LANGE LIBRARY OF EDUCATION UNIVERSITY OF CALIFORNIA BERKELEY. CALIFORNIA UNIVER- BERKELEY. . A study of eliminations, inclusions, and social and business requirements of arithmetic By Katherine Soiers B.L. 1914 Submitted in partial satisfaction of the requirements for the degree of MASTER OF ARTS in Education in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA Approved Ej AD Instructor in Charge Deposited in the University Library Date Librarian : EDUCATION OEPT. TABLE OF CONTENTS PART ONE PAGE INTRODUCTION 1- 2 THE PROBLEM THE MATERIALS THE METHOD OF APPROACH CHAPTER I — THE NEED FOR SCIENTIFIC INVESTIGATION 3- 14 IN EDUCATI 0* CHAPTER II-- AIMS IN ARITHMETIC 15-19 TRADITIONAL PRACTICAL CHAPTER III— ELIMINATIONS IN ARITHMETIC WHICH HAVE 20- 49 BEEN SUGGESTED OR MADE CHAPTER IV— MINIMUM ESSENTIALS IN ARITHMETIC 50- 63 6*ttfi23 PART TWO THE INVESTIGATION. CHAPTER I— SOURCES AND METHODS OF COLLECTING THE D SEA QUESTIONNAIRE TO THE PUBLIC QUESTIONNAIRE TO PARENTS QUESTIONNAIRE TO BUSINESS FIRMS CHAPTER II— QUESTIONNAIRE I TABLES CHARTS JUDGMENT CHAPTER III— QUESTIONNAIRE II TABLES CHART JUDGMENT CHAPTER IV— QUESTIONNAIRE III TABLES . CHARTS JUDGMENT CHAPTER V — SUMMARY I NATION OF TOPICS ESSENTIALS IN SUBJECT MATTER CONCLUSION PAGE 64- 68 69- 85 86- 91 92- 96 97-100 PAGE A SUGGESTED COURSE OF STUDY IN NUMBERING IN THE GRADES . .101-109 ESSENTIALS IN PROBLEMS BIBLIOGRAPHY 110- H 8 APPENDIX CRITICISM AND SUGGESTIVE CHANGES FOR THGRNDIKS'S ARITHMETICS. -1- INTRODUCTION. The thesis I maintain is this: Tho kind and amount of arithmetic taught in tho elemen- tary school and the method of its teaching rest upon the demands of the business and the social world. In establishing this thesis I shall approach tho prob- lem in tv;o wr.ys: first, by examining the accessible writings of recognized educators and by collecting, collating and organizing the records of what has been done by investigators up to this time in meeting the needs of society, in setting up aims, in determining essentials, in eliminating useless suV ject-mattor, in proposing methods and means of procedure, and in compiling practical courses of study; second, by first hand investigation and study of the requirements of society, I shall endeavor to search out tho minimum essentials of arithmetic. U-ing the data, records and judgments collected under the first section of this study as a basis for the comparison and testing of the later findings, I ho.->e to determine in what ways and to what extent thfl subject-matter may be further sim- plified, in how far the bulk of material may be reduced, and how great a decroase in tho time devoted to the teaching of arithme- tic may be made. By inquiring into current business practices, I Shi 11 -2- raako an effort to of ?er a harmonization of school methods and vrorld methods of procedure, and shall succ e3 t certain ii. elusions demanded by social needs. Finally, I shall include a course of study in arith- metic founded upon the facts and truths established. CHAPTER I THE HEED FOR SCIENTIFIC INVESTIGATIONS IN EDUCATION. ,v rhere is no form of knowledge go complete and final that it cannot be improved, no single human art so perfect that it cannot be made better, no form of human endeavor that does not call for further effort. For this philosophy, life Is a per- fecting, not an arriving at perfections, an 1 the joy is in the process, not in reaching and remaining at the goali* The best teaching of truth is that which seeks to define not truth itself as something already complete, but the nature of the effort required to search it out. The birth and growth of the idea of investigations, edu- cational and social surveys, and general stock-taking for every sort of institution lias been within the lifetime of the present •eneration. The idea did not spring into existence full-formed, out is developing gradually, and is marked along the way by revo- lutionary principles. This movement in the elementary schools is of special significance, for elementary school education has ful- ly entered the experimental period. Rapid changes are being made in school subjects through the influence of modern scien- tific research. The initial need is to know whither our pre:; school system is taking us, and if by critical examination it bo determined that our direction is wrong, wo ' out e uca- 1. Moore, Ernest Carroll: What la Education? 1914, page 141. -4- tional scouts to spy out direct roads to a pioneer land where we may find the truth. V/hat we have in education is an inheritance from our forefathers. It La the product of indiscriminate borrowing, modified as need forced modifications upon us. Prank experimen- tation is imperative. Our feneration must shake off tradition and organize a sound scheme for educational advancement. Changes must not be made according to the whims of individuals — the "cut-and-paste" method by which the average school superintendent compiles :i3 course of stiidy is tragically common -- nor must these changes be made by local committee? transient in character. Only after theories have been proved by investigation according to scientific method, and have been successfully tested and tried in the school of practice, may they be pronounced true and good. Many problems pertaining to the construction of course! of study already have been solved, and an encouraging body of scientific data has been secured. Many foolish things, no doubt, have been done under the shield of experimentation, but not ad- hered to by the conscientious worker in the field. He may make mistakes, but he discovers and corrects them when he te.:t3 his theories by practice. The markets arc stacked it.h books, and eager teachers, ritr more enthusiasm than wisdo , ooradic trial of this and that new idea. Opinion serves for informal n, and the disastrous results often bring disrepute upon a system of attack which is striving zealously to advance education* risons must be made between the conclusions reached in orig- inal research, and the results obtained by previous investigators alone a Ilka line before Judgments can be made. And then* that no reforms nay be offered which have not stood the toot of our own trying-ground, every fact advanced must stand the test of ap- plication; for the only effective way to determine whether the conclusions arrived at by experimental study are feasible is to see that they work out well in practice under the conditions of the classroom. There must be no gap between c ducat : L ori- montation and school practice. Fortunately, in the school world, there is a general awakening to a sense of values baaed upon sci- entific analysis. By discovery, surveys, facing and solving problems which individuals and cownamitlea have to meet, we are led away ft . ogaatical, the formal, and the untrue. Our investigations and experimentations must bo based upon sound educational aims. To make the discovery of anything worth while, we must know what human purpose is to be served by its discovery. 7,'hat is our problem? What is our purpose? What are we trying to do? What is our Aim ? Ws. must have no groping. As General Foch said, "There 13 but one manner of considering every question, that is the objective manner." V/hat is our Objec- tive ? Aims in education established by asters seen to vary: Knowledge getting, self-realization, social efficiency, culture, ■6- achicvenent, adjustment, learning to use t.ho tools which the race has found L sable, duty, growth, utility, complete living, discipline, — ho.; each aim calls to mind the ::ane of its advocate, name and aim closely locked. — Christ* s, "I an cone that they night have life, and that thoy ni it more abundantly," is an educational aim. These ideals establishc profound thinners seen so numerous one is overwhelmed v:hcn con- fronted by them. Have these scholars who have spent their years examining education arrived at diametrically different conclusions? Are self-realization, efficiency, culture, achievement, complete living, utility, growth, aims separate and apart? May they not be aspects of one great requirement? May not the notif running through all be, "'.7e serve life"? Broadly speaking, Ml nay shift advocated educational ains into three general groups: knowledge-get ti :.g, discipline, use. Knowledge getting, when knowledge fop ':.:ow- le kg ■:■ is the aim, is an old, an established aim. kith this the end of educati-n, the seeker is proscribed facts regardless of the usefulness of these facts. The schools are knowledge sarles ' -acts aro parcelled out by those who neks knowledge bestowing a business. The admonition, "Hold fast all I give you," accompanies the largess, and a strict accounting of the accumu- lated information is required. "Remember and you are educated," is the motto. Arithmetic facts, geography fasts, spelling facts, -7- history facto, literature, art and 1 f-icts, thousands and tons of thousands of thon compounded aro forced upon tho trusting ones who coma asking for that which will cure their ignorance. Urs to the past tv/enty-five years teachers universally accepted the decree of tradition, and made little effort to select aaablC from useless facts. They, with the iid of a few standardized texts, prodigally passed out f-icts. The key-word was , "Know for Ice of knowing." "Hoy/ much do you at, "What can you do with what you know?" la the shibboleth of those who lay .preme emphasis upon the acquisition of knowledge ac the end of education. Tho modern educational world is striving to overcome 1 2 this doctrir.e. The Ideals of such leaders as Hellarry, Cubberley, Dewey, ° Hall, O'Shea,^ "loore, 6 Thomdike and others, aro becom- ing directin forces. Dewey, for example, utterly opposes infor- mation as the end of education. He deprecates the accumulation of facts for recitation, and the hoarding of nowledge. "Knowledge Characteristic V/o: 1. Ue Hurry, Frank: Elementary School Standards. 1013. 2. Cubberley, Ellwood P.: Changing Conceptions of Education. 1909, 3. Dewey, John: Democracy and Education. 191G 4. Hall, . Lays Adolescence . 1904. Educational . 1911, 5. o'Chea, Vincent < Education as adjustment. 1903. Dynamic Factors in Education. 190G 6. Hoore, Ernest C: that is Education? 1914. oru.'cikc, Edward Lee: Educational Psychology. 1014. in the sense of information) neans the vorkln c i ltal, the in- dispensable resources, of further inquiry; of finding out, or learning more things. Frequently it is treated as an end in it- self, and then the goal becomes to heap it up and display it when I for. This static, cold-storage ideal of knowledge is in- imical to educative development. 1 ^" It is particularly mischievous, he holds, because the cramming of the mind with non-perti facts not only lets occasions for t . go unused, but swamps thinki . ;.."oore, beliovos that knowledge simply for knowledge sake is impractical; it Is neither warranted by facts, nor by .)lcgic 1 tests. The purpose of the investigator is to ac- quire knowledge, but when the facto arc in his possession they must meet the question, "Yftiat are they worth?" Since we arc pri- marily concerned with living r .thor than kn select those things which constitute vital knowledge. Knowledge as the oowir to act excludes all useless information. Teachers and admini; trators are profiting through an .the examination of the work done in the Francis barker, and/Horace llann schools. School" arc being looked upon as places where new ideas may bo evolved; and new discoveries made and tosted. The ideal, knowledge for knowledge s kc, vhlch 1. Dewey, John: Democracy and n, page 105. 2. Moore, Ernest C: What is Education':' Cliaptcr II. ■9- played so prominent a part in determining the curricula and the methods in our schools, which has evaluated studies according to their ability to impart information, is giving place to that knowledge which arises from use and is for use. The doctrine of general education, or formal discipline, is a second heritage from the past. The belief that training re- ceived in one line of mental activity spreads to other linos of mental activity is behind much of what we do \r\ elementary scho>ls, in high schools, and in colleges. This great assumption Ls the one upon which education has rested for many centuries. Studies, ac- cording to the advocates of this theory , have magic powers. They discipline, develop, and perfect minds . They are to be pursued not because they : ave specific values, but because t:;ey improve the mental facult5.es. Observation, emotions, reason, will, memo- i be trained in wholesale fashion. It is the schoolmaster's place to make of his classroom a mental gymnasium where his pupils' mind:.., ses upon the apparatus furnished, may be made more agile, keener, strongor. The more difficult the apoaratus is, the finer, abler, and more nearly perfect the resulting mind will be. Greek, Latin and higher mathematics Lave ranke 1 gymnasium. A Marathon throu science, a strenuous wrestling bout with gilistic encounter with algebra, daily try.: out tournaments with 3ome hundred or so of the 400,000 words of our English speech are supposed to Increase mind power. The ■10- gentler calisthenics of poetry, tart, n ic will lend | raco. Here, too, imperfect minds may be toncd-up, burnished, repaired. Tlie different types of minds are little considered and the pupils, in groups, pass with lock-step regularity from exorcise to exer- cise throughout tlie day. If we turn to experimental studies made upon this sub- ject, and judge by the writings of scholars, in the past twenty- five years, we cannot fail to recognize the error of this doc- trine. The data of those whose bias is frankly favorable to the theory seem the most conclusive against it. But old custom has imbedded it. Perhaps as many as three -fourths of the teach- f to-day, ind undoubtedly as many as nine-tenths of the edu- cated parents adhere to it as fix 1 & ngh faculty psycholc y had never been questioned. But Schools of Educat" a I research workers ars making headway. The newer I s freer of this pedantry than the older East. This confusion which began in that far-away day when the Sophists taught that if one wanted to I physician he must study rhetoric, and if he wanted to be a general he must study speech-making must be completely cleared away. 1. Dewey, John: Democracy and Education. :iooro, Ernest Carroll: What is Education'. 1914. • Rugg, II. 0.: The Experimental Determination of Mental disci- pline in School Studies, 1916 "ooro , Charles II.: The Inadequacy of ' against slplinary Values. School and Society. toe ,29 ,1917. Coover, John Edgar! Formal Discipline frc indpoint of Ba ology. 101G. ■11- hip ile says, "The problem of mental discipline, of determining under what conditions, by what methods, and to stoat extent train- inj in a given line of mental activity extends to other lines of menial activity is acknowledged to be the central problem of edu- cational psychology." Uoorc says, "Formal discipline is the cen- tral problem of educational philosophy, and the attitude which we who teach take upon this problem determines, as .. , what wo put into courser; of study, and how v/c teach that which we attempt to teach." This doctrine, undoubtedly a pmicious : alnfUl error, will ^o the way of worn out superstitions only when the ancient idol lias been completely demolished by painstaki rch. Use, the third educational aim, we hold is education's coal. Our real reason for including studies in our curriculum ia that they are indispensable helps to us in life. Let our words to our classes be, "You ar. here to learn to do certain things which the race has found that it cannot live without C Every lesson lias a specific ai. ,/ou are first to see, and then if possible to accompli ah. The question for you at all stages is, 'Can you do these socially necessary things''" , 1. Whip lo, ' . . : Preface to Ru. ' 1 na- tion of iaclpline in 'oore, E.G.: Address to Superintendents, state Normal School. Feb. 9, 1918. 3, nooro, B.C.i 'intendenti , ' tate Normal. Feb. 9, 1918. our schoolroom is a work shop where children learn to use the tools which the race has found, indispensable. It is of i-imeasur- able Importance that the things which a child 3ponds his ' upon shall he of immediate use, and shall servo in the future in such a way that he shall go on using them and increasing hll tery of them, "..e must eicaraine the needs of the adult world and fti :,jz and girls to acquire the beginnings of ledge and the skills deaanded for the future. The writer, by means of personal Interview , I ivored Lscover from the people Ives what iv e In boys and girls to school. "What do you expect the schools to do for your child?" was the essence of the quest! asked of one hundred parents in many different Wi life. College professors, alnlstere, bankers, physicians, ranchers, plumbers, firemen, grocers, shoemakers, men end women in offices, shops and 3tores, truck drivers, the scissors grinder , the gar- bage collector, and the Chinese green grocer were among those questioned. The reasons were varied; indeed, almost as numerous were they as the people who offered them: "To train hie nind." ''To learn the things he will need to know to get on in life." "To train his reasoning powers*" "To got knowledge." "So he will be cultured." "To bead) him to live wit -, and to pick up a bit of knowledge is he goes along." "To teach him to think logically." "to learn to speak - id rrlte nicely." "I don't knew. I*ve never thought about it. "very- body rends children to school. They have to." (Custom. ) "So he nay help tore, d write letters." "Ho he will know -ore and make a bet + '• ' than 1 have .r.ado." It ^as Interesting to note that U ted parents asked for knoi Ilsclpline, culture. As one vent the financial and social scale, the demand for e useful edu- cation increased. Products of colleges and schools cling to the traditional reasons for educating; the more uneducated classes j practic- 1 reasons. Tabulated the results stand: 1. Skills, and usable knowledge SO 2, General education — discipline 27 '- idge getting, 1G 4. Culture (four of these college pro- fessors.) 8 5. Social adjustment 7 C. Custom. o Tot.- 1 100 -14- Thesc data are offered ac an interesting showing of the public's reaction to a mutable question, not as proof of anything;. If we accept use as our standard, spendin . or- tional fraction of each day in the company of reading, writing, arithmetic, geography, history, et cetera, will not do. We must select the studies and the parte of the studies which we empha- size; there must be a purposive accumulation of facts. Our sole aim in learning anything must be to learn to use it. Plato has phrased it "teaching the young the knowledge which they will after- wards require for their arts." In this educational turmoil there is one confusion ee must keep elear of. '..hen we say all training shall be specific, we do not nean that education shall be narrow or circumscril , nor that subjects shall be walled off into compartments. Instru- ments in the use of which we have acquired skill must be constant- ly available: our whetted axe is a general tool. If we ar, to substitute the direct attack for the superstitious routine, our first step naxst be to gather the tools which will contribute to that result. -15- CIIAPTER II ttlE IN ARITHMETIC. Arithmetic has held a commanding place in the school curriculum for many years and lias developed for its justification a series of educational aims, broadly spo aking those may be sep- arated into two groups: 1. Traditional Aims . 2. Practical Aims , Traditional aims claim for arithmetic all-encompassing virtues. Scholarly articles have been written to show its disci - plinary T c ultural , aestheti c . and ethical values. The purely dis- ciplinary basis has justified the inclusion of hundreds of social- ly unnecessary tc ics, and thousands of archaic wad artificial problems. Claim is made that contact with absolute truth aids in setting up higher standards in all kinds of work; tliat reason, judgment and concentration are trained; tliat careful analysis is valuable in that it leads one, in considering every problen in life, to exclude the non-essentials and hold rigidly to a definite line of argument, by setting down results with clearness, and stating theories with tersenec , . I order is evolved. The cul- tural value has a certain literary claim. In one's dally reading such expressions as "a changlxi ratio," "tl ' . - trenes," "a proportional relationship" etc. do arise, but to jus- tify instruction in whole to : >icc for the sake of literary expres- ■16- context is absur". An aesthetic value is claimed. ;;ythm and orderly arrruigencnt arc supposed to inculcate an appreciation of eautiful; keen pleasure, oxalation of mind, a thrill of joy nay accompany a student's Q.E.D. , but that night '.7ell arise from the accomplishment of any completed v/ork. A careful study of arithmetic is claimed to promote ethical standards, though what there is in the study to make one treat his fellow-man better, or love his country more is difficult to say. That traditional aims dominate courses of study, and l,he v/ork mapped out is largely in terms of traditional sub- ject matter is evident in eight out of ten of the courses in use in our city and county schools. Certain California hools have only recently awakened to the fact that a new era in ari tic has dawned. The course in use in the Training School of the Los Angeles State Normal School in 1917 contained the following statements: "The cultural and ethical phases of Arithmetic must be recognize I if we are to make our subject of . educational value. A great re ponsiMllty rests upon the teachers of arithme- tic. There is const nt opportunity for encouraging clear-cut, con- cise statem jnts of conditions. Laxness and ccrelccsnes.- in speech beget inaccuracy of thought) and a disregard for logical sequence. Poverty of mind, and a patent lack of understanding of data involv- ed, are evidenced by inaccuracy of statement. CIc r, :. jt, coherent and graraaatical expression is of vital importance in 17- Ari thine tic. "Arithmetic, indeed, has a moral worth. It gives the child an opportunity for overcoming obstacles; it develops In him a broadminded, self-sufficient attitude toward difficulties. It disciplines him in persistency in effort toward accomplishment; it serves, in a degree, as a check to flabby inefficiency of thou lit , and loose vagueness of conclusion. It makes of him an invest i it- or, a questioner, a doubter, a scientist. The feelinf of certain- ty must be his if he is to attain the joy of achievement, and the intellectual delight ?^e writer is struck by the resemblance between this lit- tle book written throe and a half centuries past and her own child- hood text shell . twenty-odd years ago. The same rules, the same exprcs. ons abound. is 1 "Ho e full Branch" was on the threshold of learning by this pedantry , the writer as a "Ilo.'Cfull " r nch" net and struggled with the same rules, . ernes j Barli An old Arithmetic. Acad .y, •*. . >, -24« hours and energy upon those things, dry, dull, deadly and useless .irucl ich bind us by tradition to the past. Before 1700, we find little record of instruction in arithmetic In the United States, and such as there was was built upon English customs, weights, and measures as the early He.. Land settlers clung to the training of the Old '.orld. In 1G49 Hampton, Hew Hampshire, employed a schoolmaster "To teach to read, to write, and to east accounts if it be deslr In 1C53 Dedham, Massachusetts, had a schoolmaster who 3 teach re I , i I nd "the knowledge and art of arithmetic and the rules and practices thereof." In 1789 arithmetic was require.: by law In Massachusetts and : :hire. There had been a few English texts sparingly before the Revolution, but the manuscript, the "cyphering book" and rule book became quite common for Instruction of boys (girls exempted from deeper learning) after the act of 1789. The me- chanical, rule-of-thumb methods, however, gave then little advant- ovcr the uninstructeJ girls. The author of an old textbook published in 1795, makes an impassioned and patriotic plea for tno U3e of United States mon- ey, and the elimination of &oglish money and measures in the - Republic. lie says, "Let us, I beg you, Fellow-citizens, no longer 1. Cault, B«F.i An old text. BdttC tlon 20:279. -25- meanly follow the British intricate mode of reckoning. Let then have their v/ay and us, ours . Their mode is suited to the nonius of their government, for It seems the policy of tyrants to keep their accounts in as intricate and perplexing a method as possibl the smaller number of their subjects, then, may be able to estimat their enormous impositions and exactions. But Republican money ought to be simple and adapted to the meanest capacity." We commend the author as a f orward-1 oozing educator, for he further recommends that "Insurance, duties, commissions, indeed, anything reckoned v;ith per cents should bo calculated in one head and rule." We, in 1921, are making the same recomnendation. Some recognition of arithmetic as a lov/er school branch was made by the early colleges. In 1745 Yale required arithmetic for entrance; in 1760 Princeton required candidates "to understand the principal rules of vulgar arithmetic;" and in 1C07 Harvard's admission statements required that the entrant "be vrell instructed in the following rules of arithmetic;" namely, notation, simple and compound addition, subtraction, multiplication and division, to- gether with reduction and the single Rule of Three." o Among the early New En land texts which have loft an im- print upon our modern books for ^ood and ill is, at least, one book of English authorship. The Schoolmaster's Assistant , 1743, by Thomas Dil«orth, was used extensively in this country and held its 1. Erovm, E.E: The Making of our Middle Schools. Page 249. 2. Sources: Thomas Dilworth, The Schoolmaster's Assistant. Pri- vate library of U.C .'..'hoot, Los Angeles, California, and I). S. Bulletin no. 10. 1917 -26- popularity even after the advent of the comprehensive work of Nicholas Pike. Dilworth claims to be practical. He has nothing to say- regarding the nature and theory of number. His notation and num- eration he limits to nine digits. Pike includes seventy-eight di:its, duodecillions, dividing his periods, after the English fashion, into six digits each. Among the topics included in Dilv/orth's Master's A ssist - ant which educators are still endeavoring to eliminate are: I. Whole Numbers. 1. The Single Rule of Three direct, (proportion) - — Inverse 2. Compound interest. 3. Simple Fellowship, (partnership) 4. Compound Fellowship, (partnership) 5. Trade Discount. 6. Bxchs 7. The double Rule of Three (compound proportion) 8. Alligation. 9. Progression. II. Vulgar Fractions. 1. Compound and complex fractions. 2. Reduction of fractions,- Dilv/oi'th gives this under twelve different cases, eaeh elaborated, while addi- tion, subtraction, multiplicat' on and divieion of fracti n3 are accorded tnree scant pages with no il- lustrative examples or explanations as to the meaning or uf3c of fractions* III. Decimal Tractions. (This section Includes much subject- ir not included under that head to-day; the nc decimals was not keenly felt until Unite,: States money sneral use* ) 1. Square root. . Cube root. 3. Itole for extracting roots ere. •I. Reduction: as, Reduce 76 yd. to a decimal of a mile. • 5. Applications of Percentage as separated to ics: bar- ter, los: , , discounts, interest for years, mont 1 , purchasing freehold or real estate, annuities, Among the forty topics treated under decimals some score have already lost place in modern books; nine hold ,;ood; and the above greu [uestion, or has, in part, been eliminated. IV. Denominate Numbers. 1. Furlong in linear raea. . tod In square measure. Troy weight, (Dllworth'fl table is similar to the one used to- I 4. Clrcu urc — similar t to-day. 5. Fore ion I - Lisa "oney, In u:3e to—.: . apothecaries weight, in use to-day. 7. League. 8. Hand. 0. Gallons in a barrel, in use te- lle very many of 4 . L ' .. os given by de- arth are eliminated fr nt day texts, the .ritor finds, by comparing, practically all in the text .enty-five years aco, and further, finds nearly all in a text by David Eugene , A Grammar School Arithmetic , published in 1904, though in- tended for reference, chiefly, In the latter book. A number of other English editions 1 were printed in the United States: Crocker, Wlngate, Sough, and the "first purely arithmetical work published in the United States," an edition of Holder's Arit:motic t Boston, 1719, by J. Franklin. A rit lime tick , Vulcor and Decimal ; with the Applications thereof to a variety of Cases in Trade and Comr.erce was the first American book by an American author, Isaac Greenwood, 1729. This book found snail place in the schools and has left little imprint upon our texts for us to commend or condemn. That 1 3ces.-.ary " rt made Easy , James Ilodder; The r cho ol - easter 's ^aslstant, Nathan Daboll; The Scholar's Aritlimetic , TN an- iel Ad ms, ay be mentioned in tracing this early development of the subject in America] but the two outstanding ficuros, the ones v;ho have exerted the greatest influence upon the dovolo anent, form, content, and Instruction of arithmetic in the United States, the influence of the one of questionable benefit, the other for c oocl f arc Nicholas -'ike and Warren Collxirn, About 1779 Nicholas Pike of Ilev/buryport , Ifassachusetts published his first arithmetic, a nd in 1708 his Uew and Complete ■29- System of Arithmetic., the first book generally used in the United States, appeared. An examination of this book force- a re- luctant admiration for the genius which could conceive, order, and body-forth the content. 7,'hat shape and aims and agencies of edu- cation combined to produce him? "hat inborn ability invented this divine essence of mathematics? For it must have Bprung, "i - erva-like, from his fertile brain; practice was no forebear of it. The first 408 page* are devoted to rules, — a rule for every page, and sometimes, though rarely, a demonstrated problem to elucidate the rule — to topics, to problems, to tables, to specific directions of procedure and process; policies of insurance arc given eight cases, the Rule of Three recognizes seven complica- tions, while the Inverse Rule of Three and the Double Rule of Three are further sub-divide. 1 and distinguished from the direct eases in twelve different sections. There is no overflow or cor- relation of knowledge threatened here. Everything Is safely par- titioned and tidily niched. Following the 408 pages devoted to aritlimetic are 4 pages of "plain" geometry, 11 pages of "plain" Biometry a 45 pages of nensuration of specifics and solids in which are introduced practically all the rules of geometry, 33 pages of introduction to algel . nod for the use of academies, and 10 page;; of conic soctl The following table of contents, srr raged . ike, himself, gives a notion of the comprehensiveness of the work. 1. Pike, Nicholas: Hew and Complete System of Aritlimctic. Reprint, irivite librarv o . . beet. Los- Anrrelos. dnlifor in. ■30- Table of Contents of A Hew And Complete g. ' of Aritlmetic , Cy Nicholas Pike. Numeration: Pace Simple Addition 17 Subtraction 20 Multiplication 22 Division 23 Supplement to Contraction in Molt! plication 34 Tables in Compound Addition 42 tion 49 Problo Proa the Preceding Rules 5G Reduction Gl Vtolgar Fractions — 70 Decimal Fractions 85 1 ial Tables 99 Compound Multiplication 101 ' Division 106 Rules for Reducing all the Coins, from Canada to Georgia; also English, Irish, and French Coins Bllai , each to the par of .11 the other --- 111 ^odecimals, or Cross Multiplication — 123 Single Rule oi Three Direct 125 The Methods of Making Taxes, in ? Mote 132 Single Rule of Mhr :e Direct in Vulgar Fractions 13G To Find the Value of Cold in the Currency of England and Virginia 138 Single Rule of Three direct in Decimals 142 Rule of Three Inverse 144 Abbreviation of Vulgar Fractions 147 Double Rule of Three 147 Conjoined Proportion 153 Arbitration of Excliango.;. 155 Single Fellowship 155 Double Fellowship, or Fellowship with rime 158 Fellowship by Decimals 1C0 Practice 161 By Decimals 189 Form of a Till of Parcels 191 Tare and Trett 192 Involution 194 Evolution 195 Table of Tower. 19G Extraction of the Square Hoot 197 -31- Page Ap 11 cation and Use of the Square Root 200 Extraction of the Cube Root 203 Application and Use of the Cube Root 209 Extraction of the Biquadrate Root 210 Of the Sursolid, by Approximation 211 Of the Rootq6f all Power- 212 Proportion in General 210 Arithmetical -roportion 217 Progression 219 Geometrical -roportion 234 _- Progression 23G Simple Interest 251 By Decimals 255 Annuities, or Pensions in Arrears, at lapis Interest- 204 Present Worth of Annuities at Simple Interest 205 Discount 2GC .-_ By Decimals 209 Tarter 270 Loss and Gain 273 Equations of Payments 281 P. y Declmala 203 ion or Factorage 204 Brokerage 205 Policies of Insurance 200 Compound Interest 292 . By -.eeinals 294 . iscount at Compound Interest 299 Annuities, or Penal ens In Arrears, at Compound interest 300 Pre ent '.orth of Annuities at Compound Interest 305 Annuities in Reversion at Compound Interest 309 Purchasing Annuities forever, or Freehold Estates 314 Table shewing the amount of £1 from 1 year to 50 C10 Present worth of KL from 1 year to 40 — 319 Amount of fcl Annuity, etc. — 320 Present worth of fcl Annuity, etc. 321 Annuity, which El, will purchase, etc. 322 Circulating decimals 32;- Alllgation Iledial — - •-'•20 Single Position 334 Double Position 330 Permutations and Combinations 339 A short method of re 'uci ir fraction to a deci. al 345 Of finding the duplicate, triplicate, etc., Ratio of two numbers, wBaose difference is small — 345 To estimate t as of Objects 347 7ho "ci ht of Objects 340 Page Hi seel lane ous Questions 348 Of Gravity 357 Of the Fall of Bodies 359 — Of Pendulums 3G2 Of the Lover, or Steelyard 3G4 Of the "./hell and Axle 3G5 Of the Screw 365 -Of the Specific Gravities of 3G8 Ta les of Specific Gravities 3G9 Use of the r.arometer in Measuring Heights 375 Table of J ita of ;:oney 375 Table of Exchange 37G Ditto 377 Table of the value of sundry ;ioces in the sever:! States 378 Of Commission 379 Of the net prcceces, after the commission at 2-§ and 5 per cent arc deducted 380 Table the number of Days fr y in one th to any day in any other month 381 Table of the measure of Length of the princi al ! compared »ith the American yard - 382 Tabic direct!.: how to buy and sell by the BBndred Weight 383 Co. arison of the American Foot with the Teet of other Countries 384 Table to c '-cos or Expense* for a Year, at so much per day, week, or month 385 to find Wages, or Expenses for a month, or day, at so nach per year — - 385 I L, at 6 per cent from 1 shil- Lin^ to KL,000, and fro:.. 1 day to a year, the i. - termediate months co of 30 days c ich — 38G A Perpetual Almanack 390 Tables reducing Troy weight to Avoirdupois, and the Contrary — --- 301 An account of the Gregorian Calendar, or He. tyle — 392 Chronological Preble :s 392 -e find in which Century the last year is to be Lean year, and the contr ry 392 - find, with r ether ye , whether any year be r, or not 393 ; .or. 3. To f Domi leal Letter accord! to the Julian method according '.o the Gregor- ian Method 394 Problem 5. '^o find the *Vime or Golden i 395 Problem 6. To find the Julian Epact : 395 ore' lem 7. To find the Or .act 39G - Page To find the same, forever 397 Problem 8. To calculate the Moon's agejbn any given day - 397 ^ro' lem 9, To find the tines of the naif and full moon and first and last quarters 398 Problem 10. Having the time of the Moon's Bouthi given, to fi ge 399 Problem 11. To find th f tha Moon's southing 399 Toblcra 12. To find on what day of the week any ^iven day in any month ./ill fall 400 Problem 13. To find the Cycle of the Sun 401 Table of the Deminical Letters accord to the Cycle of Sun 402 Problem 14. To find the Year of the Dynonisian Period — 402 Problem 15. To find the year of Indiction 402 m 16. To find the Julian "eriod 403 "roblcr. 17. r, o find the Cycle of the fun, Golden Number, and Indiction for any current year 403 Problem 18. Having the Cycle of tic Sun, Golden 'lumber, and Indiction, to find the year of Christian Era — 404 Problem :.C. ater 404 Problem 20. To find on what day Easter will hap ten 404 er fro:; the Year 1753 to 4199 40C The Use of logarithm! 407 This table i3 inadequate to give the full significance of the. content a however. To illustrate: Under the Mead "Rules for reducing Federal coin and currencies of the sever 1 United States, al3o English, Irish, Canada, Nova Scotia, Livroc, Tournois, and Spanish milled Dollars each to the par of the other" Mr. Pike gives 7C rules for the exchange of coin among the Mtates alone, and then treats the foreign nations with equal consideration. As a general reference book, or, by selection, a text for special technical schools the 1 Ly valuable. S on0 of the rules are quite unintelligible to modern students . Under Trott and Tare, ike says: "Deduct the tare and trott, divide the suttle by 1G8 and the quotient ..ill be the cloff , which subtract ■:-;• from the Buttle and the remainder will be the neat." Evidently, "trett," "tare," "suttle," "cloff" and "neat" were of the vocabu- lary of th day, and thia rule for the wei hing and handling of merchandise was useful to a group of traders. One can hardly con- ceive of these rule bell g useful to the average citizen in his daily life, and even the advanced students, for whom the text was intended, could have found little with which their experience cor- responded. Yet, shall we criticize ^r. Pike's work while our own texts retain surveyor's measure and apothecaries weight? Penjamin best, in a criticism of the book says, "The volume contains, besides what is useful and necessary in the com- mon affairs of life, a great fund of amusement and entertainment. The mechanic will find In it rmich that he nay have occasion for; the lawyer, me reliant and mathematician will find an ample field for exercising their genius." This book was the gem puzzle of its day. II ny .leasing and diverting questions are included in its lists of preblei . These problems are connected with questions as to the number of changes which can be rung on a chime, how many different positions a person can assume at a dinner party, . v riations can be made of the alphabet, and many others which reflect, the influence of mediaeval dis utation. c find the forbear of our puzzle of the fox, the gect;e and the bags of corn and the complications arising from carrying then across the river two by two, which touches a rcsoonsivc association in the minds of the students of ■35- the early nineties. Then, in geometrical progression, is that ap- parently foolish bargain of tue merchant vho sold 39 y >rds of fine velvet for 2 pins for the first yard, 6 pins for the second yard, 18 pins for the third, etc. which turns out most profitable to the seller after all. One of our fairly modern books ha3 a problem wherein a crafty person offers a hundred acres of land at tvo pen- nies for the first acre, 6 for the second, etc. which the writer has used recently as a diversion to the delight and v onderment of her pupils . Pike's text ran through five edit:^ns, the fifth printed in 1832, and despite the reaction created by Warren Colburn's First Le s c ons , is the book which is most responsible for the ne- cessity for extensive eliminations to-day. The great reformer in American arithmetic was Warren Colbum. He made a protect, and a vigorous one, against the deep- root d evils of the mediaeval and English notion of teaching num- ber as symbols, and of clinging to definisions and ruler,. He en- deavored to seek out and seize upon the instincts of the child and use these as a factor in Lis educatin. On reading his masterly address delivered before the American Institute of Instruction in Boston in 1830 on Teaching of Arithmetic , one realizes he possessed 1. Reprinted in Elementary School Teacher, v. 12, (1911-12) pages 463-480. •36- an Insight into the educative process far ahead of hlfl time, and which wo are but now coining to realize. Colburn was avowedly a follower of Pestalozsi. In his preface to the second edition of his First Lessons is a lengthy tribute to Pestaloazl which gives clear evidence of his acquaint- ance of the Postaloznian plan, although he was Largely independent in his conceptions and in the carrying out of his theories. Mr. Thomas Sherwin, princinal of the hi I, Bee ten, (1030) says: "I regard Mr. Colburn as the great benefactor of his age with re- spect to tho proper development of the mathematical powers. Pesta- lozsi, indeed, first conceive.': the plan; but Mr. Colburn realized the plan j popularized it, and rendered it capable of being ap lied to the humblest mediocre oy. Indeed, I regard Firs t Lessons as the ne plus ultra of primary arithmetic." First Lessons in Intellectual Arithmetic ,' published first in 1821 achieved almost incr text. *t was followed by The f.equel to First Lessons a year later. Fiivt . . - ■ ! s is rerlly a "L'lnimum Essentials* of arlthtaetie. Mr, Ccl- burn's idea wa3 to eliminate ail useless material, to teach by the use of concrete objects, and to connect each question vith tho ac- tivity of the child himself. L. Elementary School Teacher, vol. 12, pago 424. 2. University of California Library - renrint. 3. University of California Library (original, edition of 1034) -37* A comparison of the table of contents of First Ler with that of Nicholas Pike's Nq\y and Complete System of Ari time tic is illuminatj Table of Contents of Warren Colburn 1 s First Lessons. Part I. Sec. I. Addition and subtraction. Sec. II. Multiplication. Sec. III. Division. - Idea of fraction introduced. Sec. IV. Fractions; multiplication of an integer by a fraction. Sec. V. Principle of fractions applied to larger nu i . Sec. VI. Division of an Integer by a fraction. Sec. VII. Compilations of preceding and multiplication tabic from 10 x 10 mp to 10 x 20. Sec. VIII. Reduction of fractions to higher ter Integers to fractions. Sec. IX. Uultiplical a fraction by 9r« . X. Mostly dril^on Sections IV and IX. Sec. XI. Division of one frao another. Sec. XII. Fractions written in fractional form. Sec. XIII. Reduction of fractions to a comma L lator; addi- tion and subtraction of fractions. Sec. XIV. Division of fractions by integer: I Lieation of a fraction by a fraction. Sec. XV. Mvisj »n of integer.: by fractions aa I by a fraction. Colburn specifically 3 ns the cli of The Rule of Three (proportion) of Cube Root, and oaxe :ts that Denominate Numbers, Perce: , [nterest and He ration should not bo taken as bases for separate chapters or 0VGn distinct to vie.;. 1. Colburn d Me of contents in the First Los- sons, and U sen made from a study of the material en in the various sections. -r,8- It is cl- Interesting fact to note that in 1821 '.7arren Colburn re conme nded tho elimination of Square Root from the c il tary school text and that the writer , after a lapse of one hundred years, should find it necessary to make the sane recommendation. Follov/ing Colburn came the books of Jose h Ray, books built along the line of Pile rather than Colburn, and from the waning of Colburn' z influence to the latter part of the nineteenth century, arithmetic war taught as a mental discipline. The body of natter accumulated, tr ' lng what lad o^e before, and a desire for new disciplinary material adding to the ball:. Throughout this static period people aeeepted rather than investig ted. served for information and that o . 1': jen -h, nd. It ia relate" of a learned judge .e once praise ..ring witness in these , "You are entitled to great s ir. You BUS1 ito alns with yourself* "do man could naturally . Our great" ' arithmetic COW -Tort. While there e*aa no concerted action te lifying and relating numbering to use -hiring these . i pa, one finds protests against the conventional teaching a o the dogma of formal discipline, A number of eri1 Li id pleas that schools avoid more empirical rules arid keep in view tho practical and commercial value of arithmetic t of computa- tion, rate according to trade-rules which they do not understand arc quite at loss to help ■ ■ themselves i: . :t, although, by this method (disciplinary) the scholar nay be ."oil prepared for .any computa- tions which he wlH have occasion Tor in practical life, yet lie will be quite at a loss hew to help himself whenever a case shall cone up to rhich he cannot a ply his rule exactly as he has 1 to use it." 2 Little farther was done In the way of simplifying arith- metic until 1387 when President Fr i ". lker attacked the problem in the Boston schools* and his investigations resulted in a recommendation by the school committee that the fol ubjeeti be eli from the course of study i 1. Mensuration of unusual surfaces and sc 2. Comooiind proportion. _ id Interes . -1. Equation of payments* 5. ■ G. Metric System* , 7. Comp a th twentieth century, signifi- cant rithm advocated by certain note: educat- ors throughout X and the Kiddle est. I . lr pur- pose to search ou1 racticatl In sritheetie and tc o:nit all other material. In 1003 c. "... Stone sent out s (zuestioonalre 1. Peacock, (-;-. : Educational Value of Arithmetic. London Quarterly, 1058. 2. Bernard's Journal of Education, vol.0 (18G0) e 170. 3. Ifossup, .Iter A.: Educational Research, School and Society. July 24, 1915. Page 137. •40- to business men of Indianapolis, requesting their opinions on the utilitarian value of the work ~iven in arithmetic. The replies in- dicated that certai:. to ics had absolutely no use In the business world. Dr. lurry in an addr... Ions are advisable in the present course of study, and what should be the basis for the same", delivered in 1904 befoi ti nal Department of Superintendence, recommended that the fc . topic tic: 1. 2. Tre 3. Examples ' Ltude and Tine, except the very simplest. 4. The furlong In li - Lsure. 5. The rood in square measure. G. The dram quarter In avoirdupois weight* 7. The surveyor's tal 3. Tal le on fol per. 9. All problems in reduction, asccn ! inc -:! fcsc ending, . 10. The G. C. C. as a separate topic, but not practice in detecting divisibility by 2, 3, 5, and 10. 11. All common froctior those of a very low de- nomi on 12. Ml work with the •-•. C. '". except of such very Ion i . those Ju 13. Cttons r- bo "c. 14. Compound pro 15. Perce t S separate topic, with its cases. 1G. True di ount, 17. "est problems in compound interest, and all in annual 18. Problems in partial ay ierit e of a very le hind. 19. The same for comnis~lon and broker ;ej for txample, all problems involving fractions of 20. Profit and [ i cial to ic. 21. Equation of payments - proved bankin faclll I -41- 22. Partnership - nado unnecessary, in the old oc. , by stock conpani 23. Cube Root. 24. All algebra, except cue' ace of the equa- tion as la directly helpful in Arithmetic and in other subjects in the shcool life of the pupil* In addition to all/of thee, arithmetic may be omitted r. a separate study throughout the firct year of school on the ground that there is no need of it if the number incidentally called fear In other work 1c ropcrly attended to. 1 In 1904 Joseph v. Collins of the ::tate Normal School _ .o very definite recommends.'- [a i - eussion, "The ruperintond- c:.t and th . i of Study" be aay*i ''If instruction in schools in arithmetiCj is to bs ^>ro^ ht u-j to a place \diere it will be rc- Ld it will have to include only oper- as . " He . . pendents ge throv. cxt books they nay have in ur;^ ,1^ blue pencil and sea to it Llevln omittedi 1. ... ic. . "'. . . . • nto be nit with addi- ti n of fr 3. Longitude ai j), (put .«'! 5. rrc, C. A vast deal In- ate He says, "All of . , '-• . .'.en of conpound denominate numbers shoul - 1. UcUurrV > Prank: : rocccuiivjc of the Nati nal Bducati Lation, 1004 pace.?. 10-1-2 02. ■ - lete, as also all rob lens ' clvo qjantitic. ' in more than tv/o denomination::;. Such a problem as: "Reduce 2ni. 3rd. 5yd. 8ft, 5 in. to inches" is aa absur " eto- ry. It Is evidently the product of soae school;; ' Invent . uch, ha wn ro r, ic the force of cuntom that numerous pro' lens of t o be Pound in most of the ice. of this day. 1 The superintendent should dra inell through all the tables of u;bcrc except .is uare, cubic, dry, liquid, end ' ires, all problems under them." "' i of typic analysis ^.nd proportion bciu£, "strictl; , raetleal*" i'r. Cel ! 1 I i try in the upper elements ry rade... "Let rjeomet I for p iration, and Let the \izzlo pr< laco J In 1909 G. Li. '_ . a study of the social Of aril. ... -i v.; . n published his fi. the Sixtc V r Book of l nal Society for Mid Study of I cation, do s; . "In connection .,i .1 in Arithmetic • ., Cecnersvl , a fc\7 y ;, an ct- tempt to get the judgment of the business c on a hnetlo to_ ic. . ro.-.ult of this cc . , 1. The writer ad!s, "And thi:. , ote the Cali •"■ to Text of 1919. 2. Collins, Joseph: The Superint rid the Course Study. 27. 33-09. -43- business nen of the city voted to o .it the foil les fron the aritlunotic coui^ses: 1. Troy Weight. 2. Apothecaries 3. Lonritude and Tine. 4. The surveyor's table. 5. The Greatest Common Divisor . 6. The Least Common Multiple. 7. Complex Fractions. 8. Cube Root. 9. Compound Fractions. 10. Foreign Exchange. 11. Compound Proportl 12. Time discount. 13. Case.. 2 and 3 in Percent 14. Compound Interest. 15. Partial Payments. 16. Partnership. 1 In 1910 the Baltimore Scho 1 n criticised the time expenditure and to ic 1 emphasis in arithmetic and suggested alterations, adjustments and eliminations be made to meet the needs 2 of the business world and in 1919 recommended tic so eliminations: 1. G. C. D. and L. C. H. of lar^e numbers etherwl than by factori . 2. Fractions with lar^e and unusual denominators. J. Complex and Compound fr ctions. 4. All measures not actually In use in the community at lar^e: troy, apothecaries, dr 5. Reduction of decimals to com .on fractions, and decimals beyond thousandths should receive little basis. G. Circulating decimals. The to:)ic should be studied as a part of infinite series in alcebr- . 7. Square root and cube root exc 8. Profit and loss as a separ ic. 9. True discount. " ank discount n its place entirely. Lis on, G.::.: Sixteenth Yearbook of the Rational Socioty for the study of Education. Chapter VIII, Page 128. 2. Bureau of Bducatl n Bulletin 1911, No. 4, na._.c 70. -: •- 10. Partial payments in tho forn of state rules irregular indorsemen 11. Equation of payments. 12. Compound proportion has been largely replace J by unitary analysis. Simple proporti in is of sorae importance but is beet treated as equality of tv/o fractions. 13. Business problems which do not conform to the usage of the day. 14. Large numbers and exorcises involving many nun- , bers should also be excluded as a rule. In 1911 the American Committee :io. 1 of the Internation- al Committee on the n e ChJ thematics reported on Mathe- matics in the Elementary Schools. This Committee, through its in- vestigations, found that school and business people felt an urgent for simplifying courses of study: 1. By naing small numbers in work in arithmetic. 2. By eliminating topics that are unduly confusin . By eliminating obsolete problems, topics processes* In 1912 or 1913 Dr. Walt ort of the Committee on Elimination of Subject-matter to the Io.. .'eachers As-.ociat' 2. Smith, David Eugeno: Teaching Elementary chool Subjec' . Raoecr and Others. I 207-249. -48- 6. Impractical ro luctions in denominate numbers. 7. Addition, subtracti on, multi >lication, and divis- ion of compound denominate numbci . 8. Tho present type of inverse iroblons in fracti ns and percent; 9. The various short methods of fin line interest. 10. An inverse problems of interest. 11. Partial payment . 12. Annuo. 1 intore t. 13. Undue emphasis upon the discounting of interest- be aring notes. 14. True discount. 15. Partnership. 16. Proportion as a gee - 'A\od of so 17. Foreign and domestic exchange. 13. The measurement of uncommon areas and volumes. 19. Square root and the Pythagorean Theorem. 29. The metric system. 1 This study seems to justify the conclusion that up to the twentieth century tradition tended to keep subject matter which had once been added; that net? subject matter was much more ly incorporated than obsolete matter discarded* This result- ed in a mass of material which sag cumbersome, impractical, much of it utterly useless. But v/ith the beginning of tho tv/enticth century, certain unprejudiced searchers after truth, realizing the necessity for radical change, began a vigorous campaign for the elimination of this useless material nicked up in the devel- opment and progress of the subject. There investigators claimed the air. of arithmetic is to enable one to do intelligently the common work of the world, and they proceeded to discover fro. the world what portions of the subject are serviceable in interpreting 1. Stone, John Charles: The .otic. 1918. Ilote: The r,tone-;'J.llis texts are faitliful to i 1. •49- our surroundings from a number point of vie.,, and to eliminate the rest, y forces, many men, and a coiUMendable amount of scien- tific investigation have combined to make the subject what it is to-day. '.That we have achieved thus far has serve : , ut to in- crease our interest, and to urge us to further investigations of every phase of arithmetical work. •50- CIIAPTER IV MINIMUM ES ENTIAIS IN ARITHMETIC. "The minimum essential in arithmetic is the ability on the part of the individual to do practical calculations such as are needed by the average citizen In his daily life." 1 Socrates "cursed as impious him who first separated the just from the useful," and we add, "him who first separated common sense and ari time tic." Purine the past two decades forward-looking teachers and school administrators have been concerning themselves with the Tuestion, "V/hat is essential?" and have been laboring to re- lieve the commonly accepted curriculum of its unjust burdens. Pal lie opinion itself is makinc a stronr ap teal that the arithme- tic taught La more in tune with life. In the expression of dissat- isfied business men there is very little an of let ils in the solution of the problem, but there is a stern demand for results. Schools have been tardy in response, but leaders have been studying to determine what parts of arithmetic have definite utility, and are providing special opportunities fop giving ex- clusive attention to those aspects of them. President Francis ' . alker of the Massachusetts In- stitute of Technol', the crusade in 1807. Re declared 1. Report of Committee on Course of Study In Arithmetic. Los Angeles St to Normal School, 1919. that "a f lsc arithmetic has grown u? which has largely crowded out the place of tiiie arithmetic — - The most jagged fractions such as would hardly ever be found in actual business operation, e.g., ll/20, or 13/27 are piled one on top of another, to produce an un- real and i Le difficulty; the child having been furnished with such an aritlimctical monstrosity, is set to multiplying or dividing it bj another 'compound and complex fraction' as unreal and ri liculous as itself* All this sort of thing in the teach- ing of young children is cither useless or mischievous. The Inst the existing course of study Is th t it is up of exercises which arc not exerciser, in arithme- tic at all, or principally, but are exercises in logic, and second- ly, that as exercises in logic, these are useless or mischievous General , ' ..ot universally speaking, whatever in education , ireng. I!oro than thirty years have gone by since then, and the arithmetic t -tight is still, for the most part, the Ltional thl . V.zcr found so meanl . We use a year or nor to teach fractions. Let bch him- self carefully for a month and discover when he had occasion to add 2/5 and 3/4, or subtract 2/5 fron: 3/4, or to multiply a fr c- tion by a fraction. I challenge anyone to discover an occasion in life for dividing a fraction by a fraction. 1 ^ Yet in practic 1- tlker, Francis ki Arithmetic in the J oston r.cho Is, " cadomy (Syracuse) 1087. Pago 433. 2. An Educ erintend ri cipals a f e::oericnce accepted this challei but faile '. , duri , to find le itimate oc- ly any text book used in our Ian J, we fin 1 this sort of problem: o/29 x ll/l9, 1G 5/23 -4- 15/43. Too often, even now, children are worried over the number of cubic inches In a gallon, compound proportion, cube root, aliquot parts, such problcus as, "If the principal is £u43. 87, the time 5 years 7 months and 25 days, and the amount ' U S72 9 what is the rate?" This Insane query is from the writer's childhood text: ''Ho'.' many gills in 2 hogsheads, 31 gallons, 3 quarts, and 2 pints:" A liquor dealer, retailing in gill quantities, may have found oc- casion for such computations, but we coul : hardly cl 1 an essential sir.ee the Volstead Act. From the addition, subtracti n and multiplication problems propounded, one woul I suppose our children wore all destined to become millionaires. There arc rarely any calculations which involve less than a hundred thousand. Following ir. .'alkcr's lead, Albert r . Boyden of the Bridgwater* Massachusetts, normal School, in 1894 revise.! his course of study and the methods of instruction. lie says, "Arith- metic is too often taken in a merely mechanical way, the pupils working by rule with much cypher! ig and little thinking study of arithmetic must be enriched our pupil:; shou it in less time than is usually given to the subject, an the power to think for themselves. How shall this be accomplished? Dy better teacliing, by better arrangements of subjects by use of smaller numbers, less figure work, b^ solving such -53- pr ob lens as occur in actual .li£fi. In 1902, President William R. Harper of the Unirer- sity of Chicago proposed a scheme for saving two years' tine in the completion of a college course. His suggestion was a six- year elementary course. Dr. Jolui Dewey, in disc is prob- lem says, "The proper aim of elementary tuition is to organize the instincts and impulses of children into interests and tools Six years ought to be enough to accomplish o task. Out of this ei'fort to e vital .uestion, "'.h »th knowtn a Dr. Frank Uc live red an s i i arithmetic before the Department of Superintendence of the national Educa- tional Association in 1904 which stirred educators to furthei terect in essentials. He says the content of studies should be determined, fir;;t, by social needs, second, by the child's ability to comprehend. He decries the "harmonious development of the facul- ties" theory and the puzzle problem that puzzles the teacher. He says, "Life is too full of large specific aims to be attained to allow for work that has no 1 le object." He rejects: 1.1 , "Ibort C: Education 14: page . tth 1894. 2. Dewey, John: Scho . " nuary 1903. . Ions arc Le in the pre.;en\, :ourso of Study and What should be the Basis for U Same? National Edu 1904. - - 1. liatevor cannot bo shown to have a plain relation to some re 1 need of life... 2, Whatever is not reasonably within the chil ' comprehension. 3« Whatever is unlikely to appeal to his interests* * . 4. Whatever to ics and details are so isolated or ir- relevant that they fail to be a part of any series or chain of ideas... These principles Dr. Mc Hurry pointedly applied to arithmetic, and his work became the basis for much Invest igati n and study. Guy lf« Wilson and his teachers at Connersville, Indiana in 1909, obtained from the business community an o i lion on en- ants and inclusions in arithmetic. The business community, through their merchants, bank- ers and factory superintendents, expressed themselves In favor of more attention in the ublic school:; to the following to ics: 1. Saving and loaning money. 2. Itortga 3. I. T odern " thods. 4. Building and loan associ 5. Keeping simple accounts. C . Inv .oy . 7. Bonds as Investments* G. as investments. 9. Harks of a good Investment. (It ted that the gct-rich-qaicl: concerns fleece Lo out of C60, J), 000 a ye r.) 10. Taxes, le.i , Ltures, -55- 11. Profits in different lines of business. ^ 12. Life insurance as protection and investment. In 1911, a course of study based upon those findiu was issued. This Connersville Course became the subject of s' and criticism by Dr. Jessup in the University of Iowa, and by Dr. Coffman at the University of Illinois. They conceived the idea of continuing the study of inclusions and enrichment through the sup- erintendents of all cities of the united States with a population of 4,00C and over. Through the reports made by the school superintendents, Dr. Jessup recommended that more attention be gi^en to the follow- ing topics : 1. Addition. 2. Subtraction. 3. Multiplication. 4. Division of whole numbers and fractions. 5. Saving money. 6. Public utilities. 7. Public expenditures. 8. Insurance. 9. Taxes. 10. Percent 11. Profit. 12. Build ing and Loan. 13. Investments. 14. Interest. 15. Banking. 16. Borrowing. 1. Wilson, 8. I'.: Survey of Social and Business Use of Arithme- tic, Sixteenth Yearbook, page 128. (Also In 1916 reportof Comrittee on Elimination of Subject Matter, Iowa State Teachers Association, and In A Survey of the Social and Business Usage of Arithmetic, Ph.D. Thesis, 1919.) •56- 17. Levies. ± 10. Stocks and Bonds. These same superintendents, Dr. Je3sup states, expressed them- selves as overwhelmingly in favor of jiving special attention to the fundamentals of addition, subtraction, multiplication, divis- ion, and to fractions. In 1915-1916 Professor ?/alter S. Monroe made an investi- gation of the economy of time in arithmetic* Taking as the aim in the teaching of arithmetic the equipping of the pupil (1) with the knowledge of facts, principles, and relationships existing between quantities in the solution of practical problems, and (2) with the skills which are necessary to perform these operations, he endea- vored to determine what problems are practical. His major purpose in this study was to secure lists of arithmetical problems which arise in human activities, and which possess that degree of utilitarian or socialising value \ justifies their being designated as minimal essentials of purpose. These he rroups under: 1. Occupational activities. 2. Activities of the home. 3. Personal activities. 4. Activities of school children. Dr. Calvin N. Kendall and Dr. George A. Ilrick, in ] 1. Jessup and Goffman: Supervision of Arithmetic, Attitude of Superintendents, pa.-e 15. (Also Fourteenth Yoarbo 2. Monroe, V.altor S.: Economy of Time in Arithmetic. Sixte* Yearbook, page 111. aftor a commendable narrowing of the bounds of study by elimin- ating a mass of useless material, established the* following as the le itimate field of elementary mathematics. I. Counting numbers. II. Re ad J ng numbe r s . 1. Integers - Arabic and Roman. 2. Common Fractions. 3. Decimal Fractions. 4. Denominate Numbers. III. Writing numbers. 1. Integers - Arabic and Roman. 2. Common Fractions. 3. Decimal Fractions. 4 . Denominate Numbers . IV. The Processes. 1. Addition. (a) Integers. 2. Subtraction. (b) Common Fractions. 3. Multiplication of (c) Decimal Fraction to 4. Division. three places. V. Percentage applications. 1. Trade or Commercial Discount. 2. Profit or Loss. 3. Commission. 4. Simple Interest. VI. The following subjects should be treated largely for information purposes: 1. Taxes. 2. Insurance. 3. Stocks. 4 . Bonds . 5. Bank Discount. Compound Interest. VII. Denominate Numbers in useful problems of community value . Kendall and Mirick: How to Teach the Fundamental Subjects 1915. ■. • VIII. Geometry in so far as it is required in mensuration and In making and reading working drawings in shop work. IX. Algebra in so far as the use of letters is required • In simple formulas in mensuration and in simple prob- lems solved by the equation method. Minimum essentials by David Eugene Smith, as found in a chapter on arithmetic in Teaching Elementary School Subjects , by Rapeer and others are as follows : (V.'ork for the first, second . Miss Worden.) third, and fourth grades is by 2»8, 5's, 10's, 1st Grade : (1) Count to 100 by 1' (2) The simple combinations to 10 or 12. (3) Roman numerals i seen on clock face. (4) l/2, 1/4 In concrete way. (5) Foot, inch, yard. 2nd >rade : (1) 1,000 10's. - reading, writing. (2) Counting by 2's, 3*s, 4's, 9's, 10's. (3) Remainder of 45 combinations. (4) Coin should be recognized - $ , fL. (5) l/2, l/3, l/4, l/8 applied. (6) Multiplication tables to about 5 x 10. (7) Addition of two- figure numbers not involving "carrying", and the subtraction of such numbers. (8) The square and circle. 3rd C- ratio : (1) Multiplication tables completed. (2) Sepa- rate numbers Into their prime factors and learn the simple factor* (3) Division by ono digit, using Ion; division in the latter part of the year. (4) Problem interpretation. 4th Grade: (1) Linear measure. (2) Volume. (3) Addition and subtraction of simple fractions. (4) Simple decimals. 5th Irade : (Recommended by David Eugene Smith) (1) Review four fundamentals with whole numbers. (2) Continue simple frac- tions, stressing multiplication of fractions. (3) Compound num- bers - "Happily this is becomin.: less prominent." (4) Decimal fractions are usual] a up. 6th Grade: (1) Decimal fractions. (2) The elements of per- cent' -c L.M. 1. Rapeer/and Others: Teaching Elementary School Subjects 1917. Chapters 9 and 10* -59- 7th Grade : (1) The work of this grade is civics, economics, or sociolo f ;y, not mathematics . (2) Interest (some mathematics). (3) In other countries : a. Intuitional geometry, b. Simple lin- ear equation in one unknown, c. Graphs, d. Factoring. (*■• may- hope for this") 8th Grade : (1) Same as 7th. (2) Dramatize the civics. Summary of Minimum Essentials by David. Eugene Smith. 1. Addition ) 2. Subtraction ) Whole Numbers 3. Multiplication) 4. Division ) 5. Addition ) Of decimal fractions as shown 6. Subtraction ) in the case of U.S. money. 7. The ability to find a fractional part of a number. 8. Finding of some percent of number. 9. How to multiply and divide a mixed number (£, tf) by a whole number. George Herbert Betts in his C lass-Room Method and Man- it t published in 1917, says, "The main purpose in arithmetic is concrete, direct, practical, applied. It is the business of Arithmetic to enable one to do the ordinary numbering and computing required in the common economic and social relations. The know- ledge required should be : 1. How to count objects of all kinds. How to count by nam- ing numbers only. How to count by twos , threes, etc. 2. How to read and write numbers of ten to twelvo figures. 3. The tables and processes involved in addition, subtrac- tion, multiplication and division of whole numbers. 1. BetLs, George Herbert: Classroom Methods and Management. 1917. Pages 218-219. •60- 4. Common fractions, and their addition, subtraction, mul- tiplication, and division with the use of such denominators as are con only used in business. A similar knowledge of decimals in- volving; up to three places . 5. Th6 common tables and measures employed in the ordinary life routine of the average man or woman. These are: measures of length, angle, surface, volume and capacity, quantity, weight, time, monoy, value. 6. Our monetary system, denominations, and the various busi- ness practises involving the use of checks, drafts, notes, mort- gages, etc. 7. Percentage, and its simpler applications to practical business uses. 8. Simple mensuration, applied tc lines, angles, surfaces, volumes . Attitudes to be developed: 1. A tendency not to be satisfied with guessing or approxi- mation, but to insist on finding out through the use of figures on all essential matters involving numerical values. 2. Standards of business accuracy that will result in the keeping of an accurate account of all personal or household re- ceipts and expenditures. This will make possible a proper adjust- ment of expenditure to income, and also a right balance among the different objects for vrhich money is spent. 3. Unwillingness to rely on general estimates or rough ap- proximations with reference to projects planned, as improving a home, or a farm, taking a trip, investing in an automobile, etc. 4. Insistence on detailed and accurately kept records of profits or losses from the different enterprises of farm, shop cr business . 5. The development of such a sense of values and the inev- itable lo^ic of figures as will render one proof against the -et- rich-quick schemes planned by unscrupulous promoters to catch those who through ignorance of business believe wealth to be at- tained by some kind of magic. 6. A sense of pleasure and satisfaction in the use of fig- ures and in the certainty which comos from their wise application ■61- to one's affairs. John Charles Stone in his Teaching of Arithmetic , 1918, makes the aim of arithmetic practical, and outlines the essen- tials as follows: 1. Efficiency in computation. 2. A social insight into business and industrial practices that will enable one to interpret references to such practices met in general reading or in social and business intercourse. 3. Power to express and to interpret the numerical expres- sions of the quantitative relations that come within our exper- iences. 4. The habit of seeing such relations, particularly those that are vital to our welfare. 1 Dr. Junius P. Meriam of the University of Missouri, per- haps more than any other educator, has stamped the traditional arithmetic as non-essential. He says that the best way to teach arithmetic is not to teach it at all; that there should be no re n.ilar class periods and no regular texts; that arithmetic is a school subject presented to keep the child occupied, to keep him from worse behavior, without considering the outcome of the occu- pation; that the only arithmetic worth bothering with is that which the child comes face to face with in life; that, however, boys and rrirls should, and will, have considerable to do with arithmetic as they experience quantities and measurements as they help to do those things about them, and through these will learn to handle arithmetic processes. He further recommends that we 1. Stone, John C: reaching of Arithmetic. 1918. •62- construct a school course in terms of normal activities. In his Child life and the Curricula 2 Dr. Meriam crit- icizes unfavorably the courses of study in freneral use, all text- books in part or in whole, and the methods of presentation. He says: Arithmetic is a cross section of a great variety of exper- iences in the quantitative level." "Arithmetical abilities can be measured by following pupils Into stores, shops, factories and other places of employment and there taking into account the arith- 4 metical calculations made as part of their work." "School arith- metic is strictly a form subject. It has not yet approached the study of quantitative aspects of our invironments and our real adjustments." Dr. Meriam' s attitude toward arithmetic in the elemen- tary school curriculum reminds one of that justly famous treatise upon "Snakes in Ireland", the preface of which contains the inci- dental observation that there are no snakes in Ireland. Dr. Edward Lee Thorndike states his attitude toward essentials in arithmetic in this letter: "You will find my opinion concerning what should be In and what should be left out of the elementary school course in mathematics worked out fully in the Thorndike Arithmetic. Every - 1. Personal interview with the writer, January, 1921. 2. Meriam, J. P.: Child Life and the Curriculum. 1920. 3. Ibid page 419. 4. Ibid page 467. 5. Ibid page 286. -63- tMng *n those except the few exercises marked "optional" up to page 248 of Book III should, in my opinion, be left in, together with a selection from the material on pages 249 to 286, as stated on page 249. Yours truly, E. L. Thorndike." 1 In Thorndike's newest book, New Methods in Arithmetic , 1921, a book built entirely upon the textbook material in the 2 three volumes of the Thorndike Arithmetics (now adopted as the State Text in California) he advocates useful computations as op- posed to indiscriminate ones, facility and absolute accuracy with small numbers; genuine problems; arithmetic for life. He urges that the older methods be discarded and that the newer arithme- tic, founded upon common sense, common requirements, common needs be given trial, and 1s of the belief that this will win assent and confidence on merit. Prom this examination of the works of these recognized leaders in thought, one seems justified in the conclusion that the modern aim in arithmetic is a practical one, and that throu-h re- search, teachers, superintendents, and reco/mized educators are endeavoring to find out what are the clear-cut first essentials, and what shall constitute a minimal course of ;;tudy in arithmetic. 1. Extract from personal letter received by Chairman of Commit- tee on Minimum Essentials in Arithmetic, Southern Branch Univer- sity of California. 1920. 2. See appendix. PART TWO THE INVESTIGATION •64- CHAPTER I SOURCES AND METHODS OF COLLECTING THE DATA. The initial impetus for this investigation was the common work of a Committee on Minimum Essentials in Arithmetic of which the writer is secretary. This committee was apoolnted and began its work on February 9, 1918. A group of Superintendents 2 of Southern California City Schools , and the President of the Los Angeles State Normal School (now the Southern Branch of the Uni- versity of California) banded themselves together for the better- ment of elementary school education. "Our object", said their chairman, "is to attempt, through the labors of a series of carefully selected committees, to clearly define the purpose which should regulate the teaching of each of the several elementary school studies , and in accord- ance with that purpose reduce each of these studies to its lowest terms by eliminating all lessons and parts of lessons which do not specifically contribute to that purpose, and to 3tudy the best 1. Committee: Myrtle Collier, Chairman, Southern Branch, University of Calif. Katherine Spiers, Secretary, " " Dr. A. W. Plummer, Los Angeles. Bertha R.Hunt, Santa Monica Dr. n. H. Snyder, . Ruth Smart, Long Beach. Dr. A. H. Sutherland, Los " . Rufus Mead, Pasadena. Berthile Barclay, Santa Ana. Frances Brown, Riverside. Ann Burnam, Pomona. Jessie Wilkinson, San Ber- nardino. 2. Los Angeles, Santa Ana, Pomona, Santa Monica, Long Beach, Pasadena, Riverside, Redlands, San Bernardino. ■65- ways and means of attaining that purpose in the teaching of each subject." The Committee on Arithmetic, which is still operative, met at intervals throughout the years 1918 and 1919 and much val- uable work was done. The v/riter, during the years 1919, 1920 and 1921, has extended the study and has compiled tins report which follows closely the line approved by the Superintendents of the Southern Cities and the Committee on Arithmetic. In collecting these data an effort was made to reach rep- resentative groups. The data of Questionnaires I and II were ob- tained through the cooperation of the students of the Teacher Training Department of the Southern Branch of the University of California; superintendents, principals, and teachers of Southern, p Central, and Northern California schools; schools in Arizona, and in Alaska. The large city, the small town, the rural district, and the remote outpost of civilization are represented. The data of Questionnaire III was collected through personal interviews with, and letters sent to, lar e business con- 's cerns in Los Angeles. 1. Moore, E.C: Address, Los Angeles State Normal School, Feb- ruary 9, 1918. 2. Grateful acknowledgment is made to a group of superintendents, principals, and experienced teachers, members of a seminar -roup in the Summer Session of the University of California, 1319, for valuable help ,7iven luring the years 1919,1920. With- out their help this work could not 'ave been carried on. 3. The writer thanks Dr.AIV. Plummer , Principal of the Twenty-ninth Street S c hool, Los Angeles, for permission to use data col- lected by Mm and embodied in this report. -66- The problem throughout is a positive one. It is to determine what arithmetic men and women actually do use, what op- erations are employed, what figuring is actually done, and what insight and skills are required by business men and women. Questionnaire I was sent to the general public: business and professional men and women, merchants, shop keepers, day la- borers, ranchers, miners, cattle men, the leisure class, the home keeper. The time of collecting covers a period of one and one half ye,ars . The purpose of this questionnaire was to discover the size of the numbers used in the four fundamentals with whole numbers, fractions, and decimals, and the type of arithmetic used in daily life. Questionnaire II was sent to parents of pupils In the upoer grammar grades and in high schools. The teachers of many schools, and the student-teachers of the Training School of the Southern Branch of the University of California collected these data. The questions were filled in at different ten-day inter- vals throughout a year. The purpose was to determine what flgur- Ing is actually done by people from day to day, and what arith- metical topics are in daily use. The data of Questionnaire III are compiled from the re- 1. This questionnaire follows the line of those of Dr. Jes3up and Dr. Coffman In 1913. -67- plies of such prominent Los Angeles business concerns as the following: Lewellyn Iron Works; Santa Fe Railway Company; Civil Service Commission; Los Angeles Creamery Company; Rivers Brothers, Wholesale Produce Company; H. JeWne Company, Grocers; A Ham! n and Sons, Merchants; R. L. Craig and Company, Importers and Whole- sale Grocers; Hauser Packing Company; Los Angeles Planing Mill Company; Salt Lake Route; Goodrich Rubber Company; Cudahy Packing Company; Kahn-3eck Company; Bishop and Company; S perry Flour Com- pany; Los Angeles Ice and Cold Storage Company; Newmark Brothers; Howard Brokerage Company, Farm Products; Globe Grain and Milling Company; Los Angeles Public Library; Reynold E. Blight, Certified Public Accountant, others. The purpose of th^ s questionnaire was to determine business needs in arithmetic, to invite criticism of the school product, and to ask for re commend 'it ion toward improvement in the school course of study. ■68- Questionnaire I. The tables and charts on pages 70 to84 inclusive, show, in tabular and graphic form, the results obtained from the questionnaire to the general public. Questionnaire II. The tables and chart on pages 87 to 90 inclusive, show, in tabular and graphic form, the results obtained from the questionnaire to parents. Questionnaire III. The totals and charts on pages 93 to 95 inclusive, show the judgment of a group of business men of Los Angeles. •69- CHAPTER II QUESTIONNAIRE I. (1500 printed or mimeographed copies 3ent out. 1136 replies.) To the Public: The purpose of this questionnaire is to find out how much arithmetic is used in every-day life. Do not state what you are able to use but what you actually do use. State the problem used in questions 1 to 10 inclusive. Do not sign your name. 1. Please state your occupation Answer question by underlining The numbers . 2. Do you pe r s onaTly nave occasion to you personally have occasion to add columns of 2, 5, 4, 5, 6, or more "numbers in height? ~" ~ _____ Total number persons replying, 1136 1 lumber using 2 addends 1077 i i 1 1 1 1 1056 1005 936 864 492 GRAPH. c X 5f> sefo /o Basis 1136 2 addends 3 addends . 4 addends 5 addends 6 addends .lore 1 ■78- tUEMluN III. Totuls and graphic representation of question number ill. Do you personally hare occasion to add columns of 2, 3, 4, 5, 6 or more figures in v.idth? Total number persons replying, - Number using 2 figures in width ■ 3 ■ 4 5 6 it ii 1136 lot-; 1049 964 724 039 264 c i° i5 ■>/. soi» 75 «/* /O o Basis 1136 1 2 figures mmmml^mmmm 3 figures 4 figures 5 fipures 7t^^^^^^ 6 figures More ■■■■MB 79- QUESTION IIII, Totals rj»d graphic represent ation of question rruribor ] III: io you personally liave occasion to .ultiply auribors of 2, 3, 4, f.. fir Total iiuooer persons replying 1136 Multiplicand 2 figures 1077 1 846 40C 206 °P Stf" 5C "f c 7S ¥° 100'f' Basis 1136 2 figure rnultiplicond 3 4 " « 5 " 6 :.!ore " wmmmm* -80- QUESTI01I V. Totals and graphic representation of question number V: Do you personally have oooaslon to multiply by_ numbers of 2, 3, 4, 5, ,or uore figures? total number persons reply ing Multiplier 2 figures " 3 S» 4 '» 5 B more . 113 ■1C29 • 875 ■ 611 . 364 269 OBIS. ,1<> ZOO'-JO Basis 1136 2 figure divisor " 4 '• 5 " | 6 " More " ^ ^_ •83- QOLSTIuH VIII. Totals and graphic representation of question number VIII. How many of the following fractions do you personally have occasion to use: halves , thirds , fourths , fifths, sixths , sevenths , ei -hth s, ninths , tenths , twelfths, sixteenths? Total number persons replying — 1136 TOTAIS Halves 921 Fourths 841 Thirds 687 Eighths 466 Fifths 453 Tenths 434 Sixths - 38t Twelfths 297 Sixteenths ~ 249 Sevenths 244 Ninths 222 OR'.PH Basis 1136 «/» 1 5 rf° s d» TS *f° WO Halves Fourths Thirds Eighths Fifths Tenths Sixths Twelfths Sixteenths Sevenths Ninths ■■■■■■■• - TIOH IX. Totals and graphic representation of quest in nunber IX. Do you personally Iiave occa sion to itso docimls of 2, 3, 4, or :ore places? Total nunber persons replying - 1136. GR/ ifl DM 909 599 379 — 179 Basis 1136 Decimals 2 pla 2 places 3 " 3 places 4 ■ " more " 4 places J.tore " 100°/° -84- vUKETlUN X. Totals and graphic represent Lion of question number X. Do you perso n ally have occ sion t-o compute s imple interest s Number persons replying, 1136 Y©s 724 No 412 GRy PH 00°j° (a) Yes, o, (b) 1 Mo, (c) Yes, ■o. (d) Yer, No, (e) Yes, Inda Banking Paying :n Part Payments Souare Measure Volume Board Measure Drawing to Scale Graphs Land Measure Trado Discount .87- "" "7I03HMEE *% II. cat^? ijfl+i jo 3 U«a M ft flS -do ® © 3 «T 3 c'o o OOfrjO U 85 I.JJ ii< « « ft ft wa > ks o-p o mb so Blacksmiths-- Bookkoopers" Carpenters—— Civil Engineors Creamerymon-- Electricians— — HouseviTTos— Laundrynen ~ Hail Carriers- Mochanics— -- Merchants Miscellaneous— Photographers— Plumbers— Railroadmen— Ranchers— • P.eal Estato Hen- Tailors Teachers , Students TOTAL Number of time s used in 10 days. 4 1G 2 G G 19 95 56 11 91 19 26 18 103 11 154 59 5 17 5 11 3 3 1 3 7 18 4 2 5 4 1 3 6 27 6 1 11 8 17 125 102 56 19 41 15 18 9 30 3 24 10 10 15 3 10 1 1 1 26 25 47 84 34 51 78 4 37 41 138 121 49 3 45 12 144 63 150 5G 71 61 79 40 7 50 82 4 C 5 1 26 3 15 6 13 4 41 69 49 31 10 30 1 37 21 9 55 28 2 1 5 28 15 15 10 12 1 3 10 30 7 22 16 27 13 It 1 1 -88- i -p c X o o WCC -p-p -h see a> a "S ►"'j? 7} toe So oo rH«- ,o <1 Q 1 « %■•*.* l» 2 "> " 1 '•» K ~ -M -91- JUDGMENT . Reference to the tables and graph of Questionnaire II shows that the schools aro not giving proper emphasis. Too little time is given to some topics, as, Simple Cash Accounts, or Family Expense Accounts; Cash Check, or Bills; Banking; and too much Is -iven to others, as, Division of Fractions j Volume; Proportion; Stocks, Dividends, and Bonds; Square and Board Measures. The returns from the section, "Problems used from day to day", show that wo are failing to utilize a wealth of practical 2 material for problems by following text books solely. Arit lime tic used in evory-day life . Essential topics. 1. Cash Accounts, and Children's and Family Expense Accounts. 2. Addition of Fractions. 3. Cash Checks, or Bills. 4. Multiplication of Fractions. 5. Subtraction of Fractions. 6 . Banking . 1. People often think they aro dividing by a fraction when they take a fractional part of a number. 2. See section, E ssenti als In Problems , pages 108-109 ^ this study. 3. Let the reader boar in mind that these are essential tonics, not the only topics to be used. -92- CIIAPTER IV QUESTIONNAIRE III. (100 letters or personal interviews. 51 replies in whole or in part . ) "To Business Men of Los Angeles: Will you please assist our public school teachers, and thereby help the boys and girls, by giving the following questions careful consideration and sending your reports to the Committee on Arithmetic? 1. How much arithmetic should young people know when they enter your employment? 2. In what arithmetic work do you find them weak or unsat- isfactory? 3. What suggestions do you make that may assist* in correcting mistakes? 4. So far as It comes to your attention, what work in arith- metic is being taught that is of little or no value in your business?" 1. Note 3, page 65. -93- Question 1. How much arithmetic should young people know when they enter your employment? 51 Replies oj> to-/' 20>/' st>7' 401> sof> (,0?° 70Y' 80Y' 90^' fix- Addition - 51 ' Multiplication - 50 Division , - 47 Subtraction - 45 Decimals - 42 Fractions - 40 Percentage - 32 Accuracy - 24 Mental Arithmetic - 16 Geometry - 12 _ Proportion - 8 _ „ . . Profit and Loss - 6 Discount - 5 First year algebra - 2 ■ -94- Question 2. In what arithmetic work do you find them weak or unsatisfactory? 40 Replies cW» / 70°/° eof lot"* Accuracy -40 Addition - 36 Decimals - 36 Multiplication - 30 Fractions - 28 Division - 25 Short Cuts - 21 Percentage - 20 Interest - 16 Mental Arithmetic - 10 Analysis - 9 Subtraction - 6 Thoroughness - 6 •95- Question 3. Y/hat suggestions do you make that nay assist in correct- ing mi stakes? 50 Replies )f> 30 •/> A Of 50f' tOf' 70f' 60?' thoroughness in Fundamental! Thoroughness in Decimals -46 Thoroughness in Fractions -42 Mental Arithmetic -35 Short Cuts Practical Problems ''Teach the Why" Analysis '■'■'eights and Measures Multiplication tables to 20 Question 4. T.o far as it comes to your attention, what work in arith- metic is being taught that is of little or no value in your business? '- 50 Replies oy> /of All except Fundamentals All valuable Algebra Higher Arithmet ic of- SOf' bOf 1 70 1- 60 f lot •96- JUDGMENT , Business firms demand skill. They ask for accuracy and speed in handling the fundament/ : & w^iole numbers, fractions, decimals. They require thoroughness and power to interpret prac- tical problems. Question 1. Essentials: Report of 51 Firms Addition (whole numbers) 100 % of Firms Multiplication (whole numbers) 98 % " " Division (whole numbers) 92 % " " Subtraction " 88 % " " Decimals 82 % " ■ Fractions 78 % " ■ Percentage 62 % " " Question 2. Employees weak , unskilled : Report of 40 Firms Accuracy 100 % of Firms Addition 90 % " " Decimals 90 /£ " " Multiplication 75 % n " Fractions 70 « Division _ . . 62 % " " Short Cuts 52 f a " ■ Question 3. estlons t oward improveme nt of teaching: Report of 50 Firms . Thorou-hness in fundamentals 100 % of Firms " docinal3 92 % " " " fractions --• 84 % " Teach mental arithmetic 70 % " " " short cuts 60 % " " Practical problor^ 54 " "Teach t ho Why" 50 " -97- Questi on 4. Valueless arithmetic ; Report of 50 Firms All except fundamentals 46 % of Firms All valuable 28 % " " Algebra 16 % ■ Higher arithmetic 10 % " ■ -97- SUMMAKY. ELIMINATION OF TOPICS. Through a careful comparison of the data obtained by- means of the three questionnaires, with the results, records and judgments collected under Chapter III, "Eliminations V/hich Have^ Been Suggested or Made," the following sweeping reduction in sub- ject matter is recommended: Eliminations to be made in Elementary Texts and Elementary Courses of Study. 1 1. Apothecaries Weight* 2. Troy Weight. 3. Longitude and Time. 4. Furlong in Linear Measure. 5. Hand. 6. Dram in Avoirdupois Weight. 7 . Surveyors ' Table . 8. Fathom. 9. All problems in reduction, ascending and descending, involving more than two steps. 10. G. C. D. as a separate topic. 11. All initial common fractions except halves, thirds, fourths, fifths, sixths, eighths, ninths, tenths, twelfths, sixteenths, hundredths, thousandths. (Initial fraction is the fraction -~iven for the solution of a problem.) 12. All work with L. CM. except of very low denominations. (As a separate topic.) 13. Complex fractions. 14. Compound proportion. 15. Case:; in percentage. 16. True discount. 17. Compound interest, except in si pie savings accounts as re-invested money. 18. Problems in partial payments. , 1. All of these topics are found in Texts or Courses of Study used in public schools durinf the past ten years. •98- 19. Commission and Brokerage (as applied to stocks and bonds . ) 20. Profit and Loss as a special topic. 21. Knot. 22. Partnership (as a special topic). 23. Cube Root. 24. All algebra, except such simple use of the equation as is directly helpful in arithmetic and in other sub- jects met with in the school life of the pupil. 25. Brackets, Braces, Vincula. 26. Cancellation, as a special topic. 27. Finding the whole when a fractional part is given. 28. Paper tables. 29. Gross and Great GrSss. 30. Square (100 sq . ft. used in roofing). 31. Carpeting, Lumber Measuring, Papering, Plastering, Painting, as separate subjects. 32. Surveyors' Land Measure. 33. Foreign Money. 34. Indirect problems in simple interest. (Use the equation.) 35. Bank Discount. 36. Pyramids, Cones, Spheres. 37. Metric System. (To be learned as occasion for use arises. ) 38. Initial decimal fractions of more than three places. 39. All problems v;hose content is outside the experience of the child. 40. Examples of this type: 5^-3x7^-6-4. 41. All improbable problems. 42. All problems in which the part and its fractional equiv- alent are given to find the whole. 43. All so-called problems without number. ; ESSENTIALS IN SUBJECT MATTER. If the results indicated in this study are to be attain- ed, and if the returns from the three questionnaires, which defin- itely emphasize the judgments of experimental research workers along a like line, be considered valid basis for establishing the business and social usage of arithmetj c, the following is the legitimate field of elementary arithmetic. I ^ Cr ' /Kf'i 1,2. -99- Requircmcnts : A high degree of accuracy in solving problems involving: 1. Addition, six addends, five dibits wide. 2. Subtraction, six dibits wide. 3. Multiplication, multiplicand of five digits, multiplier of four digits. 4. Division, dividend of five digits, divisor of four digits. 5. Addition, subtraction, and multiplication of these fractions: halves, thirds, fourths, fifths, eighths, tenths • 6. Addition, subtraction, multiplication, and division of decimals to threo places. 7. Simple interest. 8. Percentage, (avoid the Indirect problem.) 9. Cash accounts, and children's and family expense accounts. 10 . Banking . 11. Common measures. Schools have made arithmetic unnecessarily difficult, when, in truth, the work should be simple and easy, as an ability to do practical problems is all that is required in life. The average individual needs to know how to add, subtract, multiply, and divide whole numbers and decimals; to add and subtract simple fractions; to find a fractional part of anything; to multiply a whole number by a fraction, perhaps a friction by a fraction; to find percentages; and to be familiar with comr.ion measures. Our pupils should be able to master these necessary things, and master them thorou -hi;;, y the end of the sixth year. Studies in elim- ination and retardation which have been carried on in recent years show that a very -reat number of children do not complete the ci t-year elementary 3Chool course. In view of a f ct, it is •100- of vital importance that they be tau r ht the arithmetical opera- tions required in life as early as possible in their school career. The two years' time thus saved might with profit be utilized in r-iving boys and girls more extensive acquaintance with problems connected with social, industrial and civic life. Where desired, a unified course made up of arithmetic, algebra, and geometry, all worked out to fit life-needs, might follow the six-year arithme- tic course* In this brief and partial survey of the outstanding needs of the public in arithmetic and the relation of these needs to school instruction, many things have been omitted which deserve consideration, and many questions have been suggested to which no definite answers have been given. Much of the material here pre- sented is already familiar to the f orward-lokin 1 teacher who has made a study of arithmetic. The aim has not been to exhaust the subject, but to point out the real issues and to Troup the facts established around a center of progress. It takes 'any people working together to map out a program for the teaching of any subject. The elaboration of the following course of study or the pruning of it as the critics judge fit must be left to those in- terested teachers who will use their classrooms as trying-grounds, who are ready to study with new interest and thoroughness every aspect of the subject, and who are willing to weitfh all values on the scales of pttblic need. -102- 5. Playing dominoes, (a) Matching. (b) Counting by 5s. 6. Time, (a) Making a clock face. (b) Hours, 9, 12, 2, etc. 7. Games* scoring, (a) Bean bag. (b) Ring toss. ( c ) Nine pins . (d) Hook it. (e) Gue3sinfT games. (f ) Bui Id in • up numbers, using all possible combinations. 8. Number stories. Through such activities, the minimum number work of the pupil should be : First Grade - 1. (a) Count to 20 concretely. (b) Count to 20 abstractly. (Symbols are to be used after the numbering knowledge has been obtained by the use of objects in work and play . 2. Group objects by 2's and 5's to 20. Count by 2's and 5's to 20. (Grouping and count- ing symbolized with written words and vith digits. 3. Divide groups of objects into 2's, 3's, and 4's to 12. 4. Use term halves when groups of objects are divided into two equal parts . (Not more than 12 objects to be used. ) 5. Emphasize the relationship between quantities by means of objects. (Suggestion: Relationship be- tween inch and foot, pint and quart.) 6. Denominate numbers. Measurements - 12 inch- 1 ft. 2 pint- 1 qt. 12 things ■ 1 doz . hi « 1 nickel. 10j^ s 1 dime 2 nickels ■ 1 dime. 7. The addition of halves and halv ■103- Second Grade - 1. Review tho work of first grade. 2. Continue counting concretely by 2*s and 3's to 24; by 4*s to 40; by 5's and 10 *s to 50. 3. Present the thirty-throe combinations, concrete- ly, whose suns are 12 or less. Aim toward an abstract and sntomatic mastering of these. 4. Division of groups of objects into 2 f s, 3's, 4 ! s, 6 f s, to 24* 5. Divide groups of objects into 2, 3, 4, 6, equal parts. Maximum 12 objects. 6. Use terms halver., fourths when objects are di- vided into two or four equal parts. 7. Column addition, two digits vide. (Sum of each column less than 10.) 8. Column subtraction, two digits ride. (Each digit in the minuend to be greater than the cor- responding digit in the subtrahend. ) 9. The addition of fourths to fourths. (In hand work problems . ) 10. Denominate numbers. 25,^ ■ 1 quarter. 60 min. • 1 hour. 7 days * 1 week. Aim in the Third and Fourth Grades - Since it is in the t- ird and fourth grades that habits of accuracy or inaccuracy are formed in the basic work in mathe- matics, the aim in these grades should bo the automatic mastery of the f orty-f 5 ve combinations in addition, and the corresponding num- ber fact3 in subtraction; the multiplication tables through 10 x 10, and the corresponding number facts in division. In addition to this the child should learn column addition and subtraction In- volving the adding-ln and tho taking-from processes; rmiltiplica- tion with a multiplier of two or three digits, and division with a divisor of two dibits. -104- Third Grade - 1. The forty-five combinations in addition and subtraction made 1 automatic. 2. Column addition three digita wide, four addends. (With the adding-in process.) 3. Subtraction with Lhree digits wide. (With the taking-f rom process . ) 4. Multiplication tables - 2's, 3's, 4's, 5'3, 6's, 10' s, throTigii 6 times the number multiplied. 5. Division, corres ponding to the combinations in the multiplication tables, and also with remain- ders. 6. Denominate numbers. 3 ft. - 1 yd. 4 qt. - 1 gal. 50J? » one-half dollar. Fourth Grade - 1. Continue work of third grade. 2. Addition, four digits wide and four addends, also three digits wide and five addends. 3. Subtraction, four digits vide. 4. Complete the multiplication tables throuh the 10's. (lu x 10) Multiplicand 4 dibits wide, multiplier 2 dibits v.ide. (Using dollars and c cents . ) 5. Division: Dividend not more than five digits; divisor not more than two digits. 6. Denominate numbers. 16 02. - 1 ]b. 10 dimes = |1. 100 cents s $1. Fifth Grade - The aim in the fifth grade should be the mastery of com- mon fractions. 1. Continue drill on the four fundamentals with whole numbers . 2. Fractions. (a) Continue fraction work of the previous jes . (b) The addition of fractions in the follow- ing order of groups : Halves and halve c . -105- Halves and fourths. Halves, fourths, eighths. Halves, thirds. Halves, thirds, sixths. Halves, thirds, sixths and twelfths. Halves, tlirds, fourths, sixths and twelfths. Fifths and tenths. (c) Reduction of fractions when necessary in addition and subtraction. (d) Addition and subtraction of mixed num- bers. In addition, tv;o digits wide, three addends . (e) Multiplication of fractions - l/2, 1/3, 2/3, l/4, 3/4 each repeated a given num- ber of times. (Example, 3 x 1/2 - 3/2.) 1/2, 1/3, 2/3, 1/4, 3/4, 1/10 of a" ^roup. (Example 2/3 of 18.) Multiplication of simple mixed numbers (In playing store, etc . ) (f) Division of fractions. How many l/2's, l/3 f s, 1/4' s, 3/4's, l/io f s in whole num- bers. Example, how many 3/4's In 6, or 6 -T- 3/4 = ? Develop throu "h the concrete, use abstractly. 3. Denominate numbers . (a) Review - Continue the work of previous* grades . (b) Hew - 24 hr. r 1 day. Sixth Grs.de - 1. Drill on the fowr fundamentals ?'n whole numbers and fractions. 2. Decimal fractions. (a) The decimal Idea. (b) The four fundamentals in decimals, with a limit to tvo decimal places in initial decimals (Stress dollars and cents). 3. Per cent. 1/100 . .01 x 1%. Nothing new. 4. Find the per cent of a number. 5. Find what percent one number is of another (18 is what per cent of 20?) 6. Denominate numbe/ (a) Review denominate numbers through prob- lem work. (b) New work to be taken up through problems . •106- 100 lbs. =1 cwt. 2000 lbs. » 1 ton. 144 sq.in.« lsq. ft. 9 sq.ft.s 1 sq. yd. 160 sq.rd.» 1 Acre. 60 sec . » 1 minute . 365 days ■ 1 year. 12 mo. » 1 year. Seventh, eighth and ninth grade mathematics should be a unified course made up of arithmetic, algebra, and geometry, and presented in such a way that there will be no definite break from arithmetic into algebra. Seventh grade - 1. Drill for speed and accuracy with whole numbers, fractions and decimals. 2. The application of numbering to real life needs: (a) Social. (b) Industrial. (c) Civic. 3. Discount - a3 per cent. 4. Interest. (a) Money. (b) Investments. 5. Commission - in connection with vocational guid- ance. 6. Taxes - local; in simple form in connection with civics. 7. Personal accounts. 8. Banking; (a) How to make out a deposit slip. (b) How to write a check. (c) Hon and why to fill the stub. (d) When a check should be cashed. (e) How to stop payment on a check. (f ) How to indorse a check. (g) llo.v to indorse a note. (h) How to write a negotiable note. (i) How to compute interest. (j) Importance and purpose of savings banks. (k) Importance and purposebf commercial deposit banks. (1) How to open an account, (m) How to secure a bank draft, (n) How to use a bank draft. •107. 9. The solution of the simple equation in algebra. (a) Definition of equation. (b) Making equations. (c) Solving equations by: Adding the same amount to both sides. Subtracting the same amount from both 3ides. Multiplying both sides by the same amount. Dividing both sides by the same amount. (d) Use of the parenthesis in equations. (e) Positive and negative numbers in the equation - 10. Intuitive geometry based upon shape, size and location of objects. (a) The rectangle. (x) Perimeter, (y) Area. (b) Triangle, (x) An;;les. (y) Similar triangles - in solution of problems only, (z) Construction of similar triangles. 11. Use of the compasses in drawing straight lines, dropping a perpendicular from a point to a line, erecting a perpendicular at a given point in a line, constructing equal angles. 12. Use of the protractor in measuring angles. Eighth Grade - 1. Farther development of Seventh grade work. 2 . Graphs . 3. The right triangle - Theorem of Pythagoras. 4. Relation of opposite angles and of the angles made by a transversal cutting parallel lines. 5. Cubical contents of rectangular prisms. 6. Cubical contents of cylinders. The work in numbering must be a unit from the first grade through the eighth (Example: After fractions have been learned there is nothing new in decimals but the for 1 -! of vriting. Per cent Is only a particular common fraction. l/lOO, or a par- ticular decimal, .01 * l/lOO ■ 1%.) Much time will be saved by this unity of work. Further- •ion- more, the child will be able to master the subject of aritnmetic if this simplicity of the subject is shown. ESSENTIALS IN PROBLEMS. 1. All problem vork should be such as will develop the child's ability in numbering -- that is, the content of the problem must be within the comprehension of the child. 2. Children should make and solve problems growing out of their own experiences or environments, covering the num- ber work as outlined for the various grades, using the suggested topics of activity as a guide. 3. In problem solving the five steps are to be learned: a. State the problem clearly, or read it understand- ing ly. b. Pick out the unknown fact or facts. c. Choose the form of relating the known facts (add, subtract, multiply, or divide) In order to deter- mine the unknown fact or facts. d. Solve. (This involves the mechanics of the subject.) The child should approximate his result before solving. The result should always be checked by the pupil. 1 The returns from that section of Questionnaires I and II, " State the problem ," indicate that the stereotyped problems in the majority of textbooks in use to-day are not problems met In life. The following list is compiled from these returns, and from personal observation of the things children really do. Proble m Material : Real grocery bills, making chati'o, games, value of a cafeteria meal, measurements; inch, foot, yard, mile, (fraction of), acre, dozen, (fraction of) liquid {pints, 1. Prom "Minimum Essentials," 1921. (to be published.) ■109- quarts, gallons,) time, U. S. Money, pound, ton, averages, home garden problems, errands, cost of hnating and li -hting a home, reading meters, cyclometer, pedometer; wages, labor questions, chicken business, plotting gardens and courts for games, measuring land and city lots, cost of furnishing a home, cost of building a garage — cement, lumber, labor, paint, draw to scale, insure, — borrowing and lending monoy, commissions, insurance, local taxes, Shopping, dairy business, interest, bond coupons, banking (every phase met with by depositors), income tax, discount in shopning, household accounts, budgets, sugar and cotton industries, frc ing, fire department, city market, real estate, excavations, road and street building and repairing, transportation — street car, railroad, ship, — percentages in games , races, field meets, others provided they meet this standard: "The good textbook and the good teacher scrutinize every task they assign to make sure that it fits the pupil Tor life. They seek to find, for every aritiimetical principle and fact, the real affairs to which it aoplies and with which it should be con- nected in the pupil's mi ad* 1. Thorndike, Edward Lee: New Methods in Arithmetic. 1921. -110- BIBLIOGRAPHY ARNETT, L.D. AYRES, LEONARD P. BAGLEJC, W. C. BALL,K. And WEST,M.E. BALTIMORE COUNTY COURSE BARNES, EARLE BETTS, GEO. HERBERT BOBBITT, FRANKLIN BONSER, FREDERICK BOYDEN, ALBERT G. BRANFORD, BENCH ARA BRESLICH, E. R. BROWN, J. C. BROWN, J. C. and COFFMAN, BROWNE, CHARLES E. Counting and Adding. American Journal of Psychology. Vol. 16, pp. 327-336. Elimination of Unprofitable Subject Matter. National Education Association. 1913. pp. 243-244. Educational Values. Chapter 12. 1911. Household Arts Arithmetic. School Review, Vol. 25. Dec. 1917. pp. 722-730. OF STUDY. 1919. pp. 261-329. An Old Arithmetic. 1655. Academy 4, p. 502. Classroom Method and Management. Chapter 1-9, and Chapter 12. The Curriculum. 1918. The Elementary School Curriculum. 1920. The Essentials of Arithmetic. Education. Vol. 14, pp. 390-400. A Study of Mathematical Education, Includ- ing the Teaching of Arithmetic. Oxford, Clarendon Press. 1908. p. 392. Supervised Study as a Means of Providing Supplementary Individual Instruction in Mathematics. Thirteenth Yearbook of the National Society for the Study of Educa- tion. 1914. pp. 32-73. An Investigation of the Value of Drill Work in the Fundamentals of Arithemtic. Journal of Educational Psychology, Vol.2. 1911, pp. 81-88; Vol. 3, 1912. pp. 485- 492 and pn. 561-570. LOTUS D. How to Teach Arithmetic. 1914. The Psychology of the Simple Arithmetical Processes. American Journal of Psychology Vol. 17. Jan. 1906. pp. 1-37. -Ill- BULLETIN OP THE ST.VTE NORMAL SCHOOL. Superior, '.is., Oct. 1915. CAJORI, FLORIAN CAMERER, ALICE CHARTERS, I. W. CHASE, S. E. COLBURN, WARREN S. COLIINS, JOSEPH V. COOKE, FLORA J. COURTIS, S.A. CUBBERLY, ELWOOD P. CURRICULUM, THE DE'„EY, JOHN DIL ORTH, THOMAS History of Elementary Mathematics. 1917. pp. 215-219. What Should be the Minimum Information About Banking? Seventeenth Yearbook of the National Society for the Study of Educa- tion. 1918. pp. 18-27. Teaching the Common Branches. 1913. Chap- ter 12. Waste In Arithmetic. Teachers' College Record. Vol. 18, Sept. 1917. pp. 360-370. An Intellectual Arithmetic. 1821; Sequel to Intellectual Arithmetic; 1828; An Ad- dress Delivered in 1830 on Arithmetic. Elementary School Teacher. June 1912. The Superintendent and the Course in Arithmetic. Educational Review, Vol. 27. 1904. pp. 83-89. Minimum Requirements in Francis W. Parker School. Elementary School Teacher, Vol. 12, p. 245. Standard Tests in Arithmetic. The Changing Conception of Education. 1909, Mathematics. Fifteenth Yearbook of the National Society for the Study of Educa- tion, 1916. pp. 65-67. Democracy and Education. 1916. Schools of Tomorrow. 1915. The School and Society. 1900. Shortening the Years of Elementary School- ing. School Review, Vol. 2, Jan. 1903. Schoolmaster's Assistant. Published in England In 1743 - America in 1803. -112- DRUSHEL, J. ANDREW EDUCATIONAL RESEARCH. EDUCATIONAL RESEARCH. ELIOT, CHARLES W. FREELAMD, GEORGE E. pre:-: MAN, F. N. GAULT, P. B. GREENWOOD, J. H. QRIGGS, A. 0. HAGGERTY, MELVIN E. HALL-CiUEST, ALFRED L. HART, WALTER W. HECKERT, J. . A Study of the Amount of Arithmetic at the Command of the Higli School Graduates. Elementary School Journal, Vol. May, 1917. pp. 657-666. Los Angeles City School District. Bulle- tin No. 2. First Yearbook. July, 19 American Committee No. 1. International Commission on the Teaching of Mathematics. Mathematics in the Elementary Schools of the United States. Bulletin. 1911. United States Bureau of Education. Educational Reform. 1898. The Concrete and Practical in Modern Edu- cation. 1913. Modern Elementary School Practice. 1919* The Psychology of the Common Branches. 1916, Arithmetical Progression. Education, Vol. 20, 1900. pp. 295-297. Evolution of Arithmetic in the United States. Education, Vol. 20. 1899. pp. 193-295. Peda-^ocrv of Mathematics Seminary, Vol. 19. 1912, Pedagogical pp. 359-375. Studies in Arithmetic. Indiana Univer- sity Studies. Vol. 3. Sept. 1915, No. 32, The Textbook. 1918. Community Arithmetic. Elementary School Teacher. 1911. po. 285-295. Cleveland Survey Testa In Miami Valley. Elementary School Journal. Vol. 18. p. 447. HELMAN, J. D. and SHULTES,F .1 . A Study in Addition. Research Bulletin No. 1. , HENRY B, A Foundational Study in the Peda~o ; of Arithmetic. 1914. •113- INTE RNATIONAL COMMISSION on the Teaching of Mathematics. Report of the American Commissioners. United States Bureau of Education. Bulletin. 1912. No. 14. INTE NATIONAL COMMISSION on the Teaching of Mathematics. Mathe- matics in the Elementary Schools of the United States. United States Bureau of Education. Bulletin 1911. pp. 75-100. IOWA STATE TEACHERS ASSOCIATION. Report of Comm5tt.ee on Elimina- tion. 1916. JACKSON, L.L, The Educational Significance of Sixteenth Century Arithmetic. Teachers 1 College Record. 1901. pp. 232. JESSUP, W. A. and COFF AN, LOTUS D, 1916. JESSUP, WALTER A. Supervision of Arithmetic. School Standards and Current Practices and Society. July 24, 1915. Grade for the Introduction of a Text in Arithmetic. Elementary School Journal Vol. 15. Nov. 1914. pp. 162-166. Eliminations in Arithmetic as a Factor in the Economy of Time. National Education Association Report, pp. 209-222. Economy of Time in Arithmetic. Elementa- ry School Teacher. Vol. 14, 1914. on. 461-476. JOHNSON, GEORGE JONES, OLIVE M. JUDD, CHARLES E. KENDALL and MIRICK klappi:r, PAUL Education by Plays and Games. 1907. Teaching Children to Study. Measuring the Work of the Public School. Cleveland Foundation Survey. 1916. Introduction to Study of Education. His- tory of Mathematics. 1918. How to Teach the F\indamental Subjects. 1915 The Teaching of Arithmetic. 1916. -114- MASSACHUSETTS BOARD OF EDUCATION. A Course of Study in Arithme- tic. Boston, Massachusetts. MEAD, CYRUS D. MERIAte, Junius L. McLELI -',"' and DEWEY MCMU; \y , CHARLES A. McMUb^Y > FRANK M. An Experiment in the Fundamentals. Child Life and the Curriculum. 1920. The Psychology of Number. 1895. Special Methods in Arithmetic. 1905. What Omissions are Advisable in the Pres- ent Course of Study. Report of Proceedings of National Educational Associations. 1904. Elementary School Standards - Attention to Relative Values. 1913. pp. 115-119; Arithmetic Recommendations, p. 167. The Uniform Minimum Curriculum with Uni- form Examinations. Addresses and Proceed- ings of National Education Association. 1913. pp. 131-148. MINNESOTA TEACHERS REPORT ON ELIMINATION. 1915. Eulletin State Department of Education, St. Paul. MITCHELL , A. E. monroe, Walter scott Some Social Demands of the Course of Study in Arithmetic. The Seventeenth Yearbook of the National Society for the Study of Education. 1918. pp. 7-18. Educational Measures and Standards. 1914-1915. Derivation of Reasoning Tests in Arithme- tic. School and Society. Vol. 8. Sept. 7-14; 1918, pp. 295-299, 324-329. Warren Col burn on the Teaching of Arith- metic together with an Analysis of his Arithmetic Texts. Elementary School Teacher, Vol. 12, po. 463-480. Development of Arithnetic as a School Subject. U.S. Bureau of ;/iucation, 1- letin 1917. No. 10, pp. 1-170. •115- MONROE, WALTER SCOTT MOORE, ERNEST C Series of Diagnostic Tests in Arithmetic. Elementary School Journal, Vol. 19. April 1919. pp. 585-607. Analysis of Colburn's Arithmetics. Ele- mentary School Teacher, Vol. 13, 1913, pp. 239-246. Development of Arithmetic Teaching in United States. Elementary School Teacher. Vol. 13, pp. 17-24. Principles and Methods in Teaching Arith- metic as Derived from Scientific Investi- gation. The Eighteenth Yearbook of the National Society for the Study of Educa- tion. Part 2. A Preliminary Report of an Investigation of the Economy of Time in Arithmetic. The Sixteenth Yearbook of the National Society for the Study of Education. 1917. pp. 111-127. Analysis of Oolburn's Arithmetics 4 and 5. Elementary School Teacher, Vol. 13, pp. 239-294. Does the Study of Mathematics Train the Mind Specifically or Universally? School and Society, Vol. 7. 1918. pp. 137-140. What is Education? 1914- What the War Teaches About Education. 1919, MORRISON, H. C NA1I NAL EDUCAVI H ki Reconstructed Mathematics in the ' ' School: the adaptation of Instruction to the Needs, InterestB, and Capacities of Stu- dents . The Thirteenth Yearbook of the Nati onal Society for the Study of Educa- tion, 1314, pp. 9-32. lOCIATION REPORT. Hov a Course of Study should be Determ'ned. 1914, pp. 235-243; 223-235. NATIONAL EDUCATION ASSOCIATION. Report on the Correlation of Studies. 1895. pp. 709-714. -116- PARKER, SAMUEL PHIL: IPS, F. M. PIKE, NICHOLAS PYLE, W. H. RAPEER and OTHERS REEVE, V.. D. RICE, J. M. RUGO, HAROLD 0. SCHORL! I! G, R. , DnVID EUGENE SMITH, ARTHUR G. STAMPER, ALVA I . History of Modern Elementary Education. 1912. Value of Daily Drill in Arithmetic. Jour- nal of Educational Psycholo.-y, Vol. 4, March 1913. pp. 159-163. New and Complete System of Arithmetic. 1788. Economical Learning* Journal of Educati n- al Psychology, Vol. 4. 1913. pp. 148-158. Teaching of Elementary School Subjects. 1917. Chapter by David Eugene Smith, pp. 207-249. Unification of Mathematics in the High School School and Society. 1916. pp. 203-212. Causes of Succes and Failure in Arithmetic. Forum, 34, pp. 437-452; 281-297. Essentials in Elementary Education. Forum. 1897. Vol. 22, pp. 538-546. Statistical Methods Applied to Education. 1917. Significant Movements in Secondary Mathe- matics. Teachers' College Record. 1917. pp. 438-457. Article on Arithmetic in Monroe's Cyclo- pedia of Education. 1911. Vol. 1, po. 203- 207. The Teacing of Arithmetic. 1913. The Te-ichin - of Elementary Math matios« 1901. Mathematics in the Training for Citizenship. Teachers' College Record. 1917. pp. 211-225. Number Games and Number Rhymes. Teaching of Arithmetic. Schoul Science and Mathematics. Vol. 12. 1912. pp. 457-460. A Textbook on the Teaching of Arithmetic. 1913. •117- STARCH, D STONE, CLIFF '.VI N FIELD STONE, JOHN CH STRACHAN, JAMES SUZZALG, HENRY THORNDYKE, EDWARD L. TAYLOR, JOSEPH S. THOMPSON, THOMAS E, WILSON, G. M. Transfer of Training In Arithmetical Operations. Journal of Educational Psy- Chology, Vol. 2, pp. 306-310. Questionnaire to the Business Men of Ind- ianapolis . Arithmetic Considered as a Utilitarian Study. Elementary School Teacher. 1903. pp. 533-542. Reasoning Tests. Pub. Teachers' College, H. Y. 1916. Arithmetical Abilities and Some Factors Determining Them. 1908. The Modernization of Ar-iLVimetic. Journal of Education, Vol. 78, 1913. pp. 541-542; 548-549. Ihe Teaching of Arithmetic. 191 . Mathematics - The New Teaching. Edited by John Adams, 1918. p. 195. Primary Arithmetic. 1911. The New Methods in Arithmetic. 1921. Psychology of Arithmetic. 1921. Thorndlke Arithmetics (Three books) 1918. Subtraction by the Addition Process. Ele- mentary School Journal, Vol. 20, Nov. 1919. pp. 203-207. Teaching and Testing the Teaching of Essen- tials. National Educational ass oc^ s tion. Report, 1913. no. 56-57. Report of Committee on Elimination of Sub- ject Matter, to the Iowa State Teachers Association, Ames, Iowa. The Sixteenth Yearbook of the National Sociot;, or the Study of Education. WILSON, G. M. - A survey of the .ocial and Business U3e of Arithmetic. The Sixl jarbook of the National Society for the Study of Edu- cation. 1917. pr>. 128-142. Course of Study in Mathematics. Conner- ville, Indiana Public School. 1911. A Survey of the Social and Busines' Usage of Arithmetic - Ph.D. Thesi'g. 1919. WILSON, II. B. and G.M. Motivation of School Vork. 1916. pp. 158-182: 370-451. WINCH, V Accuracy in School Children. Does Improve- ment in Numeric?.! Accuracy Transfer to Arithmetical Reasoning? Journal of Edu- cational Psychology. Vol. 1. 1910. pp. 557-589; Vol. 2. 1911. pp. 262-271 and 534-336. VISE, C»RL T. WOOD, EKNEST K Survey of Avithmetical Problems Arising In Various Occupations. Elementary School Journal, Vol. 20. Oct. 1919. pp. 118-136. Test on Efficiency in Arithmetic. Ele- mentary School Journal. Vol. 17, ^pp* 446-453. V00DY, C. yocdm, a. Duncan YOUNG, J. ... A. Measurements of Some of the Achievements in Arithmetic. School and Society. Vol. 4, pp. 299-303. Culture, Discipline and Democracy. The Teaching of Mathematics in the Ele- mentary and Secondary School. 1907. -1- APPENDIX . CKITICISM AND SUGGESTIVE CHANGES FOR THORNDIKE'S AHITMMETICS } Book One. Part One. Third grade. Pa-e 20. No's. 12, 13, 14 have little or no meaning to 8 year old children. Same is true of No. 13 page 22. Eliminate all "counts" except from 0. See page335, 37, 79, 80, 92, 118, 100 (first 10), 120. Pages 42, 43, and 135. " which number means dol- lars etc.". There is only one number representing dollars and cents. The vord number should be replaced by part or figures . Page 44. — "So increase the 7 to 17." Impossible. "Increase" should be eliminated and what is actually done (added) stated; as, add 10 to 7 etc. Page 46. Eliminate No's. 7, 8, 9, and 10 as they are of no value to third grade pupils. Pages 65, 106, 107, 108 and 111 are not for third grade Pages 66, 70, and 71. "Write 3 In the tens column." There is no tens column. Better, — "in the tens' place." Pages 74 and 75 and No's. 14 to 18 inclusive, might do for fifth or sixth :ri c, tit not for third. Page 87. "Write 4 over the of 30." Write 5 over the of 90. 1. Note pagers, note 3. -2- Pages 88, 89, 90 are better for the fourth grade. Book One. Part Two. Fourth grade. For lower grades, especially, the form of the fraction should have a horizontal line; as,j instead of 1/3. 6t vs . 6 1/3. Page 127. No. 6, Is psychologically incorrect. Better divide by 2, 3, etc. as we would if using 875. Page 133. What is the object? Eliminate. Page 142, 143, 144, 145, 173 better be placed in Book Two - Part Two. Page 150. Why such problems for 10 year old puoils? Eliminate. Page 159. No child below the high school can ansver No's. 5 and 6, or do 14 and 15, or cut a pie into fifths. Eliminate Pare 166 No's. 5 and 8. "Write the 1 after the quotient ." 1 over the divicor is a part of the quotient. "Place 1 over the divisor as a part of the quotient" is better. Page 177. Change directions for dividing by a three figure divisor. Children cannot "think" three or ""ore finrures into a dividend. Pan;e 187. No. 2 better multiply by » first and make but o one addition. Page 191, same criticism. Why pacre 199 with Fourth grade, or any other? Eliminate. All the work of adding and subtracting mixed numbers and chan Ting Improper fractions and mixed numbers should be put in -3- fifth grade. Book Two . Part One . Fifth grade . Page 10. No's. 1, 2, 5, and 6, change "Increase" to add 1, or — etc. 8 i Page 24, "Numbers like „ --", change to "fractions like 1 8 etc. and fractions like IT etc. Page 32. Eliminate counts, No's. 1, 2, and 3. Page 72. " which numbers mean miles and." There Is only one number in the product. Should be which part or fig- ures mean miles, etc. Page 74. Eliminate No's. 17, 20, 23, 29, and 33, also Page 82 No's. 1, 2, 3, 4, and 5. Page 75. V.hy not state where to put the decimal point instead of requiring the teacher to tell where it belon.cs? The work with decimals, pages 67 to 90 and 99 to 113 incltisive should be in part two, for sixth grade, also the work with com- pound denominate numbers, pages 91 to 97 inclusive. The fifth 'rade needs more work with fractions. If it be taken from the fourth grade work and put into Book Two, Part One, the work for both grades will be improved. In multiplying a mixed number, or by a mixed number, the fraction should be used first. Book Two. Part Two. Sixth grade. Pa e 137. Eliminate complex fractions, exceet the ali- quot parts of 100, or 1000. Page 142. Eliminate No^. 3 and 6. The answer to No. 6 is nothing. Pages 200 (Commission), 201 (Gains and Losses), 202 (Fixing Prices). 205 (Sharing in percents ) , 206 (Interest) and 207, should he put into Book Three, for 7th grade. The same is true of Mensuration, pages 221 to 231. Pages 243, 111 (Commission). 244, 245, 246, and 248 shoul be placed in Book Three for 7th grade. Taking from the fifth grade and giving to the sixth, and taking from the sixth and giving to the seventh, as indicated, will make both divisions of rrork within bounds of what can be done. Book Three. Part una. Seventh grade. Page 2. " — adding thousandths — " should be number one. w 2. H -- adding hundredths — " should be number two. " 2. ■ — addi-ig ones ■ number three. " 2. ■ — adding tens " number four. Page 6. Eliminate counts . Page 13. Form of checking (no. 17) never in business. Page 15. To divide by or multiply by a " number ". What about fractions? Page 16. "Multiply the integer and numerator." Sub- stitute by_ for and . There appears but little new work for the seventh grade The percentage vork, and the mensuration riven for the sixth grade is needed for the seventh. Book Three. Part Two. Eighth grade. Work for the eighth grade is good but the quantity is out of proportion to that given for the seventh grade. E ubrm ,y of rauwjj?" UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. — ^\,,:^iii3 0MW — JUL 14 1952 LD 21-95m^l ,'50(2877sl6)476 , KoJth- y of ellmi> ■ and busineje requirement of arithrt-. ' S piers A etud 676f?23 UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY, CALIFORN/A