The Psychology of Mathematical Abilities in Schoolchildren

433 Pages·1976·19.68 MB·English

by  V. A. Krutetskii

The Psychology
of Mathematical
Abilities in
Schoolchildren
Survey ofRecent East
European Mathematical
Literature
A project conducted by
Izaak Wirszup,
Department of Mathematics,
The University of Chicago,
under agrant from the
National Science Foundation
The Psychology Translated from"’
the Russian
of Mathematical
by Joan Teller
Abilities in Edited-:by I
Jeremy Kilpatrick
Schoolchildren and Izaak Wirszup
The
University
of Chicago
Press
Chicago
V. A. Krutetskii
and London
V. A. KRUTETSKIIis deputy directorof the
Research Institute ofGeneral and
Educational Psychology at the USSR
Academy of PedagogicalSciences and is
thehead ofthe sectionon abilities.
JEREMY KILPATRICK is professoror mathematics
education atthe University ofGeorgia.
IZAAKWIRSZUPis professorof mathematics
atthe University ofChicago.
Mr. Wirszupand Mr. Kilpatrickarecoeditors
ofSovietStudies inthe Psychologyof
Learningand Teaching Mathematics
TheUniversity ofChicago Press, Chicago60637
The University ofChicago Press, Ltd.. London
9 1976 by theUniversity ofChicago
All rights reserved.Published 1976
Printed in theUnited Statesof America
80 79 78 77 76 987654321
Library ofCongress Cataloging in Publication Data
Krutefskii, Vadim Andreevich.
Thepsychology of mathematicalabilities in
schoolchildren.
(Survey ofrecentEast Europeanmathematical lit-
erature).
TranslationofPsikhologii? matematicheskikh
sposobnost'ei shkol'nikov.
Bibliography: p.
Includesindex.
1. Mathematical ability. 2. Learning, Psy—
chology of. I. Title.
QAl1.](8313 370.15'6 74-33520
ISBN0-226-45492-4
Contents
Editors' Preface vii
Preface to the Russian Edition ix
Introduction xi
1. Problem and Goals of the Study 1
1. The Theoretical andPractical Importance ofthe Problem ofMathemati-
cal Abilities in Contemporary Soviet Science andEducation 3
2. Foreign Psychological Literature on Mathematical Abilities 8
3. Prerevolutionary and Soviet Psychological Literature on Mathematical
Abilities 47
4. Statement ofthe Problem and Goals ofthe Study 60
2. Methods and Organization ofthe Study 79
5. General Method and Organization 81
6. Hypothesis concerning the Components ofMathematical Abilities 84
7. Methods Used in the Experimental Investigation 89
8. The System ofExperimental Problemsfor Investigating Schoolchildren's
Mathematical Abilities 98
9. Organization ofthe Experimental Investigation 175
3-. An Analysis of the Structure of Schoolchildren’s Mathematical
Abilities 181
10. AnAnalysis ofNonexperimentalData onthe ComponentsofSchoolchil-
drén’s Mathematical Abilities 185
11. An: Analysis ofIndividual Cases of Mathematical Giftedness in Chil-
dren 193'
12. CharacteristicsofInformation Gathering (InitialOrientation toaProb-
lem’)by Mathematically CapablePupils 224
I3. Characteristics of Information Processing during Problem-solving by
Mathematically CapablePupils 237
I4. Characteristics ofInformation Retention (Mathematical Material) by
Mathematically Capable Pupils 295
V
vi Contents
15. Some Special Issues in the Structure ofSchoolchildren’s Mathematical
Abilities 302
16. Type, Age, and Sex Differences in the Components of Mathematical
Abilities 313
17. Mathematical AbilitiesandPersonality 344
18. General Questions concerning the Structure of Mathematical
Abilities 350
References 365
Author Index 401
Subject Index 412
Editors’ Preface
The work of V. A. Krutetskii firstbecame known to Western psychologists and
mathematics educators in 1963, when Krutetskii submitted a brief paper on
mathematicalabilitiesto theSeventeenthInternationalCongressofPsychologyin
Washington (Acta Psychologica 23 [1964]:314—15). The same year saw the
publicationinEnglishofacollection ofSovietworkineducationalpsychology(B.
Simonand].Simon,eds.,EducationalPsychologyinthe USSR[Stanford,Calif.:
StanfordUniversity Press])which containedKrutetskii’sarticle, “SomeCharac-
teristics of the Thinking of Pupils with Little Capacity forMathematics.”
Krutetskii’s work aroused interest not only because he appeared to be unique
amongSovietpsychologists in investigating individual differences in mathemati-
cal abilities, but alsobecause the mathematical problems he used in his research
were so varied and ingenious. When the School Mathematics Study Group
and the Survey of Recent East European Mathematical Literature collaborated
in the late 19605 to produce a series of translations from the Soviet litera-
ture on mathematics education (Soviet Studies in the Psychology ofLearning
and Teaching Mathematics), the editors were so intrigued by Krutetskii’swork
that they devoted the entire second volume in the series to four of his articles.
Usefulasthese translationsofKrutetskii’sshortarticlesand reportshavebeen,
however, they havegiven onlya partial viewofhis research. In 1968the original
Russian edition of the present volume was published. It contained a complete
report of Krutetskii’s theoretical and experimental research of the preceding
twelve years — the first Soviet psychological monograph on schoolchildren’s
mathematical abilities. The book, published in Moscow, appeared under the
imprimaturoftheScientificCounciloftheInstituteofPsychologyoftheAcademy
of Pedagogical Sciencesof the USSR. It wasreviewed by A. I. Markushevich,
professor ofmathematics at Moscow State University and vice-presidentof the
AcademyofPedagogical Sciences;V. I. Levin,doctorofphysicsand mathemati-
cal sciences; and E. N. Kabanova-Meller, doctorof pedagogical sciences. The
book contained 431 pages and was printed in an edition of 25,000 copies.
The present edition of the book was prepared by the Survey ofRecent East
viii Editors' Preface
European Literature in School and College Mathematics of the University of
Chicago. Special acknowledgment is made to Joan Tellerforhercareful transla-
tionand helpfuleditorial work. We aregrateful tothe librarians atthe University
of Cambridge, the British Museum, the University of Chicago, and Teachers
College, Columbia University, for their assistance in checking references. Our
thanksgoalsoto ZalmanUsiskin ofthe UniversityofChicagoforhissuggestions
onthemanuscript.ThanksareduetoCarleneKilpatrick forreadingandtypingthe
manuscript and to Harvey Edelberg for his administrative and editorial help.
Finally, acknowledgment is due E. G. Begle of Stanford University, former
director ofthe School Mathematics Study Group, forinitiatingthe collaboration
that led to this project.
Jeremy Kilpatrick
Izaak Wirszup
Preface to the
Russian Edition
Thisbookreflectsthemajorproductsofourtheoreticalandexperimentalresearch,
begun in 1955, into the mathematical abilities of schoolchildren. We must em-
phasize thatour work is strictly psychological, and thereforewe shallnot—now
or in the future — pretend either to analyze current methods of mathematics
instructionortocreate anew method. Itisour deepconvictionthatthis isamatter
not for psychologists but for research mathematicians, educators, and
methodologists — that is, qualified scholars in the appropriate areas. The
psychologist can and should collaborate with them by presenting the necessary
material and conducting psychological research on particular questions.
In the future, we propose to continue ourwork in the following directions:
1. To investigate, in close cooperation withmathematicians, the mosthighly
developed structure of mathematical creativity (the abilities of a research
mathematician).
2. To attempt to discover, in collaborationwith physiologists, the physiologi-
cal nature of mathematical abilities.
3. To study, together with teachers and methodologists, optimal ways of
forming and developing mathematical abilities at school age, having clarified
beforehand to what extent the existing system ofmathematics teaching actually
molds pupils’ mathematical thinking or theirmathematical abilities.
In preparing the bookwefaced significantdifficulties, chiefamong which was
the dearthofscientificresearch onthepsychologyand pedagogyofabilities in the
Soviet literature. Inthelastfewyears(sinceabout 1959), however,interestin this
problem has increased noticeably, and several valuable theoretical and ex-
perimentalworkshavebeenpublished. Butthese worksarestillverymeagerifone
takesintoaccountthecurrentimportanceofquestionsconcerningthemultifaceted
developmentofabilitiesinyouthduringtheprocessesoflearningandmaturation.
We hope that this book will be of some help to psychologists, mathematics
teachers, mathematics methodologists, graduate students, and students in the
appropriate departmentsofinstitutes. Bynomeans do weconsider thebookto be
ix
x Preface tothe RussianEdition
free ofshortcomings. Allcomments, suggestions, andadvicefromreaderswillbe
gratefully accepted and examined most carefully.
The authorconsiders it his duty to express deep gratitudeto Professors A. A.
Smirnov, B. M. Teplov, and P. A. Shevarev, psychologists; to E. N.
Kabanova-Meller, doctorofpedagogicalsciences (inpsychology);and to Profes-
sorsA. I.MarkushevichandV. I. Levin,mathematicians,aswell astoProfessors
I. K. Andronov and E. B. Dynkin, for their attention to his work and for their
valuable suggestions.