international 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

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TABLE OF CONTENTS

Preface

Introduction

Chapter 1: An Overview of the Present Position

Chapter 2: Theoretical Considerations

Chapter 3: Practical Applications of Fundamental

Researches into the ,Problems of Mathematics Learning in Schools

Chapter 4: On the Training of Teachers

Concluding Remarks

Appendix A: Notes on Programed Learning Appendix B: Brief Report of a Reform

Bibliography to Appendix B 138

Bibliographical References to

Annotated Bibliography 163

- Chapter 1 143

- Chapter 2 145

- Chapter 3 154

- Chapter 4 157

- Appendix A 159

I-V

13

31

71

107

124

127 133

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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

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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.

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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 anced” picture of even the present state of affairs. I Whether or not the assiduous pursuit of such an ambition would have produced a more

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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

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are the following:

Should the child learn as much mathematics as possible at as early an age as possible? If so, why?

Should simple, basic mathematical notions always be introduced OS a prepara - t ion for the more complex ones, or should basic’notions sometimes be intro- duced a f t e r the more complex, for purposes of e x p I a n a t i o n ?

Might not the use of concrete materials sometimes hinder learning, by distracting the learner from essentials, instead of promoting learning, by exemplifying these essentials?

Likewise, is the principle of multiple embodiment applicable in every learning situation? May not o multiplicity of embodiments sometimes confuse the learner - or, to use the languoge of information theory, may not the “noise” sometimes drown the “message” ?

Under what conditions does Dienes’ “d eep-end” principle (requiring the introduction of the more general case prior to the more particular) apply? Where does verbalisation help in the abstraction of concepts, and where does it hinder?

Special attention should be drawn to what by many would be regarded as o maior bias in the pedagogical viewpoint from which this report has been written. Throughout the report, Dienes’ acceptance of the desirability of discovery-learning is evident. It should be pointed out that the experimental evidence is by no means unanimous in its support of learning by discovery; such as it is, the evidence from a variety of sources suggests that it is very difficult to engineer successful conditions for this kind of leorning, and that there may well be kinds of materials that ore better taught by more direct means.

Despite its point of view, which its chief compiler, Professor Dienes, claims to be biossed, and despite the many but very necessary omissions, this report is quite unique in the representativeness and range of the material it has managed to integrate. Professor Dienes, and the many other contributors, are to be congratulated for writing so useful a document, and Unesco is to be con- gratulated for producing it.

February, 1966

Adrian B. Sanford International Study Group for Mathematics Learning

John D. Williams National Foundation for Educational Research in England and Wales

AN OVERVIEW OF THE PRESENT POSITION

Chapter 1

AN OVERVIEW OF THE PRESENT POSITION

1.1 Review of reasons for change

Every country, in its efforts to adapt its economy to the 20th Century, knows that the kind of industry it can have depends on its production of profes- sional manpower and skilled labor. Such progress depends a great deal on the quality of its school mathematics.

In some countries the highest priority is on building a satisfactory sys- tem of mass education, and in such a system to produce large numbers of workers in agriculture, building and electrical trades, machine shops, offices and the like. An increase in the number of engineers, scientists and teachers is essential, even if preparation for such a future demands efficient use now of the few quali- fied leaders in schools and universities.

In other countries the changes in technology are gradually eliminating employment of unskilled labor, at the same time as they are greatly expanding the need for mathematically trained manpower at all levels. The highest priorities are on improving the quality of mathematics teaching in an existing mass educa- tion system, and on broadening the opportunities for a good mathematical educa- tion at the secondary school and college levels.

In fact, it would not be too difficult for economic and educational

PI anners, working together, to make estimates of manpower needs in any country and to translate these estimates into needs for mathematical competence in the population. This, in turn, would make possible an intelligent coordination of economic with educational planning.

In every country the citizenry participates in social decision making. Since public policy is deeply involved with science and technology, mathemati- cally literate populations are required to bring about intelligent discussion of issues and to elicit the necessary public support. In addition to the technical

competence needed for those who use mathematics in their work, there is the need to create an appreciation for mathematics, at appropriate levels, in the general population.

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Frequently in the past, energy, time, and devotion have been spent in the teaching of mathematics with questionable efficiency for the majority of students. As a result many adults have come to feel inferior with respect to the subject, convinced that it is impossible for them to achieve such abstract under- standings. There has been admitted widespread inadequacy in traditional mathe- matics teaching despite the needs current in the past and a lack of the skills aimed

at.

However, there is now a growing desire for a greater share and conse- quent self-respect by the common man for intellectual satisfactions. There is ample reason to believe that such attainments can be made, given the proper at- titudes and educational atmosphere. Just as music is accepted as a valuable part of the general education of all, with the few excelling, so in mathematics every- one can reach etrjoyment, receiving its long-valued stimulation and satisfaction.

All this is necessary in order to prepare for the rapid and unforeseeable changes which have become a permanent feature of our society.

While research in mathematics has not had the publicity or the spectac- ular effects on daily life that have resulted from advances in physics and medicine, nevertheless new knowledge of mathematics is now applied in many fields from aerodynamics to agriculture and from business management to psychology. Mathematics is indeed an active, growing field 1

Some modern research has given us new insights into the most elemen- tary mathematics taught in school. In several countries mathematicians and educa- tors are applying these insights to make great improvements in the teaching of arithmetic, algebra, and geometry. Other advances have suggested the intrcduc- tion of new subjects, such as probability and statistics.

As rapid social change forces people, more and more, to engage in learning as a life-long process, the value of mathematics in education begins to lie as much in the skills in thinking which the student acquires, as in the specific mathematical results which he masters. An effective mathematics curriculum must teach students how to attack non-routine problems which they have never faced before. The child must learn the ways in which a mathematician thinks, as these apply to even the most elementary subject matter.

We have sketched above some of the important reasons for changes in mathematics education. In the chapters which follow, detailed information is given on the kinds of changes which are being made in various places. The ex- perimentation reported here opens up exciting new possibilities as to what might be accomplished. This work indicates that children are capable of learning more mathematics with greater enjoyment than ever before. In addition, it seems likely that the problems which accompany change can be resolved. It seems fea- sible to teach almost the entire population of children a much deeper understanding of mathematics and a much greater mastery of mathematical thinking than is now being accomplished.

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While in some countries there is an ample supply of mathematicians, psychologists, and educators competent to institute these improvements, often the rigidity of the existing political and social framework makes it difficult to institute changes. It is quite possible that in countries where mass education pro- grams are just being organized, informed leadership may be able to take advantage of the new developments more rapidly.

In planning for change, it is well to begin by assessing the present state of affairs in order to determine the priority items of the new program.

1.2 Ways of assessing the position obtaining in a particular area

Such an assessment and comparison may be made at several levels of sophistication. Details of these various comparisons could easily become major documents themselves. For this reason, the questions listed below are offered as guidelines to school organizations wishing to assess their own positions at this time :

1. Mathematics content.

a) Are the properties of the number system explicitly taught in the arithmetic curriculum 3 In the algebra program 3

b) Are the reasons for the computational algorithms taught 3

c) Does instruction in geometry begin prior to the secondary school 3

d) Are proofs given in any subject other than geometry 3

2. Methodology.

a) What emphasis is given to understanding of concepts, in addition to mechanical mastery of skills 3

b) Do the students get experience with non-routine problems 7

c) Are students given independent work as appropriate 3

d) Are concrete materials used in elementary mathematics teaching 7

3. School organization.

a)

b) cl

4

4

What proportion of students is prepared mathematically for skilled trades using mathematics 3 For professions 3

How does this compare with the manpower needs of the country 3

To what extent are school administrators and educators informed regarding the uses of mathematics in modern industry, business, and government 3

To what extent are students informed regarding the mathematics needed in various trades and professions 3

To what extent are economic planners informed on these matters 7

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4. Teacher education.

a) Is information available on experimentation with mathematics teaching in other countries ?

b) Are experimental mathematics materials available for study by teachers and educators ?

c) To what extent are teachers being taught the new approaches to mathematical education ?

d) To what extent are new ideas being tried in schools ?

e) To what extent does your country’s examination system allow for experimentation with new approaches ?

f) Are teachers investigating student response to new approaches 7

5. Public understanding.

What kind of an effor. has been made to inform the public of the changing role of mathematics in the affairs of all of the world’s citizens ? Do the public realize that mathematics is the founda- tion of a technological economy ? Do they realise that the field has undergone major changes of its own 3 Do they realize that mathematics has a creative side, a poetic side and a recre- ational side, and is as worthy of study for its own sake as any discipline ?

6. Public policy.

Is there any mechanism whereby government officials can obtain competent advice on relations between economic development and mathematical education ? Are the conclusions of recent OEEC, Pan-American Union, and UNESCO Conferences avai- lable to those interested ? Have conferences and studies like this occurred in your country ? What studies of manpower needs and implications for education are being made 3

Initial steps in revision may involve conferences, institutes for teachers, experi- mentation in selected classes, and an adult education campaign designed both to inform the public and to elicit their support. Those involved in the planning should include government officials, mathematicians, psychologists, and educa- tors. These people, or their representatives, should meet periodically to eva- luate the reform undertaken and to give direction to future activity.

The following is offered as a set of alternative considerations for the evaluation of the situation in different countries.

1. To what extent does the current program in mathematics education reflect the structure of mathematics ? Is each concept taught with consideration of the whole 3 Is more arithmetic taught than is necessary for elementary computation ? Are branches of mathematics other than arithmetic presented to the students ?

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2.

3.

4.

5.

1.3

What is the nature of the methodology used ? Are mathematical topics approached from more than one point of view ? Are manipulative materials and games used to introduce abstractions 3 Has consideration been made of current research in the psycholo- gy of learning mathematics ?

How were the schools of the country providing the number of mathematicians (at all levels of training) needed to meet the current and future economic needs of the country ? Is mathema- tical training socially accepted as a field of study ? Are mathe- maticians employable ? Are school personnel familiar with the changing role of mathematics in the world ?

What part of the teachers in training are being exposed to the ideas presented in this report ? What part of the teachers in service have been so exposed ? Is information available for study by teachers or groups of teachers ?

Have the decision makers in government been informed of the mathematics revolution and its consequences ? Can positive steps for better instruction with governmental origins be enumerated 3 Are plans being prepared to support the various aspects of an effective modern program ? Have the country’s manpower needs been translated into quantitative terms so that school administra- tors can make provision for mathematical programs designed to meet these needs ?

Ways now being tried to remedy the present situation using psychological methods

This study is divided into two parts; a theoretical part, and a practi- cal part. Some of the theoretical aspects of the kind of thinking which is rele- vant to the learning of mathematics is conducted by research workers that do not have a particular eye on the applications to the study of mathematics learning. Conversely there are practical workers in the field of the study of mathematics learning as it takes place in the classroom that have no particular eye on any psychological considerations. Not a great deal will be said in this report about either of these two kinds of research. Only the kind of theoretical work which has direct relevance to the day-to-day problems of mathematics learning will be touched upon from the theoretical point of view, and only those practical projects

will be dealt with in detail that rely at least to some extent on psychological foundat ions. This will not be taken quite literally, as some mathematics projects are conducted by workers whose intuition has enabled them to organize learning situations very much in line with what would be predicted as a fruitful kind of sit- uation on theoretical grounds by those who have been engaged in such theoretical work. It should also be stated in this connection that until quite recently there was very little for a worker in the classroom to go on which the psychologist could offer him as a guide as to what sort of situations would or would not be fruitful

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from the point of view of learning mathematics. There is no known learning theory which would for example, hove predicted the success or otherwise of some of the modern methods of teaching mathematics. On the other hand, during the past decade certain attempts have been made to begin to study in detail some of the mathematical learning processes.

1.31

The school where the psychological problems underlying the learning of mathematics had been studied for the longest period, is the Geneva School headed by Jean Piaget. Jean Piaget is essentially a developmental psychologist. He and his co-workers have studied in detail the developmental stages through which certain types of concept formation pass. It is only recently that the work- ers in Geneva have started to study the process of concept formation itself while it takes place. The classical form of study has been to take a number of age groups and give a certain kind of problem to a sample of children from each age group and note the methods which the children employed to tackle the problem. From this it was inferred that the mental apparatus with which children attacked such problems developed from a somewhat primitive to a more sophisticated stage. These stages had in general terms been described already by Jean Piaget some time ago, the most important of these stages for our considerations being the con- creteoperational stage which is supposed to last from about the age of seven to about the age of eleven or twelve. But the work undertaken at Geneva which is more relevant to our study is recent work on the particular aspects of the learning of mathematics, such as the development of the idea of recurrence studied by Greco and Inhelder.

1.32

The next important source of information is the work of J.S. Bruner who has been working at Harvard on the study of thought processes as such. It is relatively recently that he has begun to look at the process of learning mathema- tics. His earliest breakthrough was reported in “A Study of Thinking” by Bruner, Goodnough and Austin. This was one of the first, if not the first, attempt to ex- ternalize the details of the thinking process. By reducing the problem to its bare bones and arranging the tasks in such a way that subjects had to pick up a card every time they thought of the next step toward solving the problem, the thought process was externalized and could, therefore, be put down, recorded and exa- mined afterwards. From this technique it was possible to arrive ot some analysis of strategies of thinking. It was found that the search for different kinds of com- binations of attributes led to different kindsofstrategies, and that different people would resort to different kinds of strategies when attacking the same prob- lem. Bruner particularly examined the strategies which subjects employed in coming to grips with conjunctive, disjunctive and relational concepts. Coniunc- tions, disjunctions and relations between concepts that were already familiar to subjects were used in the experiments. In his later work, he analyzed the process

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of mathematics learning into approximately three stages. 1) He calls them the en- active, the iconic, and the symbolic stage. Bruner believes that the child in the beginning thinks in terms of action. His methods of solving a problem are, there- fore, severely limited because if he cannot act out the solution, he cannot solve the problem. The next stage is the manipulation of images. This is what he calls the iconic stage. Images are very much more easily manipulable than actions, but nevertheless they tend to have a kind of permanence which makes them not very adapted to transformations. Mathematical thinking in particular abounds in transformat ions and, therefore, Bruner believes that fairly sophisticated mathema- tical thinking cannot take place until a child learns to think in terms of symbols and this is the third and last stage of the development of mathematical thinking.

1.33

Another important line in the study of thinking was started by Sir Frederick Bartlett. He likened thinking to the acquisition of a skill. Thinking was a high level mental skill and as such had a directionality built into it. Just

as when we are learning to perform an action there comes a time after which we cannot go back, i.e. the “direction” is then determined, in the same way in thinking when we have reached so far, the end point becomes determined. The difficulty about thinking is that very often we know or have an idea of the end point but we cannot get there. We do not know how to fill in the gaps. Or some- times we can go certain ways toward the end but we cannot see what the end is. The first kind of thinking where we have to fill in the gaps is the interpolation kind of thinking, the second kind where the end has to be filled in is the extrapo- lation kind of thinking. Bartlett makes a distinction between closed system think- ing and a more open kind of thinking. Very often in scientific or in logical think- ing we curtail ourselves considerably because of the discipline within which we are operating. If we are trying to construct a proof then we must obey certain rules of logic and we must make certain assumptions which are admissible within the parti- cular field in which we are working. For example, in the propositional calculus, if we wish to prove something we have recourse to certain axioms and rules of ma- nipulation, namely detachment and substitution. Any “formula” which can be reached from the axioms by means of these rules is colled “proved”. Such formu- lae are end points; anything which is not deducible by the admitted rules from the initial premises or axioms is not in the system. So in this kind of thinking the type of thing we are allowed to do is severely curtailed. Of course, we soon get out of a system like the propositionol calculus. We invent things like the predicate calculus and various particular kinds of predicates are introduced such as “equal”, ” more ” , etc. and arithmetic is born. Every time we introduce more assumptions and more rules of procedure we are opening the system a little more. At the oppo-

site end of the scale there is what Bartlett calls “adventurous thinking”. Here the

route is not determined, nor the end point. What is determined is the kind of

thing the end point has to be. He quotes the artist’s thinking when he is producing

1) The Study of Growth (not yet published)

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a work of art OS an example of adventurous thinking. And as likely as not the mothematicion producing his theorem might be very closely akin to this adventurous thinking because in the high reaches of mathematics the initial closures we were talking about in relation to the proposition calculus have been greatly relaxed. So quite possibly what Bartlett has to say on adventurous thinking may be highly relevant to the kind of research situations which we might wish to create in the classroom to imitote the research situation in which the mathematician finds him- self when he is trying to discover a new theorem.

1.34

Patrick Suppes has recently done some fundamental work partly on mod- el building and partly on applying his models to actual learning situations in which he would try to make certain predictions which may or may not have been volidat- ed in practice. One of his assumptions is that when a child or any organism learns something it is a process of conditioning and the response is either conditioned or it is not conditioned. What responses are considered as equivalent depends on the definition. A response can be considered OS an isolated response to a parti- cular isolated stimulus. According to this hypothesis the frequency with which the correct response is given before the response is conditioned is the some as that which we would expect cn the hypothesis of random behavior. The moment the response is conditioned then it invariably follows the stimulus, i.e. the probability of the correct response being given is one. Until the response was conditioned the probability of the correct response occurring would be that which would be ob- tained from the hypothesis of random behavior.

1.35

Richard Skemp, a psychologist at Manchester University, has been do- ing some fundamental work accompanied by practical work in the theory of mathe- matical learning. He distinguished two very different kinds of learning, rote learning and schematics learning. He also distinguishes two kinds of intelligence, sensory-motor and reflective intelligence, each of which is supposed to be activat- ed in these two different types of learning. Rote learning is defined simply as the acquisition of certain responses to certain stimuli by 1 process of conditioning. Schematic learning is defined either as the fitting of certain response patterns into alreody existing response patterns, that is into schemata already established or as the changing of the schema in order to accommodate some new responses which have been learned as correct responses but are not able to be suitably organized by the existing schemata the subject already has. So schematic learning is the fitting in of new responses into older response patterns or the changing of the old response patterns to fit with new responses. Each of these two is regarded as schematic learning by Skemp ond needs, according to him, the exercise of what he calls reflective intelligence. Mathematics, says Skemp, is essentially a structure of subordinate and superordinate concepts and, therefore, in order to achieve a schema which involved a superordinate, all the subordinates must already habe been

constructed. So no new schema can be evolved until other schemata which form

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part of the new schema have also been evolved. Therefore rote learning, i.e. learning responses to isolated stimuli is a very inefficient way of learning mathe- matics. He constructs ways in which schematic learning can be encouraged and even reflective intelligence can be induced to develop. He validates his thesis on high school children by showing that those children who score high on a test of schematic learning will score high on mothemotics examinations and conversely.

1.36

Zoltan Dienes in the post five years or so has also been conducting theoretical as well as practical investigations into the problem of mathematics learning. Dienes distinguishes between the different types of play, and claims that the energy in play could and often is harnessed in the service of creative work. This is to say that it is by plrlying around that we learn things. This is manipula- tive play. An almost equally primitive form of play is representational play, where the objects stond for other objects or ideas. A more sophisticated type of play is rule-bound ploy whereby rules are mode use of and games are made up. This ex- tends to further manipulative play, when the game itself is played with, so that the rules are changed. This kind of play intoduces the player to the ins and outs of playing around with mathematical structures.

Dienes’s view of abstraction is that it is a process of class formation. Abstract ideas are formed by classifying objects into classes through some common property which, it is discovered, is possessed by these objects. Generalization is regarded as the extension of an already formed class and, therefore, it is more of a logical operation whereas abstraction is regarded as o constructive operation. Abstraction, therefore, is likely to take place as a result of abstracting informa- tion from rather a lot of different situations in which one particular aspect, namely the structure to be learnt, is held constant. This gives rise to the principle of multiple embodiment. On the other hand the need for generality in mathematics gives rise to the principle of varying all the mathematical variables possible. The constructivity principle in its more sophisticated form is the theory of how construc- tions, abstractions, generalizations and play stand in relation to one another, as regards the process of learning. Another important principle is the principle of

contrast. In order to learn something about a relationship, this relationship must be seen not to be necessarily valid in certain other cases.

Dienes and Jeeves of Adelaide University have been working together for the last two years on some particular aspects of the thinking process, through the construction of a machine which children manipulate and it enables the process of

transfer from structure to structure, whether positive or negative, to be studied. One problem which is examined is whether it is better to stort with o more complex task or a more simple one if both of them are to be learned. Other problems are the effects of internal symmetries of structures on learning, the relationship between

. recurslon, generalisation, embeddedness and overlapping in transfer from structure

to structure. All these are being experimentally investigated in the Psychological

Laboratory at Adelaide.

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1.4 Practical ways now being tried to remedy the present situation

It will be seen that much work is going on in different parts of the world on different aspects of the problem of mathematics learning at quite a fun- damental level. It will also be appreciated that very little of this was going on even as little as five years ago. And so it would be surprising if many extremely important discoveries had taken place. But things are beginning to take shape and in this report certain of the fundamental findings which are applicable to the educational process in the mathematics classroom will be given from the various researches concluded or in progress.

Next we shall deal with the practical aspects of the problem. We shall deal only with those efforts made in the classroom with due allowance for at least certain of the results of the psychological or fundamental research being carried on in other parts of the world. These pieces of work on the practical level can be usefully subdivided into the following sections.

1.41

The first one could be called “the basic-set approach”. Workers using this approach usually deal with younger children, the assumption being that

to start at the kindergarten or first grade level is the only really profitable pro- cedure . Such workers feel that to clutterupchildren’s memory with stimulus-re- sponse mathematics hinders thinking in structures. So this approach will therefore concentrate on establishing the foundations of mathematics at the very earliest. Since mathematics as taught in schools is normally mostly to do with number and since number is an attribute or property of sets, it is clear to some people working on this kind of approach, that sets should be handled by children before numbers. So the fundamental relationships of sets to their members, of sets to other sets, such as inclusion, overlapping, distinctness and identity are dealt with. In some projects a distinction is made between a definition of set by on attribute which is applicable over a certain universal set ond the definition by enumeration of indi- vidual members. These are the intension and extension versions of the definition of sets. In some projects, the study of logic is introduced as a concomitant exert ise . Logical work has not so far been found possible in the Kindergarten at the formal level, and accordingly concrete moterials are used by those projects that attempt to introduce children to logical relationships at this age. The rela- tionships of conjunction, negation, disjunction and implication are dealt with and their relationships to the corresponding set operations are also introduced. Set operations which are relevant to mathematical work are the taking of the comple- ment, the toking of the union, the finding of the difference and the finding of the intersection. A certain amount of symbolism is usually evolved for these set opera- tions and the corresponding number operations are then built up on the explicit knowledge of the properties of these set operations. For example, unions of dis- tinct sets give rise to sums of natural numbers. The commutativity of the union operations in sets gives rise to the commutativity of the addition operation with numbers ond so on. This basic-set approach has been introduced by Patrick Supp,

by Paul Rosenbloom, by William Hull, and by Zoltan P. Dienes, at Stanford, Minnesota, Massachusetts, and Adelaide and New Guinea respectively.

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1.42

The next approach might be termed the “arithmetically oriented approach”. This can also start at a fairly young age, at six-plus or so, and per- hops the most widespread approach which could be classed under this heading is the Cuisenaire approach. Th e preponderance of numerals in certain of the text- books by Gattegno, may suggest an emphasis on arithmetic. But the attitude from the beginning is an algebraic one, the child being constantly aware of the properties of the operations involved, and not merely on the search for different names for numbers. The writings take the form of arithmetic writings, but as a result of conceptual confrontations by the students. Consequently, the student maintains an indifference towards the actual rods or the actual numbers used, ex- cept when conventional equivalent names are needed, or the numbers are applied to physical-social problems.

There are also other approaches, such as Catherine Stern’s approach, or the “Multibase Arithmetic Blocks”, and some more are coming on the market using various different kinds of materials but each approach tends to use just one kind of material. The founders and skilled users of the Cuisenaire method did not intend the rods to have only arithmetical significance, and have always rightly insisted on the algebraic ideas preceding the acquisition of purely numerical techniques. The fact remains, however, that most users of the rods appear to use them largely for purely arithmetical purposes. The ideo behind using Cuisenaire rods for learning mathematical relationships is that mathematics is simply a set of relationships not really to be derived from and not in any relationship with reality. Now if some objects can be found which exhibit in practice the same relationships as the mathematical relationships to be learned, then by a process of analogy children will learn what these mathematical relationships are by manipulating the materials which exhibit these relationships. Therefore, as the associations are made between say the rods and the corresponding mathematical structures, it is thought that in this way these mathematical structures are learned. Of course, up to a point, this is so, depending on what we mean by the word “learned”. The crucial question is transfer. Th e intelligent child will always look at every situa- tion as part of a class of situations and so, therefore, a bright child looking at the Cuisenaire rods will immediately size them up as something which is like a lot of other situations which could be built up in a similar way. Therefore, abstractions will take place by the learner bringing together the experiences with the rods and other experiences which he might have had or imagined. This does not take place with every child. The children that do notabstract in this way will merely associate and therefore if other situations arise which are not cuisenaire-constructed the problem of transfer will not be able to be solved. This can be corrected by the use of more than one kind of material. Instead of just using the Cuisenaire rods, some other equivalent material, such as beads in bags or anything else that the teacher can think of can be used and children will eventually find that which is in common between the two kinds of materials. The curious thing is that many ad- vocates of the Cuisenaire rods find such a procedure totally unacceptable, al- though again enlightened users, such as Mlle.Goutard, John Trivett and many others, do not adopt such an exclusive attitude. The protagonists of exclusive-

ness maintain, so it would seem, that if something can be associatively learned by

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means of the rods, there is no need for anything else. Certainly with almost any movement, practice or approach there will be many users who copy incorrectly rather than decide for themselves what ore the ,fundamentals. But surely to be fair we should consult those who know the true purposes and perhaps, separately, discuss why followers misunderstand. The multibase arithmetics blocks are not quite in the same associative situation because they have different versions. In these blocks there is a rectangular version and the triangular-trapezoidal version, and so children do not necessarily associate the mathematical or arithmetical situations to one physical situation, but a whole set of physical situations which are purposely made as different from each other as possible. But nevertheless the use of the Multibase Arithmetic Blocks by itself tends to be arithmeticolly-orient- ed and it tends toremain an associative rather than an abstractive approach, un- less some other more fundamental approach is joined to it. The people that have been advocating these arithmetically oriented approaches are Dr. Gattegno, and Mlle.Goutard, Dr. Roller, also Katherine Stern who designed another associa- tive set of materials whose use was based on the Gestaltist assumption of psycho-

lO9Y - Zoltan Dienes designed the Multibase Arithmetic Blocks whose use was meant to be based on the principle of mathematical variability. This is why the bases are varied in this particular material, whereas they are not varied in any of the others. Recently, the Cuisenaire Company have introduced what is sub- stantially a copy of the M.A.B. that is apart from first power pieces, second and third power pieces in the shapes of square slabs and ‘cubical blocks have been

added to the conventional sets.

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The third approach might be termed the “geometrically-oriented approach”. This has been mainly pioneered by Professor Paul Rosenbloom of the Minnesota School Mathematics Center. The idea is to foster a geometric intui- tion of the child by making him go through motions physically, by drawing and generally exploring “space” with him in the classroom. This procedure will get him to associate adding and subtracting to moving up and down in different senses along a straight line and also to associate points in a plane to ordered pairs of numbers and so on. According to Professor Rosenbloom this has the advantage that children will not have to unlearn anything when generalizations of the number system are undertaken. As the rationals are introduced, the points on the number- line merely have to be filled in. The irrationals likewise can fill up the number

I ine even more. When the complex numbers are introduced, the one line in the plane is replaced by all the points in the plane with the representatives of two imaginaries being points on another line. This approach also makes use of the idea of image manipulation. In the early grades children are told stories and in this way are given the idea of the properties of sets and they are told other stories and encouraged to invent their own stories to bringtheminto contact with proper- ties of natural numbers, directed numbers, adding and subtracting, and so on.

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1.61

The next type of approach might well be called the “Symbol Game- oriented approach” . This kind of approach is very widespread in the United States and is represented by such large organisotions OS the School Mathematics Study Group, the University of Illinois Committee of School Mathematics, the Madison Project, and the work of Dave Page. The idea being that this issomewhat similar to the Arithmetically-oriented approach in which Cuisenaire rods or other concrete materials are used, except that the games are played hot with such concrete materials, but with symbols drawn on paper or on the blackboard which have certain properties and the children ploy so to speak with these properties. Among these symbols the number-line is always included. In this way algebraical equations become playthings for children. Algebraical identities acquire a certain privileged position among equations, as the truth set of an identity is the universal set. The truth set of an equation is the set of those numbers which satis- fy the equation. Relationships between identities are also investigated and deri- vations are introduced whel-eby certain identitites can be deduced from certain other identities through the use of certain admitted rules of procedure. This is the sort of game which has been greatly encouraged by the Madison Project directed by Professor Robert Davis and certainly children seem greatly to enjoy this kind of work. Dave Page has also pioneered to o great extent the use of games with symbols and has invented o number of highly ingenious games, through the use of which children learn to enjoy the manipulation of mathematical concepts.

The Symbol game is what it says, a game that you play with symbols. In some of these games variables, place holders, or pro-numerals as you will, are indicated by certain frames of certain shapes and the operations of multiplication, addition, subtract ion, division and raising to powers are connected with these

pro-numerals and with other fixed-value numerals. Such mathematical sentences which involve pro-numerals are then studied. These are called open sentences. To an open sentence there will correspond o set of numbers, each member of such a set making the open sentence into o true sentence. The set of such numbers is called the truth set of the open sentence. Some open sentences have truth sets which are empty, others have truth sets which consist of one member, others have truth sets which consist of the universal set, that is of all the numbers. Children learn to play with these sentences; they learn for example the relationship bet- ween those sentences that have as their truth sets the universol set. These sen- tences are called identities. They even learn to derive certain identities from sets of other identities and in this way they become aware of the ways in which derivations are made. So the properties of a formal system complete with rules

of procedure begins to take shape. The Madison Project hos largely pioneered this kind of approach, though the first project to use arithmetic with frames was the University of Illinois Committee of School Mathematics. David Page has also been responsible for the invention of o large number of most exciting symbol games to the delight of the many children that he himself has personnally taught. The School Mathematics Study Group has taken up this approach, al- though possibly in ways somewhat less intriguing to children. This was thought to be inevitable, if the approach wos to be used on a large scale. If a large number of teachers and children were to be introduced to new methods, the pace was thought to be necessarily a slow one.

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1.45

Another interesting approach might be called the “Science-oriented approach” . One representative of this approach in the United States would be the Elementary Science Study, part of Educational Services Incorporated, Watertown, Massachusetts and attached to Massachusetts Institute of Technology. Here much of Mathematics is considered to be part and parcel of physics or mecha- nics, and attempts are being made to present the disciplines as one. In fact, it is thought by many workers in this Science study that mathematics as normolly taught is much too clean, whereas life is very much more messy. And so the situations in which mathematics is taught should likewise be more messy. This mess, in information theory terms “noise”, is, of course, a necessary ingredient of learning, because without the mess, children would never have a change to clear it up. In other words, the abstraction process would not hove a chance of really working if everything presented was already pre-digested and cleaned up. So accordingly to some workers in this study, even the use of the attribute blocks is suspect for the purposes of teaching logic, because the situations in which logic is used normally in life are far less clean ond very much more complicated. So they have certainly broached the problem of the child learning to cut through noise. This is simply another way of stating that children will need a lot of dif- ferent ways in which this situation arises before they can abstract that which is common. This might be called the Multiple Embodiment Theory which is impli- citly assumed to be operative by people who advocate mess in mathematical situa- tions that are presented to children.

Another important representative of the Science-oriented approach is Professor Paul Rosenbloom who has been trying also to combine the study of physics, mechanics and mathematics into one.

In Paris, there is a Mathematics teacher, Zadou-Naisky, who has invented a method of teaching many of the mathematical structures through the use of meccano and systems of wheels interlocking. The properties of fractions can be

learned by the properties of cog wheels interlocking with each other. The ratio of the numbers of rotations of one wheel to the other wheel is a physical represen- tative of a fraction and the next wheel will be in another ratio of the revolutions of the first wheel to the third wheel. These are, for example, representations of the situation in which multiplications of fractions are relevant. Rotations as well as translations are used and appropriate situations ore created in which the com- mutative, associative, and distributive lows are illustrated. It is claimed that these laws are discovered by the children through first of all building these struc- tures themselves ond secondly through manipuloting them in the classroom in prob- lem situations.

The last approach which is perhaps different from the others in many ways is the approach which might be called the “object-game oriented approach”

or the multiple embodiment approach. This is the approach advocated by Dienes.

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His ideas on promoting Mathematics learning through these principles is being put into practice in Adelaide, Australia, in the Philippines, and New Guinea, and in a few isolated places in the United States, and also to some extent in England and a few isolated places on the Continent of Europe. In the theoretical section the multiple embodiment principle wos alreody explained; for instance, an em- bodiment such as Zadou-Naisky’s could easily be added to the other embodiments that are already being used for the study of fractions or other mathematical re- lationships in the classroom. Another aspect of Dienes’s practical work is the multi-sensory one. The more of the sense modalities that can be involved, the more efficient will the learning become. If the child only hears, but does not see, he does not learn as well as if he hears and sees at the same time. Of course, if he can hear and see as well as touch at the same time that is manipulate with his fingers, he will learn all the better. If in addition to this, he can also move his body around, that is if the mathematical tasks can be embodied in such a way that the bodily movement of the child himself is part of the task, then it is hypothesized that the learning will be even more efficient.

It is suggested that not only algebraical-logical approaches should be used, but also geometrical and physical-mechanical approaches, coupled with symbolic approaches. The telling of stories and manipulating the images created thereby should also be used. One particular aspect of the approach by Dienes is what he calls the “throwing in at the deep end” policy. For example, in the case of logic or sets, instead of starting with just unions of sets or just unions of non-overlapping sets, h e would start with unions intersections, and complements from the first day. Or in the case of directed numbers, he would suggest starting with multi-dimensional vectors from the beginning. Instead of learning a multi- plication of positive and negative numbers, vorious multi-dimensional multiplica- tions, among them multiplication of complex numbers, could be used right away. The psychological justification of this is that structures ore very much more easily learned if they are embedded in other structures, so teaching a child on embedding

structure will help him in the end to learn thot which is embedded in this. Teach- ing a child complex algebra will teach him the real algebra very much more effec- tively than just teaching him the real algebra by itself, because the real algebra by itself will somehow be out on a limb and will not be inserted into anything else. So the principle would seem to involve that whatever we wish to teach children we should teach something else besides which we do not perhaps wont them to remember particularly well, but we should nevertheless teach it as on embedding structure which would help the children to orientate themselves in this new territo-

ry- The indications are from the Adelaide, Philippine and New Guinea Mathe-

matics Projects that this “deep end” policy is paying very handsome dividends.

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It must not be imagined that simply because certain approaches or certain mathematics projects have been mentioned that it is considered that the ones that are not mentioned ore of no value. It was stated in the beginning that this report wos not going to be obiective, that it was going to be definitely biassed. The bias is that it is believed by the author that the mathematical learning situations

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which are based on the consideration of the psychological and social situation OS well as on the consideration of the mathematical structures that are to be learned, are more likely to be significant from the point of view of progress made in the next decade. Much work is being done in Europe and elsewhere, e.g. in South America and in Africa, on various methods of teaching mathematics to young children. For instance, Emma Castelnuovo has been working in Rome for a very long time on new methods of teaching geometry through showing children the prop- erties of figures through movable structures and through letting them see geometry in their surroundings, in the buildings they see every day in our towns and so on. People like Nicolet have done a great deal of work in introducing the idea of the mathematical film. This was token up by a number of other people such OS Fletcher in England who has used the film for such sophisticated ideas as con-focal tonics. Television is mode use of in vorious ways in different ports of the world, but no breakthroughs of considerable importance in these areas are known to have taken place to the knowledge of the author. In the section on the practical work referred to, more detailed accounts will be given of the positions of the various projects and how the work they are doing in the opinion of the author is likely to effect future progress in tackling the problem of mathematics learning over the world.

Some important indications might be gathered from the success of the Philippine and the New Guinea Mathematics Projects as far as the underdevelop- ed countries are concerned. These will, therefore, receive special mention.

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THEORETICAL CONSIDERATIONS

I - - - - .

Chapter 2

THEORETICAL CONSIDERATIONS

The group of participants of the Stanford Conference whose task was to examine the problems of a theoretical nature were of the opinion that these problems could be divided usefully into four categories :

a) problems of a developmental nature b) problems on the nature of learning and thinking c) enquiries of an empirical or functional kind d) problems of testing and evaluation.

At the end of the Conference it was further agreed that those partici- pants that appeared to be best qualified by their research work to report in detail on these various aspects were to provide written reports, out of which the final report would then be compiled, after consultation with European workers in the various fields. Not all of these written reports were at hand at the time of writing up the second version of the report, and so it was necessary to put to- gether what is available, awaiting these contributions until possibly after the meeting with European colleagues.

2.1 Developmental research

In this connection the work of Piaget clearly stands out as the most important piece of work which is going to be relevant to the problems of learning mathematics.

2.2 Piaget’s contribution

Piaget has been doing research work on the problems of children’s thinking longer than most other psychologists. His views on the development of mathematical and logical concepts are well known. For our purposes, let it be sufficient to say that during the period with which we are concerned, the con- crete experiences of a child are of fundamental importance. He learns to operate

31

on a logical plane during this period but, so to speak, in concrete. He must have the actual situations to handle or at least images of these situations. He will not yet be able to reason hypothetically.

This has various consequences on the sorts of predictions that the odherents to this theory might make as to the ways children might attack different mathematical problems. A young child, perhaps one of five or six who hos not yet entered the concrete operational stage, assumed to last between the ages of seven and eleven approximately, will tackle a problem in an intuitive way. He will not be able, on the whole, to transform situations mentally and even to a lesser extent to transform them back to where he started. It is this idea of re- versing operations which, according to Piaget, is a crucial step in the develop- ment of the child’s thinking. If an operation cannot be reversed, then the effectiveness of this kind of operation cannot be very great. In fact, Piaget and his adherents would say that inability to reverse an operation, puts the child in a pre-operational stage. For example, a child that is encouraged to put one bead in each of two receptacles at a time and then spill the contents of one receptacle on to the table, and the contents of the other receptacle on the other table, while the two sets are both compactly arranged he will maintain that the number of members in these two sets is equal. If one set is scattered over the table ond the other is not, he will say that the scattered set is more numerous. According to Piaget this is because he is not able mentally to reconstruct the previous posi- tion of compactness from the position of scatteredness. And so he assessed the number property of a set by the perceptual properties of a set rather than by the numerosity of the set. It is the lack of the child’s ability to reverse the opera- tion of scattering into the operation of contracting that prevents him from seeing that the number of objects is not altered by scattering. This is what in the Geneva School would refer to as the pre-operational stage and it is contended that while the child is in this stage, if he is taught to operate on number, that is to add, to subtract, to multiply, and to divide numbers, this can only result in rote-learned responses. These responses cannot correspond to the mental opera- tions carried out by the child, since the conservation of quantity, i.e. the num-

ber of objects in a set, has not been achieved. Such numbers cannot effectively be operated upon because the results will not be “conserved” either.

Some of the workers in Geneva, in particular Greco and Matalon have done some detailed work on the way in which the concepts of number evolve during childhood. It is found that although for certain numbers children will realize that the next number in the series is one more than the previous number, and conversely that the one which is one more is the next number in the series, this very often is only appreciated in the case of small numbers, that is in the case of numbers that they can actually conceive as forming a part of real situa- tions, and not as extrapolations of real situations. A small child does not have actual physical experience of numbers such as seventy-seven, but he does have physical experience of numbers such as eleven, twelve, or up to twenty, and the synthesis of the next and the “one more”, i.e. of the ordinal and of the cardinal aspect of number, is achieved but gradually. This is what is known as the pro- cess of iteration, that is from N to N+l certain properties are “handed down” in the case of numbers. This children find difficult to do indefinitely. They will

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do it for the first few numbers, then they become uncertain. After a certain age, however, this uncertainty disappears and children will be able to say that the property is “handed down” to all numbers.

There is the further problem of investigating whether this synthesis is in fact able to be operated upon. For example, it is questionable, even though a child knows that the next one is one more, whether he will know that the next one after the next one is two more 3 In order to conclude this, he will have to

operate on his newly-found synthesis. His synthesis must be operational in the Genevo sense. In mathematical terms it means that although he knows that the next number to N is N+l, and, therefore, the next number to that is (N+l)+ 1, he does not know that the next after the next is N + (l+l). Or putting it even more simply, although he knows that one and one are two, he does not necessari- ly know that one more and one more are equivalent to two more, and even less so that one more and one more and one more are equivalent to three more. Greco has conducted a r-umber of experiments and found that between the ages of five and eight, there are at least five different levels through which this process passes before the final synthesis takes place.

This investigation merely touches on what is only a part of a very much larger problem, i.e., the problem of how recursive arguments based on the gene- ration of the series of natural numbers become established through the develop- ment of the child’s thinking ? The “next one” and “one more” are merely the most fundamental connections that need to be established. Other mathematical properties can be established through recursion. For example, every other number is even and every other number is odd. This may be realized by children in the case of the small numbers, but it may not be realized that this is somehow handed on through the recursive nature of the sequence. Experiments have been done on this by Greco and others in Geneva and it was found that developmental- ly there is at least a two or three year period during which the synthesis develops until finally a recursive argument can be used by children and they will say “even odd even odd even odd” and if one is even, the next one is odd, and if a certain number is odd the next one is even. Of course, experimenters had to be

careful not to use the words “even” and “odd” in their experiments because of the rote learned responses which children might have acquired during their schooling . Schooling is notoriously based on this stimulus-response type of learning and children tend to be taught to respond somewhat like parrots. So the

experimenters used expressions such as “The number of objects which can be split into two equal ports and others that cannot be split into equal parts”, or if this were the number of children who went out for an outing, could they go out in twos or would there be one left over without a friend, or some such situations

were used.

Another interesting finding was the effect which the physical realiza- tion of a one-to-one correspondence had on the ability of quite young children to use a recursive argument. In an experiment in which beads were put one by one in two separate receptacles the children were asked if after a certain number of such moves, the number of objects in the receptacles were the same. And surprisingly enough even after there were very large numbers of objects in the

33

.--. -. .-._ ___

receptacles, and even after the children did not know the number names that would denote the number of objects in the receptacles, they were still able to say with conviction that there were the same number of objects in them. It seems, therefore, that the idea of a one-to-one correspondence is an extremely power- ful teacher of the equivalence of sets which could precede the individual reoli- zation of each particular number property of particular classes of sets. This ap- proach is already being put into practice by a number of mathematics projects, such as Professor Rosenbloom’s in Minnesota and in the Adeloide and the Papua New Guinea mathematics projects, where the one-to-one correspondences are played about with considerably before the idea of number itself is introduced. It is hoped that this powerful weapon will then perhaps change the position of the developmental stages which appear to be so rigidly determined according to the findings of the Geneva school. What the Geneva experimenters found so extra- ordinary in these experiments was that even in the case of five-year-olds and even when the receptacles were different in shape, that is, if one were a low, wide one, and the other a high, thin one, the children would still insist that in spite of appearances there were the same number of obiects in each receptacle and they gave as their reasons that since they put one in each, it must be the some.

This leads one possibly to a suspicion that some of the conservation responses in the experiments conducted in Geneva might have been due to a mis- understanding on the part of the children, that is, the vocabulary used by the child might not have had the same semantic significance as the vocabulary used by the experimenter. When the child says “the same”, he might well mean “looking the same” or “having the some height”. When a situation is definitely engineered in such a way that the one-by-one joining of objects successfully builds up the sets which are to be compared with each other, then a perceptual difference does not appear to displace the realization of the one-to-one corres- pondence between the sets. The explanation in Geneva is that a temporary synthesis was made of the cardination and the ordination in this particular case, because the actions of the children actually putting the beads in, one in one receptacle and the other in the other receptacle, was a powerful influence in getting them to coordinate the order with the time element. It was clear that in this construction, to every move corresponded a one-moreness property in each receptacle. Children even said that it was impossible for there to be a different number because they never put two in one and one in the other. The Geneva experimenters found this quite a shock and found great difficulty in explaining how this could possibly have happened before what they termed conservation had taken place. It must also be added that Geneva, with all due respect to the local educational authorities, is an extremely traditional educational authority and most of the primary school learning situations are extremely standardized and most of the learning is done entirely by rote. So it is likely that in this situation possibly different results are obtained than would be obtained if the mathematical relation- ships were to be appreciated as the result of the children’s own experiences. It

remains to be seen whether in such more “advanced” child populations the same conservation problems occur as occur in Geneva and other places where the rote- learning of mathematics in the primary schools is the order of the day. In certain areas where children are beginning to be taught their number concepts through the

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use of one-to-one correspondence, this phenomenon which is regarded as an ex- ception in Geneva might become general, and the conservation problems might not arise because the one-to-one correspondence aspect of equivalence will be- come predominant over its perceptual aspect. It is quite possible that the children and their experimenters were simply speaking a different language and thinking of different things and yet using the same words.

Matalon conducted some interesting experiments on the idea of “any number”. He found that again there was a gradual comprehension of the gene- rality of the “any” over a period of development of the child. His first experi-

ment was to investigate through what stages the reolization that “2N t 1 is always odd” passes. Here again the situations were given in concrete terms so that if possible the rote-learned school responses should not interfere with the pro- gress of the experiment. The expressions “odd” or “even” were not used by the experimenters until and unless the children used them themselves in a meaningful way. The other somewhat more far-reaching experiment was to do not with “any number” but “any operation”. Matalon tried to investigate the development and the generalizatian of the idea of commutativity of multiplication. He stressed that in a product the two factors play very different roles. One of the numbers describes the property of a set and the other describes some operation which has to be performed on that set in order to get onother set. In other words one numb- er indicates state quantitatively, and the other number indicates quantitatively an operation to be performed on that state to obtain another state. An experiment was performed as follows :

A child subject was asked to put out a certain number of red counters in a pile. He was then asked to match these red counters exactly with blue

counters. In other words, the child performed the physical equivalent of the operation two times N. Then he was asked if it was possible to collect all the

members of the sets so obtained into pairs so that there would be none left over. The experimenter started the procedure for the subject by taking the first two red ones or else two blue ones, and then the red and the blue to give the child the idea that it did not matter what kind of pairs were taken. The response that

should have been given had the children realized the commutativity cf the situa- tion, was that this would be possible because N times two is the same as two

times N, N being the number of counters of each color in the set. This was ex-

tended to higher numbers, such as three, four, and five. That is, three colors, four colors and five colors were used and the generalization of the property was also investigated with the same children. An intermediate stage in the under- standing seemed to be that the evenness of the number of members in a set seemed

to be associated in the case of some children with particular ways of splitting the set. The complete equivalence of any split into two parts is o prerequisite of getting this notion to pass into an operational stage. Similarly in the study of di- visibility by three, the construction of one particular split into three equal parts is not the same as regarding this particular split as an exemplar of any split into three parts. So here again we see that there is a fairly long developmental road through which children pass from the particular to the “any”, and it is suggested that eventually children identify the idea of “any” with an act of generalization. At

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the end they are able to operate on the whole of the number series, thinking of it as one given set of data. This is a more mature stage than opet-ating on one number at a time in parts of the number series. It is this end product which gives this generality and according to the psychological results obtained at Geneva a concept of “any” does not really develop until this generalization has taken place. Again it would be possible to interfere with this developmental scheme by suitably chosen exercises through which these developmental stages could be accelerated. Some of this, no doubt, is happening already in those mathematical projects where algebra is taught through acts of actual generalization by the children them- selves and not as a set of rules to be learned by them.

There are many other detailed researches that have been recently un- dertaken by the Geneva school. It will be impossible to give an account of all these researches but in the bibliography references are made and interested read- ers can have access to this most intersting series of etudes put out with such alarming frequency and rapidity by the Geneva school.

2.3 The Work of Bruner and his Colleagues

Next we consider Professor Bruner’s work on the properties of thinking. His work is much more of a theoretical psychological character, being based on experiments and is somewhat less relevant to detailed consideration of the problems of young children learning mathematics. There are, however, some pointers which we will have to consider.

In the book entitled “A Study of Thinking” by Bruner, Goodnough and Austin, Bruner and his associates were perhaps the first to investigate experi- mentally the strategies of attaining a certain concept. They did this by putting in front of the subject an array of different cards in which a certain number of attributes were varied. For example, on some cards there was one figure, on some two, and on some of them three. On some cards there were red, on others green, and yet on others black objects. On some there were squares, on some circles, on some triangles. Some had one frame around it, some had two frames, and some three frames, and so on. The technique of the experiment was for the experimenter to say to the subject that he had thought of a certain combination such as, for example, red triangles with at least two framesaround it, and any card which satisfied those conditions was a positive instance of the concept the experimenters had chosen. The subject’s job was to pick cards in the order he wished and simply ask the experimenter if that card was or was not an exemplar of the concept. The subiect had to make use of the information in as efficient a way as possible so that in as few tries as possible he would be able to tell the experi- menter what the concept was that he was thinking about. A strategy was defined simply as a behavior pattern engaged in by the subject in making use of his infor- mation and arriving at the final goal of finding out what the experimenter’s con- cept was. It was found that subjects were much more able to make use of positive informat ion than of negative informat ion. If they picked a card which was a non-

exemplar of the experimenter’s concept, they could not make such effective deduc- tions from what information as if the answer were that the card was an exemplar of

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.

the experimenter’s concept. This had a bearing on the relationship between the ways in which subjects attacked conjunctive or disjunctive concepts in this expe- rimental situation. A conjunctive concept was defined as a concept which in- cluded the simultaneous possession of two or more attributes. For instance, a red circle was conjunctive because for the card to be a positive instance it had to be red and a circle at the same time. This is redness and circularity were con- joined into one attribute, redness and circularity. A disjunctive concept would be if these attributes were joined by the logical connective either/or. That is, for example, the experimenter might have thought that any card would be a posi- tive instance of his concept, either if it were red or if it had a circle on it, in which case any red ones would be exemplars and any circles would be exemplars. In this case it would be the negative instances that would give the most informa- tion. In every case subjects were told whether the concept they were looking for was a conjunctive one or a disjunctive one. So theoretically they should have been able to work out that a non-exemplar is an easier piece of information from which to deduce the most efficient way of proceeding next, whereas in the case of the conjunctice ones it was easier to deduce the most effective next step from a positive instance.

Several different kinds of strategies were used by the subjects. Some involved considerable cognitive strain or memory strain and others were much more effective. The most effective strategy in the case of the conjunctive concepts was what Bruner called the focusing strategy. This strategy consists in picking on a “focus card” and ticking off the attributes one after the other which are either relevant or irrelevant. If the focus card has been found to be an exemplar, then it is possible to pick another card in which only one of the attributes is varied; for example, only the shape, and the other attributes on the card are kept the same as before. If this second exemplar is also a positive instance then it follows that this particular attribute which has just been varied is irrelevant to the con- cept. If it is shape, for example, then in that case the subject will know that shape does not enter into the experimenter’s concept, so he can then go on to the number of things on the card and vory that next. Supposing there are two circles on the card, he will then be able to get an almost identical card except that there are three circles or one circle on it. If this is a non-exemplar, that is, if it is a negative instance, then the subject will know that two is a relevant port of the definition of the experimenter’s concept. So he simply ticks these off one by one and from the first positive instance he should be able to get the experimenter’s concept in four trials. This is using both the positive and negative information effectively .

Not many subjects in fact use this strategy but it is suspected that the reason for this is that not many subjects have had many dealings with purely logi- cal situations of this kind in their school or university training. This again can,

of course, be altered by introducing logical work into the schools earlier on. One interesting finding in this particular research is the great discrepancy between the ease or difficulty of the conjunctive versus the disjunctive concepts. The dis-

junctive concepts were found very much more difficult to obtain by most subjects than conjunctive ones. Bruner explains this by saying that in the case of a dis-

junctive concept the subject cannot focus on any particular thing, or combination

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of things, which is in common to all members of the class defined by the concept. In fact he gives examples whereby he tried to show that in fact when we are dealing in life situations with disjunctive concepts we tend to transform them into conjunctive ones so as to have a hold on them from the cognitive point of view.

Bruner and Dienes spent a whole academic year together working on the Harvard Mathematics Learning Project, and out of this many theoretical and practical considerations arose. Bruner and Dienes were not altogether in agreement about the conclusions and they have written up the results of the experi- ment in different ways. Readers are referred to the bibliography for details.

Bruner eventually formulated a theory of growth in which he claims that thinking passes through essentially three stages. One he calls the enactive stage in which a child has to actually act out his thoughts with his body, the next one he would call the iconic stage in which the child is no longer needing to act but he still has to manipulate concrete images. This probably corresponds to some extent to what Piaget calls the “Concrete operational stage” but the two subdivisions probably overlap. The last stage reached by the child is the symbolic stage. This is when he is able to transform and manipulate symbolic expressions such as language, or the more sophisticated type of language such as mathemati- cal symbolism. This is the most flexible of all the materials for thinking, since the images are more amenable to transformation than actions, and, of course, symbols are more amenable to transformation than images.

So, according to Bruner, the direction which development tokes, is through the enactive, passing through the iconic and finishing up with the symbol- ic stage. All along this line there is an increase in the flexibility in favor of being able to carry out more and more transformations, An action cannot be trans- formed, it has already been enacted and children thinking in terms of actions simply perform one action after the other and think in terms of juxtapositions rather than in terms of the transformations. The image is an intermediate represen- tational system between action and symbols and can be used more effectively than action for transformation. But the image is still notoriously static, lacking in generality. Bruner finds it hard to conceive that an image can be general. A generic image for him is not really a possibility and so he regards the image as an obstacle which should be done away with so that the symbolic stage can be ushered in as soon as possible. It is, of course, very possible to make use of actions and images and symbols at the same time and possibly once a child has learned how to handle symbols he will be able to handle images on a more sophisticated level and use them the same way as symbols, in fact transforming them in quite arbitrary ways to suit his purposes.

Similarly, actions can be introduced as learning situations, as has been done in a number of mathematical projects in different parts of the world. Such use of actions is on a very much more advanced and sophisticated level than the use of action by a child simply to express what he is feeling or thinking by getting some of his thoughts put into action. This can be seen sometimes in the use by children of concrete materials in which actions are necessary to achieve a result. Very often when a child is trying to solve a problem, action is necessary to do so.

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During the stage of getting more sophisticated about the action, he will very often look at the apparatus and perfor-m some minimal action with his fingers or clearly be looking at the apparatus and imagine that he is performing the actions. This is perhaps a kind of telescoping of the action-image-symbol sequence to a somewhat shortet- sequence, and not a developmental but a learning sequence. In fact, it is probably one of the differences between Bruner’s idea and Piaget’s. Piaget regardsthedevelopmental sequence as essentially a sequence of stages which organically follow one another, whereas Bruner’s conception is in terms of each learning situation going through certain stages before that particular learning cycle has been completed.

Dienes, in his eorly monograph “Concept Formation of Personality” already suggested in 1959 the possibility of interpreting Piaget’s stages in this way, so that learning cycles could be regarded as microscopic copies of the de- velopmentol cycle or developmental road which, according to Piaget and his collaborators, each child has to ga through before he reaches maturity. This is probably educationally speaking a useful conception and should receive mare attention and investigation than it has.

2.4 Open Problems in this Area

Much more detailed knowledge is required of the repertoire of con- cepts that children have as they enter school, the concepts they need in order to enter a certain stage of mathematical learning, and the background of experience which is necessary for most children to have before they can effectively develop a certain concept or circle of concepts. There is a certain amount of work being done in this field by the School Mathematics Study Group in the LJnited States (described by Gloria Leiderman) and some related work at the University of Michigan (described by Joseph Payne).

One particular aspect of the above problem relates to culturally deprived children. There is an even greater need for us to know just what kind of equip- ment some of these culturally deprived children take with them to school. What concepts can they use operationally 7 Which of these can they name and make sensible verbal statements about ? In practice how do they categorize objects into classes 7 Are their classifications radically different from those ofculturally more sophisticated children ? For example, some information could be obtained on

these points by determining whether they can recognize line drawings or photo- graphs. In the more mathematically relevant fields, one might ask what procti- cal command of cardinal and ordinal numbers they have when they enter school. In what ways does their visual imagery differ from that of others 3 What kinds of systematic recall and retention powers do they exhibit 7

Studies are freeded in the development of language and its relation- ship to conceptual thinking. The recent studies of Chomsky at M.I.T., Brown and Bruner at Harvard are promising in this respect; these studies are still in progress so that those interested in the stages r-eached by these inquiries should consult these workers directly. We know very little about how verbal command of

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language at the time of entering school is related to children’s eventual success in coming to grips with mathematical thinking and its problems. Studies of non- verbal awareness, such OS conducted by Gertrude Hendrix, should be made more systematic. Since conditional sentences begin to be used by children at about the age of five or before, it would seem that hypothetical thinking might be able to be encouraged on the non-verbal level much before the ages at which such thinking is today generally thought to be possible. Studies such as Omar Kayam Moore’s on young children and their ability to learn to read at a very early age, would appear to be relevant in this connection and deserve to be more widely known.

Another problem which will become more and more relevant in these d *. ays of rntrmate contact between different cultures and subcultures is the one of cultural conflict and its effect on the learning process. When the child’s cul- ture, as transmitted to him by his home and social background, differs widely from that which is transmitted to him at school, success at mathematical activities might indeed suffer. The possible connections between such conflicts, and mathematical learning, via the affective transformations that are likely to be taking place, will, therefore, need examination.

There are other, more immediate developmental problems that are bound to hove an impact on mathematical education: for example, the develap- ment of parent-child relationship and its effect on the acquisition of learning sets, the development of critical ond analytical attitudes typical of middle class orien- totron, particularly in the Western type cultures, which might well affect the efficacy of learning in certain particular ways or in certain areas of the mathema- tical curricula. What are, in fact, the chief environmental factors that bring about such a critical attitude ? What is known about the development or the lack of development of such attitudes in other cultures ?

There are many other problems such as the reasons why more boys than girls take an interest in mathematics in the upper grades, the reverse being gene- rally true in the lower grades. Are such phenomena due to the choice of curricu- la , gender of teachers, teaching methods, or social pressures ? These are just some of the many problems that research workers in the field will hopefully begin to tackle as soon as possible.

2.5 Research on Learning and Thinking

There is an enormous literature on learning, and it would be impossible to quote all the sources. Historically speaking the work of Pavlov is of great im- portance, as he was the first to emphasize the importance of the conditioning process as a certain form of learning. Hull’s theory was a step in the forward

direction , giving a somewhat more sophisticated picture of the process and allow- ing for the operation of more variables. Kohler and Wertheimer are also histori- cally important, as they tried to shift the emphosis from atomistic learning of chains of associations to what they called the “Gestalt” or the total configuration of what was being learned. Tolman again contributed a somewhat different type

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of learning theory, while Skinner, following in the Pavlovian tradition, ushered in the area of programmed instruction based on the hypothesis of conditioning be- ing the basis of all kinds of learning.

There are no early studies that are directly relevant to the complex learning which takes place even in the elementary school mathematics classroom, such complex learning having only been studied in detail in recent years. The reader is referred to the bibliography for information of mathematically oriented learning experiments.

The researches in this field fall approximately into three categories:

o) Those of a purely psychological nature, carried out in laboratory fashion, with appropriate controls;

b) those of psychological and laboratory nature, but which have also been accompanied by classroom experimentation;

c) those of a purely exploratory nature, undertaken iI> the classroom or with small groups of children, to explore some “hunch” of the experimenter’s OS regards children’s learn- ing behav ior.

It will probably be agreed that the work of Bartlett and Suppes will qualify under the first heading, and the work of Dienes and Skemp under the second. Some aspects of the above researches will, therefore, be described as examples of the kind of work which is being undertaken. For other work of a si- milar nature, readers are again referred to the bibliography.

2.6 Bartlett’s Contribution

Let US see now in what ways the contributions of Sir Frederick Bartlett, of Cambridge, England, are relevant to the study of the mathematical learning situations and its improvement in the classroom. Bartlett has never explicitly studied the learning of mathematics, but has studied the process of thinking as such. In his book “Thinking” he sets out his position of regarding thinking as a high level skill. The skill consists in the use of information, i.e. of evidence, picked up from the environment and in getting relevant information out of “store” and using the combination of these two kinds of information to determine the direc- tion that thinking should take to arrive at a new point. Bartlett regards thinking as invariably being endowed with some kind of direction. It has a beginning point where a problem is posed and an end point which may or may not be known. If it is known he postulates that thinking allows the thinker to orrive at his desti- nation. Accordingly, thinking, in this case, consists of a certain number of steps that follow one another in some kind of compelling logical order. He does not state that only one kind of chain of steps can lead from the beginning to the end. Nevertheless the further on the thinker moves along the road, the more and more determined his course is going to become in the some sort of way as a tennis player who is hitting the ball. By the time he is practically hitting it, he

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cannot easily retract from what he is doing, determined at that stage.

His movements are pretty much In every case, therefore,according to Bartlett, think-

ing is a kind of gap filling process and the gaps are filled in by the processing of the information which is obtained from the environment or from “store”. This kind of thinking, making use of evidence, sifting it, and arriving at conclusions, is in fact very seldom the way in which mathematics is learned in ordinary class- rooms. It would seem to follow that thinking, as Bartlett defines it, does not normally take place during the mathematics lesson. This, we might decide, is undesirable and in the practical section various attempts will be described in which thinking is now being encouraged rather than discouraged in mathematics classrooms in certain small regions of the world, which, it is hoped, will increase in size and importance as time goes on.

It is clearly impossible to summarize all of Bartlett’s work, nor is all of it relevant to our requirements. One remark he mokes, however, would seem to be of importance. He says that we must distinguish between two kinds of gene- ralization in thinking. One kind is th t a once a number of steps have been taken in a certain direction, then the remainder of the steps in that direction are more easily taken, so that the first few steps in a series which are alike in some way, make the further steps in that series more likely. This is a kind of extension pro- cess whereby a certain class of events will be extended to a larger class of events so that it includes all the steps that will enable the end point of the thinking to be reached. He then says that there is another different kind of generalization which is the process of extracting the common property from different instances and classing these together. This process of generalization is assumed to include some “stamping in” of the common properties and some “stamping out” of the uncommon properties, that is those that are not to be generalized. He has not found in his experimental work a shred of evidence that this does in fact take place in the way it is normally assumed. In other words, the classification is not in fact made and so is not able to be used in transfer unless some particular steps ore taken in order to insure this. For example, in extrapolation tasks, he found that transfer was more likely from one instance to onother if the medium employed in each instance was the same, even though the rule required was different, than if the rule re- mained the same the medium was different. We quote here from Bartlett:

1

“So far, therefore, the strong indication is that transfer, like high level generalization, is not in the least likely to occur in any of these cases unless there is active appreciation of the situation that offers it an opportunity. More than this, it seems that the appreciation must aim to make use of the structural features of the situation. The assumption that transfer of the results of training and practice can be effected merely by bring- ing together appropriate instances with like constituent items is incorrect”.

1 Sir Frederick Bartlett, Thinking. Allen and Unwin, 1958.

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This clearly has relevance to educational techniques and would seem to indicate that in the case of CI multiple embodiment technique it is not simply enough to wait patiently for abstraction to occur, but certain steps must be token in order thot the learner or thinker con explore the structural properties of each embodiment and then make the connection between the two. We shall see later on that in some mathematics projects this is done explicitly and we shall discuss this under the technique known as the “dictionary technique”; thot is, “dictionaries” or correspondences ore established in such CI way that the isomor- phism between two different situotions becomes almost compellingly obvious.

In the less closed and more odventurous kind of thinking that Bartlett has also investigated experimentally in detoil, he finds that the subjects tend to choose paths with more alternatives lather thon poths where there ore less olterna- tives. There seems to be no evidence that short cuts tend to be token by subjects. There is more evidence for subjects taking the odventurous path. This may be because subjects would possibly unconsciously turn toward a large number of pos- sibilities, as these will be more likely to contain the poths thot would leod to a correct solution. It is, however, unlikely thot such probobalistic thinking comes to the level of awareness in many thinkers. It would seem as though when the number of probabilities is large, subjects will tend to think in terms of bunches of possibilities rather than individual possibilities. A whole set of possible ways will in some way be identified and the particular way which is taken is token may- be just OS an exemplor of oil the other possible ways that could have been taken. This tendency in thinking to go for the kind of choice which is richer in olterna- tives would seem to indicate that in educotionol applications we should give our children more alternatives than they hove received. In most mathemoticol in- struction children ore told exactly what to do. In fact children receive no alter- natives at all. According to Bartlett’s evidence, a multiple choice on the whole is preferred even to the double choice by most subjects. It has certainly be found in research on motivation thot providing CI wide choice will provide a higher moti- vation in the execution of the tosks. It would, therefore, seem to be more sen- sible to provide more choice in the educational situation; the usual method of leaving no choice to children in schools would appear to be totally unnatural and not likely to leod to thinking.

2.7 Suppes’ Contribution

Potrick Suppes’ work is also of great importance in the list of pioneer- ing works into the psychological foundations of mathematics leorning. lie has

been active in building models for learning mathematical structures along with other psychologists, particularly in the United States. As we oil know, there is a whole school engoged in building up various types of mathematical learning theories. Suppes’ view about learning is that it takes place OS a conditioning process, but o certain response to o certain stimulus is either conditioned or not conditioned. On this bosis it is possible to make various predictions about the number of times a correct prediction is made before the first error, and various deductions con be made of thot kind from the mathematical model which explain in o mathematical way this theory of conditioning. It is hypothesized that once a response is con- ditioned then the probability of the correct response being given is o hundred percent.

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Until it is conditioned, it is OS though it were a random response ond the correct response is only given with the frequency that would be expected on a rondom hypothesis.

A number of studies were conducted to test the validity of this model by Suppes and his associates at Stanford. Some of these were conducted on young children with concept formation tosks of CI classical nature in which the pro- cess of abstraction wos studied, and some experiments were even done on animals. It was found that the model worked remarkably well. In fact it wasvery good by any standards. Readers should refer to the bibliogrophy for the accounts of the working of these models. Patrick Suppes is also currently engaged in some re- search on the simulation of the teacher by means of o machine which is hooked up in series with a computer. A great deal of flexibility is oble to be given to such a “teaching machine“ because of the facilities which o computer can provide. Some of the geometry is, for instance, done with o photoelectric pencil on a screen ond correctness or otherwise is immediately indicated to the subject by a response from the computer. S’ rmr or y progroms hove been written for the learn- .I I ing of deductions in the propositional calculus from given oxioms by means of ad- mitted methods of argumentotion. This, however, leads us somewhat away from the frame of reference of this report. It was stated in the introduction that only one aspect of the work was going to be seriously reported, namely the work which was going on in the psychological laboratory which had o connection with mathe- matics learning as now practised in ordinary schools or in the new experimental projects, and also those practical experimentol projects which hod some connec- tion with the psychological work that was done in the laboratory.

The theoretical work which is being done on computer simulation ond generally leading towards mechanization that is progromed leorning and teaching machines generally, is not to be reported in this booklet. Readers are referred to the bibliography for recent results in this field. For example, at the University of Illinois Committee of School Mathematics, a certain amount of work has been done on computer simulation and in this project also the common or gorden work in the classroom is being carried out in much the same way OS in any other mathema- tics project. The results ore encouraging, but this study also is relatively in its infancy and not a greot deal can yet be said obout its probable impact on the con- struction of future learning situations.

It is probably safe to soy that it would be unwise to disregard the im- portance of mechanical or programed learning, nor would it be wise to believe that mechanical learning of this type is going to replace every kind of learning that takes place now. It is reasonable to expect thot within the foreseeable future during the process of mankind, certain mental activities will be regarded 0s creative, that is to soy unpredictable, and certain others will be able to be mech- nized. These parts of mathematics that con usefully be mechanized will probably be better mechanized OS the time so saved will allow mare teacher time and more child energy to be devoted to those parts which are truly creative. Probably some compromise will be reached between the progrom construction and the creative type of mathematics leorning.

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In this report only the creative type of learning will be discussed. That is not to say that no rote learning should take place and that the rote learn- ing is of no importance. Q ‘t I I UI e c ear y certain mechanisms and certain factual relationships between operations and between numbers themselves must be memo- rised by children in order to be able to find their way about in the mathematical country. But there are easy ways of carrying out such mechanical tasks and it is not the purpose of this report to enlighten the reader in this regard.

One important experiment that has taken place over recent years which substantially contributes to our knowledge of young children’s learning processes was carried out by Suppes and Ginsberg at Stanford University. This is reported in “Science Education”. There were several experiments performed. The first one was mainly performed as a test whether the results of concept formation expe- riments could be accounted for by the simple all-or-nothing model described earlier. The experimental subiects were kindergarten and first-grade children who were required to learn the binary equivalents of the Arabic numerals four and five. That is, a one zero zero and one zero one. Of course, ones and zeros were not used, but six different stimulus displays were used, that is six different one zero zero types and six different one zero one types, in which the two objects representing the one and zero respectively were varied in six different ways. The child was required to respond by placing large fours or fives as the case may be upon the stimulus. Children were told whether they were correct or not and half the children were told to correct their errors. As might have been expected, the children who made the immediate correction performed very much better than those who did not. Also the fit of the model was remarkably good. Statistics such as expected errors before first success, expected number of success runs, expected number of error runs and so on were calculated from the binomial distribution as this gives the predicted frequencies, assuming thot the subjects were acting in a random manner, that is that they were guessing. The fit as can be seen in the report in “Science Education” referred to just now was remarkably good.

It could, of course, be objected that this kind of learning is too simi- lar to paired associate learning, and that it is not really concept learning because the subjects might simply have associated the particular features of the six different stimulus displays to the numerals four and five respectively. Since each stimulus display turned up many more times than once, the result could possibly by explain- ed on the paired associates theory; that is that they were merely associating the

“four” to six different displays, and they were associating the “five” to the six other displays. In order to avoid this difficulty in their next experiment no display was repeated. That is every exemplar of the concept to be learned was presented only once to the subject. The concepts to be learned were equivalence of sets,

identity of sets, and identity of ordered sets. The changes were rung on the order

in which these were presented ta the child subjects. Thot is, identity was learned

in some groups before equivalence, and equivalence was learned in some groups

before identity and so on. In the last experiment, equivalence was used but three

different groups were employed. In one group children simply had to say whether

the sets went together or did not go together, that is were equivalent or non-equi-

valent. In the second group the children were presented with one display set and

two other sets. They had to choose the answer which went together with the display

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set. In the third group children were presented with one display set and with three answer sets and so they had to make their choice between three possible answers.

The results of these experiments can be summa!-ized approximately as follows :

1. Learning appears to be more efficient if the child who makes an error is required to make a correct response in the presence of the stimulus to be learned.

2. Incidental leorning does not appear to be an effective method of acquisition for young children. By this is meant that in one of the experiments, color discri- mination accompanied the discrimination between the equivalent and the non- equivalent sets. The children learned to discriminate very quickly in this group, but, of course, they learned to discriminate between colors. It was conjectured that some incidental learning would take place, so that on the second day when they did not have the color aid to their learning, theil learning was conjectured to be at least somewhat bettel- than those children’s performance on the first

day who had the color aid. But this was not so. In other words, incidental learning, that is the color associations did not seem to help in sorting out the con- ceptual relationship between the stimulus displays.

In another part of the experiment, again to do with color, color differ- entiation was used in the beginning with this group. The colors were brought gra- dually closer together during the experiment so that children found it more and more difficult to make the distinction between equivalent and non-equivalent sets by the use of color. This had the effect of making the children concentrate more acutely on the stimulus material. It was found that this group learned far rrlore effectively than the other groups in this particular part of the expel iment. In the last transfer expel iment it become clear that the transfer- of a concept is more ef- fective if the learning situation requires the subject to tell whether a concept is exemplified or whether it is not, that is, whether the stimulus display is an exem- plar or a non-exemplar, than if he is required to respond differentially between an alternative of two or three possibles. At the same time in the multiple choice situation the three-response situation is more effective than the two-response situa- tion. Lastly it would seem from these experiments that a young child’s learning tends to be very specific . Prior training in one concept did not seem to effect much improvement on the learning of the related concept. That is, most children that learned the identity of sets did not seem to have any advantage in learning the equivalence of sets and so on, in spite of the fact that every two identical sets are also equivalent and so there is a considerable structural overlap. It could be, of course, conjectured that it is because of the logical relationship in one way but not the other way that makes things difficult. Piaget would probably say that the

reason there was no transfer was because the children would assume the converse of the statement that if two sets were identical then they were also equivalent. That is, they would assume that equivalent sets were also identical.

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But transfer appears to be more effective when subjects are required to recognize the presence or the absence of a concept, when this is presented in ever-changing stimulus displays keeping only the concept constant, appears to be equivalent to saying that encouraging the child expressly to abstract, that is, to bring together that which is in common to the stimulus displays and disregard that which is specific to each one is an easier thing than to get him to associate cer- tain correct responses to certain stimuli. One could argue that this argument would support the theory of abstractive as opposed to associative learning. It is also of interest that even in the case of associative learning that is, where a multiple answer situation was presented, choice between three possibilities was more effective than choice between two. In other words, throwing the child in deeper to the situation has a more beneficial effect than throwing him in at the shallow end. But it is better still to present him with the problem of abstraction by requiring him to focus on whether a certain concept is being presented or not, i.e. whether a display is or is not an exemplar of the concept.

2.8 Skemp’s Contribution

Another important experiment which has already been referred to in the introduction is one performed by Dr. Richard Skemp. This was reported in 1961 in the British Journal of Educational Psychology. Skemp in his paper de- fined different kinds of intelligence, sensory intelligence, motor intelligence, and reflective intelligence. When a child can transcend the resemblances between sensory stimuli and can think in terms of relotionships between these sensory stimu- li, then he is using sensory intelligence, according to Skemp. In Piaget’s pre- operational stage, according to this, sensory intelligence would not be fully operative because it is, for example, the look of the sets of objects rather than the relationship between their individual members that determines which of them has more in it. Motor intelligence he suggests is awareness of relationships between actions, such as between filling up and emptying, putting together and taking apart, and so on. So the child who is learning arithmetic by manipulating objects is using his motor intelligence in realizing the relationship between his actions and is using his sensory intelligence when he is realizing the relationship between the sensory input of the results of his action. Skemp goes on to say that reflective in- telligence is the ability to double back upon ourselves and realize what we were doing when using our sensory or our motor intelligence.

Skemp looks at the situation in two cycles or two signalling systems, as the Russians would have it. The environment is received by the receptors. It is organized immediately by the sensory motor intelligence and the effecters put the output back into the environment; this is the response. This sensory motor organization can again act as a stimulus to the second signalling system, that is to the reflective receptors that can again organize this and the effecters will feed back into the sensory motor organizing system which will eventually feed out into the environment through the sensory effecters. So the reflective system, accord- ing to Skemp, is in some sense built on top of the sensory motor system and ope- rates on it in the way that the sensory motor system operotes on the environment.

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As regards mathematics learning, Skemp’s hypothesis is that in the learning of arithmetic mostly the sensory motor system is involved whereas in the learning of algebra and the higher reaches of mathematics the reflective system is involved. One cannot help thinking that this hypothesis would not have been made, had arithmetic not been treated as a mechanical set of stimulus response situations, which simply had to be learned.

These days, arithmetic is no longer being strictly separated from the rest of mathematics and when we now speak about learning arithmetics we do not necessarily speak of mechanically learning how to carry out certain processes. We now often mean the learning of not only how arithmetical operations ore carried out, but why they are so carried out. In this case, even in the learning of arith- metic, we would probably need to use our reflective system to quite a little ex- tent. According to Piaget, of course, the reflective system does not really begin to develop or be fully in use until the onset of puberty, so it would seem a sen- sible assumption to make that arithmetic which is learned in the primary school, that is, during the greater part of the concrete operational stage, makes use of the sensory motor system.

It would appear to make a tidy kind of organization of the theory. Be that as it may with arithmetic, it is certainly true that it becomes more and more difficult to use merely sensory motor intelligence, the higher we move into the study of mathematics. Skemp designeda set of tests in which he was able to measure the use of the reflective intelligence by presenting the subjects with cer- tain exemplars and non-exemplars of concepts in the classical way from which they would then extract certain concepts and these concepts would then be used in certain operations which would also be taught non verbally in the test situation. The examination of combinations and reversals of the operations used were taken as superordinates which could not be learned already. The results in these tests

were then correlated with mathematical achievement. In Skemp’s test of reflec- tive intel I igence, he used three stages. Exemplars and then non-exemplars were given of a concept. Then test figures were given to subjects who had to decide whether these figures were or were not exemplars. Then operations were introduc- ed, again non verbally, by the use of figures. For instance, operation C would be turning through a right angle in the clockwise sense. This was not explained

verbally but was illustrated by examples. Similarly operation J was a reflection in a horizontal line. This was also illustrated by three examples. Operation N for example operated on a figure consisting of two different kinds of things and the number of the one kind had to be replaced by the number of the other kind, and the other way round. That is, for example, two crosses and one circle had to be replaced by one cross and two circles.

In the next part, figures were given and subjects were asked to operate on these figures by means of the operations that they had learned. Then in the last part, combinations of operations were asked for. That is, for instance, “Do an operation C first followed by an operation J on the following figures”. To make sure that the superordination which arose out of a combination of operations was what was being tested. Skemp made sure that the operations were in fact un-

derstood at this stage. That is they were explained to the subjects in case they

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had not learned them before this part of the test was given. In the concept forma- tion part, the learning of the concepts was followed by learning or “double” concepts. That is, the exemplars were only exemplars if two criteria were satis- fied. The subject had to sort out that if only one criterion were satisfied, the instance was not an exemplar of the concept, and so he had to reflect on whether in the particular instance prior consideration, only one or neither or both conditions were satisfied. Only in the latter case was it considered an exemplar of the con- cept . This is, of course, the Bruner type of conjunctive formation and it is also what happens in the Vigotsky blocks or Hull’s Arithmetic blocks when conjunctive concepts are being investigated in learning stimulations in the kindergartens.

The correlations between mathematical ability and the test scores were as follows. X was called the mathematical criterion; T the reflective activity on the concepts, that is, ability to handle conjunctions of concepts in this case; U was taken to be the ability to use the operations introduced, that is, not requiring any reflective ability; and V was the reflective ability on operations, that is the ability to combine an operation with another operation. So therefore, it was ex- pected that correlations with the mathematical criterion should be higher between T and X, and between V and X than there would be between U and X. It was hypothesized that it would be highest between V and X, as that was the highest level where operators were being combined into other operators. The correlations for fifth year Grammar school children were as follows: VTX was .58, VUX was .42, VUX was .72. The test was also given to fourth year Grammar school children, and the validation only took place a year later when these children be- came eligible to take their general certificate of education examination. The correlations were remarkably similar. VTX was .56, VUX was -48, and VUX was -73. The reliabilities of the tests on a split halves criterion ranged between .76 and .95.

It would seem, therefore, that it is amply validated that reflective ac- tivity is certainly connected with the ability to solve mathematical problems such as are encountered in Secondary School mathematics examinations. It is, of

course, not established that reflective activity is not used when handling arzthme- tical operations. One excellent feature of this experiment is that the testing was not done in any verbal sense, but the “possession of a concept” was regarded as established if this concept could in fact be used. The reflection on concepts was regarded as having taken place if such reflection did in fact result in an operation being performed which without such reflection could not have been performed. Verbal difficulties of understanding were eliminated as far as possible.

An immediate question that arises out of this study is whether reflective intelligence, if it is indeed a separate entity, can be trained. It would seem likely that one learns to reflect by reflecting and it is also fairly clear that in most of our educational steps, not a great deal of reflection takes place because all the reflection is done by the teacher. The results of his reflections are prepared in the

form of lessons which the children learn. It is somethingofa miracle that in spite

of this, children do learn to reflect. It would seem that this ability to reflect is a

necessity for the prosecution of life and probably the mathematical situations are not the only ones which encourage its formation and development.

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It is quite likely that if care were taken to induce children to reflect carefully, on what they were doing, much more rapid development of this reflect- ing activity could take place than does now. Recent researches in the use of lo- gical materials to teach young children logical relationships would indicate that this is so. The reason why young children do not use the reflective methods is be- cause they can get by without them. There are no situations important to them, which they cannot handle quite as effectively by means of childish logic. Piaget describes this as operating on concrete situations and using juxtaposition rather than implication, and SO on. If situations were created in which children were h’ hl rg y motivated to engage which for their efficient prosecution required the re- flective activity, it is quite possible that such reflective ability would thereby be helped to develop in such children.

2.9 Robinson’s Contribution

Robinson has attempted to construct yet another theory of mathematical “ability” , perhaps better termed a theory of mathematical learning, through the study of the time taken and difficulties experienced by children in coping with axiomatic structures. The axiomatic structures studied by Robinson have so far been geometrical ones, but the techniques employed by him are in all probability transferable to the study of other types of axiomatically based structures.

Practical and objective ways of measuring the length as well as the dif- ficulty of deductive arguments are proposed. If the length of a deductive argu- ment is N, and it is the time taken for a student to understand (not invent I) the argument, then it appears that the equation

t = kl ek2N

where kI and k2 are personal constants, will predict the time taken with reason- able accuracy.

It is conjectured by Robinson that there are several distinct mathemati- cal abilities. The first task of a student when confronted with an axiomatic system, is to understand it. By this is usually meant that the elements of the system have in some sense been fixated, more often than not one would suspect through the attach- ing of visual images to such elements, further that some form of permanent visua- lization (even if only of a symbolic character) of the primitive and formal operations has taken place. It is also necessary to remember some points of reference, such as some key theorems, and be aware of the possibility of carrying out formal opera- tions on these points of reference. The ability to do the above is referred to by Robinson as Level 1 Ability, and it is the time taken to engage in activities requir- ing only the use of this ability that is approximately measured by the above formula.

It appears that there is not absolute upper limit to the exercise of this ability, except that the time taken increases exponentially with the length of the argument, and so for practical reasons people engaged in such activities will even- tually have to discontinue. There are, it appears, some factors on the other hand which would produce artificial limits to the learning situations. Robinson quotes

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the following :

1) insufficient time given to learn the key points of reference

2) sufficiently long gaps in the presentation

3) immediate competition by peers, leading to guessing.

The above activities do not involve the solution of problems. There is a problem in that there is a gap in the deductive sequence between what is already known to the student and a conjectured relationship in the axiom system. In at- tempting to solve a problem within an axiom system, there are always a large number of choice points, each such choice point leading to other choice points. Under these conditions it was conjectured that each subject would reach an upper limit natural to himself, having regard to the complexity of the problem situation. Having devised a method of measuring the objective difficulty of a geometry prob- lem, such a conjecture became testable. It was in fact found that most students very quickly attained their natural upper limits in this kind of activity, and that if problems were given them which were beyond this upper limit, they systemati- cally engaged in illogical activity, ignoring the rules of the system.

The type of ability made use of in this kind of problem solving behavior was termed by Robinson Level 2 Ability. He also conjectured the presence of Level 3 Ability, which he identified with creativity, but admits that our know- ledge to date is insufficient even to formulate conjectures on the functioning of this ability.

Open Problems - -----------

If the exponential equation describing the functioning of the Level 1 Ability can be substantiated in fairly general terms, it would be possible to de- termine the values of kT and of k2 for each student and so set realistic goals for each student. Upper limits could likewise be determined for each student, and so much individual and seemingly inevitable frustration could be avoided. For ex- ample, a small value of k, would mean fast understanding of short pieces of reasoning, and there is already some evidence that small kT values are accom- panied by high intelligence test scores. There appear to be only low correlations

between KT and k2, but as yet no high correlates of k2 have been found.

Turning to Level 2 Ability, Robinson states that he has so far been un- able to find any usually considered cognitive variable which correlates highly with Level 2 activity. This might mean that problem solving ability might be more of a personality factor rather than a cognitive factor. Robinson suggests that in Level 1

activity there is continuous reinforcement by the continued understanding of the different parts of a chain of argument, and so the motivation is kept up more than

in problem solving, i.e. Level 2 behavior, where the reinforcement only comes

at the end, when the problem has been solved, and the uncertainty al iminated . mere does the high motivation of the problem solver come from, which does not

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need reinforcing until the end ? It is suggested that there might indeed be such a thing as a “mathematical personality” . Perhaps the most important immediate application of the findings on Level 2 activities is the apparent existence of an upper limit for the resolution of uncertainty. If this limit is habitually passed in the educational situation, very little profit of an analytical nature will accrue from the study of axiomatic systems such as Geometry.

Perhaps the most exciting, be it the most intractable problems, sur- round the more elusive Level 3 Abiiiif. It must be said that not a great deal is known about the functioning of “creativity” besides the fact that it undoubtedly comes into mathematical thinking.

2.10 Dienes’ Contribution

Another strand in the researches that have been taking place into the background of mathematics learning has been provided by Zoltan Dienes. This research splits roughly into three parts. The first part took place in Leicester in the late fifties where the different types of thinking of children were investigated. Constructive and analytical thinking were measured and it was found that children were very much more constructively than analytically inclined and that construc- tive thinking came much before analytical thinking. On the other hand, analy- tical thinking could be encouraged once construction of concepts had in fact ta- ken piace. Some connections were also established between different types of concept formation and children’s personalities. This led to the formation of the Leicestershire Mathematics Project, where construction was given the promi- nence whidh was its due in children’s mathematical thinking.

The success of this project resulted eventually in Dienes joining Bruner at Harvard, and in the initiation of the Harvard Mathematics Learning Project, which has already been mentioned. Much became clear during this year of joint work by Bruner and Dienes, although most of the conclusions that were drawn were somewhat provisional. There was no time to run any controlled experiments. The experiments were mostly of the Natural History type -- that is, they were taking down observations of actual mathematics learning in situ. Carefully prearranged learning sequences were administered to children. Each child had its individual observer and tape recordings and photographs were used in order to keep as true a record as possible of each child’s learning. The sessions were reasonably free ; children were not compelled to do any mathematics, but were allowed to play, rough-house, or do mathematics as they pleased. A con- siderable number of conjectures could be made, as a result of these sessions, on the connection between ploy and mathematics learning. It was reasonably evident that the initial manipulative play had to precede the rule-bound play which was seen as a more sophisticated form of play, better described as games, before much mathematical learning could take place. But it was found that without the preli- minary play, not much rule-bound ploy could usefully be ployed. The manipula- tive play is approximately, perhaps, what Bruner would have called the enactive stage. After which followed the manipulation of images. This was also investi- gated in some parts of this project by telling children stories which had a mathema- tical structure and children were asked to manipulate the situations in the stories

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and so solve mathematical problems. Such stories were tailored to mathematical structures, such as group and vector spaces and the results were very gratifying.

A distinction was made between abstraction and generalization, much in the same way as Bartlett made his distinction between the two kinds of genera- l izat ion. Abstraction was regarded as a constructive activity, whereby the child would isolate the class of events and so would form an actual class by recog- nizing a common property to the members of this class. This was class format ion. Generalization was regarded as a class extension; that is a class having been formed, children at times realized that there was a more extensive class with the same or similar properties. So general ization corresponds, in Bartlett’s terms, to trying to perceive a goal in the general direction in which the thinking was taking place and abstraction was the high order generalization, as Bartlett puts it, where the common essence was being extracted from o number of different situations.

In both the Leicestershire Mathematics Project andthe Harvard Mathematics Learning Project, the multiple embodiment principle was employed

SO that artificial abstraction exercises would in fact be studied. “Similarity exert ises”, in other words, artificial abstraction exercises, were also interspersed in the sessions SO that it could be tested whether the structures were in fact trans- ferring from one embodiment to another. Since it was thought that abstraction was a constructive process, being a building up of a class out of its members, this was assumed to be the more natural process for children to engage in than genera- lization which was a much harder process because it made use of the inclusion relationship between two classes, and so appeared to be a logical or analytical process. This was born out by the experiments; abstractions took place with greater ease than generalizations, although some generalizations were not able to be transferred to other situations. These cases where the generalizations took place on one embodiment only, resulted in what was called a perceptual block. This is a danger which could occur if only one kind of mathematics material is used in learning situation. It wos, therefore, thought that the use of one ma- terial, either Cuisenaire, or Katherine Stern, or Multibase Arithmetic Blocks, or any other type of materials would be not as good as using several materials whereby an abstraction could be encouraged.

A further aspect of the Harvard Mathematics Learning Project was the study of symbol manipulation. Once the children had reached a certain stage in their abstraction, they were able to manipulate symbols. They had their tables painted with chalk-board paint and were each provided with chalk so that they could doodle away their thoughts on symbolism of their own choice on their own desks. This, in one or two cases, resulted in some creative use of symbo-

lism. In the case of one child at least a mathematical problem relating to identity of quadratic functions wos turned by the subject into a problem on functional re- lationships in the quadratic functions, and the only reason given by this subject for this change was that he thought this was a more interesting problem. This particular subject was only eight years old. It is seen, therefore, that the Bruner stages of Enactive, Iconic, and Symbolic could be gone through in a reasonably short time. The time taken by this particular subject was less than five weeks.

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Bruner in these sessions tended to emphasize the importance of the symbols, where- as Dienes tended to emphasize the importance of the experience leading to the concept formation, which then was able to be expressed in the symbols. The dis- agreement was expressed in philosophical terms by Bruner suggesting that in the beginning was the word, while Dienes suggested that in the beginning was chaos. The chaos had to be turned into order through manipulating whereas, according to Bruner, the word was there all the time and merely had to be understood, and the road towards symbols was almost inevitable.

Of course, in some sense this is the classical quarrel between what we might call the Realists and the Idealists, that is the Aristotelians and the Platonists. The Realists believe in the reality content of ideas. Ideas according to Realists are merely distillations, a processing of information received ; whereas the Idealists believe that the structure is there already in the world, only to be found. Possibly Bruner would not agree himself to be branded as an Idealist, but nevertheless his emphasis on symbols as opposed to the experience is probably tending to make him rather that way than the other.

It was necessary after this stage to get down to some experimental work on the actual processes by which the abstraction of mathematical ideas and forma- tions of structures, transfer of structures, and so on, took place. Experiments were designed jointly by Jeeves and Dienes in Adelaide as a result of the work at Leicester and Harvard, which is reported in Thinking in Structures, and in the work entitled Generalization and Embeddedness in Transfer (see the Biblio-

graphy).

In the first part of this work, mathematical groups were used as an ex- perimental paradigm and it was presented to the subjects in the form of a game whereby the subjects and the experimenter played cards and the next card played by the experimenter was decided by the previous card played by the subject and that played by the experimenter. These two cards together determined the next card played by the experimenter and the game consisted in the subject predicting what the experimenter was going to play. He went on predicting until he got the whole idea of which two cards together, that is subject’s and experimenter’s cards, together produced which particular card played next by the experimenter. So the binary operation was embodied in the form of a card game. This enabled the ef- fect of relationships between structures on learning to be studied. The efficiency of learning a simpler or a more complex structure first which were embedded in each other was tested, the result being that it was in every case more profitable to start with a four element group followed by o two element group than to start with a two element group followed by a four element group. It was also found that in the case of children the more symmetric type of structure was more easily learned than the less symmetric ones. Symmetry of a structure was defined by the number of automorphisms, that is self mappings, in the structure. The strategies of at- tacking the problem were also investigated and operational pattern and memory strategies were distinguished. The operational strategy is one whereby the sub- ject would regord the card he had played as operating on the card that the experi- menter had played. The pattern strategy was regarded as the formulating of the whole structure in terms of areas of the defining matrix of the group,

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and the subject and the experimenter were not regarded as being on different levels. The memory subjects were those that exhibited no particular tendency one way or another towards operotional or pattern play and insisted at the question- ing at the end of the games that they had merely remembered the combinations. It was found that adults tended to use operational strategies more than children and on the whole it was found that by far the most popular strategy with both children and with adults was the pattern strategy. More children used memory strategies than adults.

The next part of the study concerned itself with the extension of the problems to more complex structures. The effect of recursion or generalization as opposed to the effect of embeddedness would have on the whole a more beneficial effect than pure general ization. The generalization series were tested on groups of three, five and seven elements. The embeddedness was tested on groups with three, six and nine elements. Both the cyclic group with nine elements and the nine group which is a direct product of the three group which is a direct product of the three group by itself were used. In the latter, of course, there are a very large number of automorphisms. These researches are only in their initial stages and it would probably take several years for the details of the learning of struc- tures to become clear. What is evident is that it is not efficient to start from a simple structure and pass on to a more complex one. It is conjectured that this is because children are less able to generalize thon to abstract. That is, a child is more able to construct a three group and generalize this to a six group, in which case he has to undo the closure he has made on the three group. Therefore, there is a certain amount of negative transfer when he is trying to generalize from the three group to the six group. On the other hand, having learned the six group first, he merely hos to look at his structure and discover that there is a three group in it.

Another result is on the symmetry question. That is, subjects will predict very much more often in a less symmetrical structure as though the struc- ture were more symmetrical, than when learning a symmetrical structure predict- ing as though it were less symmetical. The first comparison was made between the Klein group and the Cyclic group with four elements. In the learning situation in which the Cyclic group is being learned, during the learning, there were found to be twice as many predictions as though they were learning the Klein

grouPI in spite of the evidence for the Cyclic group than the other way around. The Klein group has five non-trivial automorphisms whereas the Cyclic group has

only one. lt would seem that there is a gestaltist kind of tendency towards symme-

try. lf this is so in general, then this would have a very important application on the choice of structures that we give children to learn when we start them with a new subject. Up to quite recently the education of children wasnot able to be based on anything but hit-and-miss notions about what we just happened to think was a suitable way in which to put things, or of a suitable set of concepts or cur-

ricula through which to put them. Through the researches taking place today, it is hoped that within a few years it is going to be possible to say with o certain a- mount of scientific authority that certain sequences are more effectively learned in certain orders than in certain other orders.

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There has been o great deal of ad hoc classroom experimentation, some not very purposeful, some extremely inspired. Under this heading would be classed some of the work done in the classroom by people such as Robert Davis, David Page, and William Hull. The “existence theorems” proved in these in- vestigations are now leading to the possibility of asking the sorts of questions to which answers will in all probability be obtained through the carrying out of controlled experiments. This is where the professional psychologist moves in, once the adventurer in the mathematical classroom has unearthed interesting men- tal processes for him to investigate and bind together in causal relationships to each other. As an example of such work, we shall give a short account of the work of William Hull who has been experimenting with the possibility of getting very young children to apply logical thought processes to game situations which they get highly motivated to engage in.

2.1 1 William Hull’s Contribution

William Hull, working at Shady Hill School in Cambridge, Massa- chusetts, and later working at the Elementary Science Study of the Educational Services Incorporated in Watertown, Massachusetts, was the first to show in concrete situations, that is with actual kindergorten age children that some quite high powered logical thinking could take place as a result of gomes being devised which children enjoyed playing and which involved for their successful playing the use of logical principles. Some of these games involved three or even four way Venn diagrams, that is, would involve the simultaneous consideration of three to four separate attributes and their negations as applied to one and the same object. In some cases it was found that upon first explaining the task, some four or five year old children put the attribute blocks out in the right places without making any errors, or perhaps just one or two out of thirty-six possible pieces.

Various other games were devised, such as the twenty question game which is reminiscent of the Bruner type of concept attainment, where children were simply asked questions about a particular block thot another child has thought of. And by a process of elimination they would eventually find in the least number of questions possible, the block that the other child has thought of. The attributes that we used were color, shape, and size; sometimes thickness was used as well. There were two sizes, and there were sometimes four colors and four shapes, sometimes three colors and three shapes, as the case may be. These materials are constantly being developed, to allow children to expand into greater and greater degrees of complexity.

Dienes in Adelaide has olso been working with similar types of attri- bute blocks which are really a slight extension of the Vigotsky Blocks but used in learning as well as in testing situations. Dienes extended Hull’s work in the classroom to different age groups as well as introducing it into the Papua, New Guinea mathematics project as a part of a new curriculum for native children. The extensions consisted mainly in adding disjunctive games to the conjunctive ones, which were then also turned into implication games. From a disjunctive

pile, implications had to be deduced by the children. For example, if in a pile

were put all the blocks which were either red or not squares, then if any square were picked out of this pile then it was bound to be red. But if a rred one was picked, it was not bound to be square. So the non-invertibility of implications was one of the aims of this learning as well OS the forming of the contra-positives. That is, “If square then red” but you could also say, “If not red, then not square”. That is if you picked one out of the pile, one which was not red, it was bound not to be a square because had it been a squore it would have been red.

At the eight-year-old level it was also tried whether the work on dis- junctions and their implications was oble to be transferred to verbal situations. This was difficult at the six and seven-year-old level, but became very much easier at the eight-year-old level, Children are asked to think of “if-then” sit- uations in their own experience and see if their “if-then” situations were always true, what other “if-then” situations would follow from them.

It would seem, therefore, that according to the results of these experi- ments, whot Skemp has called a reflective activity is certainly present in younger children, if only we give children problems which are easy to solve but which they cannot solve without such activity. This goes in some sense counter to Piaget’s insistence that this kind of logical analytical thinking does not begin to take place until the beginning of puberty. Of course, Piaget’s developmental theory does not necessarily mean that the age groups have to remain constant. The development does have to go through certain stages and it may be that by ad- ministering certain particular structuredsituations to the children the rapidity with which children pass throughthessdevelopmental stages might greatly be in- creased.

2.12 Open Problems in this Area

As has already been remarked, there is an immediate need for a body of sustained experimental work on the learning of complex mathematical concepts by children. The present literature on learning is mainly restricted to quite simple, elementary mathematical concepts and does not begin to cover the range of con- cepts planned for the elementary school curriculum now in use or the range of the future.

With more experimentation into “what il

oes on” there wilt arise a need to coordinate the new findings into more or less co erent theories. These theories will have to be much richer in structurethan the present ones, so as to be able to account for what goes on in mathematical learning. Present theories of

learning, whether behaviorol or cognitive in orientation, do not provide an ade- quate scientific explanation of any appreciable portion of the child’s learning of school mathematics.

There is need for more experimentation in children’s perceptual leor-n- ing with particular reference to the learning of geometry. The present studies in learning and thinking on these matters are restricted and inadequate from the

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standpoint of their relevance to questions concerning the learning of geometry. It was noted by many participants of the Stanford Conference that one advantage of such studies would be that they could be mode less verbal in character and, there- fore, could get at the perceptual processes more directly than can be the case with studies concerning the learning of abstract concepts not capable of visual representation in any easy way. It wos also noted that the detailed study of perception has until recently been somewhat neglected in the learning literature generally. This neglect is independent of questions of relevance of such studies to the mathematics curriculum.

There is need for more studies on transfer in the classical manner, with particular reference to positive or negative transfer in moving from one con- cept to another, where such concepts are in some prearranged relationship to each other. Work on such problems is progressing at Adelaide University, and a start has also been made at the University of Minnesota. There is the fundamental question of whether transfer depends more on the specific or more on the structural similarities and differences between structures. The eventual, even partial, solution of this problem will enable curriculum planners to do some more intelli- gent sequencing of topics to replace the present hit and miss variety of curl iculum planning.

2.13 Enquiries of an Empirical and Functional Kind

Brownell carried out a large number of empirical studies, possibly the most important one of these being the last one, entitled : “Arithmetic Abstractions, The movement towards conceptual maturity under different systems of instruction”. A short report of this work is, therefore, given, the reader being again referred to the references at the end of the report for other works of similar nature.

2.14 Report of a recent Study by Brownell

The purpose of the study was to find what progress children in British Infant schools make toward abstraction and maturity of arithmetical concepts after being exposed for three years to eithet- the Cuisenaire program, the Dienes pro- gram, or the Conventional program.

By interview technique data was collected from 478 children enrolled in Scottish schools and from 928 enrolled in English schools from responses on 8 number combinations, 2 in each operation, and on 12 verbal problems, 3 in each operation. Interview data obtained from questioning the child gave information on the child’s thought processes OS he dealt with each number task and observation and evaluation of responses in terms of such factors as correctness and quickness gave information on quality of performance.

Data was obtained from the Scottish children on the Cuisenaire and on Conventional programs which were the two programs widely used in Scotland at that time. In English schools, the Cuisenaire and the Conventional programs were

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widely taught for all three years of the Infant schools and the Dienes program from the middle of the second year in school through the third year.

Findings from analysis of data revealed that in Scottish schools, the Cuisenaire program was considerably more effective than the Conventional pro- gram in promoting conceptional maturity. The evidence indicated just the oppo- sit in the English schools. The Conventional program was far more effective than either the Cuisenaire or the Dienes programs in moving children to steadily more mature (meaningfully abstract) ways of thinking. The Cuisenoire program was slightly more effective than the Dienes proslam. Data on quality of perfor- mance followed the same pattern as the data for conceptual maturity in both Scottish and English schools. There was little difference in the ability of sub- jects in the three groups to explain solutions to verbal problems.

The Dienes program was most effective in meeting the social aim of mathematics as evidenced by the superiority of the Dienes subjects to solve prob- lems. The Conventional program showed a weakness in this respect.

In discussing the results Dr. Brownell cites evidence to show that the quality of teaching account for the diverse results found in Scottish and English schools. In each country the better taught program was most effective in promot- ing conceptual maturity. Analysis of responses indicated that all programs were weak in teaching the ability to explain solutions of subtractions and division problems.

Dr. Brownell grants that the study did not measure the totality of con- ceptual maturity of any program. He readily admits that the goals of the three programs vary considerably and that the interview items measure only a small part of each total program. Dr. Brownell feels the following implications for the teachingof mathematics to children in the primary schools in the United States are justified on the basis of his research :

1) No drastic changes in the teaching of mathematics of the early grades is indicated. Rather continuous evolving improvement of the program is in order.

2) Enumeration and counting of sets of objects should continue as beginning methods of obtaining concepts of number and to serve as setting from which the combinations can be dis- covered.

3) The Dienes Multi-Base-Blocks employed later and more slowly can contribute greatly to the intelligent learning of arithmetic.

4) The measurement approach (Cuisenaire) and the Conventional program should be combined in programs for the early years. Modification of the Conventional program in the direction of the Cuisenaire program would be advisable.

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__

5) To gain intelligent mastery of the addition and sub- traction facts by the end of grade three and of the multiplication facts by the middle of grade four is a worthy and reasonable goal as a fundamental part of a mathematics program.

It will be noted that the conclusions from an empirical study will vary according to the aims one considers mathematical education to have. The Brow- nell study quoted above was clearly arithmetically oriented, and the conclusions are influenced by this.

2.15 Report of a Recent Investigation into Methods of Teaching Arithmetic in the Primary Schools in England and Wales

The aim of the investigation carried out by the National Foundation for Educational Research in England and Wales was to evaluate methods of teach- ing arithmetic that have recently been devised and that purport to bring about a better understanding of, and more favorable attitudes toward, arithmetic and ma- themot its in general.

The framework of this investigation was provided by a theory of cogni- tive learning in which teaching method is conceived both as a means of imparting information to the child, and also as a meons by which the information so impart- ed actually affects more or less permanent and increasingly adaptive changes in the child’s internalised conceptual structures (codes) and information-processing strategies (coding routines). Effective teaching - consistent with the teacher’s own aims - is that which enables the child to develop efficient and enduring cod- ing routines. Learning so carried out is strongly motivated positively; negative emotionality is created when existing coding structures are incapable of dealing with the immediate input.

Current methods of teaching were categorised in terms of the tech- niques used to construct and to implement coding routines. Traditional methods are defined as those that stress rote learning of symbolic data and in which little attempt is made to illustrate and to make understood the logical structure of arith- metic. Such methods are characterised by much drill, mechanical work, extrin- sic reward systems, an authoritarion mode of presentation, and so forth. Struc- tural methods are those in which logical structures are meant to be illustrated by reference to concrete models or analogies. Such methods fall into two kinds: uni-model, such as the Cuisenaire and Stern, in which one set of concrete models is used (rods or blocks) and which provide the child with a provisional, concrete, coding system; and multi-model such as the Dienes materials, in which several qualitative different concrete analogies ore used and out of which the child at least has the opportunity of constructing an abstract coding system. Motivational methods are those that do not use specifically structured physical models as such, but rely upon “real life” environmental situotions and that em- phasis arousing and maintaining the child’s interest in number situations. Ideally (although this did not appear to be the invariable practice) structural and motiva- tional methods should be utilised in a permissive and stress-free atmosphere.

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Several predictions about the effect the different method groups would have under varying conditions were made. The main predictions were that children taught traditionally may be superior in mechanical arithmetic but those that were taught by both kinds of structural method, and especially multi-model methods, would be superior in terms of conceptual understanding of arithmetic and of attitudes toward arithmetic. Motivational methods were expected to be valuable in the infant department but not in the junior school. Further it was ex- pected that uni-model methods like the Cuisenaire would be most suitable for more intelligent children, and multi-model methods such as the Dienes for all children but especially for the less intelligent,

In the major part of the investigation, involving over 4,500 ten year olds attending primary schools throughout England and Wales, traditional uni-mo- del structural and motivational methods were compared with each other in terms of performance on mechanical, problem and concept tests, and of emotional reac- tions to arithmetic as ossessed by a questionnaire. Method interactions with in- telligence and with certain school characteristics (such as streaming, amount of formal work taught in the infants’ department) were also investigated.

As predicted, the best uni-model results came from children of the highest intelligence (this was limited to boys only), otherwise there was virtually no difference between uni-model and traditional performance and attitude scores. Motivational methods were as effective as the other methods when the former were used only in the infant school, but the consistent use of motivational methods throughout the infant and junior departments produced very poor performance scores.

A multi-model method (Dienes) was investigated in only one school but the cumulative effects of the method were investigated over a two-year period. The children in this school were matched individually with control children who had been traditionally taught. The Dienes-taught children were found to be no different from the controls in verbal ability but were significantly superior to the latter in mechanical and concept arithmetic, and also had much more favorable attitudes to the subject. As predicted, dull children were found to derive greatest benefit from the method. A significant (predicted) interaction with extent of use of the method and sex was also found: boys made steady pro- gress with the method at each year of testing whereas girls at first made worse pro- gress than the controls and only after c further year’s experience did they show marked improvement (which more than made up for their initial loss and which, in fact, finally equalled to the boys’ overall improvement).

This general pattern of the results fitted the theoretical assumptions very well, and within the limitations of the survey type of research and subject to confirmation under more rigorously controlled condition?, the following con- clusions seem warranted :

1) Uni-model methods (the Cuisenaire and the Stern) do not seem to provide the conditions necessary for a conceptual understanding of arithmetic any better than those provided under already existing traditional schemes; neither do they appear to bring about emotional attitudes to the subject

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that are any more favorable than those created under traditional methods. Although there was some evi- dence that the provision of a single concrete coding system was useful computationally, this kind of structuring does not appear to be radical enough to lead to the development of understanding except in those favorable cases of boys of high intelligence.

2) The cultivation of generalised mathematical coding systems, and thus of genuine understanding, seems to depend, in average and particularly in dull children, upon experiences that are best provided by multi-model methods, i.e. those in which a variety of perceptually and qualitatively different but logically isomorphic experiences are provided.

2.16 Open Problems in this Area

The aim of the empirical studies should be to fill the gap between the practical problems of mathematids education on the one hand, and the theoretical and systematic studies on the other hand.

One problem needing empirical investigation in the classroom is the one of how to handle individual differences in the classroom. How can we apply in the practical classroom situation what we know about individual differences ? How can classroomsaccommodate the demands of individual differences ? The participants of the Stanford Conference felt that the studies to date had not given enough guidelines on this problem.

More studies in motivation are needed. How does motivation vary in strength and kind from oge group to age group ? A type of motivation which will help greatly at one age group might be quite unsuitable at others. Empirical studies could be extremely useful at this point.

There is clearly a great need to extend the Brownell type of systema- tic study to the investigation of the learning of the new type of concepts that are being introduced into the curricula in different parts of the world; there is also need for deeper studies of more traditional topics, such as mental arithmetic, and response latency times.

Experimental studies of behavior in the face of the introduction of new curricula with well defined behavioral criteria would also be a welcome aid to the planners of such curricula.

Much more needs to be discovered on the role of sensory and motor modalities of learning, in particular how the situation varies in the case of GUI-

turally deprived children.

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2.17 Testing and Evaluation

2.171 Some Considerations on Testing

A program of testing is directed at one or more of the following groups:

1) The individual

2) A collection of instructional materiols (a curriculum)

3) Quality control of 1) and 2)

All testing programs are related to a specific collection of objectives. These objectives may relate to decision making devices concerning population screening, the acquisition of a specific collection of facts, or the acquisition of special behaviors. Cronboch’s discussion of “Course improvement through evalua- tion” is especially recommended.

Many testing programs appear to foster a narrow set of objectives. This is especially observable among those programs in which there is testing for in- dividual student acceptance or rejection for entrance into the next educational sequence. instruction is often reduced to the repetition of those items which appear most frequently in the testing instrument. The learning-testing environ- ment created by such a program employs repetition and mimicry as the measure of success. Success with the instructional materials often becomes confused with a large number of correctly repeated responses. If the instructional program is to undergo a change which is more than momentary under these circumstances, then the image of success as well as the form of the assessment instrument must change; conversely, if the assessment of success is to undergo change, then the instruction- al program must be altered. The location of effective entrance into this closed network is not the critical aspect. It is vital, however, for one to recognize that the success of any alteration in either aspect is dependent upon a subsequent change in the other aspect.

The extent of this change may, of course, be a nominal one -- altering the items on species names from one plant to another. Or the change may be a monumental one -- a choice of instructional materials followed by an ap- praisal of behaviors independent of the materials selected.

The testing programs should be adapted to the particular country and culture. The instructional materials associated with recognizing the commonly observed vegetation for African students should not necessitate the importation of the vegetation common in England. Nor should the assessment instrument include items related to English plants. Such events may well be observed among those testing African students simply because the original English instrument contained such items.

The specification af desired outcomes (the objectives) of o curriculum or the effects of a particular collection of materials should be independent of the

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language common to those materials. In the test, the objective of materials or progr-urns should be stated in behavioral language which in turn enables the assessment of those obiectives to be written in the language of the particular in- structional materials used for each individual. The unifying frame of reference has its or igin in psychology rather than one special language of a given discipline.

There is available in the literature a number of attempts to specify this more general psychological framework. Of special interest are those of Bloom with reference to his efforts in the construction of a taxonomy of educational ob- jectives and the task analysis strategy as evidenced by the learning set and search set hierarchies of Gagne. However, the theory of the testing, as it has develop-

ed, is almost wholly oriented toward grading individuals. Other uses of tests still lack a basic philosophy and the consequent determination of optimal (or nearly optimal) test-making and scoring procedures.

If one does not assume that the assessment procedures are restricted to classifying individuals, but that the measure of the group is of some interest, then a testing program is free to adopt quite different procedures from those pre- sently employed. For example, it is possible to obtain valid, reliable estimates of group performance by administering only a sample of the assessment items from a given instrument to each student. It is of interest to note that a strong case can be made against any kind of formal evaluation.

There is in process the International Study of Educational Achievement, directed by Bloom and other-s in several countries, whose aim is to establish means by which children of various cultures and backgrounds con be measured for edu- cational proficiency on an international standard. It is to be hoped that evidence of these means will become available in the near future.

2.172 Attempts at Testing Understanding

1. Some American Attempts

Dutton (1964) reports several attempts by American investigators to relate performance on so-called tests of understanding to performance on more con- ventional tests. A brief account of some of these will be given.

Brueckner (1934) found that a quantitative relationships test correlated only -576 with a computation test, and .661 with a problem-solving test, when administered in the fourth and fifth grades. Spainhour (1936) likewise found a far from complete correlation between understanding tests and convention- al tests of computation and problem-solving. Dutton (1964) similarly com- plains that his test of understanding correlates by no means highly with the California Achievement Test reasoning or fundamentals test.

It is not entirely clear why these investigators should be so concerned at the levels of correlation they dicovered. It is difficult to find justification for assuming that their understanding tests were comparable with the tests of more conventional kinds. Normally, when we hope to make a meaningful statement

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about the correlation obtaining in a certain context between two tests, we have information about their correlation in a reference context. This infor- mation gives us some idea of the significance or otherwise of the correlation found.

Another set of studies reported by Dutton concerned the relatively slow de- velopment of arithmetical understandings, compared with the computational skills acquired by children. Glennon (1948), for example, developed a test of arithmetical understanding based on the concepts underlying computa- tional processes taught in grades 1 - 6. He found that in grade 7 only 12.5% of the understandings had been grasped on an average, and that even in grade 2 no more than 37% had been grasped. Flournoy (1963) similarly found that the understanding of underlying principles lagged behind computational acti- vity. Rapaport also devised a test, and, it seems, arbitrarily assuming that a score of less than 50% was inadequate, he also decided that under- standing was not present to support the computation techniques being learnt.

Perhaps it is rather presumptuous of these investigators to assume that their tests did measure what can be called “understanding”. Without the develop- ment of such tests through use in a variety of circumstances, it is difficult to know whot meaning to ascribe to them. For example, we could say that the unsuccessful children tested did have understanding of the concepts that un- derlay the computation that they were able to perform, but, perhaps through lack of intelligence, could not apply this understanding in the particular situations presented to them in the insight tests used.

2. The Use of less formal Methods

Buswell (1961) outlined several techniques for assessing the understanding of a child in an interview situation. Weaver (1955) taking his cue from Buswell, tried to obtain information about the level of children’s thinking by asking them to think out aloud. He found this a very effective way of aster - taining understanding and recommended that teachers should use this technique regularlytoobtain this kind of information about their pupils.

Hotyat (1957) interviewed French children between the ages of 11 and 13 in order to discover the difficulties that they found in working arithmetical problems. He found that the most common mistakes were made because of a failure to understand problems from a practical point of view, and an inabili- ty to appreciate the mathematical relationships that held between the ele- ments of the problems.

Perhaps we can put greater trust in the conclusions of these last two investi- gators, but perhaps this is only because their conclusions are rather unam- bit ious.

3. Work by the National Foundation for Educational Research (England) on the Development of Insight Tests

It is relatively easy to define and construct tests of the ability to compute, for this is little more than the ability to perform certain specificable kinds of

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operation - an ability which we can test by requiring the performance of a representative sample of such operations. However, as we at present use the term, “understanding” is more difficult to define and it is more difficult to know when we are managing to test it, for we have not decided upon the precise behavioral implications of imputing it to a performance or upon the kinds of achievement that we can take as its token. An attempt to do some of this spadework is being made, however, and in a series of articles on understanding, Williams (1963, 1964) examines some aspects of its accepted and useful meanings, and some of the aspects of arithmetic that can profitab- ly be “understood”.

Probably, the definition and testing of understanding will follow the course that was taken by those of intelligence : tests seeming to divine what seems to be what we mean by understanding will be constructed, and those of such tests which eventually prove useful, both practically and in relation to our theoretical structures, will form the criteria for a behavioral definition of this concept. However, at present, there are no such commonly-accepted tests of understanding, so, for use in research projects, and examinations, the N.F.E.R. has had to construct its own tests of understanding.

How can we say what these tests measure ? To a certain extent we can expect this to be self-evident, for the test items will certainly be novel to the pupil, certainly involve the appreciation of mathematical principles (two conditions that can be expected to require the exercise of understanding) and, since they do not rely heavily upon computation, are unlikely to be contaminated by this last aspect of arithmetical performance. As a means of ensuring that these items measured what they appeared to measure, a sample of pupils was questioned after performance on the test in draft form, in order to ascertain whether the items had been correctly performed by virtue of an understanding or incorrectly performed by virtue of a lack of it. This is in many ways an unsatisfactory basis upon which to lay a claim for the validity of a test, but at this point in the development of tests of understanding, little more could be done.

The tests that have been developed have been used for three kinds of purpose:

a) Experimental Research

In connection with two projects which have been mentioned else- where in this report, and which were concerned with assessing the relative merits of different methods of teaching arithmetic, two tests of mathematical understanding were developed, The first of these was divided into two parts, the first of which was based on Piagetian situations, while the second of which was what might be described as an “eclectic” selection, of items devised accord- ing to the criteria of generality and freedom from complicated computations that were mentioned above. The second test of in- sight was in a sense diagnostic, for it included items that could be divided into five kinds: for measuring appreciation of equi- valence of groups after division or multiplication; for measuring

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b)

c)

equivalence of groups after addition or subtraction; for measuring appreciation of fractions ; for measuring appreciation of ordinal relations; for measuring appreciation of positional notation. This division of items into specific categories was related to the particular experiment01 interests in the outcome of using certain methods of teaching.

A Nationol Survey of Attainment

For this purpose tests were used at the primary school and secondary school levels. An encouraging feature of the results of one of the insight tests (at the lo+ level) was that where the examinee failed on one item in a set based on the same concept, he tended to foil also on other items in that set. This certainly indicated that a determinant of success on these items is the grasp of a concept that is common to them all.

Standard Tests

For use in such examinations as the 1 l+, special versions of the N.F.E.R .‘s usual tests hove been developed. These have tended to include quite a number of items that can be expected to test insight rather than either the capacity that can be expected to underly performance on the usual arithmetical “problem”, or computational ability. Such tests are proving very popular with Local Education Authorities in England. It is hoped that they may have a backwash on primary school teaching, and that this conduces to the use of teaching methods that will en- courage understanding .

4. Testing of “pre-mathematical ” Concepts

In the course of his experiments on the development of the child’s conceptual processes, Piaget devised several ingenious situations which would reveal whether, in certain respects, children had reached certain levels of con- ceptual maturity. It has been recommended by various authors (see Williams, A ., 1958, Churchill, 1961, and Williams, J.D., 1963) that these Piagetian tests should be used as tests of the child’s ability to understand con- cepts that can be regarded as basic to his ability to cope with arithmetical techniques. In other words, it has been recommended that they should be used as “readiness” tests.

There are powerful objections that can be made to the concept of “readiness” testing, and particularly in this case. Readiness to develop a certain kind of skill or to acquire certain kinds of information must always be relative to the manner in which the skill is to be developed or in which the information is to be assimilated, ond, moreover, to the learning experiences that lead up to this stage in learning. A point of view that is currently enjoiying a great deal of acceptance, is that there is always some legitimate way of introduc- ing any topic at any age (see Bruner, 1962).

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From the Geneva school itself, there comes evidence that the ability to perform many of the Piagetian tasks could be forced to develop much earlier than Piagetians had previously believed possible. Where children have been allowed to develop an appreciation of the principle of 1 to 1 correspondence, it has been found that they ore able at a very early age to behave as if they could conserve discrete quantity (see Dienes, 1961).

The validity of the concept of readiness testing can be questioned in other ways, also. While it is clear that without a grasp of the notions that Piaget regards as fundamental to the understanding of arithmetical operations, a child could not be said to understand these operations completely, it is at the same time possible to assume that he might from some points of view and to some useful extent be said to understand.

Again, what experimental evidence is there that it is injurious to learn techniques of computation before understanding them 3

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PRACTICAL APPLICATIONS OF FUNDAMENTAL RESEARCHES INTO THE PROBLEMS OF MATHEMATICS LEARNING IN SCHOOLS

Chapter 3

PRACTICAL APPLICATIONS OF FUNDAMENTAL RESEARCHES INTO THE PROBLEMS OF MATHEMATICS LEARNING IN SCHOOLS

3.1

In this chapter we shall survey some of the concerted and individual efforts that are now being made in different parts of the world to apply what we know about the processes of thinking and learning to the practical classroom situa- tions as they present themselves to the teacher in charge. In the last section a number of studies were reported, some of which were of a carefully controlled character ; readers will be able to decide for themselves the extent to which the points referred to have been experimentally proven by referring in detail to the literature cited in the bibliography. The work referred to in the chapter is on the whole of a more tentative character, and readers should not imagine that a great deal of strictly controlled experimentation can be cited in support of the view- points put forward by the different types of “mathematics projects” listed below. Many years will elapse before such studies can be concluded and firm results established about the relative efficacy of one type of suggested methodology or even curriculum over another. Th is is bound to be so since the finished product of the processes to be compared is the adult person who has completed his educa- tion. Less ambitious studies are obviously possible, and some such studies have already been carried out. Some of these are also given in the bibliography. It was unanimously agreed by the participants of the Stanford Conference of December 1964 that one of the steps which must be taken to improve mathematical learning in extent as well as in quality is the greater personal involvement of each child in the process of learning through the much wider use of concrete ma- teriols than is now the custom. The situations in which concrete materials are presented should be as varied as possible; it wos stressed by many participants that there are bound to be great individual differences in the time that would need to be spent by different children in each situation, so that a strict sequentially de- termined set of steps would be cr most inefficient method of organizing such con- crete learning situations, at any rate until such time as computer-based learning became economically possible in any numbers. It follows that o great deal of future mathematical learning will have to take place in small groups through joint efforts by such groups at grappling with the problems presented in the situations,

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_I_- __- - - . _ - -

-

or even by children working entirely by themselves. This is not to say that class lessons will not be useful at times or that discussions between teacher and class cannot fulfil1 on extremely useful role. It is suggested, however, that to the interaction between teacher and children, should be added interaction between the children themselves, as well as the interaction between the children and an artificially created mathematical environment which will be responsive to enquiry. It was also emphasized that concrete materials should not be regarded as essential- ly for younger or slower children. Mathematics laboratories should be able to fulfil1 the sort of role which Science laboratories fulfil1 in the learning of Sciences. It is certainly suggested that new ideas should be introduced in situational form, so as to enable children to form the ideas themselves as a result of an organic concept formation process, and not merely as a norrowly defined stimulus-res- ponse pattern.

3.2

In the next section six different types of mathematics projects will be described, the types being given their “names” because of some particularly strong thread or principle which runs through the theory and the practice of these pro- jects. It will be appreciated that the classifications given are largely arbitrary, and that many other equally useful and viable classifications could have been given. It should also be remembered that many programs will “belong” to more than one classification. Also some programs are intended to provide a full mathe- matical program, while others are definitely of a supplementary nature.

In the discussions among the contributors to this report it was agreed that all the mathematics projects which can be classified in one or the other or several of the classifications below, had a certain fairly large number of common features. These common features were thought to be the following :

1. Emphasis on structure, 2. emphasis on “meaning”, “understanding”, and “discovery”

as opposed to rote learning of specific processes, ’ 3. the creation of a positive attitude towards or even emotional

involvement in mathematical activity OS a basis for moti- vating mathematical learning,

4. the content of the curriculum to be broadened far beyond the conventional four rules,

5. induction to precede deduction wherever possible.

3.3

We have already said in the introduction that the practical applica- tion of the researches that we have just adumbrated in the theoretical section can

1 The term “discovery” refers to the learner’s acquisition of knowledge new to him. Here and seriatim, the term is used as “guided discovery”.

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be grouped approximately into the following groups :

1.

2.

3.

4.

5.

6.

Those interested in the basic-set approach; that is, in creating situations where sets, set operations, logic, etc. are learned at the early stages from kindergarten through to second grude.

The arithmetically-oriented approach according to which the main object of primary school mathematics is the understandingofondproficiency at perform- ing arithmetical operations.

The geometl-ically-oriented approach where the number concepts and gene- rally mathematical concepts are tied very closely to spatial relationships.

The symbol-game-or iented approach, where it is assumed that the symbols already carry the mathematical information to a very large extent for children in much the same way as words themselves will carry such information for them. In other words, it is assumed by advocates of this type of approach that symbols can be used as playthings so that through playing with symbols the properties of that which is symbolised can be established.

The science-oriented approach in which, on the one hand, the natural sci- ences and mathematics are regarded really as one discipline. And on the other physical embodiments making use of physical laws are employed to make clear to children the workings of some mathematical laws.

The object-game oriented approach in which it is hypothesised that it is bet- ter to provide a multiplicity of situations embodying the same concept in order to encourage abstractions, and that, if something has to be learned, it is very often better to start not at the simplest case but at a somewhat more complex than the simplest case, in order that the simpler case shall be con- sidered in context by the child.

3.4 Work on the basic-set approach

In the United States, probably Patrick Suppes, Paul Rosenbloom and William Hull, have been operating the fundamental approach and Dienes in various parts of the world was contributing to this.

Also the Creater Cleveland Math Project, the School Mathematics Study Group, and any programs that grew out of the Ball State Math Project ore based on this type of beginning.

What these attempts all have in common is that the workers in these projects believe that the foundation on which the idea of number is based is expli- cit knowledge of the properties of sets, because it is assumed that since number is

a property of sets, the fundamental notions relating to sets must be learned first because number is superordinate to set and therefore number cannot properly be understood without the subordinate concepts of set being understood first.

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The most fundamental property of sets .is thot of membership. Members make up the membership of a set. COI lect ing the members together.

There are two fundamentally different ways of One is by simply enumerating which members

are going by definitions to belong to the set and the other to state certain proper- ties which members must have. If the second method is employed for defining sets, then a universal set must at first be defined over which the property is operat- ing. For example, if the property is blue, unless we are going to consider all the blue objects in the whole universe, which would hardly be a well-defined set, tie would have to consider a section of the whole universe; there may be some pieces of wood in a box or maybe the furniture in a certain room and so on, which for the purposes of the exercise would be considered the universal set and then a property such as blue would define those members of the universe which do possess this property. This procedure has the advantage that the complementary set is thereby immediately defined. The complementary set will be the set of those members of the universal set which do not possess this property, If we use the enumeration approach, then the complementary set is not defined because we would have to include in non-exemplars the entire universe, excluding the members that we have enumerated. Another difficulty about enumeration is that as soon as the membership of sets can consist of abstract ideas such as, for exom- ple, numbers, or vectors, or transformations such as translations, reflections and the like, then these cannot in fact be enumerated because there tend to be more than a finite number of such mathematical entities. And so, for example, even if we try to enumerate the natural numbers and write 1, 2, 3, . . . what we mean by the . . . is that we are superimposing a certain rule of generation and so we are really stating a property. The rule of generation in this case being N to N + 1. So, mathematically speaking, it is difficult to see how we can get away with enumeration definitions of sets for very long.

Having introduced sets and members of sets, the equality of sets can next be introduced. Two sets are equal if they consist of exactly the same ele- ments, not merely of similar elements. Elements that look almost exactly alike are not necessarily exactly the same elements. Here a certain difficulty must be got over and it is the difference between representation and reality. For exam- ple, we might draw the representative of a set and put in a pair of curly paren- theses, a house and a tree, and we might put in another pair of curly parentheses, o house and a tree. Whether we can put an equal sign between these two symbo- lized sets, depends on whether the two drawings are symbols for the very same house and whether the two trees are symbols for the very same tree. And if this is

so for both the house and the tree, then we can put an equal sign between them.

If this is not so, we cannot. But, of course, we could have drawn the tree first and then the house next in one representation and the house first and the tree next in the other representation. So the order in which we present the member-

ship of a set is immaterial. This leads to the idea of equality of sets. It must be stated that some projects do and others do not emphasize the idea of intersections

of sets at the beginning. The idea of equality of sets is a difficult one OS was

found by Suppes in his computer-based mathematics learning laboratory. It is important that children should come across physically realized representation of the “sets” situations : this is likely to accelerate rate of learning and increase the

I ikel ihood of transfer.

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Another very important idea is the idea of subset, which must be dis- tinguished from the idea of a member. “TO be a member of” is the defining prop- erty of a set and “to be a subset of” is stating a logical relationship between the subset and the set, namely that of inclusion. In the projects in which sets are introduced, kindergarten and first grade children play a great number of games with members and sets, picking out sets from their environment and playing games of whether certain children do or do not belong to this set, whether they belong to the complement, whether a certain set is a subset of a certain other set or whether it is not, and so on.

Having acquired the notions of membership in relation to sets, equality of sets and subsets of sets, operations on sets could be introduced. The most im- portant operations are union, intersection and difference, of which a complement is a particular case.

The union of two sets is the collection of all the members of either set. That is, if we have the boys in the class in one set, and the girls in the class in the other set, then the union of the two sets is the set of all the boys put together with the set of all the girls in the class. Now in some fundamental approaches, such as Patrick Suppes’s, the union operation is introduced only for sets with empty intersections. The reason for this is that the set equivalent of addition is the union of’sets which have no common members. C I ear I y if we i o i ned two sets together which hod some members in common, then the number of members in the union would not be the sum of the numbers of members in the separate sets which were u n it ed, so it is considered by the directors of some projects that this difficulty should not be presented to children in the beginning. It might, on the other hand, be argued that if distinctness of sets is to be appreciated by children, then undis- tinctness has also to be leorned. Since adding is based on the union of distinct sets, then it would seem that the concept of “distinct” is a subordinate concept of adding and, therefore, should be learned first. It is difficult to see how dis- tinctness can be learned without examples of sets which are not distinct. In Paul Rosenbloom’s and Dienes’s approach, the children are immediately presented with sets which are not distinct and such sets are united. It is not found in either of these projects that children encounter undue difficulty in handling the problem arising. It does mean, of course, that the set operation of intersection has to be taught concurrently with the set operation of union. This is in line with the “throwing them in at the deep-end” policy, where the more complex situation is sometimes better presented first, from which the simple situation can then be de- duced. Intersection is the operation of finding those members of the universal set which are members af two sets. This operation does not have an arithmetical equivalent and possibly this is why it is not introduced immediately in some projects.

Another set operation that is introduced is that of “finding the differ- ence” but again in most projects only the difference set between o set and one of its sub-sets is considered, whereas differences between any two sets could be defined by saying that difference between set A and set B thot is “set A remove

set B”, would be the part of set A which was not in set B, or “set B remove set A” would be the part of set B which was not in set A. TO the author’s know- ledge, there are not mathematics projects in which this form of the difference set

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is introduced at the stort. It is ogain clear that it is on the forming of the differ- ence set between sets and sub-sets that the arithmetical operation of subtraction is based. So subtraction is superordinate to the operation if finding the difference sets. Therefore, learning the difference sets should precede the learning of sub- trcict ion. A porticulor case of finding the difference set is, of course, the find- ing of the complement in which case we ore finding the difference set between the universal set and any particular set. For example, if all the children in the class form the universal set, then the complement of the set of boys is the set of girls. The set of girls is the difference set between the universol set and the set of boys. If not&ion has been developed for the difference set and one hos been agreed on for the universol set then a notation for the complement could be uni- versal set followed by the removal sign followed by the symbol for the set. In

this way no new symbolisms need to be introduced for complements. In Poul Rosenbloom’s project in Minnesota, these insights into the relationships between sets and into the operations on sets ore helped along by a collection of stories and exercises based on these stories. These stories hove been carefully constructed to appeol to children ot the kindergarten and first grade levels. But in every one of these projects, it is suggested that concrete activity with actual physical sets of objects should ploy o considerable part in the process of learning. In the Rosen- bloom project, to such manipulation is added the delight of monipuloting the imoges which ore created in the children’s minds by the telling ond re-telling ond reshaping of vorious different kinds of stories involving sets.

Hoving estoblished sets ond set operations it is then possible to go on to equivalent sets. In fact, it is not strictly necessary to go on to unions ond intersections and differences before going on to equivolent sets. The equivalence of sets is o relationship between sets and is, therefore, superordinate to sets them- selves. So equivalence of sets con be taught possibly parallel with the teaching of the set operations which ore also at the some level. In Paul Rosenbloom’s project o great deal of activity is devoted to the construction of one-to-one cor- respondences. This would seem on essentiol pre-requisite before a child con reolize whot is meont by “equally numerous” or “equivalent”. If two sets con

be put into one-to-one correspondence, they ore called equivalent. So obility to put sets into one-to-one correspondence with one onother,. arranges sets into equivalence classes. Thot is, certain sets are equivalent to each other and others ore not. The set or class of oil those sets which ore equivolent to each other form on equivalence class. Th e common property of belonging to this class is the number property which each of the sets belonging to such o class possesses. So, therefore, the numerals ore then introduced at this stage indicating to children that it is the name that is given to thot property. For instance, the set whose members are my house and my tree, con be put into one-to-one correspondence with the set whose members ore my pencil and my pen, ond with o number of other sets which consist of pairs of obiects. And th e common property of all the sets belonging to this equivalence class is denoted by the numeral 2. In this woy the cardinal numbers ore built up ond numerals will then indicate the cardinality or the cardinul number property of ony set within one particular equivalence class of sets. We now hove reached the cardinal numbers. A considerable omount of experience will hove to be put into associate with these the ordinals as well. Exercises OS described in Greco’s and Matalon’s experiments might be ways in which we con help children to establish the connection between the cardinal and the ordinal

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aspects of natural number. New Guinea, Projects.

This is being done in the Adelaide and the Papua,

It will be realized that there is a complete isomorphism between set operations and logical operations. No mention has so far been made of the logi- cal operations, but in the work on mathematics learning in the classroom conduct- ed by William Hull in Cambridge, by Zoltan Dienes in Adelaide, by Ron Carlyle in the Phillipines, by Biemel in Paris, the logical counterpart of the structure of set operations is also introduced at the very outset. The way this is done is through the use of so-called attribute blocks first used by William Hull in Cambridge. These are a set of colored blocks of three or four different colors, three or four different shapes, two different sizes and sometimes two different thicknesses. This clearly lends itself to playing various games of a logical charac- ter, conjunctions, disjunctions, implications and the like. It will be appre- ciated that intersections of sets correspond to the conjunctions of the defining attributes of these sets and unions of sets correspond to the disjunction of the de- fining attributes of these sets. Negation corresponds to the defining attribute of the complementary set, so it would seem that having played the logical games in which the logical operations of conjunctions, disjunctions and negations and im- plications, are played with, ot one stage the isomorphism between this logical structure and the set structure should be established. This is, in fact, what is attempted in the above-mentioned Mathematics projects. The time when this can usefully be done seems to be about the 5-7 year level. The advantage of joining the logical work with the set work is not merely in order to be mathematically puristic; the logical work has been found to be highly motivating for young children. They love playing the games and also playing the logical games appears to have some considerable “brain-sharpening” effect, having appreciable trans- fer value into other parts of the school learning situation, as has been reported by many teachers taking part in these projects. For example, in Papuo, New Guinea, where in some of the local languages there is not even a word for either/ or, and owing to some rigid tribal structures choices are not very often given to children, the idea of choice has for the first time been introduced and teachers have reported that children have for the first time used the word “because”. They have said to each other “You can’t put that there because it is like this or like that”. If this is so obviously so in the case of children living in primitive culture, it is probably likewise the case with more sophisticated children, only the situa- tion there is so overlaid with other different factors that it is more difficult to de- tect the immediate effect of playing these logical games on these children.

In the projects operating the basic-set approach, once the minimum facility in handling sets has been reached, i.e. once the union of distinct sets and the forming of difference sets have been learned additions and subtractions can be introduced. Addition is simply the operation of computing the cardinal number of the union of two distinct sets once the cardinal number of each of these two sets is given. Similarly subtraction is the process of computing the cardinal number of the difference set between a set and one of its subsets, once the cardi- nal numbers of this set and its subset are given. So a considerable amount of physical as well as symbolic exercise of the embodiments of these arithmetical operations will be usublly necessary before the connection is made, i.e. before

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it is realized that the operations on numbers although similar to the operation on sets are at a different level. In order to emphasize this, different words are used. In the case of sets we do not add a set to another set, we join a set to an- other set and we add a number to another number. We remove a set from another set to find the difference set but we subtract a number from another number to find the difference. Of course, the commutative property and the associative property of unions of sets will lead to the corresponding commutative and associa- tive properties of addition. This is not alwaysdone in this way. Some mathema- tics projects use aids such as Cuisenaire rods, others merely use the symbols for numbers, i.e. written numerals and rely on children’s previous experience through which they will have realized that five and three is the same as three and five and 6ut of these particular cases they come to the generalization that it is imma- terial in which order they add two numbers.

Multiplication as well as division con naturally be based on corres- ponding operations on sets, and this is done to a greater or to a lesser extent by workers in the mathematics projects that can be classed under the name of basic- set approach .

used : There are basically two approaches to multiplication that have been

a) the union of equivalent sets, and b) the Cartesian product.

In the first approach the starting point is a set F, whose members are sets, all of which are equivalent to each other, and F2 is any one of these sets, or a further set which is equivalent to each of them. SO

F1 = (F2,1r F2,2, F2,3r . . . . . . . 1

and F2 is any set which is equivalent to each one of the mutually equivalent sets

F2,1. The union U1F2,1 = P is formed.

The cardinal number of the set P so obtained is termed the product of the cardinal numbers of the sets F1 and of F2.

In the second approach, any two sets F1 and F2, each with a finite number of elements, are considered. If

F1 = (a1,a2,a3,aql . ...) and F2 = (bl,b2,b3,bqr . . ...)

then the set of all possible pairs aibi are formed, where aibi is considered to be the same pair as b-a;. product of the car cl-

The cardinal number of this set of pairs is said to be the tnal numbers of F1 and F2.

Both the above approaches are feasible, having been tried extensively

in elementary school classrooms. Naturally the material will not be presented as

above, but in real situations as for example all the possible ways in which boys in a set can ask girls in another set for a dance, or all the possible ways in which the

chairs and the tables in the classroom can be paired, or all the possible ways in which a set of coplanar points (no three of which are collinear) can generate lines

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by joining the points in pairs, etc. If the union of equivalent sets is chosen as the approach, it is best to precede the formal work with exercises of one-,to-one correspondences between sets of objects and sets of sets of obiects, as, for exam-

ple, in a class of 40 children, ten tables can be set up and around each table four children can be asked to sit, thereby physically creating, through contiguity, a one-to-one correspondence between the set of sets of four children, and the set of tables put to use.

Detailed work with sets has led to another very interesting aspect of the work that has been going on at the foundations of mathematics learning. It is the work into the logical thinking of young children. William Hull’s work in Cambridge, Massachusetts, at Shady Hill school and later at the Elementary Science Study of the Educational Servieces Incorporated in Watertown, Massa- chusetts, as well as Dienes’work in Adelaide have already been mentioned in connection with sets and logic under Section 2.11.

Children were asked to think of “If-then” situations in their own ex- perience and see, if their “If-then” situations were always true, what other “If-then” situations would follow from them. It would seem, therefore, that according to the results of these experiments, what Skemp has called a reflective activity is certainly present in younger children, if only we give children prob- lems which are easy to solve but which they cannot solve without such activity. This goes in some sense counter to Piaget’s insistence that this kind of logical analytical thinking does not begin to take place until the beginning of puberty. Of course, Piaget’s development theory does not necessarily mean that the age groups have to remain constant. The development does have to go through certain stages and it may be that by administeringcertainparticular structured situations to the children the rapidity with which children pass through this developmental stage might greatly be increased.

3.5 The arithmetically-oriented Approaches

The original aim of these projects would seem to be perfecting of arithmetical understanding and techniques of children under their care. It would

probably be correct to say that in the arithmetically oriented approaches, all that has been said up to now about sets is taken for granted to have token place. That is, it is taken for granted that children realize that subtraction is the arith- metical activity corresponding to the finding of difference sets between sets and subsets and so on. It is not assumed that this has been made explicit by the children, but it is assumed that by playing about with objects and sets of objects, children will have developed the idea of cardinality, and so the ideas of adding and subtracting can then immediately be built on that. In other words, o great

deal is taken for granted which might indeed have taken place with some children but it is quite possible that others will not be so fortunately placed. The founda- tions are taken for granted and various methods of devices are introduced whereby the arithmetical operations are supposed to be better understood. Let us consider

again for a moment what is meant by being “better understood”. Here perhaps

there might be a divergence of opinion about what mathematics really is and how

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it is based. In the formalist conception mathematics is regarded as completely empty, that is an empty shell with just structural properties and no reality con- tent. This is the supposed end product from the pure mathematical point of view of a long process of abstraction which takes place possibly over years or even de- cades. It could be argued that mathematical relationships are not just empty, but are expressions of that which is common to various quantitative situations that we find around US in the world, and this common quantitative essence is expressed by the abstraction of mathematics. As opposed to this, the formalist attitude would deny any reality content and would not have anything to do with the actual process of abstraction from the concrete to the final pure product. It would seem then from the formalist point of view that if a one-to-one correspondence could be established between the properties of a certain set of physical situations and actions and a mathematical structure and if children were able easily to perform and learn the relationships between their actions and the percepts which result from their actions, then they would have learned the mathematics because the mathematics, so to speak, is all there, and oil you have to do is to establish the connection between something which is easy to learn and this ideolized, ubi- quitous, etel-nal mathematics. This is essentially what happens when materials like the Cuisenaire rods or Catherine Stern’s material or even the MultibaseArith- metic Blocks are used, if they are used in isolation. All these materials owe their origin to the early pioneering work of Maria Montessori. It is a credit to her genius that her materials were of a greater variety than many now in use. What we have termed associative learning takes place when one material is used exclusively. Abstractive learning is the result of abstracting that which is in common to many different situations and takes place in the case of those children who can avail themselves of other information they may have in store. It must be assumed then that information which is usable in connection with the learning in progress is available and secondly thot such information can be taken out of store by a recognition of its relevance. These are requirements that are met only by certain children probably by children that are termed “more intelligent” than others who are not so fortunotely placed. It would seem that in the case of the less intelligent children, the learning apparently takes place in just the same way, but with the difference that the abstraction will not have taken place, therefore, very little transfer will be noticeable. In a number of try-outs recent- ly conducted in Adelaide and Papua, New Guinea, it has been found that Cuisenaire-taught children were not able to transfer their knowledge of the commuta- tive principle learned with the rods even to situotions where the same principle had to be demonstrated using only the centimetre cubes, let alone with groups of children or chairs or tables. There is much in the single material opprooch, of which the Cuisenaire approach is the main one, which appears to employ similar psychological principles to those employed by the other mathematics project which use the basic-set type of approach. Mlle Goutard in her dealings with the Education Department in the province of Quebec in Canada has put these togeth- er very neatly in o little book entitled “Talks for Primary School Teachers”. The points are approximately as follows:

The importance of free play is emphasized. Children should be given the rods to play with, to do what they like with. Many children, particularly children who have been to school for some while, are probably starved of this kind of play. And as we have seen in the theoretical section, it is very probable

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that manipulative play or even representational play is the starting point of most of the later more advanced cognitive work. It is suggested that in the beginning as little interference os possible should take place by the teacher. One way of formulating this is that there should be less teaching and more learning. The task of the teacher is to create a learning situation in which the child is able to use his own creativity to begin to constl-uct his concepts. Of course, it is not pos- sible to leave children entirely free indefinitely. It can hai-dly be expected that children will discover the whole of mathematics out of ten little colorcd rods, but once children exhaust their own inventiveness and begin to get fractious, cer- tain games can be played which have certain rules. Some of these ore explained in Book 1 “Arithmetic with numbers in color” by Gattegno. So as is suggested in Dienes’s theory, fr-ee play should give rise to rule-bound play after abstraction has taken place. The properties of the material can then be played with. One important suggestion is made by most people using Cuisenaire rods and other allied materials is that the action should precede verbalizing and that the verbalising should precede writing. At no time should children be worried to record octivi- ties in which they are engaged. The action is itself a teaching-learning situation and, of tout-se, it can be discussed verbally while the action is taking place, but writing can be left till later. Mlle Goutord even suggests that once the writing has taken place the rods should not be used at all. In other words, while the rods are still necessary, writing is not really going to be very useful. It is also good to throw a spanner into the works by asking awkward questions. For ex- ample, would what we have done just now have been possible with another rod 3 Does it mean that what we are doing is general, or is it a particular property of a particular rod ? Sometimes children make wrong generalizations and so the situa- tions must be engineered cleverly enough so that either they do not make these or if they do, the teacher must create situations which will torpedo these generali-

zations. Counter examples which would show children that their generalizations are incorrect should always be up the sleeves of teachers in the mathematics classroom.

The Cuisenaire rods are particularly suitable for making clear the order relations between numbers. This is done by constructing various stairways in the

beginning. In the beginning the rods “increase” in even intervals of either one

centimetre cube at a time or several centimetre cubes at a time, later they “increase” at irregular intervals. The longer ones and the shorter ones can re-

present the idea of greater than, or less than, again in an associative way, tem- porarily identifying length with quantity. The lengths or the colors of the rods

are acting as associates of the corresponding numbers. The inequality of numbers

can be represented physically by the juxtaposition in parallel positions of rods corresponding to these quantities. Also equivalence can likewise be represented

by the equality of lengths. If two different “trains” are made, with different

numbers of different colored rods, if one train con be put next to the other and

both ends coincide, then this can be a perceptual peg on which to hang the idea

of equivalence. This is the counterpart of what in the basic set approach was be- ing done by the one-to-one correspondences between members of differ-ent sets.

Of course, equivalence can also be perceptually recognized by not only looking at lengths but also by looking at areas and eventually volumes. The areas ore eventually semi-symbolically indicated by crossing a rod over on to another one,

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thus indicating the product of the number corresponding to one rod by the number corresponding to the other rod. The number of unit pieces that we should obtain, were we to put down as many rows in a rectangle as there were units making up one rod, and as many unit cubes in each row as there were unit cubes making up the other rod, would be represented by such “crossing over”. By analogy, triple products are represented by putting three rods on top of one another at right angles and so powers are represented by the same colar rods being put across one another in the form of crosses, as many rods being placed in these towers OS the number in the exponent which is being used. For instance, 125 would be represented by three red rods being put on top of one another, each rod being at right angles to the one below it. This would indicate that each time we put a rod across another, we take as many times what is below this new rod as the number of unit pieces that make up that particular rod. That is each “floor” of the tower represents a multiplication by the amount which is the number of the units in that “floor” of the part of the tower that is underneath thot “floor”. It is easy to see that the properties of exponents can be readily obtained by playing about with such towers, i.e. by building them up and dismantling them.

Work in groups and work by individuals are bath encouraged, but class teaching is not excluded. It is thought that newcomers to the class can be inte- grated into the work, by these newcomers joining existing groups where the most able children will be able to help the newcomers to establish themselves mathema- tically by handling the rods along with their new friends. The written work follows, as we have said, as a direct consequence of having learned how to handle materials, having discussed the problems and having hopefully extracted the mathematical content of the activities. Fractions and certain other algebraic identities can also be learned in this way. It is almost certain that some children will learn in an associative way.

The rods are useful in making clear to children that numbers can be ab- toined in a great number of different ways by carrying out multiplications, addi- tions, subtractions, divisions and so an. Different routes can lead ta the same final number. These activities can eventuolly be symbolized by representing numbers as results of different series of operations: Like 7 times 4 plus 2 leads to the some end-point as 4 times 7 plus 2. Fractional and exponential notations are introduced as soon as notation becomes possible as a result of the children having learned to handle the corresponding situations with the rods. It, therefore, becomes possible far children to handle quite complex symbolic manipulations which would normally be quite beyond them, had they not hod the lead-in through learning how the colored rods behave and inferring that this is how the correspond- ing mathematical entities behove.

The function of memory is brought into focus as it is explained that al- though certain facts might be the optimum set of facts to remember by certain

people, other people may find it more politic ta remember different sets of facts. When we want to find something out in arithmetic, we can do so in two ways:

1) We can refer to facts we remember; 2) we can refer to some relotionships and deduce some

further relationships or facts.

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In some countries such OS India, multiplication tables up to twenty must be learned by all children and in others only tables up to twelve or up to ten are required. It seems that the more we learn the less reliance we place on our ability to deduce more information out of less data. For each of us there must be an optimum equilibrium position between what we can usefully remember and what we can work out. The time spent on rapid calculation might be less than the time spent on possibly ineffectually remembering a great number of disconnected facts. So it is suggested that no hard and fast rules be laid down about facts children should or should not remember. If they can very quickly work out a result which they do not in fact remember, this could be just as efficient or more so than try- ing to remember a lot of facts, possibly to no avail.

In introducing fractions, care is taken that fractions be regarded both as operators and as states. The operator angle is that which has to be done to some state to obtain another state. SO this other state is a fraction of the first state. This is the operator angle. Of course, if we applied the operator to the unit state, whatever that may be, the final state is, of course, the fractional state corresponding to the fractional operator which has been used to operate on the unit state. It is not, needless to say, explained in these terms to the children but the actions to be performed on the situations represented by rods are again associated equivalents to operators and the configurations of rods of various kinds are the perceptual equivalents or associates of states. In this way again isomor- phic situations are built up out of actions and percepts to aperotors and states, so that according to the formalist view of mathematics, the empty structure of ma- thematics is, in fact, learned without having regard to any real situation which might be describable or representable thereby.

It is true that many children are able to retrace their steps and find their way from the symbolized rod-manipulation which is their mathematics, to some corresponding structure exhibited in problems. As we have already adumbrat- ed, this is o pedagogic difficulty, and its presence is being realized now by a number of advocates of the Cuisenaire materials. Mlle Goutard herself says that

the materials are not intended to be sole ones, but that other materials should also be used, but mainly for the learning of those ports of mathematics for which the rods are not themselves suitable. For example, it would be ludicrous to study the properties of circles by means of rectangular rods !

Much of what has been said concerning the Cuisenaire Rods applies equally well to materials such as Catherine Stern’s or Unifix or the Multibase Arithmetic Blocks, or any material used in isolation by itself. All these mate- rials have their advantages but it would seem that their advantages would be greatly increased if they were used in conjunction with each other, or in con- junction with physical situations, so that the transfer from manipulation with ma- terials to these physical situations were artificially encouraged by creating such situations in the classroom first. Cuisenaire rods are used in a great number of countries in the world now. They originated in Belgium and many of the elemen- tary schools in Belgium use them, as they are also used in a number of areas in

the world. Mr. Roller of Geneva, lnstitut Rousseau, edits a special iournol,

“Nombres en Couleur”, which gives an account of the various activities in

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different parts of the world where Cuisenaire rods are being used. Readers are referred to the Bibliography for other information on this and other structured ma- terials. There is one important difference between the Cuisenaire, Stern, IJni- fix and other materials and the Multibase Arithmetic Blocks. The Multibase Arithmetic Blocks use explicitly a number of different systems of numeration. Although other materials, such as the Cuisenaire rods, can also be used for mul- tibase work, as can sets of arbitrary objects as described in the basic set approach. The multibase blocks are, however, specially constructed so that from a unit piece upwards, there are first power, second power, third power, fourth, fifth and sixth power pieces in a number of different bases. Naturally the sixth power pieces would be impracticable in the higher bases, so in the lower bases the material proceeds to higher powers than in the higher bases. In one form of the Multibase Arithmetic Blocks or MAB the three spatial dimensions are used for representing the first, second, and third powers. The unit cubes ore put together end to end to make a piece of the first power much in the same way as in the case of the Cuisenaire rods. For the physical representation of the second power, they are put in the form of a squire slab, and for the representation of the third power they are put in the form of a cube. For the fourth power, these cubes are again put end to end and the process begins again, until the pieces get too big for classroom use. The use of the different bases is an application of the “throwing them in at the deep end” principle which, as we have seen, is begin- ning to be validated as a result of a number of experiments in different parts of the wor Id. In the new version of the Multibase Arithmetic Blocks, instead of providing only one kind of unit, many different kinds of units are provided and the different power pieces are built up according to different geometrical principles. Some of the units are small squares. For example, in base 4, bigger and bigger squares are built so only two-dimensional extensions are used. In bases three and six, triangles, trapezoids and hexagons are used. The purpose of this variation is to provide children with perceptually different situations, so thot they do not get “stuck” to one particular set of properties and one particular set of materials such as possibly the color of the rods, or their shapes, or their sizes. In other words, an attempt has been made to turn the material into the kind of material in which associative learning is less likely and abstractive learning is more likely to take place. Various other kinds of situations are also suggested in the Teachers’ Handbooks, such as binary and ternary genealogical trees at which there are either two branch points at each branch or three branch points at each branch and

so on, on which two-base and three-base and even higher base games can be played which will eventually be seen to be equivalent to the games played with the blocks themselves. In some mathematics projects, multibase blocks are used in an arithmetically-oriented way. These are the Leicestershire Mathematics Projects and the Surrey Mathematics Research Group. On the other hand, in the Adelaide, Papua, New Guinea, and the Phillipines mathematics projects, the basic-set approach described in the first section of this chapter are also put

into practice. That is, by the time children come to use the blocks for the pur- poses of learning arithmetical operations, they will have been through a thorough

training in the properties of sets, of one-to-one correspondences and so on. Naturally such a preliminary would be equally possible if Cuisenaire rods were

used. Such an arithmetically-oriented approach can be broadened and enriched by other mathematical activities taking place at the same time. It is now generally

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agreed tl\~t quite a considt:inble amount of yeometry con be learned very early in the cot-eel- of children. 11 <lr>sfot motions, geometry as well as mathematical groups can be played witlh hum the second grade onwards and even isomorphisms be-

tween different representations of mathematical groups can be played with, as early as second and thiti-d g~crdes. The wider use of transformations would seem to be an advantage because it will teach children the generality of the idea of opera- tor and state in mathematics.

If there are two light switches in the classroom, controlling the lights in different parts of the room, q uite a few tronsformations of the lighting effects in the room con be performed by flicking one or both of these switches. For example, if switch A lights the left part of the room, switch B the right part of the room, then there are four states of the room:

1) Both parts dark ; 2) Left lighted up, right dark; 3) Right lighted up, left dark; 4) Both parts lighted up. 5) Both parts lighted up.

The operations or transformations can be :

(N) Do nothing; (A) Flick switch A; (B) Flick switch B; (AB) Flick both switches.

Chains of states, transformed into each other by means of successive transformations, can then be generated, such as for example :

1) - (A) - 2) - (AB) - 3) - (A) - 4) - (B) - 2) - (A) - 1).

Or the states of the game can be determined by one of the children standing up and facing in one of the four cardinal directions, i.e., North, South, East or West. The transformations could be :

(N) Do nothing; (R) Turn through a quarter turn toward your right hand; (L) Turn through a quarter turn toward your left hand; (H) Make a half turn.

Chains of states can then be generated such as for example :

North - (R) - East - (H) - West - (R) - North - (L) - West - (L) - South.

When they do an adding they will know that this is an operator operat- ing on a state much OS it might have been a certain action which they might per- form on their own bodies when their state was standing in a certain place in the classroom. These games can be structured round mathematical groups or possibly semi-groups or directed numbers as for example in the case of a number line drown

on the floor. It is not even necessary to hove numerals on them, the positions can simply be marked by colors or letters.

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If the purpose of the game were the teaching of the combination of an operator with another operator as being equivalent to some third single operator, then the actions can be carried out one ofter the other and it can be verified that from whatever state the child starts, the succession of such two actions con at times be seen to be equivalent to one single action. For example, in the first chain it would be possible to go from state 1) to state 3) by applying the trans- formation (B), i.e., the first part of the chain could be shortened into 1) - (B) -

3).

Or in the second chain, instead of passing from East to North via West, North could be reached from East by (L) nnd so the part East - (H) - West - (R) - North could be replaced by East - (L) - North.

The above experiences do not, however, convince the player of these games that these “replacements” can be carried out independently of the state with which we start. So replacements of the succession of (A) followed by (AB) by the single transformation (B) would need to be experienced many times, starting with any possible state, similarly replacements of the succession of (H) followed by (R) by the single transformation (L) would I’k I ewise need to be experienced in mony situations before it is realized that the two operators one after the other ore equi- valent to a single opeartor independently of the state at which we first start operat- ing . But when this is realized we have reached what Skemp has called the re- flective stage. We have acted reflectively on what we have discovered how to do, i.e., we have reflected on the operator whose concept we have acquired. This can and does take place as early as the eighth or ninth year and it would seem that it is unduly pessimistic to say that we need to wait until the onset of puberty before we can get children to engage in reflective activity in the schools.

3.6 The geometrically-oriented Approach

The most thorough going example of this kind of approach is the Minne- sota School Mathematics Center directed by Professor Paul Rosenbloom. By thorough going is meant that although there ore other mathematics projects, in- dividuals and school systems that use the geometrical approach to some extent as an aid to the understanding of number properties, Professor Rosenbloom uses it as

a fundamental framework on which to hang all the properties of numbers.

He uses the isomorphism which corn be established between points in o line and real numbers or between points in cr plane and ordered pairs of real numbers, to help children to come to grips with the properties of numbers and of functions. At first only natural numbers are used by children, and so only certain points in the line or certain points in the plane are used. But later on when children come across fractions or negative numbers or complex numbers, there will be no need to unlearn anything and the model is right there and merely needs filling in. The

use of points in o plane gives the classical model for the idea of function and, therefore, in particular for the elementary arithmetical operations, Additions

and subtractions can be done by jumping up and down the number line in a one- dimensional space OS suggested by the UICSM, or the SMSG, or the Madison project and Dave Page’s exercises. But in Professor Rosenbloom’s project the

idea is taken much further and he suggests that, for example, in the case of multiplication and division, they use geometrical constructions in the number plane as the isomorphs of these arithmetical operations. For instance, multipli- cation by two would be represented by the function whose equation is Y = 2X. And division Y =1X and

by three would be represented by the function whose equation is so on. In this case, operations can be superimposed upon one

onoth%r and so, for example, fractions can be looked at eventually as superim- positions of two operations in much the same way as a superimposition of opera- tors in matrix

P Igebra. For example, we may have the “dividing by three”

function Y = 3X and the “multiplying by two” function 5

= 2Y, the superim- position of these two operations leads to the function Z = 3X. This graph can then be plotted and the operation of two-thirds can be obtarned as a composite of the multiplication by two and the division by three and the geometrical model is there ready for computing any multiplications by two-thirds once the value of X is given. The value of X is the state to which the operator two-thirds is appl ied, and Z will be the state which is obtained when the operation has been carried out. If and when negative numbers are introduced, the points are there ready and waiting for the children on the opposite sides of the axes and instead of a quarter of the plane the whole of the plane will then be filled up. Similar- ly irrational “points” or eventually “complex points” will likewise be filled up. Naturally many exercises are suggested which will tie these operations with everyday situations, a variety of which children are familiar with in their ordina- ry surroundings. In this way they learn to interpret natural and social phenomena in terms of the quantitative data which they themselves have gathered. Paul Rosenbloom has said that he will be making attempts to relate geometry with art and arithmetic with work in music. Of course, the differences between the notes in the scale are multiplicative differences as far as the frequencies are con- cerned. And so this should be an interesting example to practise the idea of powers and roots. To obtain the frequency of a note a semitone above a certain note you have to multiply the frequency of this note by the twelfth root of two, because going up an octave we have to multiply the frequency by two, and the semitone is one-twelfth of an octave.

In the geometrical tradition of Italian mathematics, Emrnn Castel- nuovo of Rome, suggests various similar devices for the study of fractions. For example, the use of similitudes in blowing figures up in certain proportions and shrinking them down in certain others and the combination of such similitudes will naturally lead to the application of fractional operators. She also suggests the use of colored rectangles, colored in different colors on each side as represen- tatives of the products of positive and negative numbers. The passage from one combination of plus and minus to another can be obtained by simply folding the rectangle over from one quadrant of the coordinate system to another one. Each such fold will, of course, change the colors. And if you start in the positive quadrant with the positive color then folding it over to one or another of the ad- jacent quadrants will produce a negative color. Folding it again will produce a positive color. This again gives a geometrical representation of the law of mul- tiplication of positive and negative numbers.

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Some of the schools in Europe, tradition of Decroly,

notably in Belgium that follow the follow in a sense the same tradition except that the follow-

ers of Decroly strongly believe that children should learn their mathematics from phenomena OS they occur in nature. Since most natural phenomena are continu- ous phenomena he suggests that the continuous change is more fundamental. Later this should be broken down into discontinuous change. The growth of plants, the passage of time, the movement of bodies across space, are fundamental global experiences. There is a certain amount of psychological evidence that such global perceptions come before perception of the detail of such phenomena. It is the examination of minute detail which give rise to measurement and accord- ing to Decroly is a more sophisticated part of the business. So it might be said that, according to Decroly the child should start with a global appreciation and then analyse it. There is a certain amount of evidence that children find this rather difficult to do. Readers should consult the Bibliography .

According to Montessori, the process should be in a sense reversed, and children should start with discreet operations, that is, not with continuous change but with discontinuous change. So she suggests that children should play with concrete material and that mathematics should grow out of constructing different structures with beads or rods or various other materials which Montessori has placed in the classroom. Montessori is a forerunner, therefore, of the

present run of systems that advocate what have now come to be known as structur- ed materials. The psychological theory behind such practices is that children can think constructively and so will build up their concepts and finish with an abstraction rather than perceive an immediate complex and analyse this into its components. Possibly the difficulty of the conflict between the two schools lies in that Decroly takes his cue from perception while Montessori perhaps thinks more in terms of concept, formation, to use a modern term. There is a certain amount of evidence that perception does take place more in the large, that is globally, in early childhood, and this is analyzed later, but if instead of per- ception we consider the formation of concepts the child does not start off with a fully formed concept, but he must construct this inductively out of his experience. So perhaps if a distinction is made between a percept and a concept the conflict between the two theories might largely disappear.

It must be mentioned here that almost all mathematics projects work- ing in different parts of the world, if they use geometry as a part of their curri- culum, they make attempts to look at it from the modern point of view, that is, from the point of view of transformations. Geometry is considered to be the set of those properties of figures which remain unaltered when a certain group of transformations is used. Euclidian geometry is, therefore, considered to be the set of those properties which remain unaltered through displacement. In the case

of two-dimensions, this can always take place through rotations, translations and reflections. For example, any displacement can be effected by means of not more than three successive reflections. Affine geometry is considered to be the set of those properties which remain unaltered under parallel projection. Affine geometry, therefore, no longer contains the study of individual lengths or the study of angles. On the other hand, parallels remain parallel, also ratios of lengths on parallel lines remain unaltered under parallel projection.

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Such geometry can be studied, in countries where the climate permits, through holding up figures in the sunlight. Wooden shapes, or shapes constructed out of Meccano pieces can be help up in the sun, turned into various positions and the behavior of the shadow studied. For example, it will be found that o square corn only have a parallelogram as a shadow, it cannot have a general quadrilateral which is not a parallelogram because parallels are preserved. The diagonals of the square present an instructive example. It can be seen that diagonals of o square bisect one another by means of symmetries. The succession of two sym- metries whose axes are at right angles is equivalent to a point symmetry or ct half rotation about an axis perpendicular to the plane of the square through its centre. This property of the centre of symmetry being at the mid points of the diagonals remains true in the case of parallel projection. To say that a figure has point symmetry means that if we join any point A of the figure to the centre 0 of sym- metry and produce this line, it will cut the figure again at the point A’ such that AO’ = OA’ and this is true for every point A of the figure. And so every line through 0 is a line which cuts the figure in two points, the segment joining these two points being bisected by the centre of symmetry. Since ratios of seg- ments on parallel and, therefore, on the same lines remain unaltered, the point of symmetry must be reserved in the shadow. So the centre of the shadow will also be a centre of symmetry of the shadow. This means that the diagonals bisect one another. Considerations of this kind lead to a very much more general ap- preciation of properties of figures than isolated theorems in which simply one theorem follows the other. In the conventional treatment the theorem that the diagonals of a parallelogram bisect one another would not be differentiated from theorems to do with angles which are not affine theorems, unless they are to do with angles which remain equal on account of parallel properties which are trans- mitted in parallel projection. One affine property which gives rise to a consider- able amount of useful learning is convexity. A convex figure is projected by parallel rays into another convex figure, and so it is certainly an affine property. Convexity and non-convexity of boundaries of regions can be related to the turns right and left as somebody walks around the boundary. The boundaries could, of course, be polygons, or curves, or a combination of polygons and curves.

Affine geometry is also studied in some geometrically inclined pro- jects through the use of square grids. These are best painted on the floor to en- able children actually to carry out the movements which correspond to the affine transformations. Particular affine transformations can be studied in which a square, for example, is transformed into another square but somewhat larger and turned through a certain angle, or other transformations which distort squares in-

to rectangles that are not squares, and yet others which transform squares into rhombi or into general parallelograms. Particular cases such as “uniform blowing

up” or “uniform shrinking down”, pure rotations, stretching or shrinking in one

direction only, etc. can be introduced, and the particular properties of these

particular types of transformation studied. On one occasion recently while some such affine games were being played by some children in the New Guinea High- lands, several children exclaimed at once : “This is like the shadow game!” The multiple embodiment approach had clearly worked and the abstraction of the common geometrical structure of the “shadow game” and the “grid game” sudden- ly occurred simultaneously in the minds of a number of participants.

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The projective properties of figures ore also studied. These can be studied by studying shadows of a point source,’ such OS a light. Of course, on ordinary bulb is not Q good point source, and possibly a projector projecting on a screen would be perhaps o better illustration; or holding pencils or rulers close up to the eye is onother way of introducing children to point source projection. At the for end of the scale ore theotopological properties. These ore very easy to teach, even second grade children will find it greot fun to discover properties of connectivity of regions and the like.

Practical devices used for getting children to arrive inductively at some of the Euclidion, offine , projective or topological relationships ore possib- ly the use of ropes on the floor for topological relotionships or hoops which con be put inside one onother, Moebius strips con eosily be constructed, ond cut down the middle by means of o pair of scissors with the usually unexpected result. For offine and projective properties, OS we hove seen, sunlight and the point source of light form excellent apparatus, and for Euclidian properties mirrors ore o very good device. In particular rotating mirrors facing each other, so that the angle con be altered ot will. Figures con be either drown or objects placed in between the mirrors, the angle varied and the reflections investigated inside the mirrors. It is surprising how difficult it is for children to foretell how the objects will be apparently situoted “inside” the mirrors, before they hove seen them. They will not reolize that the mirror itself will be reflected in the other mirror ond, there- fore, there will be o reflection of the reflection ond if there is a mirror inside that reflection, there will be o reflection of CI reflection of o reflection ond so

on. By making the angle between the mirrors very small, o very lorge number of reflections can be illustrated. Naturally tracing a figure over o fold on to the same piece of paper by means of carbon is o good way of establishing the proper- ties of reflection or symmetries. Or transparent plastic figures con be used on which figures con be drown so thot when we have turned them over on to their reverse sides, the figures can still be seen, but they con be seen to be reversed from right to left, or turned upside down as the case may be. An interesting game is the sorting of capital letters into four different categories. Imagine a mirror stonding up beneath the letter ond imagine whot its reflection will be. It will either alter the look of the letter or it will not. We con imagine the mirror to the right or to the left of the letter and we con see whether the letter is altered. It will either be altered or it will not. Hoving drawn the reflected letter we con

then use the other reflection, in which case we will hove 01 transformation of the

letter but through o point symmetry. So there will be four kinds of letters. There will be some thot ore completely unaffected by placing the mirror either to the right or to the left, or beneuth and above the letters. There will be others that are affected both ways, and the ones that ore affected only one way. From this the properties of the combinations of reflections ond rotations con be gathered seeing, of course, that the learning does not stop at the study of figures with

only lines of symmetry. Children should naturally study, for example, not only rectangles when studying figures with two lines of symmetry, but also figures in the alphabet, or numerols that hove two lines of symmetry.

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In Belgium, Professor Papy, of the Free University in Brussels, could also perhaps be considered as on experimenter in the geometrical approach. He makes use of the very powerful symbolism of grophs in order to establish the vorious properties of the different kinds of relations in which sets of elements can stand to one onother. At first, he considers the relations that con exist between the elements of o single set and later on he goes on to the mapping of one set on to other sets. His first exomple is to get children to stand up in class ond every person is osked to point at another person whose Christian name begins with the some letter OS the pointer’s surname. If your surname and your Christian nome begin with the some letter you must point ot yourself. It may be that you hove to point at several other people in the class. In that case this cannot be physicolly carried out, so you hove to point at them at different times or perhaps use orrows on the floor. If on element is transformed into itself by the relationship then you draw o loop, that is on arrow storting from CI point and doubling bock upon itself. In on osymmetricol relationship such OS inequality, there ore no loops. In on equivalence relotionship every point hos a loop, and so on. Popy has found that it is possible to teach children quite rigorous mathematical proofs of theorems such as Bernstein’s theorem, without hoving recourse to ony mathematical sym- bolism apart from the drawing of grophs. He also has introduced the technique of making o “mathematical film”. Such CI film would, in effect, be the proof of certain relationships. The film is not to be taken Iiterolly. It consists simply of o sequence of graphs and each graph is obtained from the previous groph by some olterotion or some addition to the previous graph. These ore drown by the pupils on different sheets of paper and then ore stuck together with scotch tape and CI great long roll is mode up which is the film. Such o “scroll“ represents o proof of o theorem. These proofs con be unrolled and discussed by the class. Some- body might suggest that CI certuin port of the film could be cut out or be shortened or in some woys altered. Looking ot these films is a very powerful method of getting children to reflect on their own intellectual activities ond sharpen their rnathemot icol thinking. Papy himself hos not tried this approach with children under the age of twelve but it would seem from the account given that many of the games he suggested of Q relational type could be ployed with much younger

age groups.

3.7 The Science-oriented Approach

As we have said, the Elementary Science Study ot Watertown, Massachusetts, is one group whose adherents feel strongly the necessity for con- necting child research into science and child research into mothemoticol struc- tures. Paul Rosenbloom’s work in the Minnesota project known OS Minimost also tries to unite these two disciplines, estoblishing a healthy balance between scientific ond mathematical work. One of the ardent European exponents of the

sc.ience-oriented approach is M. Zadou Noisky who is working ot Lyc6e Mongeron, just outside Paris, with wheels and pieces of meccano, through the use of which he tries to simulote the properties of numbers. Translations ore used as addition ond rotations OS multiplications, ond the distributive, the com- mutative and the associative lows ore all illustroted by mecms of physical-mechan- ical situotions which con all be personolly built up by the children. If o child

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does not have such deft hands, he can simply handle the mechanisms that have been built up by the other children. Great success has been achieved by Zadou Naisky in this kind of work and certainly the children In his classes thoroughly enjoy their mathematics lessons in which, in common with other projects, dis- covery of the mathematical relationships is encouraged rather than any straight teaching of mathematical know-how being administered.

The University of California Elementary Science Project does also coordinate mathematical work with children in discovery of Science. See : “Coordinates: An Introduction to the Use of Graphs and Equations to describe Physical Behavior”.

3.8 The Symbol-game-oriented Approach

Let us remind ourselves that for a symbol game to be effective, the symbol that is being played with must in fact efficiently symbolize the mathema- tical entity for which it stands. All symbol games stand or fall by that assump- tion. It must’be assumed that a considerable amount of painstaking work has taken place beforehand of a fundamental kind, or of the arithmetically-oriented kind, before children can either enjoy symbol games or profit from them. Most of the projects in the United States including such large ones as the University of Illinois Committee for School Mathematics, the Madison Project and the former University of Illinois Arithmetic Project, etc., are based on the idea that symbolically expressed mathematical ideas can be profitably played with.

The first kind of symbolism introduced for playing with is the open sentence. Frames are used in which any numeral can be inserted provided that the same numeral is always inserted in the same shaped framewithinagiven sen- tence. This is the rule of substitution and non-adherence to this rule is referred to by the Madison Project as an “illegal” activity. It does not mean that an

illegal activity will lead to a false statement, nor is a false statement always the result of an illegal activity. The set of all numbers which when substituted for a particular open sentence make that open sentence true, is the truth set of that open sentence. Of course, the members of the truth set will be single numbers if there is only one kind of frame in the open sentence; they will be ordered poirs of numbers if there are two kinds, they will be ordered triads of numbers if there are three different kinds of frames, and so on. Some truth sets are empty,

for example, the open sentence LJ + 1 .= [::-I] + 2 has an empty truth set. There are no numerals which we could legally insert in the frame which would make that open sentence into a true sentence. Likewise there are some open sentences whose truth set is the universal set, i.e. which is true for all legal substitutions. Such an open sentence is: : + ! 1 = 2 x [ j. Whatever numerol we substituted in those frames, as long as every frame had the same numeral in it, we would end up with a true statement. An open sentence whose truth set is the

universal set is called an identity. ldeniities form a fairly large part of the symbol game approach. Open sentences can, of course, be played with by putting the problem of finding the truth set. There is no point in finding the truth set of identities, because it has been found already. But various other games can

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be played with identities. For example, other identities can be deduced from them or lists of identities can be made and then the problem can be posed of

shortening this I ist .

The work of shortening a list of identities to the shol,test possible one

which contains the same information is a practical problem leading to the idea of

axiom systems. Shortening means, of course, to prove in some sense that a cer-

tain identity could be left out of the list and the some information was nevertlle-

less contained in the rest of the list, This entails “proving” that this is so.

Th is in turn entails the idea of a derivation. In derivations to do with identities

the properties of equivalences are used. These are :

1-A A

2. IfA 8, then B A

3. If A B, and if B C, then A C.

In the Madison Project work, a den-ivation begins with a trivial iden-

tity that is with an identity in which exactly the same symbols in exactly the

some order are written on each side. These are allowed in virtue of the first

property cited. Children become quite adept at deriving identities from other

identities using the above rules. The problem of proving when a list is Ireally the

shortest possible, i.e., when no further identity can be stl-uck out by any further

derivation, has not to the author’s knowledge been broached yet in any mathema-

tics project. This will have to await the more thorough logical training in expli-

cit ways of elementary school children. This has recently begun, both in the

Adelaide and the Papua New Guinea Mathematics Projects, and eventually a

certain amount of formal logical thinking will become possible as o result. Inde-

pendence of axioms will be able to be handled by suggesting truth tables with

more than two values, as is done in the conventional proofs, as, for instance,

given in “Grundlagen der Mathematik” by Hilbert and Bet-nays. It must be

emphasized, however, that in the majority of mathematical projects care is tak-

en that identities and axioms remain an end-point of a creative thought process

on the part of the children and not a beginning on which to base a formal system.

This is recognized by all people who have recently given thought to the pedagogi-

cal problems of mathematics learning and shown in resolutions of international

meetings, such as OEEC and UNESCO meetings that have recently taken place

on mathematics teaching.

The symbol games goes, of course, very much further than playing

with identities. This is only one kind of mathematical game that can be played

with symbols. As has already been said, open sentences can be played with,

the main game being transforming open sentences by “admitted” transformations,

i.e., by transformations which would leave the truth set unaltered. By defini-

tion an open sentence will be called equivalent to another open sentence if

their truth sets are identical. This leads to the techniques of the solutions of

equations. Again care is taken not to provide fixed algorithms for children but

to let them play about with the symbols themselves, so that they will find algo-

rithms of their own. Any method which a child has started himself will become

important to him and in some sense his own personal property of which he is proud.

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This will make mathematics a personally meaningful activity to him rather than something imposed from above. The UICSM have been from the very beginning of their work in 1952 keen to establish this style of teaching, as a result of which the children are given very carefully selected mathematical situations, so that they will construct just as carefully although freely the corresponding mathema- tical constructs. Jack Easley likens the procedure adopted by the UICSM to that of the experimental psychologist. When a psychologist prepares a learning ex- periment, the conditions of the learning to be examined must be very carefully thought out so that the required phenomena shall take place. Of course, there are differences because a certain amount of latitude in the form of permitted play activity must be allowed so that the first stage, the play stage of the concept formation should have an opportunity to become operative.

Symbol games can be played in many ways. For example, Dave Page formerly of the University of Illinois Arithmetic Project, now of ECI Watertown, Massachusetts, often plays the following game with a class of children. He writes the numerals, one, two, three, four, five, six, seven, eight, nine, ten, in one row, and eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, immediately underneath, and so on, until he reaches a hundred. There are then ten rows of ten numerals each. Then he introduces the operation of “left to right arrow”. The “left to right arrow” does what it is intuitively obvious it should do. It generates a numeral from a previous numeral, namely the one which is immediately to its right. It is fairly quickly learned that the “left to right arrow” generates the next numeral in the sequence, with, of course, the exception of ten, a twenty, a thirty and so on; that is, at the end of the array when it does not generate anything because there is not anything to the right of ten, nor to the right of twenty and so on. It is left to the children to generalize to this case and most children will respond by suggesting that in this particular case the left to right arrow should, in fact, go to the beginning of the next row, or some children will fill in an eleventh column with elevens, twenty-ones, thirty-ones and so on, and say, “Oh, it doesn’t matter if you have certain numerals twice”. You come to this numeral in the eleventh column if you happen to start with a “left to right arrow” from the tenth column. After this, up and down arrows are introduced. Again, if playing around with “down arrows” it becomes clear quite soon that the opera- tion represents an addition of ten. If you put a down arrow at thirteen it will be pointing at the numeral twenty-three and so on. After this a combination of these operations is broached. For example, what happens if you have a right arrow followed by a down arrow ? Some classes suggest, having discovered what happens, that you move up one and down one at the same time, so you might as well go down diagonally one step, that is, take one step the way the Bishop moves in Chess. These can be combined with “right to left arrows” and “down to up arrows” and the combinations of these operations can then be played with. This sort of game takes children out of their conventionally conditioned arithme- tical situation and allows them to think of states and operators as such. It allows them to conceive of a certain class of activity as a certain kind of operator and also allows them to operate on the operators by ordering a certain kind of joint operator to any succession of two other operators. So games are being played with the mathematical notions assuming, of course, that all the symbols that are being used are fully meaningful for all the participants. Such is probably the

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case in the kind of game that is played by, say, fourth, or fifth grade children in elementary schools. The use of desk-calculators has been tried in a number of places and it has been found that not only arithmetical techniques can be improved by their use, but greater degrees of understanding can be achieved, as was shown recently by a study carried out by Fehr. Desk calculators have also been used for many years by the schools in Leicestershire, England.

Another good symbol game is the machine game introduced by UICSM. In every machine, there is always an input and an output. You do something to the machine, and then it does something for you by performing an action. So the mathematical operator is regarded as a machine and numbers can be fed into the machine and then other numbers will be generated at the output end of the machine. First very simple machines like add one, or subtract two, multiply by three, can be played with and inverse machines can be investigated and com- bined machines of adding and subtracting, multiplying and dividing can even- tually be brought out. The first use of the machine is to put an input in and then start wondering what the input must be to reach that same output. They soon dis- cover that they can work a machine backwards by inverse operations. They will, therefore, construct the inverse machine by simply putting in the reverse order all the inverse operations of the original machine. The states in the case of the younger children are represented by concrete representations of the corresponding numbers. In the symbolic expression of the machines the intermediate states are given place holders besides the place holders for the input and output states. If place holders are used only for the input and output, it is not very clear that children will understand what the superimposition of several operators means.

A “machine” might look like this: (OP : Operator)

Input Op State A Op State B Op Stote C Op output

Needless to say, the “legality rule” for substitution does not apply here. To make it applicable all “frames” would need to be drawn as different shapes. If the intermediate states A,B and C were left out, it might not be clear to some children how the “machine” was put together. Perhaps they will

need to be reminded for some time that after every operator a state would be gen- erated and the next operator would have to operate on this state. This particular

kind of symbol game is particularly useful in introducing children to the idea of fraction. A fraction can be regarded simply as a succession of a multiplying and of a dividing machine, or else as a succession of a dividing and of a multiplying machine. It will take children some while to discover that they can inter-change the order in which they apply the multiplying and the dividing machines. They will likewise soon discover that they cannot interchange the order of a multiply- ing and an adding machine, and this will be a lead-in to the distributive law of multiplication over addition. It will also be useful to distinguish between unary

and binary operations or singularly and binary operations OS the expression is used by UICSM in these cases. A unary operation operates on one state and creates

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another state. Then each of the arithmetical operations can be considered either as a unary or as a binary operation. For example, if adding two, is regarded simply as a machine which operates on, say, the state three and generates the state five, in which case we have a unary machine. Or the process can be regard- ed as the states two and three being fed into a binary machine. The handle is turned, and the state five is turned out. In this case it will be seen that addition and multiplication are commutative, but division ond subtraction are not. On the other hand, when the interchange-ability of the machines themselves is con- sidered, then division is commutative. This is to say that, for instance, divid- ing by two followed by dividing by three is equivalent to the machine which di- vides by three first and then divides by two. But on the other hand it is not true to say that a state two divided by three gives the same state as the state three di- vided by two. So here the distinction between the binary and the unary operation will become very important. This will also be important when addition of frac- tions is introduced. Having arrived at the state of say two-thirds, i.e., having operated by the multiply-by-two machine and by the divide-by-three machine on a chosen unit state, and then having perhaps obtained another fractional state such as three-quarters by the use of the multiply-by-three machine followed by the divide-by-four machine, then these two states must be fed into a binary ma- chine in order to obtain the sum of these two fractional states. It would, of course, be possible to use one of them as a fractional additive and unary operator acting on another fractional state yielding a further fractional state, though it would seem that this would be an artificial way of looking at the problem.

A symbol game can provide an extremely powerful mathematical learning situation. There are, however, some drawbacks and difficulties. One difficulty, for instance, when played between teacher and children in a closs- room has the obvious disadvantage that the entire class can hardly ever partici- pate. Although UICSM have made a great point in plugging away at a certain relationship and not going further until practically every child in the class is ready to respond, this is clearly an ideal situation and practically unobtainable. Some children will either have to be held back and told not to tell the other children so that these other children do not get done out of their discoveries and the pleasure thereof, or if the level of excitement in the act of discovery is to be keptupbythosechildren that are quickest in getting there, then clearly some part of the class will have to be left out of the exercise; these “slower” ones will simply not follow what the other children are talking about. This is the inevit- able consequence of taking a class in a class discussion. It can be remedied to some extent by not always taking the whole class for a discussion but the teacher can perhaps select a certain section of the class that he or she believes to be ready for a certain new departure in mathematics. The rest of the children can be gainfully occupied in some other mathematical or even non-mathematical ac- tivity on their own.

The other difficulty, which has already been adumbrated earlier, is that beyond a certain point the efficiency with which mathematical symbols carry mathematical information in the case of children or indeed even in the case of adults, becomes much less efficient than the corresponding efficiency of verbal symbols in ordinary language. It is assumed in playing the symbol game that the mathematical language which is being used in the games has the same information

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carrying capacity as ordinary language has in ordinary situations. Although this is true in the case of fairly simple games, once the superordinate concepts are constructed and then superordinates of superordinates if no experiential component comes into the learning to bolster up the concept formation, it is quite possible that a lot of children will be left out on a limb and will not really have any very abstract idea of what is happening. We must not confuse symbolic work with abstract work. In the case of elementary school children who are mostly in the concrete operational stage, their abstraction still has to take place very largely from concrete experiences, and although they can manipulate images, be it con- crete images, once they have got on to manipulating symbols, when they start manipulating symbols for the manipulation of symbols, they might easily break the thread which ties them to their concrete roots. An eventual concrete basis is after all the root of all scientific, including mathematical inquiry. In some psy- chological experimentation that has been taking place lately, it has been found that it i s possible for children in the concrete operational stage to build ma- thematical constructions out of previously constructed abstractions, but these ma- thematical constructions have so far only shown to be possible to build as a result of accompanied representotive concrete experiences. That is, for example, children are able to build the concept of algebra once they have built their con- cepts of a vector space and of a mathematical group. They can synthesize the idea of a two-dimensional vector space with a cyclic group with four elements and construct the complex algebra. They can handle this in a number of different embodiments, and eventually abstract the mathematical construction of the com- plex algebra as a result of their activities. To what extent they will be able to engage in such mathematical constructions without the aids of artificial concrete situations, only further research will tell. On the other hand, as we have seen in the case of the subjects in the Harvard Learning Mathematics Project in 1960-1961, it is possible for eight or nine-year-olds to use the symbolism of ma- thematics recently acquired in a creative way. It must be stated though that the cases where this has occurred has been in children of extremely high intelligence. In the case of children of not such high intelligence sometimes it may be noticed that the presentation of a concept in a symbolic and mathematically honest way loses a certain proportion of the children. Reframing the same concept in more concrete terms and exploiting the personal experiences of children might render the same concept within the reach of practically every child.

3.9 The Object-game-oriented or Multiple-embodiment Approach

Let us now consider those mathematics projects which pay attention to the set approach as well as to the multiple-embodiment principle in building up the mathematical abstractions of the children under their care. There are quite a

number of such projects now. In Europe the work in Leicestershire and in Surrey follows approximately these lines. There is one school in Italy, Scuola Cittb

Pestalozzi in Florence where the approach is being tried. In the Budapest Ma- thematics Project several classes are now working. The treatment of logic and

of sets is followed by the use of multibase materials as an application of the “deep end” principle. The multiple embodiment principle is resorted to for the creation

of abstractive learning situations. In Australia, the Adelaide Mathematics

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Project is operating on this basis. There are also a number of schools in Victoria and New South Wales and Queensland working in co-operation with Adelaide, as well as the Papua New Guinea Mathematics Project started early in 1964. There is also the Philippines Mathematics Project started in 1962, working on similar principles. The basic-set approach as developed by Patrick Suppes and Paul Rosenbloom in the United States, is being used in all these projects with the exception of the Leicestershire and Surrey ones. There the logical approach using attribute blocks is being tried and it is hoped that the sets approach will eventually accompany this. It is impossible to describe the workings of this ap- proach in all aspects of the mathematical situation that are being created, but let it be said that the dynamic principle of allowing the children free play in the beginning now commonly recognized by many projects as essential, is being given the first priority in all these projects. Also the multiplicity of experience of the kind as well as of the amount is being given considerable attention in all of them. The multiplicity of the kind of experience is calculated to lead to greater depths of abstraction and the variation of the different mathematical variables is calcu- lated to lead to greater extents of generalization on the part of the children.

We shall describe briefly how these principles are put into practice, for example, in the creation of a learning situation where directed numbers are learned. These situations can be created with children of nine and over. The mathematical basis is that a directed number may be considered to be the common property of a certain class of ordered pairs of natural numbers. Naturally, children are not told this but are asked to sortout piles of objects in which there are some of each of two kinds as for instance, pens and pencils, or cups and saucers. They sort these out into piles in which there is always the same number more of one kind of object than of the other kind. Some square blocks, for ex- ample, have been painted red and green respectively and children will put out “two more piles” as an exercise for the getting to grips with the equivalence of “two more” situations . For instance, they might put three green squares and one of the red squares in the pile; or else perhaps ten green squares and eight red squares in the pile and so on. That is, whatever pile they make in that exercise there will be two more green squares in each pile than red squares. These piles will all have the property “two more green than red”, or plus two as applied to green squares. They naturally also make other piles in which not the “two rnoreness” but say the “three moreness” or the “five moreness” properties are established, Of course, it is not always the green ones that there are more of, but sometimes there will be more red ones. It is quite curious how children hate to call a pile a “less pile”,they prefer to call a pile in which there are “two less green than red”, “two more red than green”. This is being allowed for quite a long time. In fact a terminology has been evolved by the children of calling the piles green piles and red piles depending on whether there are more green than red or more red than green. This immediately leads to the idea of zero piles where the zero does not mean nothing, but it means that there are the same number of each color in the pile. This establishes the physical equivalence of the classes of the ordered pairs which are equivalent and whose number properties are known as positive two or negative three, etc.

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Additions are introduced by uniting piles and subtraction by removing subsets of the piles from piles. It must be understood, of course, by the children that any time during the operations a pile can be replaced by an equivalent pile. That is, for example, if there is a five more green than red pile and you need to remove two more red than green from it, there might not be two red pieces in that pile. This can always be remedied by putting say five red ones and five green ones in the pile, which does not alter the number-name of that pile. Here children are getting used to operating on a different universe of discourse, that is a different universal set, where numbers now are no longer properties of sets. But they ore properties of pairs of natural numbers whose concrete representations are accessible to them on their school tables. So additions and subtractions can easily be executed by unions and differences as before, but the equivalence re- lations are now different from before. Lest children get used to the idea too quickly and begin to think that these sorts of judgments can only be made in one kind of situation, they are thrown almost immediately into situations where there are not just red and green squares but also red and green triangles and probably red and green circles. It will be seen that by doing this, multi-dimensional vec- tor spaces will begin to be represented in concrete situations. These will not be representatives of vector spaces until multiplications by scalars are also represented but this also comes later on, once the group with two elements has been studied in other situations, too. The representative of three-dimensional vector will in the particular embodiment be, for example, a pile of red and green squares, a pile of red and green triangles, and a pile of red ond green circles. And the vector name of such a pile is going to be an ordered set of directed numbers and each directed number is the common property of a whole class of ordered pairs of natural numbers. So we are building really very fast here. We are throwing them in

truly at the deep end.

It would be unpardonable to use only this one kind of representation because what we described earlier as a perceptual block might easily develop, and all except more gifted children might consider vector spaces as essentially to do with red and green pieces of various shapes. So, for example, the conven- tional spatial arrangement can also be introduced and a grid can be pointed in the school yard or maybe just the floorboards in the classroom itself can be used. Chalk marks can be drawn across the floorboards and the floorboards themselves can be used as units of displacement, which can be counted. So we can have across steps and floorboard steps. The across steps are taken when you walk across the floorboards and the floorboard steps are taken when you walk along them. While moving along the across steps, on one side of the classroom there might be a window and on the other side there might be a door. So we can speak of a window step which is towards the window and door steps which are towards the door. Walking along the floorboards we can speak about chalkboard steps and locker steps if the floorboards stretch between two such objects. So we can speak of window steps which are towards the window and we can speak of door steps which are towords the door. Walking along the floorboard, we can speak about chalkboard steps which are towards the chalkboard and the locker steps which are towards the locker. Children can start from any point on the grid and

go for walks. Somebody could make a record of all the steps that he makes in all

the various directions. At the end of the walk the history of his walk has been

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recorded. It can be established whether he walked more steps towards the black- boards than towards the locker or the other way round, or whether he walked more towards the door than towards the window or the other way round. Supposing he had walked more to the blackboard than towards the locker and more towards the window than towards the door, such a walk will be called a blackboard-window walk because the blackboard steps exceed in number the locker steps and the window steps exceed in number the door steps. Of course, the equivalence ex- ercises will have to be practised in this situation, too. Children will have to go for a number of different walks with the same walk name. The name of the walk, therefore, can remain the same for a number of different walks. And so all the possible walks are brought into equivalence classes on the principle of the excess number of walks in one and the other of the two dimensions that are being used. Again walks can be, so to speak, added and subtracted, that is, put together and removed from each other. The joining of the two walks is not very difficult. At the end of the first walk you can start on another walk, and simply consider that the whole walk is one. “Taking away” is a little difficult. One can per- haps pretend that half way through a walk one got on a bicycle or a motor car and did the rest of the journey by using a mechanical device. So being given the entire walk and its name, and being given the part of the walk that was done by mechanical devices and its name, the problem is, what is the name of the walk done on foot ?

So this is another representation. Many other representations can be found. There is, for example, the story of the dance hall which is often told to represent two or many dimensional vectors. The story is approximately this. Boys and girls go to a dance. The rules of the establishment are that children are only allowed to dance with children and adults are only allowed to dance with adults, and it is only permitted to dance with a member of the opposite sex. Also it is compulsory to dance if you have a partner to dance with. If you have not a partner that you are allowed to dance with, then you go to the refreshment room, and drink a cocacola. Now it will be seen that if there is nobody in the refreshment room, then there must be the same number of boys as girls and the same number of men as women. This is the zero situation. So this is a zero party and if a zero party comes to the dance hall, then it does not affect who is in the refreshment room. The people at any moment drinking coke in the refresh- ment room are the representatives of a two-dimensional vector. The opposite sexes are the opposite senses in each direction and the children and the adults represent the two dimensions , of which this particular vector space is built up. Parties can be joined together to practise addition exercises and parties can go home or various accidents or mishaps can happen to the dance hall so that boys and girls and men and women are required to do repairs and it can be asked just how many people are left in the refreshment room while the repairs are being carried out. This, apart from giving another embodiment and furthering the abstraction process, also makes use of perhaps a somewhat untapped pedagogical device for mathematics learning and that is the mathematical story which relies on the mani- pulation of images created by the telling of the story. In Paul Rosenbloom’s project much of this is done, and whole teams of story writers get together and study the kinds of stories which young children will find stimulating so as to get them into the situation where they can manipulate fairly severely constructed

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images for the purpose of learning the corresponding mathematical structures. In projects that use multiple embodiments, image manipulation is part of the learning process set up for the children. Various other situations are also used, napkins and napkin rings, or balance beams with metal washers on hooks that are placed at regular intervals from the fulcrum on either sideandsoon, for establishing the ideas of directed number and vector. In the case of the balance beam the over- balancing on one side or the other side are the representatives. In the case of the tables, the excess number of knives to forks or the excess number of forks to knives or else the same excess numbers in the cases of cups and saucers, napkins and napkin rings or any other utensil that can be put on tables that can be used either as physical or as image representatives of multi-dimensional vector spaces. It is considered important that the geometrical representation should not be the only re- presentation of vectors. This would necessarily confine children’s thinking to, at best, three dimensions. Children that have been brought up on the multiple embodiment and image manipulation technique will not find it difficult to think in any number of dimensions because for them a dimension is not necessarily confined to movements in space which are independent of each other but are reb- garded as measurable situations in life generally, which are independent of each other. Of course, dependent situations can be artificially introduced. For example, we might say that one day the manager at the dance hall said that in the refreshment room there must be the same number of males as females. In that case, if this rule is violated, people simply cannot come in. This will create a one-dimensional sub-space out of a two-dimensional space represented by the re- freshment room. This can naturally be applied to the other embodiments of the vector space situation.

The next step is to build a multiplication situation out of the vector spaces. The group with two elements is the structure which is used for multiply- ing the vectors by scalars. Since the group with two elements is so common in all parts of mathematics a great deal of time is used for finding situations in which this is embodied, but it has been found, following the deep end principle, that it is easier to teach the group with two elements by first introducing a group such as the Klein group or the Cyclic group. The Klein group has been found the easiest to teach, partly because it has three sub groups each of which being the group with two elements and partly because it is psychologically a more sym- metrical structure. It has a larger number of automorphisms, that is self-mappings than the Cyclic group with four elements, more, of course, than the group with two elements which only has the trivial automorphism in it. So, for various psy- chological and practical pedagogical reasons the problem of establishing multi- plication in a vector space is broached by introducing a number of situations from which the properties of the Klein group can be abstracted. Again the multiple embodiment principle is used so that not only one game but several games are played in which the rules correspond exactly to the rules in the Klein group. For example, movements along the sides or across the diagonal of a rectangle will provide such a situation, or the flicking of one or both or none of a set of two light switches and the corresponding transformations in the lighting or otherwise of the classroom will also provide such a situation. Two children facing the class-

room and turning round one by one, or both of them or none of them, to face the blackboard or the class, also provides situations of the same kind. Having

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extracted the properties of the group with two elements from such games, it will be realized by children that the group of two elements has a neutral element which they often describe as the non-changer and the other one which they some- times describe as the turner-abouter, or the changer. This structure can be used again in story form or in mechanical form in games with structured materials. For example, an operator pile can be put next to a state pile in the case of red and green pieces. If a number of green ones is put next to a vector then that means that the pile has to be taken as many times as the number of green objects that have been put in the operator pile. If red objects are put in the operator pile then such creation of multiple piles will still have to be carried out, except that the color of every piece will have to be changed around. We can choose, for example, the green as the non-changing color and the red as the changing color.

so, for example, we can put two red objects in the operator pile. They can be two red counters or two red matches. They will have the effect of doubling each pile of the representative pieces for our vector but also changing the color of each object in the pile. So we have now introduced a multiplication by a scalar or its physical equivalent in the form of a game and so we now have a complete vector space. Of course, we also have an algebra, if we only use a one dimensional space. We need only use squares and then in the operator pile, we likewise put red or green squares. This situation parallels exactly the situation of the multiplication of directed numbers.

Following the “deep end principle” it may not be sufficient to stop at this. Children could be encouraged to build algebras of this kind with different kinds of operators. Supposing we have red and green triangles, and red and green squares, then the green square could be picked as the non-changing operator, the red square as the color changing operator, then children will nearly always suggest that the green triangle should be the shape changing operator and the red triangle should be the shape and color changing operator. This leads to an alge- bra which is not the complex algebra, but very close to it. It will be seen that in this case the unit element has four square roots instead of two as in the complex algebra and the red square, that is the “minus one”, does not have any square roots. To introduce the complex algebra, instead of the Klein group which is represented by the present suggestions for the operators, we could use the proper- ties of thecyclic group of four elements as the properties of the operators. A good embodiment would simply be the use of arrows drawn on pieces of cardboard. We could put the arrows either to point from left to right or right to left or away from us or towards us. And then the operators would be defined as follows: An arrow pointing away from us could be the non-changer and the arrow pointing towards us could be the changer of the direction to its opposite. An arrow point- ing to our right could be a turner-through-a-right-angle towards the right, that is in the clockwise direction and an arrow pointing towards the left could be a turner-towards-the-left, that is in the anti-clockwise direction. The properties of this game will give children the properties of complex algebra. The arrow pointing away will be the representative of the unit element. The arrow pointing towards you will be the representative of what is conventionally called negative -

1, and the arrow to the right will correspond with positive - i, and the arrow to the left will correspond to negative - i, or the other way round.

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According to the multipleembodiment principle abstraction can take place as incidental learning as a result of coming across different forms of the same structure. It was seen in experiments conducted by Suppes, however, that at least in the case of young children this did not take place to any extent. If particular attention is not drawn to the similarities in the different embodiments of the structure such incidental learning does not take place at all. One way to draw particular attention to the similarities in the stimulus material, is to give children special tasks of identifying the properties of one embodiment with the properties of the other. For example, between any two physical representations of the Klein group, we can get children to establish a one-to-one correspondence between the respective representations. This one-to-one correspondence can then be “played with” to see whether it does in fact transform true sentences which can be constructed in one embodiment into true sentences in the other em- bodiment . This is, of course, always possible and, in fact, in the case of the Klein group there are always six possible such one-to-one correspondences, usually calleddictionaries by the children, corresponding to the six automorphisms in the Klein group. In the case of the Cyclic group with four elements, there are only two suchdictionaries possible. Dictionaries should likewise be establish- ed between representations of the vector spaces. It will take some children quite a while to realize that a necessary and sufficient condition for a dictionary to work is that opposites should be made to correspond to opposites so that as long as this condition is satisfied the dictionary can be constructed in any w(1)’ at all.

These games of “making dictionaries” which are really isomorphism exercises can then lead to problems in which it is impossible to construct an iso-

morphism. It was found possible in one experiment conducted at Sherbrooke in early 1964 with some second grade children for one of them to discover sponta- neously that there was no dictionary possible between a representation of a Klein group an one of the Cyclic group. This child, after about a quarter of an hour of trials, simply went to the blackboard and told the class that they were wasting their time because the whole thing was impossible and produced immediately a classical reductio ad absurdum argument. The age of this child was only eight. This would appear to show that at least in the case of some children the technique of playing around with isomorphisms provided such children with a mathematical tool which encourages the development of quite mature mathematical thinking in

the very young.

3.10 Next Steps

It would seem that as a result of the work of the projects mentioned, the application of the “deep end” policy, that is of the policy of teaching a

structure by also teaching at the same time a superordinate structure including the one to be taught, is a pedagogical policy which pays off. Likewise the multiple

embodiment principle has been in use for a sufficiently long time for us to know that it does lead to fruitful mathematical learning. Experiments are now in pro- gress to try and test the hypothesis that there are significant differences in the concept attainment of those children who are given a multiple embodiment treat- ment as against those who are given a one-embodiment plus over-learning on this

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embodiment. Much more research in the study of the learning of structures will have to be carried out before we can build up a truly rational basis for the teach- ing of mathematical concepts. Much is coming to light but very much more still needs to be discovered. It would be idle to say that today we have any solid knowledge on which we could base a sci- entific pedagogy of the teaching of mathematics. On the other hand there are sufficient indications that we can use what knowledge is available in the setting upof mathematical learning situations, which are very much more efficient than the traditional ones which are still the rule of the day in most countries in the world. It is very necessary that psychological researchers, mathematicians and active teachers in the field, should begin to cooperate in mapping out the problems and their solutions. As soon as solutions are available, teachers should begin to apply them in the field. This is the task of every mathematics project which has been discussed. It is o credit to the workers who have been engaged in the field that such co-operation between professional ma- thematicians , professional psychologists and professional teachers is now attempted in a great number of centres in different parts of the world. There are still, however, many learned institutions where psychologists will not soil their hand with work in the classroom, but will put on their white overalls and go to the laboratory and run rats in mazes with much greater equanimity. Similarly mathematicians in the Mathematics Departments of Universities are finding it difficult to associate with their teaching colleagues working in secondary, primary and kindergarten classes. Fortunately, however, the tide is now turning the other way and there Is hope of far more co-operation in the future than has been in the past. Enough has been said to make it clear that such co-operation is essential. If the founda- tions of mathematics are to be taught to our children, this must begin at the very beginning when they first come to school. The foundations have to be laid at the beginning and, therefore, the foundations of mathematics must have relevance to the foundations of learning mathematics. Research mathematicians, who are conversant with the modern workings of the foundations of mathematics, should be encouraged to have as much contact as possible with the present mathematical knowledge and awareness of up-to-date educational practices among teachers in different parts of the world. This leads to the very great problem of teacher training. Teachers, particularly in elementary grades, are not well prepared by their own mathematical education, to handle the new mathematical learning situations created by the new mathematics projects. It will be necessary to put across in most parts of the world a mathematical re-education of teachers, SO as to give them at least a bird’s eye view of how the mathematical foundations hang

together.

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ON THE TRAINING OF TEACHERS

Chapter 4

ON THE TRAINING OF TEACHERS 4.1 Methods of Teacher Training

It is quite clear that the kind of reform of mathematics teaching which has been reported here will be impossible without some form of re-training of the teaching personnel. Neither the mathematical nor the pedagogical training of our present teachers for the most part makes them well adapted for coping with the requirements of the sort of mathematical classroom situations which would have to be created in order to produce in large numbers mathematically well in- formed and I iterate people.

There are essentially three ways of handling the problem of teacher training :

1. To provide materials which teachers can read. They could for example learn their mathematics out of books, and the new teaching techniques out of written suggest ions.

2. Courses or workshops could be organized where teachers could learn both mathematical content through lectures, demonstrations or mathematical la- boratory situations, and also have practical expel iences. Each such course might extend its value by training one or two apprentices.

3. To create the learning situations within the actual schools where the teachers are practising. This means establishing new types of classes and thus training the teachers on the classroom floor. In this way, perhaps with a little added reading, teachers would have the chance to learn the necessary mathematics and pick up incidentally the ways and means of creating and developing mathematical situations which are pedagogically useful.

4. Have teachers participate in various programs in roles other than that of a learner. It would be an especially valuable bit of training, for example, if they were to work as research assistants in learning experiments. They could be trained to serve as observers, collecting information about the learn- ing process of students. Other appropriate roles which have strengthened

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some of the best teachers are writing or evaluating new curriculum materials and teaching others with less mathematical knowledge.

All these methods and combinations of them have been tried in different parts of the world in the various different mathematics projects here described as well as in those not mentioned. the combination of all three.

Probably the most effective method of training is

4.2 Mathematical Training

Let us consider first the problems presented by the requirement to re- train elementary school teachers in mathematics. For the most part elementary

school teachers the world over have had little or no mathematical training. Quite a large proportion .of teachers have not studied mathematics beyond the age of 14. This means that apart from certain processes which they have learned to carry out, such teachers would be completely ignorant of what mathematics is, not to men- tion any knowledge of detail in its foundation or of later developments. These teachers who have had secondary training in Mathematics have for the most part had the traditional training in which algebraical manipulations and the rote- learning of the geometrical theorems have been prominent. The aim of such learning would have been the passing of examinations through the rote-learning of solutions to certain types of problems frequently occurring in these examinations. Mathematicians will agree that such training very seldom results in what they would call a knowledge of mathematics. The learning of the properties of struc- tures and of how they hang together is not achieved by rote-learning. Teachers

who are in possession of some mathematical knowledge will in some sense be even less fortunately placed. They will have to be convinced that what they learned is not on the whole what is now considered as mathematics. They will have to change their attitude to mathematics as such and look at it from a structural point of view rather than from the point of view of getting answers to certain questions.

Let us now consider those teachers who have been even more fortunate and have had a certain amount of training in what a mathematician might call Mathematics, that is, who have penetrated into the problems surrounding the re- lationships between abstract constructions. For the most part the teaching of such real mathematics would have been carried out in a highly logical manner, very seldom in an inductive manner. If mathematical structures are presented to young children in a completely logical predigested symbolic way, it goes without say- ing that such mathematics will not be assimilated into the thought pattern of such young children. So even these most fortunately placed teachers who do, in fact, understand to some extent or even to quite a considerable extent the structural properties of mathematics, parts of which they are called upon to teach children, will have to change their attitudes and realize that deductive rigor and logical analysis are not the beginning but the end point of mathematical attainment.

It is only those mathematicians who are on an even higher rung of the ladder, that is, those that have actually contributed themselves to the discovery of mathematical relations not previously known, who would realize the extent to

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which the inductive method is used in the search for new relationships. In new research anything and everything is allowed and it is only when a hypothesis is to be tested and proved or disproved that absolute rigor is required of himself by the practising mathematician. It is, or course, very rare that a researching mathe- matician would be found teaching young children in the elementary school ciass- room. On the other hand there is no need for too many such research mathema- ticians to be teaching elementary schoolchildren, so long as there ase some who can then inspire the teachers that watch such teaching. An enthusiastic mathe- matician can excite children with the beauty of mathematical structures that he is able to get them to play with. The excitement that such abstract games can cause among quite young, unsophisticated, and not particularly gifted children is unbelievable.

It would seem, therefore, that the research mathematician is in a position of great responsibility if he is to take an interest in the development of the understanding of mathematics in the child population of the world. This is being gradually better and better understood by research mathematicians all over the world and such people are now getting together and building syllabuses, manning the staffs of mathematics projects and doing demonstration teaching, creating learning situations of various kinds. More and more of the mathematical skill of such people is being put at the disposal of both the teacher training and of the child-learning situations. Even so, there are clearly not enough research mathematicians who are interested enough and able enough to cope with the prob- lems of the elementary school classrooms, to man all the necessary posts the world over where such work will be necessary. So some kind of delegation of their abilities and powers would seem to be necessary.

There are two ways of doing this. One is by writing books, and the other is by holding workshops and persuading other practising teachers to learn the appropriate mathematics and the techniques of imparting such mathematics to children. A number of first rate mathematicians have given up their time to write books in which mathematical thought and the handling of mathematical structures is explained to the laymen. Such books are, for instance, Courant and Robbins, “What is Mathematics ?” and a work by Rozsa Peter “Playing with Infinity”. One could quote a whole string of others from which interested educationalists could, if they wished, gather to quite a large extent what the modern strands of mathematical thoughts were. It would, therefore, seem necessary to encourage more research teachers to read them. Libraries should be prominently stocked

with such works and possibly discussion groups could be held during which mathe- matical ideas contained in such books could be discussed. Such groups could be led by members of mathematics departments of local Universities. Such work needs not to be confined to teachers. Very often parents become interested in new work of this kind and parents and teachers and sometimes children as well have been known to join discussion groups for several months on end studying pure math- ematics for its own sake. The popularizing of mathematical thought is just as much a necessity in today’s world as a hundred years ago the popularizing of the ability to read and write was a necessity. Any way in which the spread of the knowledge of how to handle mathematical structures can be encouraged should be

taken.

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4.3 Educational Training

It is not really possible to separate the mathematical from the pedago- gical province. Particularly is this almost impossible at the elementary level. On the whole an exposition by a mathematician of some pure mathematical theo- rems will leave an elementary school teacher cold, but watching a class of high- ly motivated and excited children enjoying a mathematical theorem will have quite the opposite effect on such a school teacher. If an elementary school teacher can be shown such general interest being aroused in pure mathematics among his own students, he will then become activated to learn the mathematics as he will realize that without knowing this, he will not be able to create these exciting situations.

If the truth were known, probably most, if not all the mathematics projects functioning the world over began at first in one or two classrooms. The original participators intended to show to the teachers and their administrators in charge that the learning of abstract mathematics was a practical possibility for young children, and that it was found exciting by them. It is more often than not out of such humble beginnings on the classroom floor that most of the mathe- matics work with children now going on in different parts of the world began. Of course, in practice it is impossible for a mathematics project to reach every single teacher individually and, therefore, the change often takes the form of seminars and discussions followed by or accompanied by the observation of demonstration classes in schools where the new methods are already in action. If possible such training culminates in active participation by the teacher who is given the chance to get the feel of how the new situations handle.

In spite of what has been said about personal contact being desirable in such teacher training, not all teachers may get the chance of actively participat- ing in discussion and in handling situations. It is necessary, therefore, to have the written word available to which teachers during and after training could refer. This has been the practice in the mathematics projects in the United States such as the School Mathematics Study Group, the Minnesota School Mathematics Center and also in the Southhampton Mathematics Project in England, which has in some sense modelled itself on the American projects.

The practice has been for a writing group toget together for several weeks during the Summer to write the books for the following year through discussion and argument, after which these books would be tried out in a number of trial schools during the following year. Feedback would be requested so that during the year after a revised version of the same or of new mathematical materials could be prepared by next Summer’s writing group. For example, in the UICSM, they are now bringing in the eighth version of some of their materials. This means that they have been very alive to the constant need for reflecting the opinions of the users. They also take note of the feedback to further their own understanding of the problems through constant discussion among themselves.

Although it may be that mathematics can be learned to some extent from books, the characteristic style of teaching which appears to have developed

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as a result of the inception of the new mathematics projects is almost impossible to learn from a book. The UICSM h ave pioneered from the very beginning discovery style of teaching and this was made possible by Max Bedermarfs active participa- tion in the teaching situation himself. Naturally the deep insights which such teaching situations constantly provide for the teacher in charge have been instru- mental in suggesting the kind of methods which have since been found very profit- able in the teaching of the new kind of structured mathematics.

Robert Davis has remarked that another important aspect of the new teaching is not merely the aspect of the discovery being left to the child, but also what he has called the light touch. It appears that the learning situation is a somewhat delicate psychological situation. Teachers , probably without mean- ing to, very often tend to bulldoze through such rather delicate situations of which they possibly are not aware. A child will very often not dare to respond in case he gets something wrong. Even the very word “wrong” seems to be inap- propriate. A moral stigma appears to be attached to making an error. Perhaps the words correct and incorrect would be very much better employed in such a situation.

Also if a child makes an incorrect response, the new methods would suggest that instead of telling this child that he is incorrect and telling him the correct answer, a question would be put and the whole thing would be discussed by the class. The teacher would suggest that John thinks this and George thinks that, and well, who can defend his answer ? And so a debate can take place on the pros and cons of the situation and eventually,of course, the teacher as the Chairman can direct the discussion to such mathematically relevant situations which would eventually lead to the correct solution of the problem through class discussion . Such discussion would not make the person who committed the error feel inferior and, in fact, he might feel that if it had not been for his error there would not have been an interesting discussion.

Teachers often need reminding that a child who has pages and pages of correct exercises to his credit has not been doing any learning. During any learn- ing situation while something is still being learned, errors are bound to be committ-

ed. Once no further errors are being committed, no further learning is taking place, except, of course, overlearning. As has already been said the relation- ship between overlearning and abstroctive learning is not yet clear, and awaits experimental investigation.

It must not be imagined that the pedagogical efforts in developing a new style of teaching has been confined to the United States. Much has been accomplished in Europe, particularly in Belgium by Papy and Servois. Paw, at the Free University in Brussels, is responsible for a very large systematic effort in the secondary sector of Belgian education in which the teacher has assumed o very important part. There ore, in fact, in Belgium two official syllabuses, the

old one and the modern one. And schools have to apply for permission to use the new syllabus. Such permission is not given unless a certain number of their staff have been to the now hallowed workshop at Arlon and have satisfied the di- rectors of the mathematics project that they will be able to handle the new learn- ing situations created by the new syllabus.

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Much hos been accomplished by many individual workers in different parts of Europe, such OS Puig Adam in Spain, Pescarini in Italy, Gattegno al- most all over the world, Krygowska in Krakow, Poland, Varga in Budapest, Mlle Felix in France, all of whom and many others have pioneered the discovery approach and the light touch. A number of these individual workers have banded themselves together into what they have termed the International Commission for the Improvement of the Teaching of Mathematics, and they still hold yearly meetings in different parts of Europe which are completely unsubsidized and where every participant pays his or her own expenses. At these meetings discussions are held. No lectures are on the whole given as such, but discussions are followed by demonstration lessons and again discussions. The result is a very stimulating exchange of ideas on the problems of the teaching of mathematics. Teachers of mathematics and elementary school teachers in their host country in every case have enormously benefitted from these meetings and gone back to their classrooms with renewed vigor and determination to put into practice the new mathodology of mathematics teaching.

It would seem that the banding together of interested individuals and organizations for the purpose of discussing and promoting the better understanding of mathematics ond teaching should be encouraged. Of course, the International Study Group for Mathematics Learning, under whose auspices this present booklet is being prepared, is another such international organization which is open to individuals and research centres interested in the promotion of the better under- standing of mathematics among children, and the improvement in the teaching and the better understonding of the problems of teaching or learning of mathematics under different conditions. Th ere is a great deal of interplay between the vorious projects that subscribe to the International Study Group, each being willing and eager to learn from the others and conversely to put results achieved at the dispo- sal of other projects. In this way it is hoped that a greater degree of international co-operation is being achieved and it is made less likely that duplication of effort will take place. In every part of the world, as many people as possible interested in the same problems are going to profit from even partial solutions to these prob- lems achieved in other parts of the world.

4.4 Some Problems in Under-developed Countries

It will be of interest to note that even in some under-developed countries it has been possible to institute a certain amount of teacher training, which will enable native teachers of such countries to handle some of the rather delicate mathematics learning situations. Perhaps one of the leost developed countries in the world is the Territory of Papua, New Guinea. Parts of this country have not even been opened up and are closed for communication by Euro- peans. Many natives have never seen a European. It is only since the end of the second world war that great efforts have really been made to bring a certain amount of culture and civilization to the population of the Territory. A mathe-

matics project was begun in the early part of 1964 at the suggestion of the Feder.81 Government of Australia. This has resulted in a number of key schools being visited very frequently by Dienes, who in the course of his visits attempted to

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train the head teachers, who were Europeans, as well as the others, who were partly native, and partly European on the staffs of the schools. It must be rnen- tioned that some of the teachers in the schools would only have had six elementary grades of education followed by one year of teacher training. Others would have had secondary school experience, but none of the teachers involved in the pro- ject have tertiary training. Even the European teachers who are involved are only now attempting to get University degrees while working on the job.

In the case of the native teachers with the minimum amount of training, it has been found necessary to map out to a considerable extent the exact procedure which they will be called upon to execute in the classrooms, and the head teachers would have to discuss each day in detail with these teachers what the procedure has been ond what it would be during the next day or two. These instructions would be put down on duplicated sheets and the teachers would follow them. This does not mean that the teaching was stereotyped. The children are being given mathematical games included in the new syllabus. In other words the technique

of training the teachers on the classroom floor by motivating them through getting children to engage directly in mathematics learning was again the beginning of this mathematics project. A training program has been instituted for schools oper- ating in 1965. This is done in connection with the existing schools and the exist- ing staff participating in the training.

From results up to date, there i:, 83 evidence to suggest that native children are in any sense less capable of learning mathematics than any other children. It does appear, however, that native teachers are finding it more dif-

ficult to learn this than the European teachers, as they have already acquired very fixed attitudes and these attitudes are difficult to change. Possibly the reo- son why the idea has got about that New Guinea children are not able to learn things in the some way as European children is because we have tried to push them into a framework which is entirely foreign to them. The formal work which hos nearlyalways accompanied the learning of mathematics in the past is not at all suited to the lives of these children, which are full of actions. It has been found that the careful selection of games in which children take an active part, that is, in which the movement of their own.bodies form an essential part of the mathemati-

cally structured games, has resulted in the mathematics being just as easily learned

by these children as by any other children. It is hoped that through the introduc- tion of mathematical activity on the physical plane, New Guinea natives will be able to be brought to the same level of culture as far as mathematics is concerned OS any other people in other ports of the world.

Perhaps it would be unfoir to describe the Republic of the Phillipines as an under-developed country, but it is certainly less developed than say most of the Western European countries or the United States of America or Canada. Yet during the past two years, largely through the energies of Mr. Ron Carlisle, the mathematical situation - particularly in the primary schools in the Islands - has radically changed. Literally dozens of workshops hove taken place. Experi- mental classes are springing up everywhere. The University of the Phillipines and the Laboratory School for which they are responsible are themselves now

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interested in the new possibilities opened up by the experiential approach. The same methods are used in the Phillipines learning situations as in Papua, New Guinea. In fact, there is now a considerable amount of interplay between the two projects, results being interchanged frequently and it is hoped that exchanges of staff will be possible very shortly.

It will be interesting to watch during the next year or so what becomes of these two Mathematics projects and how the children that have participated in the new mathematical learning situations acquit themselves when they reach high school and are possibly subjected to the rigor of more sophisticated mathematical thinking. If the foundations that have been laid are sound, we should see then the development of mathematical situations which could possibly have repercussions in other less developed or under-developed countries in different parts of the world as in parts of Asia and Africa.

4.5 Criteria for Successful Teacher Education

In the present changing world circumstances it is very difficult to state criteria for success in the education of teachers. Today’s requirements are not tomorrow’s, and what we regard as adequate training today, will in all probabili- ty be iudged inadequate in a few years’ time. It is certain that no teacher has ever finished his training. In the same way as other professional persons such as lawyers and doctors, teachers will have to accustom themselves to the necessity of keeping themselves constantly up to date in a world of rapidly changing needs. Clearly much more mathematical and methodological training is necessary than is now given to the vast majority of teachers. The question is how much more such training can be realistically envisaged as an addition to the present forms of training. How much mathematics should an elementary school teacher know ? How much psychological knowledge of the development of children, of the methods of thinking of children should he be familiar with ? Some attempts are made below in giving some criteria, but these must be regarded as merely guidelines.

1. Assuming some such content as envisaged by the Cambridge Conference of 1963, the question then must be asked how well the teacher is conversant with these topics. This means that the following topics must be familiar to elementary school teachers :

Sets, operations on sets, one-to-one correspondence and order properties leading to natural number, groups, rings, fields, vector spaces, including the building up of the real number system, complex numbers, at least two matrices, including scalar products, etc., geometry of transformations, sym- metries, rotations and translations at least in the plane, porallel projections (affine properties), point projections,topological ideas such OS regions, boundaries, joins intersections, insides, outsides,simple and multiple curves, multiple points, convexity (leading to projective ideas from topological ones), not to mention the properties of the elementary arithmetical operations with the full understanding of all the relevant algorithms involving such re- lationships as the laws of exponents, etc. One rough criterion might be

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2.

whether a teacher can follow and be quite at home with the UICSM High School Mathematics Course 1, or Patrick Suppes’ Logic for Elementary Schools.

All the above imply a high degree of skill in making both inductive and de- ductive inferences. Elementary school children will for the most part be working on the inductive level, at least in the immediately foreseeable fu- ture, but this does not mean that the teacher in charge should not possess an overview of the deductive structure within which the children under his care ore making their inductive enquiries. For example, conventional proofs in geometry should be recognized as abbreviated and logically incomplete. The presupposition that the logical volidity of steps in such a proof can be almost immediately seen by readers of the proof should be understood. Skill in de- monstrating the validity of such steps should be recognized as unnecessary but helpful in case of doubt. The possibility of obtaining a completely mechani- cal eleboration of each step ina proof should be realized so that one need never have a feeling that proofs depend for their clarity on some mysterious ability to follow mathematical reasoning.

Apart from the criterion of actually being in possession of certain pieces of mathematical knowledge, we must also consider the criterion of willingness and ability to acquire more such knowledge. Mathematics is a living Science, and if a teacher is to keep himself abreast of developments, he must be pre- pared to do some reading from time to time. Adler’s “The New Mathematics“ or Courant and Robins “What is Mathematics ?” might be quoted as the kind of standard of reading that teachers might be expected to engage in. The mathematical games and puzzles section of the “Scientific American”, also Martin Garner’s books growing out of this section, provide another kind of

example.

Another question we might ask is how well a teacher is prepared to help de- velop creative mathematical enquiry in his students. We might ask about the teacher’s attitude to guessing, for example. An authoritative teacher will frown on any kind of guessing, whereas a teacher bent on fostering the crea- tive side of mathematical thinking will regard guessing as an integral part of the mathematical thinking cycle. Collecting examples with a view to making generalizations, or checking previous generalizations, the constructing of proofs in however primitive and awkward ways, are further examples or crea- tive avenues that can be opened up for children by teachers. Books such as Lakatos’ ” Proofs and Refutations” give ample examples of the function of proofs in mathematical enquiry, and such books should be widely read by ele- mentary school teachers. In other words, teachers should have at their fin- gertips many avenues through the encouragement of which children can be shown at quite an early stage that mathematics is an inductive and creative

enterprise, and thot the proofs and other organizational details of mathema- tics are used to tidy up all the exciting new things that are constantly being discovered in this ever growing field of knowledge.

3. Another relevant criteria1 question would be: How well is the teacher prepar- ed to obtain insight into how children learn ? In our present state of know- ledge on this subject, probably the most important single point to bear in mind

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is still the development of skills on the part of the teacher in working with children on mathematical ideas. Most teachers will have plenty of oppor- tunity to develop such skills in their daily dealings with the children they teach, and probably a storehouse of information gathered first hand in this way by the teacher himself is still the most important single asset that could be listed under this section. Such a teacher must also be able to communicate his findings to other teachers, some indeed should make a point of describing their findings in local or national teachers’ iournals. The small amount of relevant knowledge, or indications we seem to have from the growing body of reported experiences of teachers and research workers, on how children in fact learn, is also essential for teachers to get to know about.

The literature teachers should be familiar with in this sort of field is hard to specify . However, if teachers are going to acquire sufficient analytical know-how to handle content adumbrated under 1 ., it is hard to resist the suggestion that they should be able to read the writings of Piaget, Dienes, Skemp, and others with a good deal more comprehension than most educators and psychologists are able to at present.

4. It is also relevant to ask to what extent teachers are able to understand the purposes of mathematical education, and to what extent they are able to ex- plain these purposes and the methods by means of which these purposes are served, to such people as their own pupils, other teachers, parents, adminis- trators, and the like. This means that teachers should be aware of the kinds of applications that are being made of mathematics in the society in which they live. Furthermore, the potential needs of society have influenced the methods of teaching, so it is necessary to understand the long range implica- tions of the new approaches: Teachers should be able to assess curriculum suggestions in the light of utility, as well as in the light of completeness of the whole mathematical picture which the particular curriculum suggestions offer to students.

It may well be objected that the criteria proposed are unrealistic because:

a) They might well be taken as descriptions of the innovators themselves, or at least of the assistants to the principal innovators in the field. Perhaps an unrealistically high ability is being demanded of teachers, if they are to fulfil1 all the above requirements.

This may well be so, but this does not mean that it is not wise to set our sights high. It will clearly be impossible to create large numbers of teachers of the kind that possess even the majority of the attributes enumerated in the near future, but it is not impossible to create some. These will, hopefully, become leaders and in turn create other more highly trained and qualified teachers, and possibly by means of a kind of snowball effect, the majority of teachers will be reached in a fore- seeable period of time.

b) It might be objected that teachers who are as well prepared for their jobs OS is suggested here will not stay in teaching, but will move to more attractive and lucrative jobs.

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This again may well be so. The answer is obvious. The status of the teaching profession must be raised to the same status as the other pro- fessions such as doctors, lawyers, etc. This means a stiffening of the training, of the selection, and an immediate raise by a very substontial proport ion, of the generally poor pay received by teachers in most countries in the world. It is unrealistic to assume that a first class job can be done by people who do not receive first class recognition for their services, in salary as well as in respect for their extremely important work.

c) It might also be objected that elementary school teachers cannot specialize so highly in one subject, as they are expected to know about other fields to the same extent.

In most of the present situations this would appear to be a valid ob- jection. There are, however, some partial answers. Possibly one consideration might be that the skills required in teaching such a delicately structured subject as mathematics may be greater than those required for the teaching of certain other subjects.

Another plan advocated by some is specialization or departmentalization in the elementary school. This is done in some places, and if a “Home room” is kept where children stay for at least two hours during the day, it appears that no great psychological disadvantages accrue out of departmentalization. Or in larger schools sometimes one teacher at each grade level might specialize in the mathe- matics allotted to that particular grade level, while other teachers of the same grade level might specialize in other subiects. These “semi-specialists” would lend an unofficial hand to those of their colleagues who happen to be in need during short periods, during which the classes for which they are responsible, can be left working on some tasks by themselves. It must be stated, however, that there is no consistent evidence that departmentalization in the elementary school results in more effective learning.

The best answer to all the above objections, however, is that ele- mentary school teachers in many countries of the world are finding it possible to put over programs and methods such as have been described. A dedicated teacher will find a way to do what he feels to be essential for the progress of the children under ‘his care.

4.6 Methods of Teacher Education

Many methods are possible for the improvement of teachers and pro- spect ive teachers. It is our intention here to suggest some means that might prove feasible in a given locality but have not been tried there. This is, therefore, only a suggestive list of possibilities and not a systematic analysis of the problem.

1. Consultants to the teachers of an elementary school may be obtained from a variety of different sources and used in a variety of different ways. The

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possibility of finding someone who con work out a relotionship with a group of teachers that provides real learning on their part with minimum threat or sense of competition should be sought whenever possible.

a) Possible sources of such consultants in math include universities and colleges, high school math teachers, personnel from mathematics cur- riculum projects, curriculum specialists of the school system, unem- ployed, but well-educated wives (school parents perhaps), volunteer workers from organ izat ions I ike the U. S . Peace Corps.

b) The many ways in which such consultants could work with teachers are suggested below in discussing methods of teacher education. However, some less formal ways should be described briefly. Any way in which the consultant can teach classes in the school while observed by teach- ers and also observe them teaching that does not create anxieties on their part is to be sought. Teachers should be encouraged to volunteer for such participation as “guinea pigs” or perhaps just to shore classes with a guest teacher for a few weeks. Communicating about teaching is at best a very tricky business and sometimes, when direct confronta- tion with the act of teaching (as exemplar or as object of critical’dis- cussion) are not possible, there is little effect of the visit of an expert on the actual practice of teachers. One important consideration is al- lowing time for teachers to go over in detail with the consultant o lesson observed, either his or theirs.

2. Local study groups of teachers con be formed (with or without consultants) which study both mathematics and pedagogy. This requires that some source of ideas be available, and the references suggested above moy be appropriate for such purposes. Sometimes, however, a new text may be appropriately used by a study group before undertaking to try it with children. Core should be taken to avoid letting such a study group become too formal so that the teachers involved lose sight of their role in helping children learn and con- centrate exclusively on their new role as students of mathematics trying to solve problems.

3. Pilot classes may be established in an elementary school in order to try out or demonstrate some new approach. Such classes might be taught either by a local teacher, an outside consultant, or by a combination of the two as sug- gested above. In any case, opportunities should be made available for the members of the teaching staff to visit the class frequently and to participate in the discussion of the teaching. Too often schools wish to decide on a change in the curriculum on an across-the-board basis. Although there are understandable reasons for doing so, the difficulties encountered in such a method of changing the curriculum of a school are typically so great (espe- cially teacher dissatifaction) that a reversal is made later. Pilot classes per- mit a realistic appraisal of what a change might mean by the staff, and the administrative problems involved should be tackled seriously. Typically, parents should be informed of such a class and the reasons for it in advance and given a chance to withdraw their child if they should object.

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4.

5.

6.

7.

8.

9.

10.

University and college courses, whether taken in residence, via extension or correspondence, are SO c,bvious a means as scarcely to deserve mention. However, there are some new possibilities that may have been overlooked both in the way of mathematics courses, courses dealing with pedagogy, and combinations of mathematics and pedagogy. Some of these will be mentioned briefly in the last section.

Another approach to teacher education is to have one or more teachers in residence in a school, without regular class assignment, to work on the fea- sibility of some change. Presumably pilot classes and other means of com- municating with the regular teachers will be developed in such cases.

Television and movies can sometimes be arranged, and several teacher training programs have been prepared for these media. In areas without such facilities, radio might well be used for the sarne purpose.

Self-study materials sllould not be overlooked. Some teachers, particularly are motivated to work individually better than in a group situation. A varie- ty of materials can be had which can contribute grecttly to a teacher’s know- ledge and teaching techniques. These include programed texts, either for teachers especially or student texts which illustrate teaching techniques that often can be used with a class as well as on an individual basis in programed form. Film strips and a variety of new mathematics texts are available. There are some rather poorly conceived materials on the market, too, ad- vertised as new mathematics materials. Some source of relioble guidance should be made available to teachers engaged in self-study.

Workshops for elementary school teachers led by a consultant are popular means of in-service education. They vary greatly in effectiveness and should be considered carefully by teachers and administrators alike before investing very large amounts of time in them. In mathematics, it seems likely that teachers should expect to cover significant amounts of new content in such workshops as well as discuss means of communicating it to children. It is possible to have quite large workshops if the leade,r has a number of well trained assistants.

Professional organizations are giving a good deal of attention to assisting teachers in keeping up with new ideas in mathematics education. The jour- nals dealing with elementary school mathematics teaching (and also those for secondary school math teachers) are worthy of careful attention by teach- ers.

Centres can sometimes be established in regions within eosy distance of a number of schools around. These may be formal operations with outside sup-

port, or they may be simply the result of one or two teachers who have ac- quired some advanced ideas and are willing to share their experience. School systems accessible to such demonstration centres should attempt to make it possible for teachers to take a day’s leave from time to time to visit these centres.

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11.

12.

13.

4.7

Teachers often become isolated from each other so that they know only quite vaguely what others are doing. lnnovat ions cannot spread effective- ly in such a situation and various means should be tried to break down the barriers that make each teacher’s classroom his own private kingdom. Several forms of team teaching hove been used which do involve frequent conferences between teachers about the instructional program and about the progress of individual pupils. Grade level meetings can also serve this purpose. However, these arrangements generally do not permit the teaching of one teacher to be openly examined by others, a prerequisite to effective diffusion of innovations in content and method throughout a school system. Co-teacher arrangements can be effective here, especially if teachers can be paired off for that portion of the instruction they share so that each has a definite contribution to make to the operation. Another arrangement calls for teachers to work in pairs with each observing the otherrs teaching. Any such arrangement is likely to consume more time than the usual isolation pattern. However, learning new content and pe- dagogy can proceed much faster when there is another person present with whom to discuss and analyze one’s lessons.

A teacher can analyze his own lessons to a very useful degree if he has a tape recorder and can take the time to play it back for self-criticism. Of course, the procedure of criticizing taped lessons can be useful in group instruction in the art of teaching as well as by individuals. In some research projects, protocols (records) of the classroom dialogue have been transcribed and can then be used for discussion and study by teachers in training. In- structional programs with student responses entered can also be used in this

way. One most valuable function is to permit teachers to focus attention of pupils with learning difficulties. Responses which cannot be interpreted appropriately during the class session can often be analyzed better after- wards, and then more appropriate remedial action can be planned.

Apprentice teachers form the backbone of student teaching in most countries. However, there are opportunities for some teachers to work in a kind of advanced apprenticeship to developers of curriculum materials or to profes- sors engaged in elementary teacher training operations. Typically, the project will be trying out a new approach and need teachers willing to learn it and use it in a classroom. These kinds of roles certainly tend to have a beneficial effect on the skill of the teacher - primarily perhaps because the teaching is scrutinized so carefully. Universities and colleges may well wish to consider establishing more formal advanced practicums (say in the teaching of mathematics) for credit.

A Sample of Teacher Education Programs worthy of Note

After describing briefly each of 11 different teacher education programs

both pre-service and in-service, which offer some promise in meeting the challenge

of curricular innovation in mathematics, a more detailed description is given of

one in-service program that has been used effectively to introduce new mathema- tics materials into a school system.

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1. Mathematics departments in a number of colleges and universities have in- stituted special courses in mathematics for elementary school teachers. This enterprise has received considerable inspiration and guidance from the Com- mittee on the Undergraduate Program in Mathematics of the Mathematical Association of America. Syllabi can be obtained from this organization which are actually being followed in a number of American institutions of higher education.

2. A number of colleges and universities have instituted a subject-matter major for elementary school teachers in preparation. This is in contrast to the prevailing pattern of education majors for these students. Although such graduates are usually expected to be able to handle any subjects in any grade of an elementary school, the depth obtained in one area is supposed to pro- vide a source of strength in that area for the entire faculty and to provide a greater sense of perspective in teaching any subiect. It would appear likely that such persons may also gravitate to the positions of curriculum specialists in the areas of their major fields.

3. Along a somewhat different line, several schools have experimented with laboratory courses in teaching for undergraduates. For example, at the Hilo Campus of the University of Hawaii, Easley and Miyashiro used the freshman orientation course in education as an orientation to the art of teaching math- ematics and physical science in elementary and junior high schools were judged appropriate subjects for all prospective teachers to work in for this introductory laboratory experience because new curricula were available which were designed to facilitate thinking on the part of pupils as opposed to rote-learning. Several mid-western universities have also experimented with having freshmen interested in teaching do a small amount of actual teaching.

4. At Webster College near St. Louis, MO., students preparing to become curriculum specialists for elementary school undertake a program which com- bines several interesting features in a way that promises to turn out excep- tionally well-equipped graduates. During each of their four years these students work in a school setting in a variety of roles. They take an under- graduate major in one of four fields, mathematics, natural science, social sciences or English. There exists at Webster College a national curriculum project in each of these fields. (For example, in mathematics it is the Madison Project, directed by Robert Dovis.) The students in this progrom each spend a part of their undergraduate program working with the personnel of the project in their major field to learn how content and pedagogy are related.

5. Another interesting idea combines pre-service and in-service training pro- grams in one where geographical situations permit: This has been developed particularly in some of the new nations of South East Asia to increase the rate of teacher improvement. Pre-service trainees have the advantage of more up to date content information and some new pedagogical ideas. In- service trainees have the advantage of classroom experience. When they

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can be made to recognize the mutuality of these advantages they can work together effectively teaching each other in a way that a single teacher trainer cannot accomplish.

6. The U.S. Peace Corps launched a massive program of educational oid in the Philippines, sending some 900 volunteer teacher-aides to work in village schools. Although the major thrust of this program was in science and English instruction, mathematics had a significant role. The aid was typi- cally untrained in any specialty but received intensive training in an infor- mal approach to working with teachers and children with materials that per- mitted them and their co-teachers and pupils to discover interesting scienti- fic and mathematical ideas. In mathematics, the materials used included Cuisenaire rods and Dienes Multibase Arithmetic Blocks. The volunteers thus provided a channel through which innovations could flow to the village school from a large variety of sources.

7. In 1964, the Chicago School System held a workshop for some 600 - 700 elementary school teachers in which they were introduced to a new mathema- tics program which they were to try in their own classes. Patrick Suppes led the group at the primary level and Robert Davis the group at the upper elementary level. Each of these leaders was assisted by a group of teachers who were experienced in the use of their materials. The reports from parti- cipants in this workshop received by the leaders after the teachers had begun to try the materials in their own classes were quite encouraging.

8. In 1963, the State of Illinois established a program of grants to local school districts who would establish demonstration centres to show a program for gifted youth. Each district proposed its own program to the State Depart- ment of Education which reviewed them and awarded the gronts for their operation. The centres have been highly successful as means of bringing to the attention of administrators and faculty new curriculum materials and teaching techniques. Many of the centres have employed new curriculum materiols in mathematics (as well as in other subiects) and obtained with the funds granted them, expert advice from University specialists in these programs.

A plan for an in-service progrom in a school or district.

1. First year :

a) Select teachers with high probability of success in teaching some of the new material along with old material ;

b) obtain a math consultant to work with them;

c) establish an in-service program for principals and resource people led by consultant;

d) arrange for all teachers to receive materials used by the teachers selected.

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2. Second year :

a) Repeated and expanded (teachers and topics);

b) principals and resource people meet, but not quite as often;

c) committee appointed from experienced people to recommend curriculum materials for next year by end of first semester;

d) public relations with parents and communify;

e) by end of first semester, every teacher involved in in-service training.

3. Third year:

a) Use new materials ;

b) in-service with own trained people;

c) trained people work with consultant.

4. Fourth year:

a) Continue with local resource people (consultants less frequently);

b) develop independence and add it ional innovation.

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CONCLUDING REMARKS

It is very clear that the crux of the whole problem of progress lies in the problems presented by teacher training. We have seen that there are various avenues along which the necessary mathematical and educational information may reach the teachers in the elementary schools who are in such great need of such informat ion. All these avenues should noturally be used, including possibly mass media of communication, such as radio, the Press and television. In the Sep- tember 1964 issue of the “Scientific American” a whole series of most excellent articles are to be found in which experts in the field explain to the intelligent layman what modern mathematics is about. There are also radio and television programs. The University of Washington at Seattle runs a mathematics program for teachers, as does the French television network in a twice weekly program at peak listening time.

Apart from teacher training, it is very important to bring the public in general up-to-date. Frequent reports in the Press of activities of mathematics projects, popular articles of the ins and outs of mathematics as distinct from a set of tricks learned to pass certain examinations, are all necessary parts of the pro- gram of mass education which will have to be undertaken to bring about a mathe- matically literate population of the world within one generation.

It is hoped that this report will be of some help in placing the most up-to-date information in the hands of those who need it regarding the develop- ment of the mathematics learning situation in at least some parts of the world. No such report could possibly hope to be either objective or comprehensive. Apologies are offered to the many individuals and centres that have not been di- rectly referred to in this report, or whose work has not been described. It is hoped that through organizations such as the International Study Group, more interested workers will be able to obtain information as well as disseminate it to others through central clearing houses or publications in iournals.

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APPENDICES

__. , I ._ - . . . - -___- . .

Appendix A

NOTES ON PROGRAMED LEARNING

The past decade has seen the rise, especially in the United States, of a teaching method known as “programed learning”. Effectiveness of the method has been tested in a number of investigations with generally positive results where care was exercised to adapt the type of programming to the subject matter and to the learner. Hence, it can generally be assumed that programmed learning has considerable promise as an instructional tool.

1. Origins

Origins of the method ore in the psychology of learning. Psycholo- gists who study the learning process have long been aware that the techniques of general education frequently fail to apply principles of learning which have been established in laboratory studies. In the typical lecture-textbook teaching situa- tion there are usually long delays between the behaviour of the student and its pertinent consequences, namely reinforcement or knowledge of results; yet it is a reliable laboratory finding that the efficiency of learning is diminished by such delays. The attention of the student is compelled by extraneous means, if at all, although it is well-known that attention is a crucial factor in learning. Further- more, the student is too frequently a passive agent in the process of his education, for traditional methods often require only his physical presence rather than his positive interaction with the educational materials. Even when his active parti- cipation is elicited via problem sets, recitations, workbooks, etc., the elicita- tion is sporadic instead of continuous. Finally, individual differences in learning rate are always present, and these are only superficially taken care of by splitting the child population into slow and fast learners.

At a deeper level there is often a failure of motive and purpose in classroom instruction. Despite considerable emphasis by pedagogues on the neces- sity for each instructor to follow a well-designed, p lanned route to a set of clearly defined educational obiectives, it is often the case that neither teacher nor student

know either the route or the goal. Even the most careful and preceptive teacher is usually unable to state his educational objectives clearly in behavioural terms.

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The list given above of discrepencies between our partial knowledge of conditions for learning and our traditional practices is admittedly incomplete and is offered only to illustrate the basic point. These and other difficulties are, of course, common knowledge among educators. It is sometimes a long step between diagnosis and remedy. A possible (and perhaps only partial) remedy has been put forward by psychologists, most notably in the beginning by B.F. Skinner of Harvard University, the method of programed learning.

2. The Method

Despite its youth, the method has developed so rapidly that even its taxonomy exceeds the reach of the present brief discussion. Instead of attempting a complete description of the various forms which programed learning has taken, only a concise characterization of what a program is will be attempted.

Basically a program is a sequence of single tasks, called “frames”. Each frame is a stimulus (usually a brief statement) to which the student must make a response. The programmer tries to make each frame unitary, that is, readily comprehensible, unambiguous (especially with respect to the response), and of single purpose. A frame should present a new association to be learned, or provide repetition and review of previously learned material, or provoke discri- mination between associated or stimuli and responses. It should attempt one and only one of these tasks at a time. Following the subject’s response, an accept- able answer,is exhibited so that he may compare his response immediately with the acceptable one. Then he proceeds to the next frame.

Thus : A program is minimally a text set up in a format which insures the con- stant interaction of the reader with the text and which provides immediate confirmation or correction of his every response. Maximally, the program can be an intricate arrangement in which the student is linked with o com- puter so that the student-computer interaction may be directed according to complex instructions which allow for great individual variation.

---.-

Even in minimal form, the program has other potential advantages over a conventional text. The materials can be written in conjunction with a group of students. Great educational efficiency can be achieved in the writing, for as soon as the material is poorly presented, the pilot group of students will begin to make excessive numbers of inappropriate or incorrect responses. A study of the structure of the responses will steer the writer to the source of the difficulty and show him how to revise it. After several such revisions, each involving an inde- pendent pilot group of students, a pedagogical document of strength and viability can be achieved by a creative programmer.

Moreover, a program cannot be written without careful and explicit definition of educational objectives. For the programmer accepts the responsibi- lity of shaping unambiguous responses within his individual frames. This cannot be done unless the objectives are completely worked out and expressed ultimately in terms of a reduction to the class of individual responses which occur. In many

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areas of study the responses may themselves be complex or delicate, yet the programmer must elicit them unambiguously and in proper relation to his large ob- ject ives.

3. Uses of Programs

There are a number of potential uses of programs in the educational enterprise . Some of these are :

a)

b)

c)

a) For the complete teaching of a course,

b) For remedial study,

c) For supplementary study, or course enrichment.

Courses have successfully been taught entirely via programs, success being defined as subiects on the whole eventually giving the acceptable responses. Advantages of the program over lecture-discussion-textbook teaching are most apparent in the domain of individual differences. Each student may proceed at his own rate without regard for the rates of fellow students. In addition, it is possible to design so-called “branching” programs in which each student gets an appropriate amount of repetition and review. In these programs, diagnostic frames appear. The response to diagnostic items determines whether each student should continue in a straight line or detour onto o side road for additional review.

A major disadvantage of the programed course relates to student motivation. With rare exceptions, programs presently available are not sufficiently com-

*pelling to hold the sustained interest of all students. Hopefully, gradual improvement in literary quality may help. Otherwise some invention of na- tural motivating devices which can be incorporated into programs is most de- sirable.

Remedial uses of programed materials are beginning. In most cases, an entire course, such as remedial English or mathematics, is given. There is need

for short programs over limited material. For example, students of college physics may not have mastered trigonometry; a short program over triangle trigonometry and its application to the composition of forces would have great

value.

Supplementary uses of programed materials also often consist of the use of a program for the entire course. The program is completed in addition to the

regular course work. Here again short programs seem to be most effective in use. A teacher, for example, often presents an important concept and illus- trates it by means of two or three examples. Some students need far more examples, but for others the two or three given are already too many. Short

.programs, each covering a crucial concept in a given field of study, would

be helpful.

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4. The Scope and Level of Programs

Programs seem to be most easily constructed via the written word. Hence the programming effort was initially directed toward adults and late adoles- cents. Programmers have, however, devised straight forward techniques for in- troducing visual materials, and there is currently a fruitful extension of programed materials into the elementary grades. Examples of mathematical materials at elementary levels in programed and near-programed format are cited in the refer- ences to Spooner, Suppes, etc. A by-product of the programming effort at ele- mentary levels has been the confirmation that mathematical concepts previously thought to be advanced and of considerable difficults can be taught successfully to children who are quite young.

The construction of a program involved breaking the major objectives down into a set of individual behaviours. This is most readily done in fields like mathematics or physical science where there is already a detailed, accepted structure. There is no a priori reason why other fields, such as those in the hum- anities, cannot be programed. What is required is a detailed statement of ob- jectives, a behavioural description of the objectives, and a program of sufficient flexibility to permit alternative routes through the field.

Programs are susceptible to continuous improvement. Both the program and the responses of students are potentially matters of public record. Analysis of the responses for the purpose of program improvement can be carried on as long as the program is in use.

5. Implications in Programed Instruction

One fundamental implication of programed instruction is that any learn- ing to be programed can be considered as divisible into a finite number of successive frames, such that by learning the acceptable response or responses to each stimulus in each frame along the succession, the learning will have been achieved. This immediately raises some questions:

1. What kinds of concept materials in mathematics, or what kinds of techniques of computation,can be subdivided into a linear succession of steps, so that the successful passing through of these steps guarantees that the concept materials or techniques have been learned ?

2. What kind of criterion is acceptable in deciding whether the learning has or has not been successful ?

There are no easy answers to the above questions. Clearly the second question would need a satisfactory answer beforewe could begin to answer the first one. The only type of behavioural definition of success in learning a concept so far advanced is the ability of a student to solve problems of quite unfamiliar character by recognizing the relevance of the concept to such problems without

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such relevance having been previously explicitly taught for such problems. Again the question arises, how unfamiliar should the criterion problem situation be? How much “noise” should be thrown in ? And most important of all, how are we to measure in any scientifically valid manner the extent of “closeness” of a certain type of problem to another type ? Of course, the question of success in learning is of the heart of all efforts to improve instructions. (See Section 2.17, Testing and Evaluation.)

Once a generally acceptable answer to the second question is forth- coming, the first question will be gradually answered by practical work on pro- grams. it is very likely thot a great deal of the mechanical techniques so neces- sary both in arithmetic and in algebra, and indeed in some of the “higher” branches of mathematics, will be able to be satisfactorily programed. This is likely partly because the criterion for proficiency in applying conceptual strup- tures which are supposed to be built up during mathematical learning of an ab- stract kind. It is also clearly easier to break down a complex technique into easier sub-techniques than to break down a concept into easier sub-concepts. The former is largely a technical problem, the latter is the combination of a psycho- logical and a mathematical problem.

It is difficult to see how other diagnoses besides success or failure at responding appropriately to a particular stimulus can be built into a program. This means that at the bifurcation points of a program the decision whether a student takes the “straight ahead” or the byway, depends just on his success or failure at one or several points along the succession of stimuli. It is becoming more and more clear through classroom experimentation that different children have different styles of learning and thinking, and quite possibly failure at one point might be due to the student having been taken along a path which he finds too foreign to his ways of thinking. Making such a student go through a review might eventually “stamp in” the appropriate kinds of response, but is unlikely to solve his problem for the next bifurcation point. Much research is needed to establish reasons for failure at particular points in programs.

Possibly the most fruitful attitude to adopt at this stage to the place of programed instruction in the educational scene is one of readiness to employ pro- grams, possibly a large number of short ones, to achieve proficiency along some of the mechanical stretches of the mathematical road, and a healthy doubt or “wait and see” attitude where it concerns the building up of abstractions and their relationships to each other.

6. Materials and Sources

Available programs in the United States are most comprehensively listed in the following :

Programs ‘63: A Guide to Programmed Instructional Materials. U.S. Office of Education, 3401543.

Programs ‘64: A Guide to Programmed Instructional Moterials. U.S. Office of Education, 34015-65.

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Appendix B

BRIEF REPORT OF A REFORM

by F. PAPY (Brussels)

The present report has no other aim than to show how a reform of mathematics teaching, which was undertaken in Belgium, achieved success.

Any attempt at reform should take into account local conditions which can vary from one country to another. Nevertheless, reformers in all countries find common ditticulties which can be overcome by using the same methods.

The Belgian reform has so far only been applied to teaching in secon-

dary schools. It is true that we have carried out a great many isolated experi- ments in primary education. These prove that a whole series of notions connected with sets, relation, binary enumeration and even groups, could be given a place in the reform of primary teaching.

Any reform of mathematics teaching involves above all instruction of the teaching staff. The Belgian experiment could not have succeeded without the organisation of courses by the Centre Belge de Pedagogic de la Mothematique (Belgian Centre for Mathematics Education). This year about 3000 secondary school teachers took these courses in twenty towns, twenty afternoons a year.

For the organization of such courses it is necessary to have a teaching staff whose members are mainly secondary school teachers. This staff should

itself be prepared by special courses of instruction.

The difficulty is much greater when it comes to primary school teach- ing simply because of the enormous number of teachers.

It was possible to retrain secondary school teachers who took part in the experiment, but the staff of the Centre was not large enough to deal with the number of primary school teachers. This wos the main reoson for limiting the

Belgian experiment to the secondary school level.

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The second phase of the reform will be aimed at primary teaching which will bring in its train a revision of the secondary syllabus.

Our research has always been carried out, however, with all levels of teaching in mind.

The reform was begun in 1958 by the experimental adoption by the Minister of Education and Culture of a modern mathematics syllabus compiled by Madame Lenger and Monsieur Servais and designed for pupils training to be primary teachers. It was intended to show future kindergarten teachers of children from 3 to 6 the importance of some of the basic concepts of modern mathematics.

These originate from common knowledge and may be perceived in the thought processes of very young children.

It is, of course, not a matter of attributing superior knowledge to them, but just of presenting by means of children’ games situations favourable to the birth and development of’ notions. We have moreover established that teachers who had followed this new syllabus had a completely new attitude towards children.

Experiments begun in the classes of future Froebelian school teachers have been very fruitful. This method of teaching designed for pupils in no way specialists in mathematics brought to light some basic educational techniques of the modern teaching of mathematics. Mention should be made, for example, of the systematic use of Venn’s diagrams, the discovery of multi-coloured graphs, and of the educational value of the binary system, the attempts at rapid introduc- tion to real numbers and the beginnings of geometry based on the notions of set, relation and axial symmetry.

In July 1959 at the end of the first year of this experiment, the first workshops at Arlon were organized, where 150 teachers gathered together. We had thought, it is true, that we would be able to reveal the basic notions of sets and topology to the teachers in two comprehensive lessons.

These lectures passed over the heads of many of the participants. The program was too ambitious and the lecturer had not understood that the acquisition of new ideas can only be achieved gradually and in suitable situations, irrespec- tive of the age of the audience.

Luckily, on the last day, a practical lesson for 15 year-old pupils introduced the notion of open sets. The audience, who had found the lectures tedious, were won over by the practical lesson. They blamed the lecturer for not having treated them with the same care as the pupils. “You should speak to us as you do to them”, claimed certain of the obiectors.

We have since had ample opportunity to verify the basic truth of this principle. Ideas are acquired at all ages in the same situations, through the same problems and by the same exercises. This fact is of great use in that it allows the teachers to absorb at the same time both the elements of mathematics and the

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educational techniques employed in its teaching.

Numerous visits to experimental classes gradually changed the climate of thought in the teaching profession, which thus became increasingly receptive to the idea of reform.

In 1960-1961, Fr6d6rique Papy began to teach a modern mathematics syllabus to training college students intending to teach children from 12 - 15 years of age (regents). This undeniably modern syllabus has at present been adopted provisionally by the official teaching body and the Catholic teaching profession. About 50% of the training colleges have already chosen the new syllabus and are preparing particularly competent future teachers for the new teaching.

From 1958-1961 the Commission for Reform set up by the Ministry of Education and Culture organized numerous conferences for mathematics teachers. The Arlon lectures, concentrating with increasing success on the needs of teachers, attracted an ever-growing public. We had become strong enough to begin an ex- periment on a larger scale. To this end we compiled the “suggestions for a new syllabus for 6th grade (12 to 13 years of age)“. These met with a favourable re- ception on the part of Monsieur Levarlet, who was then Director General at the Ministry of Education and Culture, and the mathematics inspectors. The syllabus was adopted as an experiment and taught in about 20 school classes. It was thought fitting to help the teachers who had agreed to teach this bold new sylla- bus by providing them with continuous teaching. Depth of action was necessary if the experiment was to become as widespread as was hoped. To this end the Belgian Centre for Mathematics Teaching was created with two essential aims in view:

1) to carry out research in mathematics teaching 2) to inform teachers.

From 1961-1962 working groups were organized in several Belgian towns. Willing and competent teachers introduced their colleagues both to the elements of modern mathematics and to the educational techniques employed in its teaching. We were thus faced with an usual state of affairs: some of our colleagues teaching the same elements of new mathematics to pupils of 12, students of 18 - 20 in teacher training colleges, and finally teachers from 25 to 60 in working groups. This time, too, it was seen that the same mistakes were made by all those whowere learning and that the same situations and exercises favoured the acquisition ofnotions.

In 1961-1962, in the hope of speeding up the reform, we took a science class of pupils aged 15-16 to see whether or not it would be possible to begin the new teaching at this age without the preliminaries of a new syllabus based on that of the lower class.

This experiment taught us much, particularly in the case of the teach- ing of geometry, which we had based on Artin’s expansions in order to reach the vector plane as quickly as possible.

We were, however, forced to admit that a great many of the efforts of teachers were spent in deconditioning pupils who had been inadequately prepared

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for simple understanding of the process of new mathematics. It was proved thus that true reform of secondary teaching had to begin at the lower level. From a social point of view, it is in fact indispensible because it is the only way of teach- ing new mathematics to all the children ot the secondary level.

Thanks to the support of the Minister it was possible to make the experiment general in all 12-year-old classes and to widen its scope from year to year into the secondary level classes.

Today the experiment has reached the level of higher education.

It is important for those who teach children from 6 to 12 years to be acquainted with the modern syllabuses that can be taught from 12 to 16. This is why we shall now give a broad outline of the experimental syllabus which was introduced so successfully in Belgium.

12 to 13years ------- ----

Sets: Use of Venn’s diagrams, exercises in the use of sets. Relation : Use of coloured graphs, possible properties, functions,

equivalence, order, association of the composition of relations.

Binary numeration. The ring of rational integers. The beginnings of affine geometry founded on original intuitive axioms.

Demonstration in very simple logical situations. Group of translations of the plane. Notion of group: calculation of group, cyclical group.

It must be emphasized that this teaching was given to 12 year olds for social reasons. Dienes’ experiments showed that many of these ideas may be taught at a much earlier age, which would, moreover, be desirable.

When the necessary time has been devoted to the teaching of the notion of sets and relation the pupils seem to gain a firm grasp of them - which is a great help to further education.

In connection with tyching in the 12 to 13 year old class the follow- ing works might be consulted :

The 13 to 14 year old class is devoted essentially to a strict introduc- tion at the pupils’ own level of the field of real numbers and the vector plane. This course culminates in the discovery of the vector plane and terminates in the systematic use of this structure to prove geometric propositions. Pupils thus begin

to realize that the vector plane is a real tool which can help to make their work much easier.

1) Die ersten Elemente der modernen Mathematik and (MM 1 I : see bibliography

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In connection with teaching in this class F 1 (see bibliography) might be consulted from the teachers’ point of view and MM 2 (see bibliography) from that of the pupils.

Teaching 14 to 15 year olds involves metric geometry from the notion of axial symmetry onwards. This course culminates in the introduction of the vector plane on a strictly positive scale and ends in its systematic use in establish- ing propositions of metric geometry. In connection with this the reader’s atten- tion is drawn to GP and MM 3 (see also bibliography).

In the 15 to 16 year old class a psychological return is effected. The pupils make less and less use of original axioms in favour of axioms of definition of the real and Euclidean vector plane. Thus, there is a gradual abandoning of the progressive axiomatic introduced in the first three classes as the new exhaustive axiomatic becomes the starting point for reasoning.

For more details on this the reader is referred to the first three chapters of F 2 (see bibliography) written from the teachers’ point of view and the second part of GP (see bibliography).

This syllabus offers no originality where material covered is concerned since all the conferences dealing with mathematics teaching underline the neces- sity of introducing as soon as possible real numbers, vector plane structures, and the notion of sets.

The Belgian experiment offers an organization of subject matter which enables it to be taught to children from 12 to 16 owing to new educational techniques.

The Belgian Ministry of Education and Culture has just made the ex- periment general and these modern syllabuses compulsory in all classes, from 1968 onwards.

Brussels, June 1965

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Bibliography to Appendix 8

MM1

MW 1

Mathematique Moderne - vol 1 (Editions Didier, Bruxelles-Paris, 1963, pp. 468)

Moderne Wiskunde - 1 (Editions Didier, Bruxelles-Paris, 1965)

MM2 Mathbmatique Moderne - vol 2 (Editions Didier, Bruxelles-Paris, 1965)

“Erste Elemente der modernen Mathematik” (Schriftenreihe zur Mathematik, Hefte 10 - 11, Otto Salle Verlag, Frankfurt-Hamburg, 1962-1963)

G “Groupes” (Presses Universitaires de Bruxelles. Dunod, Paris, 1961, pp. 249)

” Groups” (Mat Millan, London, 1964, pp. 220 + 32 illustrations)

” I Gruppi” (Feltrinelli Editore, Milano, 1964, pp. 263)

F 1 “GBomBtrie affine plane et nombres reels” avec la collaboration de Pierre Debbaut. (Collection Fr6d6rique no. 1, Presses Universitaires de Bruxelles, 1962, pp. 65)

“Ebene affine Geometrie und reelle Zahlen” unter Mitwirkung von P. Debbaut. (Vandenhoeck ,und Ruprecht, Gettingen, 1965, pp.70)

F2 “Initiation aux espaces vectoriels” (Collection Fr6d6rique no. 2, Presses Universitaires de Bruxelles, 1963, pp. 88)

“Einfuhrung in die Vektorraumlehre” (Vandenhoeck und Ruprecht, GiSttingen, 1965)

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“GrotpoYdes” (Labor, Bruxelles. Presses Universitaires de France, Paris, 1965)

GP “GBometrie Plane” (Labor, Bruxelles. Presses Universitaires de France, Paris, b parahre).

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BIBLIOGRAPHICAL REFERENCES

Chapter 1

LAKATOS, 1.: Essays in the Logic of Mathematical Discovery. Ph.D. Dissertotion. Cambridge 1961.

LAKATOS, I. : Infinite Regress and the Foundations of Mathematics. Aristotelian Society Supplementary Volume 36, 1962, pp. 169-191.

GOALS FOR SCHOOL MATHEMATICS: The Report of the Cambridge Conference on School Mathematics. Published for Educational Services Incorporated. Houghton Mifflin Company, 2 Park Street, Boston, Massachusetts 02107, 1963, pp. 102.

HARTUNG, M. - VAN ENGEN, Henry - KNOWLES, Lois - GIBB, E. Glenadine : A Program Overview for Grades 1 - 6. 433 East Erie Street, Chicago, Illinois 60611, Scott Foresman and Company. 1960. pp. 176.

JUDD, Charles H.: Education as Cultivation of the Higher Mental Processes. New York. The Macmillan Company, 1936. pp. 206

DIENES, Z. P. - JEEVES, M.A.: Thinking in Structures. Vol. I, Psychological Monographs on Cognitive Processes, 128 pp., 1965, Hutchinson Educational Ltd., 178-202 Great Portland Street, London W. 1, England.

CONFERENCE ON MATHEMATICS EDUCATION FOR BELOW AVERAGE ACHIEVERS: School Mathematics Study Group. 367 South Pasadena, California. A.C. Vroman, Inc., April 1964. pp. 130.

FREGE, Gottlob: The Foundations of Arithmetic, translated by J.L. Austin. New York, N.Y. Harper Torchbooks, The Science Library, Harper and Brothers, 1960. pp. 119.

143

PIAGET, Jean - INHELDER, Barbel - SZEMINSKA, A.: The Child’s Conception of Geometry. 49 East 33rd Street, New York, New York 10816, The Academy Library. pp. 408 9

DAVIS, Robert 8. : ‘The Madison Project’s Approach to Theory of Instruction, Journal of Research in Science Teaching, vohrme December 1~64, pp. 146-162. 20th and Northhampton Streets, Easton, Pennsylvania 18042.

144

Chapter 2

BARTLETT, Sir Frederick : Thinking: An Experimental ond Social Study. New York, Basic Books, Inc. Publishers, 1958. pp. 203.

BARTLETT, Sir Frederick : Remembering: A Study in Experimental and Social Psychology. 32 E. 57th Street, New York. Cambridge University Press. 1961. pp. 317.

BROWNELL. William A. : Arithmetic Abstractions: The Movement Toward Conceptual Maturity under Different Systems, 1964. Cooperative Research Project No. 1676, OE No. 2-10-103. U.S. Office of Education, Washington, D.C.

WASHBURNE, C.W. : The Grade Placement of Arithmetic Topics: A Committee of Seven Investigations. Report of the Society’s Committee on Arithmetic, Twenty-ninth Yearbook, Part II. National Society for the Study of Education. Bloomington, II I inois, Public School Publishing Co., 1930. pp. 641-70.

BARAKAT, M. K. : Factors Underlying the Mathematical Abilities of Grammor School Pupils. British Journal of Educational Psychology, 21. November 1951. pp. 239-40.

SCANDURA, Joseph M.: An Analysis of Exposition and Discovery Modes of Problem Solving Instruction, The Journal of Experimental Education, Volume 33, No.2, Winter 1964.

BRUECKNER, L.J. : Intercorrelations of Arithmetic Abilities, Journal of Experimental Education, 3, September 1934, pp. 42-44.

BRUNER, J.S. : The Process of Education. Cambridge, Harvard University Press, 1962.

COXFORD, Arthur F., Jr. : The effects of instruction on the stage placement of children in Piaget’s seriation experiments, The Arithmetic Teacher, Januory 1964.

145

FUNKENSTEIN, Daniel H.: Mathematics, quantitative aptitudes and the masculine role, Diseases of the Nervous System, Monograph Supplement, Volume 24, No.4, April 1963.

KESSEN, Wm.- YUHLMAN, Clementina (Ed. ): Thought in the Young Child. Report of a Conference on lntellective Development with particular Attention to the Work of Jeon Pioget. Monogrophs of the Society for Research in Child Development. Serial no. 83, 1962. Volume 27, no. 2, 176 pages, including bibliography .

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LUNZER, E.A. : Recent Studies in Britain based on the Work of Jean Piaget. Notional Foundation for Educational Research in England and Wales. Occasional Publication No.4. 1960.

SMEDSLUND, Jan: Concrete Reasoning : A Study of Intellectual Development. Monographs of the Society for Research in Child Development. Seriol no. 93, 1964. Volume 29, no. 2, pp.39, including bibliography.

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CALLANTINE, Mary F. - WARREN, J.M.: Learning Sets in Human Concept Formation. Psychological Reports, 1955.1, pp. 363-367.

DUNCAN, C.P. : Description of learning to learn in human subiects. American Journal of Psychology, 1960, 73, pp. 108-114.

DUNCAN, C.P. : Transfer ofter training with single versus multiple tosks. Journol of Experimental Psychology, 1958, 55, pp. 63-72.

MORRISETT, L. - HOVLAND, C.I.: A comparison of three varieties of training in human problem solving. Journal of Experimental Psychology, 1959, 58, pp. 52-55.

RESTLE, F. : Toward a quontitative description of learning set data. Psychological Review, 1958, 65, pp. 77-91.

146

AEBLI, Hans: Development of Intelligence in the Child. Institute of Child Welfare, University of Minnesota, August 1950.

BERLYNO, D.E. : Recent Developments in Piaget’s Work. British Journal of Educational Psychology, 1957, 17, pp. 1-12.

BRUNER, J.S. - INHELDER, B. - PIAGET, J.: The Growth of Logical Thinking. J.A. Psychologist ‘s Viewpoint. British Journal of Psychology, 1959, 50, pp. 363-370.

CHURCHILL, Eileen: A New Look at Piaget’s Theory of Stages: Some Experimental Evidence. Bulletin British Psychological Society, 1918, 34, pp. 66-67.

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INHELDER, B. : Developmental Psychology. Annual Review Psychology, 1957, 8, pp. 139-162.

INHELDER, B. : Some Aspects of Piaget’s Genetic Approach to Cognition, in : Thought in the Young Child : Report of a Conference in Intellectual Development with Particular Attention to the Work of Jean Pioget . 1940.

ISSACS, N.: Piaget’s Work and Progressive Education. In : National Froebel Foundation, 1955, pp. 32-45. (b).

ISAACS, N.: Piaget and Educational Theory. Bulletin British Psychological Society, 1957, 33. 26.

ISAACS, N.: About “The Child’s Conception of Number” by Jean Piaget. In: Nationol Froebel Foundation, Some Aspects of Piaget’s Work, London 1955.

LAURENDEAN, M.- PINARD, A. : Causal Thinking in the Child. Inter- national Universities Press, 1962.

LOVELL, K . : A Fallow-up Study of Some Aspects of the Work of Piaget and lnhelder on the Child’s Conception of Space. British Journal of Educational Psychology, 1959, 29. pp. 104-117.

LUNZER, E.A. : Recent Studies in Britain based on the Work of Jean Piaget. London. Notional Found&ion for Educational Research in England and Wales. 1960 (a).

LUNZER, E.A. : Some Points of Piagetian Theory in the Light of Experimental

Criticism. Journal of Child Psychology and Psychiatry 1960,l.

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NATIONAL FROEBEL FOUNDATION. Some Aspects of Piaget’s Work. London 1955.

PIAGET, J. - INHELDER, B.: Diagnosis of Mental Operations and Theory of the Intel1 igence. American Journal of Mental Deficiency. 1947, 51, pp. 401-406.

PIAGET, J. : The Right to Education in the Modern World. In UNESLP, Freedom and Culture, New York. Columbia University Press, 1951, pp. 67-l 16.

PIAGET, J. : General Problems of the Psychological Development of the Child pp. 3-l 1 and Equilibration Structures pp. 98-105 in: J.M. Tanner and B. Inhelder (Eds) Discussions on Child Development. New York. International University Press. 1956, Volume 4.

PIAGET, J. : The Child and Modern Physics. Scientific American. 1957, 196 (3).

PIAGET, J. : Piaget Rediscovered. A Report of the Conference on Cognitive Studies and Curriculum Development, M. 1964. Edited by Richard E. Ripple and Verne N. Rockcastle, School of Education,

PIAGET, J. : How Children form Mathematical Concepts. Selected Reodings on the Leorning Process. Ed. Harris and Schwahn, Oxford University Press, New York 1961. pp. 358-365.

PIAGET, J. : Principal Factors determining Intellectual Evolution from Childhood to Adult Life. In: D. Boreport (Ed .)Organization and Pathology of Thought. New York. Columbia University Press, 1951. pp. 154-175.

PIAGET, J. : Principal Factors determining Intellectual Evolution from Childhood to Adult Life. 1n:E.L. Jartley and R.E. Hartley (Eds) Outside Reodings in Psychology (2nd Ed.) New York. Crowell, 1958, pp. 43-55 (b).

PIAGET, J. - INHELDER, B.: The Child’s Conception of Space. 1956.

PIAGET, 1. : The Right to Education in the Modern World. In: UNESCO, Freedom and Culture, New York. Columbia University Press. 1951, pp. 67-116.

SCIENTlFlC AMERICAN: The Child and Modern Physics. 1957, 46-31.

148

SLATER, G.W. : A Study of the Influence which Environment plays in determining the Rate at which a Child ottains Piaget’s Operational Level in his early Number Concepts. Unpublished Dissertation, Birminghom University, 1958. Used by Smedelund in the Conservation of Substance and Weight.

WOODWARD, Mary : Concepts of Number of the Mentolly Subnormal studied by Piaget’s Method. Journal of Child Psychiatry, 1961, 2.2, pp. 249-259.

WRIGHT, John C. : From Piaget to Pedagogy. (A Review) Contemporary Psychology. 1963. 8. pp. 353-354.

DIENES, Z. P.: Concept Formation and Personality. Leicester University Press. 1959.

DIENES, Z. P.: An Experimental Study of Mathematics Learning. Hutchinson. 1963.

DIENES, Z. P. - JEEVES, M.A.: Thinking in Structures. Hutchinson, London 1965.

BRUNER - GOODNOW - AUSTIN: A Study of Thinking. New York. Wiley. 1956.

GAGNE, R.M. - PARADISE, N.E.: Abilities and Learning Sets in Knowledge Acquisition. Psychological Monographss, 1961. Volume 75, no. 14.

SKEMP, Richard R. : Reflective Intelligence and Mathematics. The British Journal of Psychology, Volume XXXI, Part l., February 1961,

BRUNER, Jerome S. : Some Theorems on Instruction illustrated with Reference to Mathematics. From the section Theories of Learning and Instruction. Sixty-third Yearbook of the Notional Society for the Study of Education, Part 1. 1964. Chapter XIII, Chicago, Illinois, U.S.A.

KATONA, G. : Organizing and Memorizing. New York. Columbia Univer- sity Press. 1940.

KERSH, B.Y. : The Adequacy of Meaning as an Explanation for the Superiority of Learning by Independent Discovery. Journal of Educational Psychology. 49. 1958. pp. 282-292.

KITTELL, J. E.: An Experimental Study of the Effect of External Direction during Learning on Transfer and Retention of Principles. Journal of Educational Psychology, 48. 1957. pp. 391-485.

149

McCONNEL, T. R.: Discovery versus’ Authoritative Identification in the Learning of Children. University of Iowa Studies in Education, 9: 11-62, 1934.

MacLATCHY, Josephine H. : The Preschool Child’s Familiarity with Measurement. Education 71 : April 1951. pp. 479-82.

McLELLAN, James A. - DEWEY, John: The Psychology of Number. New York. D . Appleton Century Company. 1895. pp. 309.

OSLER, Sonja, and others: The Effect of Stimulus Complexity on Concept Development for Two Levels of Intelligence. This study wos summarized in the volume by Palerma and Lipsit. Research Readings in Child Psychology. Holt, Rinehart and Winston.

THE SCOTTISH COUNCIL FOR RESEARCH IN EDUCATION: Its Aims and Activities. A Publication of the Scottish Council for Research in Education. No. XXIV. Revised Edition. University of London Press. 1953. p.23, 18-19.

SWENSON, Esther J. : Organizotion and Generalization as Factors in Learning, Transfer, and Retroactive Inhibition. Learning Theory in School Situations. University of Minnesota, Studies in Education, No. 2. University of Minnesota Press. 1949.

pp. 9-39.

THIELE, C. Louis: Contribution of Generalization to the Addition Facts. Contributions to Education, No. 673. New York. Bureau of Publications, Teachers College, Columbia University. 1938.

PP. 84.

WEIR, Morton W. - STEVENSON, Harold W. : The Effect of Verbalization in Children’s Learning as a Function of Chronological Age. Child Development, 30: 143-149. 1959.

BROWNELL, William A. : A Critique of the Committee of Seven’s Investigations on the Grade Placement of Arithmetic Topics. Elementary School Journal 38: 495-508. March 1938.

BROWNELL, William A. : The Development of Children’s Number Ideas in the Primary Grades. Supplementary Educational Monographs, No. 35. Chicago. University of Chicago Press. August 1928.

BROWNELL, William A. : The Effects of Practising a Complex Arithmetical Skill upon Proficiency in its Constituent Skills. Journal of Educational Psychology, 44: 65-81. February 1953.

150

BROWNELL, William A. : Psychological Considerotions in the Learning and the Teaching of Arithmetic. The Teaching of Arithmetic. Tenth Yearbook, Notional Council of Teachers of Mathematics. Washington, D.C.,The Council, o Department of the National Education Association. 1935. pp. 1-31.

BROWNELL, William A.- CARPER, Doris V.: Learning the Multiplication Combinations. Duke University Research Studies in Education. No. 7. Durham, N.C. Duke University Press. 1943. pp.177.

BROWNELL, William A. - CHAZAL, Charlotte B.: The Effects of Premoture Drill in Third-Grade Arithmetic. Journal of Educational Research, 29: 17-28. September 1935.

BROWNELL, William A. - HENDRICKSON, Gordon: How Children learn Information, Concepts, and Generolizotions. Leorning and Instruction. Forty-ninth Yearbook, Part I, Notionol Society for the Study of Education. Chicago. University of Chicago Press. 1950. pp. 92-128.

BROWNELL, William A. - MOSER, Harold E.: Meaningful versus Mechanical Learning: A Study in Grade II Subtraction. Duke University Research Studies in Education. No. 8. Durham, N.C. Duke University Press. 1949. pp. 207.

BROWNELL, William A. - STRETCH, Lorena B.: The Effect of Unfamiliar Settings on Problem Solving. Duke University Research Studies in Education. No. 1. Durham, N.C. Duke University Press. 1931. pp. 86.

BUSWELL, G .T. : Comparison of Achievement in England and California. The Arithmetic Teacher, 5: l-9. February 1958.

CRAIG, H. C.: Directed versus Independent Discovery of Established Relations. Journal of Educational Psychology 47: 223-234.

1956.

HAGGARD, Ernest A. : Socialization, Personality and Academic Achievement

in Gifted Children. The School Review (The University of

Chicago Press) 65: 388-414. Winter 1957. Copyright 1957

by the University of Chicago.

BUSWELL, G. T. : The Psychology of Learning in Relation to the Teaching of Arithmetic. The Teaching of Arithmetic. Fiftieth Yearbook.

Part II. National Society for the Study of Education.

Chicago. University of Chicago Press. 1951.

CHURCHILL, E.M. : Counting and Measuring. Toronto. University of Toronto

Press. 1961.

151

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DIENES, Z. P.: A Short Introduction to the Use of Algebraic Experience Materials. Nationol Foundation for Educational Research in England and Wales. 1961.

DUTTON, W. H.: Evaluating Pupils’ Understanding of Arithmetic. Prentice-Hal I Inc., Englewood Cliffs, New Jersey, U.S.A. 1964.

FLOURNOY, F., ond others: Pupil Understanding of the Numeration System. Arithmetic Teacher. 10. No. 2. February 1963.

GLENNON, V. J.: A Study of the Growth and Mastery of Certain Basic Mathemoticol Understandings on seven Educational Levels. Unpublished Doctoral Dissertation, Groduate School of Education. Harvard University. 1948.

HOTYAT, F. A. : French-speaking Countries: Belgium, France, Switzerlond. Review of Educational Research. 27, February 1957.

SKEMP, R. R.: Reflective Intelligence and Mathematics. British Journal of Educational Psychology, 31. 1960.

SPAINHOUR, R. E.: The Relationship between Arithmetical Understanding and Ability in Problem-solving and Computation. Unpublished Master’s Thesis in Education. Duke University. 1936.

WEAVER, J. F.: Big Dividends from little Interviews. Arithmetic Teacher, 2, April 1955.

WILLIAMS, J.A.: Number Readiness. Educational Review. Volume II, No.1. 1958. (A useful summary of the scope of this kind of test.)

6

WILLIAMS, J.D.: Ready for Arithmetic. Teaching Arithmetic. Volume 1, No. 3. Autumn Term 1963.

WILLIAMS, J.D. : Understanding and Arithmetic - 1 : The importance of understanding. Educational Research. Volume VI, No. 3.

June 1963.

WILLIAMS, J.D. : Understanding and Arithmetic - II: Some remarks on the nature of understanding. Educational Research. Volume VII, No. 1. November 1964.

ADKINS, Dorothy C.: Construction and Analysis of Achievement Tests. Washington, D .C . U.S. Government Printing Office 1947.

BLOOM, Benjamin 5. : Quality Control in Education. Tomorrow’s Teaching. Oklahoma City, Frontiers of Science Foundation of Oklahoma Inc. 1961. pp. 54-61

152

GLASER, Robert - KLAUS, David J.: Proficiency Measurement : Assessing Humon Performance. In: Robert M. Gagne (Ed .) Psychological Principles in System Development. New York. Halt, Rinehart and Winston. 1962. pp. 419-474.

BLOOM, Benjamin S. ( Ed .) : Taxonomy of Educational Objectives: Affective Domain.

BLOOM, Benjamin S. (Ed.): T axonomy of Educational Objectives: The Classification of Education Goals: Cognitive Domain. New York. Longmans, Green. 1956.

GAGNE, Robert M.: The Acquisition of Knowledge. Psychological Review. 1962. Volume 69, No. 4. pp. 355-365.

GAGNE, Robert M. - PARADISE, Noel E.: Abilities ond Leorning Sets. Psychological Monograph. 1960. Volume 75, No. 14. (Whole No. 518.)

GAGNE, Robert M. - MAYCX, John R. - GARSTENS, Helen L. - PARADISE, Noel E.: Factors in Acquiring Knowledge of o Mathematical Task. Psychological Monograph. 1962. Volume 76, No. 7. (Whole No. 526.)

LORD, Frederic M. : Further Problems in the Measurement of Growth. Educational Psychology Meosurement. 1958, 18, pp.437-454.

MYERS, Sheldon S.: Questions illustrating the Kinds of Thinking required in Current Mathematics Tests. Educational Testing Service. 1962.

BLOOM, Benjamin S. - BRODER, Lois J. : Problem-solving Processes of College Students. Chicago. University of Chicago Press. 1950.

HALL, G. Stanley : Contents of Children’s Minds on Entering School. Pedagogical Seminary 1: 139-73. 1891.

HASELRUD, G. M. - MEYERS, Shirley: The Transfer Value of Given and Individually Derived Principles. Journal of Educational Psychology, 49 : 293-298. 1958.

ANDERSON, G. L.: Quantitative Thinking as Developed under Connectionist and Field Theories of Learning. Learning Theory in School Situations. University of Minnesota Studies in Education.

Minneapolis. University of Minnesota Press. 1949. pp. 40-73.

153

Chapter 3

ELEMENTARY MATHEMATICS SERIES, K-6. Greoter Cleveland Mathematics Program. Educational Research Council of Greater Cleveland.

BRUNER, Jerome S.: The Process of Education. 1960. A report of the Woods Hole Conference of 1959. pp. 97. New York.

Random House, Vintage Books.

MINNEMATH curriculum materiols, mathematics units developed for K- 3 not intended for use with children in school classes. Of interest to teachers, school administrators, mathematics supervisors, et. al. (1963). University of Minnesota Press. 2037 University Avenue, S. E ., Minneapolis, Minnesota 55455.

MINNEMATH CENTER REPORTS (periodical). Free quorterly indicating current studies and materials being developed in elementary school mathematics. Minnemath Center Reports, Minnesota School Mathematics and Science Center, University of Minnesota, Minneapolis, Minnesota 55455.

NEW SCHOOL MATHEMATICS (periodical). Bimonthly in-service bulletin on the learning of mathematics on the elementary school. Philippine Mathematics Study Group, c/o Banaag Press, 1739 Felix Huertas, Manila, Philippine Islands.

LOVELL, K. : The Growth of Basic Mathematical and Scientific Concepts in Children. Philosaphicol Library 1962.

SKEMP, Richard: Understanding Mathematics.

DIENES, Z. P.: Building up Mathematics. London. Hutchinson. 1960.

DIENES, Z. P.: The Power of Mathematics. London. Hutchinson. 1963.

DIENES, Z. P.: The Formation of Mathematical Concepts in Children through Experience. Educational Research. London. December 1959.

BEBERMAN, M. : Improving High School Mathematics Teaching. Educational Leadership. 1959. Volume XVII, No. 3, pp. 162-166.

154

BEBERMAN, M. : Searching for Patterns. Urbana, Illinois. University of Illinois. 1963.

BEBERMAN, M. - HART, A.G. : New Approach to Mathematics. Discovery. March 1962.

GOLDEN, W. : UICSM in Its Second Decade. Journol of Research in Science Teaching. 1963. Volume 1. pp. 265-269.

HALE, W. T.: UICSM’s Decade of Experimentation. The Mathematics Teacher. 1961. Volume LIV, No. 8. pp. 613-619.

HENDRIX, G. : Learning by Discovery. The Mathematics Teacher. 1961. Volume LIV, No. 5. pp. 290-299.

HENDRIX, G. : The Psychological Appeal of Deductive Proof. The Mathematical Teacher. 1961. Volume LIV, No.7, pp.515-20.

Coordinates: An Introduction to the Use of Graphs and Equations to describe Physical Behavior. (Teacher’s Manuals) (Grades 2-6). Dr. S. P. Diliberto, Elementary School Science Project, University of Colifornia, Berkeley, California.

COHEN, Donald: A Lesson on Absolute Value. December 1964 - from the Arithmetic Teacher.

A Report on Some Experiments in the Use of Desk Calculators in Education. Worship Street, London E.C. 2, Addo Limited, Educational Bulletin No. 5. pp. 47-51.

KERR, E.: Desk Calculators. Mathematics Teaching. No. 22. 1963.

FEHR, H. F.: Report on the Use of Desk Calculators. Munro Calculating Machines. 1956.

GUNDLACH, B. H.: Elementary Number Games and Puzzles. pp. 160 + appendices. 1959. (out-of-print)

CUISENAIRE - GATTEGNO: Numbers in Colour. 1954.

GATTEGNO, C. : From Actions to Operations.

GATTEGNO, C. : Teaching Mathematics to Deaf Children.

GATTEGNO, C. : Mathematics with Numbers in Colar. Textbooks A,B,C,D, 12345678910. 11111,111

GATTEGNO, C. : Arithmetic.

155

GATTEGND, C. : Teacher’s Introduction to the Cuisenaire-Gattegno Method of Teaching Arithmetic.

GATTEGNO, C. : Now Johnny can da Arithmetic.

CHAMBERS, C. E.: The Cuisenaire-Gattegno Method of Teaching Arithmetic.

GOUTARD, M. : Talks for Primary School Teachers.

GATTEGNO, C. : Modern Mathematics.

GOUTARD, M. : Mathemat its and Children.

GATTEGNO, C. : For the Teaching of Mathematics. Volumes 1,2,3. (All of the above books concerning the Cuisenaire Method are published by Cuisenoire Co., Reading, England.)

156

Chapter 4

In-Service Mathematics Program for Elementary Teachers. Experimental Edition. 95 pages. Instructor’s Manual pp. 103. 1964. Minnesota School Mathematics and Science Teaching Project. Minnesota School Mathematics and Science Center. University of Minnesota, Minneapolis, Minnesota 55455.

CUISENAIRE, G. - GATTEGNO, C.: Modern Mathematics with Numbers in Colour. 99 Great Russell Street, London W.C. 1. William Heinemann Ltd.

PAPY : Mathematique Moderne. Bruxelles 1963.

DIENES, Z. P.: Modern Mathematics for Young Children. ESA. London 1964.

DIENES, Z. P.: Mathematics in the Primary School. Melbourne, Macmillan. 1964.

DIENES, Z. P. - JEEVES, M. A.: Thinking in Structures. London. Hutchinson. 1965.

DIENES, Z. P. - GOLDING, E. W.: Beginning Mathematics.

Paris. OCDL. 1965.

SEALEY, L. G. W.: The Creative Use of Mathematics in the Junior School.

Oxford. Basil Blackwell. 1960.

WARD - HARGRAVE: Fundamentals of Elementary Mathematics. Addison Wesley. 1964.

29th Yearbook of the National Council of Teachers of Mathematics: Topics in Mathematics for Elementary School Teachers. National Council of Teachers of Mathematics. 1201 Sixteenth Street, N. W., Washington, D.C. 20036.

BANKS, J. Houston: Learning and Teaching Arithmetic. Second Ed ition. Boston, Massachusetts, Allyn and Bacon Inc. 1964. pp. 448.

157

OHMER, Merlin M. - AUCOIN, Clayton V. - CORTEZ, Marion J.: Elementary Contemporary Mathematics. 72 Fifth Avenue, New York 10011. Blaisdell Publishing Co. 1964. pp. 380.

PETERSON, John A. - HASHISAKI, Joseph: The Theory of Arithmetic. 605 Third Avenue, New York, N.Y. 10016. John Wiley and Sons, Inc. 1963. pp. 303.

DIENES, Z. P.: Modern Mathematics for Young Children. Pinnacles, Harlow, Essex, England. The Educational Supply Association. Ltd., School Materials Division. 1965. pp.90.

Mathematics : Its Contents, Methods and Meaning. (Three volumes) Translated from the Russian. 1963.

PAGE, David A.: Number Lines, Functions, and Fundamental Topics. 60 Fifth Avenue, New York, New York 10011. The Macmillan Company. 1964. pp. 283.

BERNE, Bernard I., Principal : Introductory Handbook for Teachers. Nl:w Mathematics for Elementary Schools. Bronx, N.Y. 1965. pp. 56.

DUTTON, Wilbur H. - ADAMS, L.J.: Arithmetic for Teachers. Englewood Cliffs, New Jersey 07632. Prentice Hall Inc. 1963. pp. 370.

Basic Concepts of Mathematics: an Introductory Text for Teachers. (for use in Training Colleges in English-speaking Africa) Watertown, Massachusetts. Educational Services, Inc. 1963.

In-service Mathematics Program for Elementary Teachers. Experimental Edition pp. 95. Instructor’s Manual,pp. 103. 1964.

Minnesota School Mathematics and Science Teaching Project, Minnesota School Mathematics and Science Center, University Minnesota, Minneapolis, Minnesota 55455.

STERN, Catherine : Mastery in Mathematics. Teaching Arithmetic. Vol. 3, no.1. pp. 24-32. New York, N.Y. 10022. Pergamon Press. Spring Term 1965.

Conferenceson the Training of Teachers of Elementary School Mathematics. (free) (CUPM Report No. 7, 1962), (CUPM Report NO. 9, T963), (CUPM Report No 11 , 1964), Committee on the Undergraduate Program in Mathematics, CUPM Office P.O.Box 1024, Berkeley, California 94701.

MITCHELL, Benjamin E. - COHEN, Haskell: A New Look at Elementary School Mathematics. Englewood Cliffs, N.J. 07632, Prentice- Hall, Inc. 1965. pp. 354.

158

Appendix A

Lessons for Self-Instruction in Basic Skills. (Arithmetic) (Programed materials for use in grades 3-8) California Test Bureau, Del Monte Research Park, Monterey, California.

SPOONER, George: Mathematics Enrichment Programs: A,B,C, 1962. Three programed texts for grades 4-6. New York. Harcourt, Brance and World.

FINCHER, Dr. Glen E.: The Construction and Experimental Application of a Programed Course on the Addition and Subtraction of Fractions for Grade Five. 1963. Ph.D. Dissertation, Ohio University.

SCHRAMM, Wilbur: The Research on Programed Instruction: annotated biblio- graphy (Washington: U.S. Office of Education, 1964, pp.1 14), U.S. Government Printing Office, Division of Public Documents, Washington, D .C .

EASLEY, J. A. - GELDER, H. M. - GOLDEN, W. M.: A Plato Program for Instruction and Data Collection in Mathematical Problem Solving. Urbana, Illinois. Coordinated Science Laboratory. 1964. CSL Report R-185.

An Experimental Study of Programed versus Traditional Elementary School Mathematics. Research Report no. 61-01. Mr. John C. McLaulin, Norfolk City Schools, 402 E. Charlotte Street, Norfolk, Va., 23510.

Programed Instruction (Periodical). The Center for Programmed Instruction, Institute of Educational Technology. Teachers College. Columbia University, 525 West 120 Street, New York, N.Y. 10027.

159

ANNOTATED BIBLIOGRAPHY

An Analysis of New Mathematics Programs. 1964. pp. 72. An analysis of eight programs for content and method. National Council of Teachers of Mathematics. 1201 Sixteenth Street, N.W., Washington, D.C. 20036.

An Intuitive Development of the Real Number System and Related Topics. A CUPM-level text with problems for use in Teacher Training (NSF Grant G 23795) by George Baldwin and Ralph Crouch. 1963 University Bookstore, New Mexico State University, University Park, New Mexico.

DEANS, Edwina : Elementary School Mathematics: New Directions. Bulletin 1963, No. 13 (OE-29042), pp. 116. U.S. Government Printing Office, Washington, D.C. Factual Descriptions of Major Experimental Mathematics Projects.

FRASER, D.C. : The Concept of Skill : First of a series of articles appearing in “Continuous Learning”, Volume 3, No. 6. November- December 1964. The Canadian Association for Adult Education. 113 St. George Street, Toronto, Ontario, Canada. In the first article, the author discusses Sir Frederick Bartlett’s analysis of skill as a sequential development in space and time. pp. 257-260.

LAKATOS, I. : Proofs and Refutations. Volume XIV, Nos. 53-56, 1963-64. pp. 117. The British Journal for the Philosophy of Science. Thomas Nelson and Sons, Ltd., Edinburgh 9. Mathematics is shown to grow not by establishing many theorems, but by the logic of proofs and refutations.

Leadership Role of State Supervisors of Mathematics. (OE-29032), Bulletin 1962, No. 1, pp. 107. The evolving mathematics curriculum: its improvement, in-service and pre-service training, research and evaluation. U.S. Government Printing Office, Division of Public Documents, Washington, D.C. 20025.

SAWYER, W. W.: Introducing Mathematics: 1. Vision in Elementary Mathematics. 1964. pp. 346. The first of four volumes in the series, concerned with traditional mathematical topics treated in terms of recent aspects. Penguin Books, Inc., 3300 Clipper Mill Rocd, Baltimore, Maryland 21211.

LOVELL, Dr. K.: The Growth of Basic Mathematical and Scientific Concepts in Children. An overview of present knowledge concerning the ways in which children develop mathematical and scientific concepts. Warwick Square, London E.C. 4. University of

London Press, Ltd., 1962. pp. 154.

163

The Revolution in School Mathematics. A report of regional orientation conferences in mathematics. 1201 Sixteenth Street, N. W., Washington, D.C. 20036. National Council of Teachers of Mathematics. pp. 90.

Bulletin of the Nuffield Foundation Mathematics Teaching Project. (First issue, November 19&l), containinq activities designed for organizing a mathematics course for children 5-13.

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Typed by: lngrid Schwanda on IBM machine Xerographed by : G .M.L. Wittenborn Sahne, Hamburg 13