DOCUMENT RESUME ED 096 171 SE 018 471 AUTHOP Peeve, W. D., TITLF The Teaching of Arithmetic. National Council of Teachers of Mathematics, Yearbook 10 [1935]. INSTITUTION National Council of Teachers of Mathematics,Inc., Washington, D.C. PUB DATE 35 NOTE 298p. AVAILABLE FROM National Council of Teachers of Mathematics,Inc., 1906 Association Drive, Reston, Virginia 22091 EPRS PRICE MF-$0.75 HC Not Available from EDRS. PLUS POSTAGE nESCRIPTORS Activity Learning; Arithmetic; Cognitive Development; *Elementary School Mathematics; *Instruction; *Learning Theories; *Mathematics Education; Psychology; Reference Books; Research; Teacher Education; Transfer of Training; *Yearbooks IDENTIFIERS computation ABSTPACT In the first chapter Brownell critically examines the psychological bases of the three mostcommon theories of arithmetic instruction: drill, incidental learningp and meaning.In chapter 2 the results of a nation-widesurvey of actual teaching practices are reported. Chapter 3 presents a contrast between "informational arithmetic" and "computational arithmetic." In chapter4 "social u+ility" is defined much more broadly than just"computationally useful," and implications for arithmetic instructionare discussed. Following this is a survey of opportunitiesfor use of arithmetic in an activity program, with specific examples. Chapter 6 discusses practices in the teaching of fractions and decimals,followed by the report of a study on transfer by Overman. Reported in chapter7 are the results of a survey on current practices inteacher-training courses in arithmetic. Two chapters are devoted to transfer of training in arithmetic and types of Qrill; usingthese as a basis, David Eugene smith expounds on the past, present,and future of instruction :n arithmetic. In chapter 11 arithmetic isconsidered from a mathematical viewpoint as contrasted toa pedagogical viewpoint. Next Gestalt psychology is discussed, withimplications for mathematics teaching. The last chaptercompares the efficiebcy of different methods for division. (LS) THE NATIONAL COUNCIL TEACHERS OF MATHEMATICS 1201 SIXTEENTH 5.11tE11:7. N. W. WASHINGIoN 6, D. C. 144 14-4 4) THE NATIONAL COUNCIL OF 0 TEACHERS OF MATHEMATICS La THE TENTH YEARBOOK THE TEACHING OF ARITHMETIC of IlkolAtOti '0 U I 01PARTMINt WP t Pe.% s4IALTm rgorll,G..tFpMOO. II.A1 OUCAtiON Pic otLy HA% op .4 sr mice* NATIONAL INSTITUT, OP AN? p KY ODUCAtiom . t. uMF14 Prole° "LAC'. v A. tiFtf h *MOM &hp ()Wt./4,114A r,C, '"I ;4' %CINI <1,4 int Nr.kitit 4 At WI l mg 01.14A 40 Pr% I % v I h ( tk.0141 ,04414dro t t,1 nI to ttif 4A r`r) NO' %II , 'AM , V 41 I t,4 r'14If g ()I.( A hood toolf Wt vv%vl M t;IN 0. ,01/4 Mt y., Ml n1 4M,% I r)..t A' r4 n0+.' ,t`4 :91 potv a41 tItiv btu 041 t7,1141 BUREAU OF PUBLICATIONS teatbero College, Columbia Illittbevoitp NEW YORK CITY 1935 CoPYriebt Ms by TUB NATIONAL COUNCIL OFMCKIM OF MATIMATICII Imommelma Correopeadeace Matins to ud'Mete for additloul nide of the Trask Yearbook sad earlierYearbooks *add be addressedto SUMO OP PUBLWATIONI, TIMMSCOLLIN. Comment Unsveasire, SU WestWO Street, New York City h EDITOR'S PREFACE TH:s is the tenth of a series of Yearbookswhich the National Council of Teachers of Mathematics beganto publish in 1936. The titles of the preceding Yearbooksare u follows: I. A Survey of Progress in the Past Twenty-five Years. a. Curriculum Problems in Teaching Mathematics, 3. Selected Topics in the Teaching of Mathematics. 4. Significant Changes and Trends in the Teaching of Mathe- matics Throughout the World Since1950. s. The Teaching of Geometry. 6. Mathematics in Modern Life. 7. The Teaching of Algebra. 8. The Teaching of Mathematics in the Secondary School, 9. Relational and Functional Thinking in Mathematics. Bound copies of all except the firsttwo Yearbooks can be secured from the Bureau of Publications, Teachers College,Columbia Uni- versity, New York, N. Y., for P.73 each postpaid.The first Year- book is now out of print Ind the second is obtainableonly in paper covers ($1.25 postpaid). A complete set of Yearbooks (Numbers to so inclusive) will be sent postpaid for Pais°,a saving of practically so per cent. The purpose of the Tenth Yearbook isto present some of the most important ideas and proposals concerning theteaching of arithmetic in the schools.Through this Yearbook the National Council of Teachers of Mathematics wishesto express its interest in the more elementary phases of mathematicsand in the teachers who present it in the classroom. I wish to express my personal appreciationas well as that of the National Council of Teachers of Mathematics to all of thecontribu- tors to this volume who have given so freely of their timeand interest in helping to make this Yearbook worthwhile. W. D. Rstvx THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS OFFICE'S PresidentJ. 0. HAW" University of Oklahoma, Norman, Okla. Pint Vke.hesidentRaux BIATLIV, Harvard University, caw bridge, Mass. Second VicePresidentAugts R. CONODON, University of Nebraska, Lincoln, Neb. Secretary-TreasurerEown4 W. Scunarnaa, Western Illinois State Teachers College, Macomb, Ill. EDITORIAL COMMITTEE Editor-in-Ckiel and Arnim: Mornrier of t.te Mathematics Teacher and the YearbookW. D. Ruin., Teachers College, Columbia University, New York, N. V. Associate EditorsVssA SANFORD, State Normal School, Oneonta, N, Y.; Hamar E. Swam:, University of Chicago, Chicago, Ill. ADDITIONAL MEMBERS ON THE BOARD OF DIRECTORS JOHN P. EVERITT, Kalamazoo, Mich. II 2933 One Year ELSIE P. JOHNSON, Oak Park, III.. II II 193$ RALEIGH SCHLORING, Ann Arbor, Mich. 1933 HARRY r. BARBER, Boston, Mass.. 1936 Two Years MARTHA HILDRBRANDT, Chicago, 1936 ( W. S. SCHLAJCH, New York, N. Y.. 1936 War, Ban, Rochester, N. Y. 1937 Three Years H. C. CHRISTOIFERSON, Oxford, Ohio. 1937 Elam Wousair, Minneapolis, Minn.. 1937 The National Council of Teachers of Mathematics is a national or ganization of mathematics teachers whose purpose is to: I, Create and s,aintain interest in the teaching of mathematics, 2. Keep the v:dues of mathematics before the educational world. 3. Help the inexperienced teacher to become a good teacher. 4. Help teachers in service to become better teachers, S. Raise the F;eneral level of instruction in mathematics, vi THE TENTH YEARBOOK Anyone interested In mathematicsIs eligible to membershipIn the Council upon payment of the annualdues of Sim. The dues also include subscription to the official journal of the Nati AnalCouncil (The Matheasatia Teacher). AUcorrespondence relatine,to editorial busi- ness matters should be addressedto The Math. oaks Tacker,sts West saoth Street, New York City. a CONTENTS OSAPTIM PASS I. PSYCHOLOGICAL CONSIDERATIONS IN THE LEARNING AND THE TEACHING OF ARITHMETIC WiUIons A. Brownell : II. AN ANALYSIS OF INSTRUCTIONAL PRACTICES IN TYPICAL CLASSES U4 SCHOOLS OP TIM UNITED STATES Leo 1. Brueckner 32 III, INFORMATIONAL ARITHMETIC B. R. Buckingham 51 IV. THE RELATION OF SOCIAL ARITHMETIC TO COMPUTATIONAL ARITHMETIC G. T. Burwell 74 V. OPPORTUNITIES FOR THE US... OF ARITHMETIC IN AN AC- TIVITY PROGRAM Paul Z? Norma and Others 85 VI. ECONOMY IN TEACHING ARITHMETIC J. T. Johnson 121 VII, CURRENT PRACTICES IN TEACHER-TRAINING COURSES IN ARITHMETIC Ramie Dustin Judd and Robert Lee Morton157 VIII. THE PROBLEM OP 'TRANSFER IN ARITHMETIC James Robert Overman173 IX, TYPES OF DRILL IN ARITHMETIC Austin C. Repoz86 X. RETROSPECT, INTROSPECT, PROSPECTDavid Eugene Smith20I XI. THE MATHEMATICAL VIEWPOINT APPLIED TO THE TEACH- ING OF ELEMENTARY SCHOOL ARITHMETIC C. L. Thiele212 XII. THE NEW PSYCHOLOGY OF LEARNING Raymond bolder litheeler233 XIII, MAKING LONG DIVISION AUTOMATIC Clifford B. Upton251 V 1 I THE TEACHING OF ARITHMETIC vii PSYCHOLOGICAL CONSIDERATIONS IN THE LEARNING AND THE TEACHING OF ARITHMETIC By WILLIAM A. BROWNELL Doh thawsky Aatriseattc is singularly unfortunate In the language Ibich has come to be used to describe the processes by which its subject mat- ter is to be learned and to be taught. One continually encounters such terms as "the number facts," "skills," "consumers' arithmetic," "incidentallearning,""automatic associations " "fixed habits," "bonds," "z00% accuracy," "crutches," "meaningful experiences," "drills," etc., etc.These terms have a significance which is seldom sufficiently recognised, for in one way or another they imply certain theories regarding the psychology and pedagogy of arithmetic. More Important, perhaps, they lead directly to the adoption of in- structional practices of varying degrees of merit. Analysis of ambiguous and misleading terms would repay the time required.Since space limitations forbid, however, another method is employed to get before the readersome of the crucial psycho- logical aspects of learning and teaching arithmetic.This method consists in examining critically the psychological bases of the three commonest theories with respect to arithmetic instruction. To this examination the remainder of this chapter is devoted. The reader should be warned that a theory of arithmetic instruction is rarely, If ever, practiced in pure form. The theory held to predominantly by a teacher will determine the points of emphasis in her teaching practice, but the practice itself will reveal the influence of other and perhaps conflicting theories.The three theories, as they are here isolated for analysis, are not so isolated in practice.While it is probable that a particular teacher can be classifiedas adhering in general to one theory, she will not be found to be a one-hundred- per-cent practitioner of that theory.Rather, she tends, wittingly or unwittingly,