Table Of Contentinternational studies i n education
YATHEMATICS
IN PRIMARY EDUCATION
Learning of mathematics
by young children
prepared by the
International Study Group
for Mathematics Learning,
Palo Alto, California, and
compiled by Dr. 2. P. Dienes,
Professor of Education,
University of Adelaide,
Australia, on behalf of the
International Study Group
and published with the permission of Unesco
1966
unesco institute far education, hamburg
The Unesco Institute for Education (Hamburg) is an institute of international
character financed by a trust fund into which are paid contributions, including
that of the Government of the Federal Republic of Germany. While the
programmes of the Institute are established in consultation with the Director-
General of Unesco, the publications of the Institute for Education are issued
by the Institute under its sole responsibility, and Unesco as an organisation
is not responsible for their content.
The points of view, selection of facts and opinions expressed are those of the
authors, and do not necessarily coincide with official positions of the Unesco
Institute for Education, Hamburg.
8 Unesco 1966
-.- -..-- -.-. -----.-____l--
TABLE OF CONTENTS
Preface
Introduction I-V
Chapter 1:
An Overview of the Present Position 13
Chapter 2:
Theoretical Considerations 31
Chapter 3:
Practical Applications of Fundamental
Researches into the ,Problems of
Mathematics Learning in Schools 71
Chapter 4:
On the Training of Teachers 107
Concluding Remarks 124
Appendix A: Notes on Programed Learning 127
Appendix B: Brief Report of a Reform 133
Bibliography to Appendix B 138
Bibliographical References to - Chapter 1 143
- Chapter 2 145
- Chapter 3 154
- Chapter 4 157
- Appendix A 159
Annotated Bibliography 163
_---.--
PREFACE
PREFACE
Classroom experiments in various countries and research work by educa-
tional psychologists have led to the conclusion that the ability of young children
to learn is greater than had been believed in the past. In fact, this capacity for
learning or, more precisely, for thinking and reasoning among primary school
children has paved the way to new teaching practices.
This is particularly true in the teaching of mathematics where new
developments have prompted a rethinking of the scope and nature of the teaching
of this subject. The old pattern of arithmetic lessons is gradually being replaced
by approaches which involve an integrated mathematical course comprising the
classical subjects of arithmetic, algebra and geometry.
A great deal of the information on this subject is scattered and often
difficult to obtain. The present report on the learning of mathematics by young
children, which has been prepared by the International Study Group for Mathe-
matics Learning (I .S.G.M.L.) and is published by the Unesco Institute for
Education, Hamburg, will, it is hoped, constitute a useful source of information
on this subject .’ It is a valuable contribution to the activities in UNESCO’S
programme aiming at the improvement of methods of instruction. The report
examines the theoretical problems of learning mathematics and describes some
of the practical applications of the fundamental researches into the problems
of mathematics-learning in school. It also includes suggestions on the problem
of training or rather re-training of teachers, which is necessary if these new
methods are to be applied successfully.
It is believed that this study will encourage new research activities
which in turn will lead to further improvement of teaching practices. UNESCO
wishes to express gratitude to I.S.G.M.L. and the Institute for Education,
Hamburg, for this contribution to the execution of its programme.
INTRODUCTION
.- -
INTRODUCTION
In primary school mathematics education we are on the threshold of some
remarkable changes - both in what is to be learnt and in how this learning is to
take place.
For some years, continuing movement towards the reform of secondary school
curricula has been evident in many countries. In the fields of mathematics and
science, the outcome of this movement has been tangible and convincing. New
approaches to the curriculum have produced some excellent teaching-texts, to-
gether with appropriate teaching-aids for providing audio-visual experience and
laboratory activity. While thn secondary level has been chosen as the first point
of attack on the traditional curriculum, changes at this level prerequire changes
at the primary level, and entail changes at the tertiary level. The secondary school
reform has highlighted the need for a total overhaul of the curriculum.
The interdependence of the primary and secondary stages in a child’s learning
career should be self;evident, and reform is now being directed to the modification
of basic teaching in the primory school in order to prepare the child for the new
patterns of secondary education. In particular, revolutionary mathematical topics
are being introduced at this level, and are displacing the traditional preoccupation
with mechanically-learnt arithmetic. In both special instructional situations and
everyday classroom practice, educational innovators are finding that, given the
right kind of teaching; children can grasp at an early age many of the important
concepts of mathematics.
These changes in the content of curricula have been accompanied by experi-
ments in the development of new teachingrmethods. In many parts of the world,
mathematics-teachers and educational research workers have been engaged in evolving
new approaches to the teaching of mathematics in primary education. A host of
didactic inventions has resulted from this work; these range from concrete apparatus
which provides manipulative and sensory experience, to symbol games of various
kinds. The new approaches appear to succeed not only in bringing the young learner
to an understanding of mathematics, but in eliciting from him a much more positive
involvement in this subject than has been apparent in the traditional learning situation.
While the possibilities of reform - of both curriculum and teaching-method -
have by no means been fully exploited, it is thought that enough material is now
available for some kind of statement to be made concerning the directions in which
primal-y scl~ool mathematics education might odvonce. A summary of some of the
results that are now ovoilable should pr-eve particularly useful to the growing number
of notional outhorities who are in the process of refer-ming tlleir educational system.
As part of its pr-ogramme for international study and advancement of education,
Unesco ha:. boon collecting and documenting information on various aspects of the
educational dcvclopmcnts toking place in its Member States. The present report is
the result of r-1 contract bctwcen Unesco and the International Study Group for
Mathematics Learning (I.S.G.M.L.)(l) which wos commissioned both to review
researches into the learning of mathematics by pupils 6 - 12 year-s of age, and to
describe significant classroom and curricular projects of interest to educators,
mathemoticions and psychologists who ore concerned with mothemotics-learning.
First discussed in 1963, tlris report is the result of the contributions of many
people rrepre,,c,nting different countries and viewpoints. That it offers such o coherent
picture of the internotional scene in mothemotics lear-ning is a tribute to the willing-
ness of the contributors to provide information and to discuss in conference the
features of this report as it hos evolved through three versions.
The original version of the report wos prepared by Professor Zoltan P. Dienes
of the University of Adelaide and a member of the Advisory Council of the I .S .G .M.L.
That version served as the basis for study and deliberation by participants in the first
study conference held at Stanford University in December 1964.
Participants were invited from all sections of the United States. An effort
was made to include those persons who represented the most advanced thinking in the
areas of mathematics, psychology, and education, so that there would be a proper
balance among these disciplines. In addition, an attempt was made to include those
persons who represented major kinds of institutions and agencies, namely government,
university, school and private agency. The conference, convened for two days, set
as its objectives the critical review and revision of the initial report. Committees at
the conference dealt with various sections of the report and made suggestions and
revisions as necessary, after reporting to the entire conference group and gaining a
consensus.
Professor Dienes incorporated the suggestions for revision in a second version
of the report, which was then duplicated and made available to participants of a
second conference, held in Paris in April, 1965, as well as being sent to all members
of the Stanford conference for their further review.
Following the suggestions of the participants of the second conference and
with further additions contributed by them and others, the report again underwent
extensive revision.
(1) The I.S.G.M.L. is an orgonisation which wos established in 1962 in order to
promote mathematics learning by cncouroging the investigation of the processes
by which it is achieved and by facilitating exchange and disseminotion of teaching
techniques and materials. To these ends it pr-ovides an informotion service, it en-
couroges the pooling of resources and the initiation of cooperative ventures, and it
publishes o quarterly bulletin on recently completed and ongoing work.
ii
The present document - the third, and final version of the report - was
scrutinised yet again by the participants of a conference held at the Unesco Institute
for Education, Hamburg, in January, 1966. At this conference, decisions were
made concerning the final form and scope of the report.
The Hamburg conference was attended by an international group of specialists
in mathematics education who used the report as a working document for discussions
of action that might profitably be taken in the future in the field of primary school
mathematics education. One important outcome of the conference was the decision
that the third version of the report should be published, so that the many interesting
ideas that it contains should be made available to countries facing the problems of
reform in mathematics education. A second report, covering the conference itself,
will be prepared. This will contain further accounts of curricular reforms, and
observations concerning the implementation of such reforms, thus supplementing the
present document. Further conferences and documents are planned, for the purposes
of apprising educationists in different countries of the reform activities occurring in
other countries, and of providing some guidance in this difficult area. It is hoped
that, eventually, it will be possible to set up an international system for coordinating
reform activities in different countries.
The greater part of the present report is devoted to four aspects of the problems
involved in mathematics learning in the primary school:
(i) The abundant theorising and experimenting on mathematics learning that those
concerned with its psychological aspects have contributed.
(ii) Practical applications of this fundamental research, in the form of six general
kinds of approach to the teaching of mathematics.
(iii) The crucial problem of arming teachers with knowledge of the new curricular
content and appropriate teaching techniques.
(iv) Some of the strategies involved in the implementation of reform. Relevant to
this aspect is an appendix written by G. Papy on a reform in progress in Belgium.
A chapter-by-chapter bibliography is provided at the end of the report, to-
gether with a briefly annotated bibliography of works that are likely to provide useful
leads on the subjects considered in the report.
It is not the aim of this report to provide a comprehensive account of the
topics it tackles. We are at a stage in educational experimentation and reform at
which things are still growing rapidly. At this stage communication is very
necessary for further growth, but naturally cannot anticipate the final shape of
things. We envisage that the present report will be followed by others that not only
complement, but to some extent supersede it. The report points the way for
continuing survey and analysis. In particular, it is hoped that as a result of this
description of reforms, others may be brought to light, so that on some future
occasion a more complete account may be compiled.
Moreover, as chief compiler of this report, Professor Dienes disclaims any
ambition to have produced a “b a I anced” picture of even the present state of affairs.
Whether or not the assiduous pursuit of such an ambition would have produced a more
iii
useful report is difficult to say. However, we are entitled to suspect that withorrt
the complexion with which Dienes’ personal involvement in this field colours his
account of it, this report might lack the coherence and persuasiveness that
characterise it. Certainly, such reports often gain from being written by one pen -
- especially when this pen belongs to a commentator who, like Professor Dienes,
has been in such close contact with so much of the relevant activity that has been
taking place in varibus parts of the world. Indeed, it would be surprising if a
report on mathematics learning were not somewhat biassed by the opinions of
Dienes - since so much of the work in this field actually der ives from them.
At the Hamburg conference some surprise was expressed at the amount
of emphasis this document places on the views of psychologists. It was pointed
out that the findings of psychologists tend to be too vague, too general, or
insufficiently related to mathematical learning situations to be of very much use
in influencing the form that mathematics teaching should take. Some participants
had found that psychologists lacked the mathematical expertise to make a
significant contribution to mathematics-teaching, and that the most useful
contributions had come from mathematicians and practising teachers. In defence
of the inclusion of psychological material the following points were made :
(i) Although the theories of psycho!ogists are in themselves insufficiently precise
or specific, they provide a general framework within which particular solutions
to the problems of mathematics learning moy be found by those who are con-
fronted with these problems.
(ii) Whatever the degree of success that psychologists have enjoyed hitherto,
mathematics-learning certainly brings with it problems of a psychological nature,
that need a closer examination than most teachers are equipped to provide.
(iii) Perhaps, if mathematicians, teachers and psychologists were better oc-
quainted withone another’sdisciplines, each kind of expert could contribute the
better to the construction of theoretical models that would embody the weolth
of all three disciplines. Th e inclusion of psychologists’ views in the present
report may be regarded as a step in this direction.
Many questions concerning psychology and teaching are explicitly raised
in the report, for example:
How is the child’s receptiveness to mathematical ideas affected by such cultural
factors as cultural deprivation, language structure, cultural conflict, sex role,
etc? If the strategies of reform are going to be adjusted to o variety of cultural
environments, such questions will need answers.
Do children at different levels of development require different kinds of motivation?
Witherto, only simple mathematical ideas have been involved in research into
mathematics learning. Will researches into the learning of co m p I e x ideas produce
interestingly different results?
However, perhaps these are not the only questions that we are not yet in
a position to answer. In this area, few issues have been finally settled; and
among the more important in the trail of questions that this report leaves behind it
iv
