The Teaching of Arithmetic. National Council of In the first chapter Brownell critically examines the

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DOCUMENT RESUME
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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 most
common theories of arithmetic
instruction: drill, incidental learningp and meaning.
In chapter 2
the results of a nation-wide
survey of actual teaching practices are
reported. Chapter 3 presents a contrast between "informational
arithmetic" and "computational arithmetic." In chapter
4 "social
u+ility" is defined much more broadly than just
"computationally
useful," and implications for arithmetic instruction
are discussed.
Following this is a survey of opportunities
for 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 chapter
7 are
the results of a survey on current practices in
teacher-training
courses in arithmetic. Two chapters are devoted to transfer of
training in arithmetic and types of Qrill; using
these as a basis,
David Eugene smith expounds on the past, present,
and future of
instruction :n arithmetic. In chapter 11 arithmetic is
considered
from a mathematical viewpoint as contrasted to
a pedagogical
viewpoint. Next Gestalt psychology is discussed, with
implications
for mathematics teaching. The last chapter
compares the efficiebcy of
different methods for division.
(LS)
THE NATIONAL COUNCIL
TEACHERS OF MATHEMATICS
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BUREAU OF PUBLICATIONS
teatbero College, Columbia Illittbevoitp
NEW YORK CITY
1935
CoPYriebt Ms by
MCKIM
TUB NATIONAL COUNCIL OF
OF MATIMATICII
Imommelma
Correopeadeace Matins to ud
'Mete for additloul nide of the
Trask Yearbook sad earlier
Yearbooks *add be addressed
to
SUMO OP PUBLWATIONI, TIMMS
COLLIN.
Comment Unsveasire, SU West
WO Street, New York City
h
EDITOR'S PREFACE
TH:s is the tenth of a series of Yearbooks
which the National
Council of Teachers of Mathematics began
to publish in 1936. The
titles of the preceding Yearbooks
are 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 Since
1950.
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 first
two 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 obtainable
only in paper
covers ($1.25 postpaid). A complete set of Yearbooks (Numbers
to so inclusive) will be sent postpaid for Pais°,
a saving of
practically so per cent.
The purpose of the Tenth Yearbook is
to present some of the
most important ideas and proposals concerning the
teaching of
arithmetic in the schools.
Through this Yearbook the National
Council of Teachers of Mathematics wishes
to express its interest
in the more elementary phases of mathematics
and in the teachers
who present it in the classroom.
I wish to express my personal appreciation
as well as that of the
National Council of Teachers of Mathematics to all of the
contribu-
tors to this volume who have given so freely of their time
and
interest in helping to make this Yearbook worthwhile.
W. D. Rstvx
THE NATIONAL COUNCIL OF TEACHERS
OF MATHEMATICS
OFFICE'S
PresidentJ. 0. HAW" University of Oklahoma, Norman, Okla.
Pint Vke.hesidentRaux BIATLIV, Harvard University, caw
bridge, Mass.
Second VicePresidentAugts R. CONODON, University of Nebraska,
Lincoln, Neb.
Secretary-TreasurerEown4 W. Scunarnaa, Western Illinois State
Teachers College, Macomb, Ill.
EDITORIAL COMMITTEE
Editor-in-Ckiel and Arnim: Mornrier of t.te Mathematics Teacher and
the YearbookW. D. Ruin., Teachers College, Columbia University,
New York, N. V.
Associate EditorsVssA SANFORD, State Normal School, Oneonta,
N, Y.; Hamar E. Swam:, University of Chicago, Chicago, Ill.
ADDITIONAL MEMBERS ON THE BOARD OF DIRECTORS
JOHN P. EVERITT, Kalamazoo, Mich.
2933
II
One Year
ELSIE P. JOHNSON, Oak Park, III.
193$
. .
II II
1933
RALEIGH SCHLORING, Ann Arbor, Mich.
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 mathematics
Is eligible to membership
In the
Council upon payment of the annual
dues of Sim. The dues also
include subscription
to the official journal of the Nati Anal
Council (The
Matheasatia Teacher). AU
correspondence relatine,
to editorial busi-
ness matters should be addressed
to The Math. oaks Tacker,
sts
West saoth Street, New York City.
a
CONTENTS
OSAPTIM
PASS
I. PSYCHOLOGICAL CONSIDERATIONS IN THE LEARNING AND
:
WiUIons A. Brownell
THE TEACHING OF ARITHMETIC
II. AN ANALYSIS OF INSTRUCTIONAL PRACTICES IN TYPICAL
CLASSES U4 SCHOOLS OP TIM UNITED STATES
Leo 1. Brueckner
32
B. R. Buckingham
III, INFORMATIONAL ARITHMETIC
51
IV. THE RELATION OF SOCIAL ARITHMETIC TO COMPUTATIONAL
G. T. Burwell
ARITHMETIC
74
V. OPPORTUNITIES FOR THE US... OF ARITHMETIC IN AN AC-
Paul Z? Norma and Others
TIVITY PROGRAM
85
J. T. Johnson
VI. ECONOMY IN TEACHING ARITHMETIC
121
VII, CURRENT PRACTICES IN TEACHER-TRAINING COURSES IN
Ramie Dustin Judd and Robert Lee Morton
ARITHMETIC
157
VIII. THE PROBLEM OP 'TRANSFER IN ARITHMETIC
James Robert Overman
173
IX, TYPES OF DRILL IN ARITHMETIC
Austin C. Repo
z86
David Eugene Smith
X. RETROSPECT, INTROSPECT, PROSPECT
20I
XI. THE MATHEMATICAL VIEWPOINT APPLIED TO THE TEACH-
C. L. Thiele
ING OF ELEMENTARY SCHOOL ARITHMETIC
212
XII. THE NEW PSYCHOLOGY OF LEARNING
Raymond bolder litheeler
233
Clifford B. Upton
XIII, MAKING LONG DIVISION AUTOMATIC
251
V 1 I
THE TEACHING OF ARITHMETIC
ii
v
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,"
"incidental
learning,"
"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 reader
some 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 classified
as 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, to employ various features of two or even three of