1914, Tabulation Test I — Addilion Attempts Rights soxjrce of Scores Detroit Boston General Probable June Standard Detroit Boston General Probable June Standard Number in 1 Groups J 1,31s 20,441 3,618 1,31s 20,441 3,618 Grade 3 3-6 4.6 0.7 2.0 4 S-4 S-3 4-7 6.0 2.7 2.6 1.9 3-0 S 6.7 7.2 7-1 7-S 3-9 3-7 3-9 4.0 6 8.4 8.3 8.0 9.0 4.6 4-9 4-4 S-O 7 9.2 9.2 8.9 lo-S S-4 S-6 4-7 6.S 8 10.2 II.O 9-7 12.0 6.7 7.8 S.6 8.0 TESTS AND RESULTS 213 Tesl 2 — Subtraction Grade 3 ■3.8 4.0 0.9 I.O 4 5.6 S-S 5-7 6.0 3-1 2-5 1.2 30 S 8.0 7.6 b.S 8.0 5-S 4.9 4-5 5-S 6 8.8 9.0 8.9 lo.o 6.2 6-3 6.1 7.0 7 9.8 1 0.0 10.2 "•5 7-3 7-3 6.9 8.5 8 12.3 11.4 11.7 12-5 9-5 8.6 8.4 lO.O Test J — Multiplication Grade 4 3-6 3-9 3-9 4-S 1.0 1-3 1-3 I'S S 6.4 S-8 6.0 7.0 3-8 3-3 2.6 4.0 6 7-4 6.9 7.2 8.S 4.8 4.8 4-5 5-S 7 9.6 8.0 8.4 lO.O 6.0 5-1 5-2 6.5 8 lO-S 9-S 9.9 ii-S 7-5 6-5 6.4 8.0 Test 4 — Division Grade 4 1.9 2.6 3-1 3-5 0.7 0.7 0.7 1.0 S 4-9 4-5 4-5 S-o 2.7 2.0 2-3 3-0 6 6.4 5-8 5-8 6-5 4-4 3-3 4-3 5-0 7 8.6 6.9 7.6 8.5 7-1 5-1 5-8 7.0 8 10.3 8.8 9.2 lO-S 8.8 6.9 6-3 9.0 Later Standards After giving almost half a million series B tests to children in forty-two different states, Mr. Courtis found the following to be the approximate median score for June. These standards, or medians, are for speed only, as he did not include problems inaccurately solved. 214 THE SXJPERVISION OF ARITHMETIC STANDARD (TIME) SCORE SERIES B: TESTS June Standard Individual ScoitE in the 4 Operations with Whole Numbers Test i Test 2 Test 3 Test 4 Addition * Subtraction Multiplication Division 3 3 4 3 2 4 S 6 S 4 S 7 8 7 6 6 9 1 10 9 8 7 II II 10 10 8 12 12 II 1 II Time allowance, Minutes, 8 4 6 8 345 487 631 20s 3479127468 4179 67)61707 943 1867396737 36 683 8S9 17s 794 "Translated into words the table means that in June the graduate of a grammar school should be able to work correctly in eight minutes twelve examples like that under Test I ; in four minutes twelve examples like that under Test 2, etc." Mr. Courtis says: "The scores given in Table I represent approximately the median speed of work for the different grades and are based upon returns that are nearly nation-wide in scope, The range of variation in TESTS AND RESULTS 215 schools in different cities and states is approximately four examples above and below the median; i.e., in some schools the median eighth-grade scores wiU rise as high as i6 examples in addition and others go as low as 8 examples. Not more than five eighth-grade classes per hundred will exceed these limits, except as very peculiar and special conditions prevail. On the other hand, the range of speed of work in individuals varies from a score of but two or three examples to scores of twenty-four examples, the limit of the test." Supervision Important The experimentation of Rice, Stone, and Courtis all suggest the fact that the most important single factor in effective arithmetical instruction is that of close supervision. Mr. Rice was convinced that the most important single factor was that of tests being given by the supervisor. The preHrninary investigation of Mr. Stone and the extended investigation of Mr. Courtis are such as to enable a supervisor at the present time to cooperate with other supervisors in the matter of arriving at standards of excellence which we have a right to expect of children in addition, subtraction, etc., in the different grades. No supervisor at the present time can afford to neglect the opportunity of securing , the cooperation of others in regard to this important phase of work. 2l6 THE SUPERVISION OF ARITHMETIC The investigations which have been made in the past confirm the thesis that the supervisor who knows the results to be expected from grade to grade in arithmetic, and who definitely tests the progress of the teachers in attaining these results, will be able to arrive at a satis- factory standard of eflficiency in the teaching of arith- metic under supgrvision. Mr. Courtis's tests are effective agencies in deter- mining the station of an individual student, a class, or a school. By means of them any student can discover his relative strehgths and weaknesses in the abilities tested, and any teacher or supervisor can determine those particular phases of arithmetic that the children should receive additional instruction in as well as those that may for the time being be neglected or dropped altogether. Certainly there can be no justification for continuing practice upon those things that the pupils have already acquired greater skill in than the world needs. A superintendent can also use such standards as these for estimating the efficiency of his teachers. The appli- cation of the standards will not reveal to him the causes of weakness in his teachers, but will reveal the points at which weakness occurs. The superintendent and teacher are then free to try new devices at these points for the purpose of securing greater efficiency. Unlike some of our present-day reformers who are TESTS AND RESULTS 217 wedded to the doctrine of individual indifferences and who advocate a course of study for each child, Mr. Courtis contends that there are certain habits, skills, and knowledges which should be acqmred by all the children. Put in his epigrammatic way, he says that instead of giving uniform material in a uniform way and getting varying results (as we have in the past) we should give varying material to varjdng children to get uniform results. This, we beheve, to be an end devoutly to be desired. It calls for a study of the in- dividual capacities and abilities of the pupils and for an intelligent application of materials to suit their needs so that they all attain that uniform standard, — a standard determined by the social serviceableness of the material in the ordinary walks of hfe. APPENDIX A Table Showing Variations in Time Schedttle Akron, Ohio bffers 25 min less in grades 5 and 6 than in 4, 7 & 8 Albia, Iowa " 25 " )j )) )» 7 " 8 ' "6 ^ Altoona, Pa. " 50. " " " " 8 ' " 7 .. Ann Arbor, Mich. " 15 " )) )) 77 6 ' " S Ann Arbor, Mich. 30 " » )> ;7 6 ' "7 Atlanta, Ga. 50 " " " " 7 ' " 6 & 8 Berkeley, Cal. " 20 " JJ }} 5, 6, 7 & 8 ' "4 Birmingham, Ala. 50 " " " 7 ' "6 Boston, Mass. " 40 " )) )> )) 6 1 ' "5 Boston, Mass. " 20 " 77 )J )( 7&8 ' "s Brockton, Mass. 60 " ■ M » 3> 8&9 ' "7 Chicago, 111. " 100 " J) JJ J> 5, 6&7 ' "4 Chicago, 111. " 145 " 8 ' "4 Cleveland, Ohio " 15 " " " " 5 ' "4 Cleveland, Ohio " 20 " " " " 7 ' "6 Danbury, Conn. 25 " " " " 7&8 ' "6 Davenport, Iowa " 25 " >J )> J) 6, 7&8 ' "4 Davenport, Iowa " 50 " » 7> JI 5 ' "4 East Saginaw, Mich. 30 " " " " 8 ' "7 Emporia, Kan. " 15 " )> )} >> 8 ' " 7 Dover, N. H. " 125 " " " " 8 • "7 Dunkirk, N. Y. " 75 " " " " 7&8 ' "6 Gloversville, N. Y. " 23 " )> J) » 6 ' "4 Gloversville, N. Y. 50 " " " " 5 ' "4 Jpliet, 111. 30 " " " " 7&8 ' "6 Kalamazoo, Mich. " 20 " )> )> .) 7&8 ' "6 Keokuk, Iowa 33 " " " " 4 ' "3 Keokuk, Iowa " 38 " , " " " 5 ' "3 Keokuk, Iowa " 21 " " " " 6 ' "3 Keokuk, Iowa 17 " " " " 7 ' "3 Los Angeles, Cal. / Jj JQ )j " " " 7&8 ' "6 Manchester, N. H. 50 " " " " 6&7 ' " S Memphis, Tenn. " 10 " " » " 8 ' "7 Minneapolis, Minn. " 45 " )» " )J 7&8 ■ "6 Muscatine, Iowa 45 " " " " 7 ' "6 Newton, Mass. " 30 " » Jt 1> 7&8 ' "6 NUes, Ohio " 100 " 8 ' " 7 Oakland, Cal. " 120 " " " " 7&8 ' "6 218 APPEN DI} C A 21 Olean, N. Y. offers 25 min. less in grade 8 than in 7 Plainfield, N. J. ' 25 " ' ' 4 ' " 3 Plainfield, N. J. ' 125 " ' ' 6 ' " S & 4 Plainfield, N. J. ■ ISO " ' ' 7 ' "S&4 Portland, Maine ' ICO " ' 8 ' " 5 & 4 Portland, Maine ' ' 50 " ' ' 7&8 ' "6 Riverside, Cal. 5 " ' 7&8 ' "5 St. Cloud, Minn. 25 " ■ 5&7 ' " 5 St. Cloud, Minn. ' 20 " ' 8 ' " S Savannah, Ga. SO " ' ' 6 ' " S Savannah, Ga. ' 100 " ' ' 7 " S Savannah, Ga. ' 75 " ' 8 ' " S Schenectady, N. Y. 25 " ' 5&6 ' "4 Schenectady, N. Y. 70 " ' 7 ' " 4 Schenectady, N. Y. 100 " ' ' 8 ■ "4 Spokane, Wash. ' IS " ' ' 7 ' "6 Spokane, Wash. ' 10 " 8 ' "6 Springfield, Mass. 25 " ' 5, 6, 7 & 8 ' " 4 Tacoma, Wash. ' 25 " 8 ' "7 Watertown, N. Y. 50 " ' ' 6 ' "S&7 Total cities . . . . 40 Total cases S9 Cases of reduction in grades 7 anc 18 . . 40 '(>' OS H W CM H o g o o Cn o 8 o c o O o oo )H 1 f ^ g G S ol M P- o 8 • * o a • • *T30'T30^o^n>tin^n o (^ o f* o n> n ™ o n n fti S "> S " n ^ S "> S " a 3 3 B 13 a p+ r^ r+ ft- s> s. 2, 2, a. a. o S" o o FT FT p p p p> s Ol oo<» to Cm to CM M to IH to M OJ M W M Cri 4k 4k Cn Cn NO C3S ^ H Crt w « 3 Oa Ca Ji. O o -^ Os4i nO Cn Cn t-i ^ at M M t-f O O M Ol Ca K> W ONCn O 4^ Ol ^ H OJ M M M M g O^ Cm ^J C/1 00 Cn 00 On oo M 4^ Oo S5 » Os M H to to to to CM to to o Oj -4 -O -O to Cn M OJ O NO OS S t t CM OJ to Oo s. M OO Cn CM 3 to H OS ^ 1 O H M O 1-4 1 ^ W to OJ ^ O w C>i OO O 00 O OOsO Os 59 S j_i M W )H W OJ Oi On to 00 K> 00 00 w > OO Si a Oa -a lO H 4^ M *0 t/i ^J 4i. o O OO-vj 00 O £3 M Cm »H M M w cn M OJ O O^Ca K> O -J cn NO cn M S9 M M M e t/i OS to M Cm k:i lO to lO «H (O On -^ 00 O Cn CO 00 o -^ S9 CM to Cm to to M 4^ „ o w On Ol Crt M (yi M to O ^J ^ M M to M OS O -^l M O O -J W to O M 0o oo cn O Cn Cm ^ ^ C/i 00 CM CM H ■t^ ^ O S5 OS O Ji' to Cn CM O to M CM i-f H M t~t w to t-« Cm CM trt M M -O 4- O •. o^-p»- Cm to Cn M S5 s H M M M M O to lo Cn Ocn -o O vO 4^ 55 Ol ^t cn ^ Cn CM -^ w e O Ca *-J Cm NO H W M Ol to M to M to M M OsO o o o -t*- \o -^ Cn M S9 W M M e Cm Cm Cm CM CM 4^ p:S OS On NO •vj Cn to C/l M M ^ CM CM K> to M S5 d M i-i o ^ o ^ nO ^ o ^ 00 Cn ss Ol JD, M 4k 10 cn 4>. cn CM Cn CM cn tH ^ \0 'O »-< -o O H M c^cn CM to i-t M Cm to CM to CM "5 ^ O^ H *J O cn 4". 53 M CM to to HI e oo to 9" Cm If M -^ M . to On M ^ OsCM OsC/i N cn 4*- fe9 Ca ^ Ca ^ Ui 4^ -J tn ^ Cn ^ Cn Oo CM Ol ONCn 00 O S3 10 -^ CM to H M -o > o to M M H lO to i--2 S d o > o cn North At c . South At 2 . North Ct I . South Ce Western . . No. of cases Per cent of total Cases Per cent of total Cases Per cent of total Cases Per cent of total Cases Per cent of total Cases Per cent of total Cm to Cn lOMCMtHK(-^M CMOn OOtO OnOnO tOCMCM>ONO si M M OS IH IH (O to M to ■'-jCm OnOOCm OsOnm Ol -3 trf Ol 5959 M •o O On M tOMMCMCM tOCM -j-' 00 CMMM CMOQM tOCM CnCn to OO^cnCioCM >h~o cn OS 4^ to M 4- OS cn to 00 Cm Cn H CM to 1 4" OJ K) 4^ to Oo 4^- O Cn O -^ O s CM OO M CO Cn CmmiowCmOO CmOn O^J-- 04^ -t^ 5959 M to CN OO IH tH Cm )H IH nOJi-Cm 004* OsCmCmnO*^ -^ 4- M to tHCn4>'CMN04^»0 to to-^ Cn O 4i. 4- Ol 4^ CN 00 O Oo 4>- to Oo Cn Oo M M to On to Cn OCM4>-vOcnOo 0000 M 5959 to 00 Cn OO CMMtOMWONCM CMCn OCMNO ooc^c^lH-J o+- 01 -a M 01 59^ CM On to Cm iH4kK)Oo 004* mCm Os io4'4i' OOCn-^ C0'HCn4ik 8g 5959 H M On M Cn M IH CM M M 04^ O CNCio M 004>'NO On 69 59 OCMtOMto4-'H Cn to Cn 4»- Cn 4i. OS ^j Cn to NO l-f M to -^ M M M Cm M 10 4-OsOOn4'On- OjCm (^ ^s S5^ < to On Cm CMtHioiHto-^ Oocn O^jO OOnO too to 0-*» -a Ol 59 I: lO IH Cm iH4kto4^o Onih4uoo On C3sCm -vJ Oo OO ih 4k On to s -^ O M MMIHOotO MtO IH Cn w O 4* 4»- On On Cm 4»- ha 01 53 M Oo M M 4^ M M X 4^- Cn cn 4^ 4" Cm 4». to to CTn w Cn if w so M to Cn Cm 00 Cm On 1 59 < ^ M o to to M to Cn 11 Cm mnO to4^ M I04^IH-^H 3 59 Cn Cn CM s Cn to-^^t^^ CTNHCn 4i-oo 0004^ OsCnCnOCn 3 59 to to H to 4- M tOMtOCnOO IOCm Oo C>0 OOtO OsM-^ 0*^ k9 01 59 M to MM MM 0>FO Oo NO 4^ Oo CM Cm 4»- to Cn 00 to to 1 T to H •. OOtO tocncn (J 00 Ol 55 M CM M Cm ^ CM M 4k (o to M Cm Cn 00 00 nO to l-f M Oo to M M M 5 ED M tn to 00 M NOCMCnCMCn 104* tH4^ 00 59 Oo 00 (0 H IH ^M4^tOCMOtO 4"^ CmO h OsOnO MCaO h s 59 Cn Cn Cm M ^ MCnOocn4k-o MCnO cmnO »h to oo4^4'"-oCMCn 59 to CM MMCMtOM4k CMOn e H" to to to to to M lO CO Oo H -^ 4" 11= H Z S 6? ^ ^ M H liOOO PO « to CO t^ C^ M -+ w W «0 »-« W 9 a O i—i 03 t— I > < I— I M-) < O o w o o H o I— < o « o u u < Q o a u < w o X « 3 Q > U o H > o •z > Q o ss 6S 00 Tfw ■^« lOfO^OtOO' O^ OO OO lO S -o 9 Pi 6? ^.'5?*^^'^ "^OOO »*5-?fo. vO CO OO fO S€ 6S toco t^Mt-tWtHtO ^ o «e ■ o -^o O O Tj- 0» O M lo r^ o -^ lo T^ lo O i-rfMMOtPOO fe?&? 3 a; O O (0 <0 M ^O(O»HT^00cO'Nr^^-IfO ■e th S^ r^r* n c>^Cf*^-hio z M a o o o <; a M 00 lo t^ \0 O r^ o M lOO t^M w cOThxO OvV>iO HI M M 6? s (0« lor^O >-• w lor^t^Ht^o S rOl->C4HIC*IHIHMC4vOC4 MMior^wMwroo.-* 6? s v4 H M l-( Ol tl Oi « M « « C« l-i W C< CI o Tj-t^ MM(Mror<-r« 6? IS OOOO O O Oooio^O Or--*"^ tOroio-*to-o M w lor^Ci « fO^t^t^ i IH M CI HI il M O 0» t-( t^ HI (O »0 CI CO « « o MM CI (O >-« lO 0000 O co-^w^O r-Hi iotJ-o co« OW mO 0»hi C>Ov tH tOCOlOCOlO-'d'XOC* -*« Tj- M 6? HI W HI W s *4 HI M « eotO'ia-^ Mioci St3 HI lo t>. Oi vO 'O tJ- (O to <0 CO CO e « » „ H^iOI>-0■*0^0 O O O vO W HiMHiCIMMCICIlOC) 6? low M HI HI c-^i-ioeor-. lOO HI tN.toiO'«t»o-*ioeo-*'*io g ■^ O O r^ O »o\0 O lo « « o» HIHIHIHIHIHICIHIMIOHI s v4 HI loci co-^O TfiOHi H* r>.co H< Ti- lo 00 w lo lo ^ ^ ^ Tj- rh e W M « M lO *f5t--0 "4-Ot-« Hiiococo MC4HIHI HIHIH»C0HI 6? Ooooo O coiow -«!hoo^ HI TtHt rocO<(or^o>« HI lOHi ii->io\o HI ^i-ic* w i 1- O'^r^Ti-O'co ClMC»MrOH.CI_C)U-)(M II O O H. CJ fO CO ■^ rj lo lo vo \o e HI voioOr^O ■^loO'Oirijn INDEX Abilities of Children in the 6A grade in the fundamentals, 168-176 Addition, practice experiments, 185- 194 Adequate standards of judging text- books, 134 Algebra : in grades, 136 topics taught in elementary grade, 142 Algebraic symbols, views of present- day writers on use of, 143 Analysis, grammar-grade texts, 132 Appreciation of arithmetical situations, a phase of arithmetic, 5 Arithmetic : two phases of, 4 value of socialization of, 149 varying the emphasis upon, 39 Arithmetical abiUties, relationship be- tween time expenditure and, 172, 173 _ Arithmetical relationships in public and private life, 5 Arithmetical terms, recurrence of, 130 Arithmetical topics : factors influencing choice and se- quence of, 2 1 investigations, 22 Arithmetics : comparison of five elementary, 122 order of topics in five elementary, 124 Articulation of subjects with life, 14s Baltimore report on topics in arith- metic, 24 Bonser, Dr. Frederick G., 199-201 Brown: experiment on the value of daily drill, 194-199 report of variation* in time given to arithmetic, 44 Business appUcations of arithmetic, 21 Changing conceptions of arithmetical values, 3 Chart : difference in variation and the central tendency in practice, 5 r elimination of topics, 10, 11 grade of introduction of text, ro3 increased attention to commercial topics, IS order of teaching multiplication tables in geographical divisions, 83 per cent of recitation to drill, 95 spiral, topical and combined method, 68 variation of subtraction methods in different areas, 75 Commercial activity, generated de- mand for arithmetic, 39 Commercial topics, per cent of super- intendents who favor increased attention to, 14, 15 Comparison of five elementary arith- metics, 122 Concrete problems of different periods, 114 Constants and variables in percentage, 115 Course of study : principles underlying making, 108 social character of material in the Indianapohs course of study, 152 textbook as a basis for, 106 Courtis investigations, 208-212 Courtis test : definition of aim, 211 diagnosis and remedy of individual defects, 212 effective agencies in determining station of individual student, etc., 216 limitation of training, 212 specialization of training, 212 series B, 214 Current criticisms, j. 222 INDEX Daily drill, Brown's experiment on the value of, 194-199 Dangers inherent in localization, 148 Detroit pubUc school investigations, 208-212 Development of arithmetic : in academy period, 40 in grammar school period, 40 in Massachusetts, 39, 40 little attention in early schools, 39 with establishment of high schools, 40 Dictionary of terms and names, 125 Division, practice experiments, 185-194 Drill, a phase of arithmetic, 4 Drill in arithmetic : relative value of periods for, 185- 194 results of drill, 196 tables showing median per cent of ftime favored for, 93, 94 variation in, 92-96 Early grammar school, arithmetic in, 40 Elimination of topics : graphic representations of, 1 1 summary of, 20 table showing per cent of superin- tendents who favor, 9 Emphasis, varying the, upon arith- metic, 39 European countries, recitation time given to arithmetic, 47 Evolution of subjects of study, 146 Forms of mathematical experience with which children should be familiarized, 3 Frequency of abilities by ages, 205, by grade, 202 Fundamental operations, value of short drill exercises in, i94-r99 Fundamentals and reasoning, tests for, 168, 176 Geometry : grade occurrences of, r44 in grades, 136 Giles, tests for fundamentals and reasoning, 176-181 Grade ; in which arithmetic text is intro- duced, loi-ios in which arithmetical topics are taught, investigations of, 23-28 in which topics are taught, sum- mary of, 28-30 Grade Occurrence : of geometry, 144 of seven specified topics, 31-38 Grades : algebra and geometry in, 136 algebraic topics taught in, 142 frequency of abilities by, 202 median ability and variability for, 203 Grammar grade texts, analysis of, 132 Graphic representations of eUmina- tions, ir Indianapolis course of study, 152 Integration of the separate units, 113 Introduction of text, grade for, 97- 105 Investigation in regard to develop- ment of standard tests, 208 Investigations : of Baltimore commission, 24-25 , of Dr. Payne, 23-23, 43-54 Van Houten, 25 Kirby, practice in the case of school children, 185-194 Kirkpatrick, experiment on ways of teaching multiplication tables, ■ 182-185 Localization of subject matter, 148 Longitude and time, 12, 13 Mathematical judgments : ability of children to express, 199 test for, 200, 201 Median abiUty and variabihty; for each grade, 203 for each sex, 206 Median scores, series B, 212 Median time expenditure, 54 Mensuration, shifting content of, 117-122 INDEX 223 Method : of subtraction, 73-80 spiral and topical in different cities. 69-71 spiral and topical, in different geo- graphical regions, 67-69 topical vs. spiral, 65-67 Multiplication tables : economical way of teaching, 181-185 effect of educational dissatisfaction on sequence of, 85-86 experiment in gaining facility in use of, 181-185 influence of tradition in sequence of, 79 undermining of tradition in sequence of, 80-85 Obsolete material, elimination of, 8 Oral drill, chart showing per cent of recitation time to, 95 recitation time favored for in differ- ent cities, 94-96 Oral work ; median per cent of recitation time proposed for, in geographical divisions, 88 median per cent of recitation time proposed for, in cities, 89-91 Payne : grade distribution of topics, 23- 25 investigation of recitation periods^ 43-54 Percentage, constants and variables in, rrs Practice in the case of school children, 185-194 Problems, related to business life, 145 Reasoning ability of children, Bon- ser's investigations, 200 Recitations, per cent of time given to arithmetic, 43-63 Recommendations : of Committee of Ten, 136 subjects given attention in elemen- tary course, 12 Recurrence of arithmetical terms, 130 Relating problems to industries, etc.. Relative emphasis of given topics, 124 Rehance upon textbooks, 109 Revised standard scores for series A, 210 Rice: analysis of conditions of teaching arithmetic, 41-42 conclusions on results of tests, 162- 167 experimentation of, 215 results of tests given to pupils; 161 tests given to grade pupils, 155-167 Skill, demand far greater in funda- mental operations, 5 Social character of material in the Indianapolis course of study, 152 Socialization of arithmetic, 149 Spiral and topical method : in different cities, 69-71 in geographical regions, 67-69 sanction of usage, 71-73 Spiral plan, failure of, 112 Springfield tests, r54-i55 Standard score, series B, 214 Stone : experimentation of relation be- tween time allotment and achieve- ment in arithmetic, 172 investigation in arithmetical abili- ties, 42-43 relationship between different abili- ties, 168 tests of abilities of children in the 6A grade, r68-i76 Subjects of study, evolution of, 146 Subtraction : methods of, 73 variation of methods in, 74-78 Superintendents : attitude of, on subject matter, 8 attitude with reference to use of algebraic symbols, 137 functions of, 166-167 per cent who favor elimination, 9, 10 per cent who favor increased atten- tion to commercial topics, r4, 15 Supervision, importance of, 215 224 INDEX Supervisor, results of Rice's tests of interest to, 164-167 Supervisory problem, 6 Table: achievements of the systems of tests, 170-171 arithmetical terms, 131 Baltimore report on arithmetical topics, 24 Baltimore survey of recitation time, 46 elimination of topics, 9 frequency abilities by age, 205 frequency grade occurrences, 37 frequency of abilities by grades, 202 grade occurrence of arithmetical topics, 26 grade occurrences of geometry, 144 grade of introduction of text in cities, loi grade of introduction of text in geographical divisions, 98 historical variation of percentage and its applications, in increased attention to commercial topics, 14 median ability and variability for each sex, 206 median score, series B, 212 median time expenditure, 54 number of items in addition, etc., 126 number of minutes per week in grades, 48, 49 oral work in arithmetic, 88-89 order of teaching multiplication tables in cities, 83, 84 order of teaching multiplication tables in geographical divisions, 87 per cent of recitation time by cities, 45 per cent of recitation time by grades, 44 per cent of recitation time in grades in foreign countries, 47 percentile distribution of total school time devoted to arithmetic, 63, 64 Table — Continued recitation time favored for drill in different cities, 94 recitation time for drill work in different geographical divisions, ' 93 revised standj^rd scores for series A, 210 specific cases of elimination of topics, 17-19 spiral and topical method, 67-68 spiral and topical method in dif- ferent cities, 70 time expenditure on arithmetic and on other grade subjects, 56- 61 time saved by relating problems to industries, occupations, etc., 150, 151 topics in arithmetic, 23 topics taught in elementary grade algebra, 142 topics taught in geometry, 144 use of algebraic symbols, 137, 140 various items in five elementary arithmetics, 123 variation of subtraction methods in cities, 77 variation of subtraction method in different geographical areas, 74 year in which seven specified topics are treated, 32 Teachers' familiarity with commercial phases, 6 Tests : Giles', for fundamentals and reason- ing, 176-181 given by Rice, 155-167 relative value of drill periods of different lengths, 185-194 the Springfield, 154-155 Textbooks : a basis for the course of study, 106 adequate standards of judging, 134 attention in, to commercial phases, 7 changing character of, no reliance upon, 109 Texts, analysis of grammar-grade, 132 Time: amount given to arithmetic, 43 INDEX 225 Time — Continued economy of, in recitations, 52 percentile distribution of total school, devoted to arithmetic, 63, 64 relationship between expenditure and arithmetical abilities, 172- 173 relative expenditure, 54-64 Stone's investigation as to amount given, 42-43 Topical vs. spiral method, 65-67 Topics : algebraic topics taught in grades, 142 demanding increased emphasis, 14 relative emphasis of, 124 taught in geometry, 144 Two phases of arithmetic : definition of, 4 Two phases of arithmetic — Continued teachers' familiarity with, 6 Value of socialization of arithmetic, 149 Van Houten investigation : critical study of time distribution of arithmetic, 55 ehmination specially mentioned, 16- 19 grade occurrence of arithmetic topics, 26, 37 Variation of: subtraction methods in different cities, 77-78 subtraction methods in different geographical areas, 74-76 View of present-day writers on use of algebraic symbols, 143 Printed in the United States of America. 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