For the Teaching of Mathematics Volume 1

For the Teaching of Mathematics Volume 1

Caleb Gattegno

Educational Solutions Worldwide Inc.

First published in 1963. Reprinted in 2011.

Copyright Š 1963-2011 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-338-8 Educational Solutions Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

In This Series 1

What is Modern Mathematics?

by Gustave Choquet

2 For the Teaching of Mathematics

by Caleb Gattegno Volume 1 Part I Mathematics and the child Part II Pedagogical discussions Volume 2 Part III Psychological Studies Part IV On films Volume 3 Part V Elementary mathematics Volume 4 Part VI Miscellaneous Topics Part VII Mathematics Teaching and Society Book reviews 3 Talks for Primary School Teachers

by Madeleine Goutard 4 Mathematics and Children: A Reappraisal of Our Attitude

by Madeleine Goutard

Table of Contents Preface ........................................................................ 1 Introduction ................................................................3 Part I Mathematics and the Child.................................9 1 Logics................................................................................... 11 2 Substitutes and Examples ..................................................21 3 The Use of Mistakes in the Teaching of Mathematics ...... 33 Part II Pedagogical Discussions .................................43 1 Note on Pythagoras’ Theorem............................................ 45 2 A Note on the Teaching of Mathematics ............................55 3 Notes on Intuition in Mathematics ....................................73 4 Some Problems Involved in the Teaching of Mathematics ......................................................................... 87 5 Reforming Mathematics Teaching ...................................103 6 Mathematics and the Needs of Society ............................ 113 7 What Matters Most ...........................................................127 8 Why Study Mathematics?.................................................139 9 Formalization and Sterilization........................................145 The Sources of This Volume..................................... 153 Index ....................................................................... 155

Preface

This book is a collection of my articles that appeared between 1947 and 1963 in journals in a number of countries. I should like to express my gratitude to the Editors concerned for permission to reproduce them here.

I have occasionally added a text which, though accepted by a journal, was not subsequently printed for a variety of reasons (for example, the journal stopped publishing, or the editor learnt of some more acceptable author for the purpose in mind), or which was not even submitted for publication but nevertheless seems to me likely to be of interest for the readers of such a collection.

As the amount of material available was considerable, I first subdivided the articles according to subject matter, deciding to publish four volumes respectively concerned with (a) general pedagogical ideas, (b) psychological contributions and mathematical films, (c) elementary school mathematics, and (d) miscellaneous topics; second, I discarded a number of scripts which might have detracted from the main purpose of the book,

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For the Teaching of Mathematics

as they were either too bulky or written in a language leaving many points obscure or too specialized.

My reasons for publishing this book are, first, that it now appears that I have finally left the study of mathematics teaching behind me, being engaged in other projects that make any serious return to this study during my lifetime most unlikely; also that as my views have changed a great deal since I began to work professionally on mathematics education it might be of help to students of this subject to have a display of research projects and tools produced for them. Moreover, since the end of the second world war, I have, in addition to carrying out research, formed a number of associations in various countries, founded journals and study-groups, generated interest in the challenges met in mathematics education, and put into circulation certain ideas now in common use. It seemed that I could help students of the history of ideas in education by bringing together some sources no longer considered worthy of bibliographical mention.

The following introduction attempts to present a general view of the work done over a quarter of a century, and is the only article written especially for this volume.

I am deeply indebted to Jeremy Steele, who assisted me in the preparation of the scripts for publication. Improvements on the originals were permitted only where the substance was not touched.

C. GATTEGNO Reading, March 1963.

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Introduction

In the choice of the title of this work the word For needs some explanation.

Mathematics teaching has been my occupation since 1927, while my mathematics learning goes back further than I can remember. At school I was not particularly distinguished, but some aspects of numbers fascinated me. I believe that I even noticed some relationships between numbers that I retained because of their charm on my mind.

For the first ten years as a mathematics teacher I was arrogant and demanding, not noticing the faults originating from my own defects and preconceived ideas. I was successful with only a very few of my pupils, those who could perhaps more easily than others bridge the gaps I left. But in 1937 a teacher at an infants school put some pertinent questions to me that showed me where I was wrong and forced me to revise my basic premises.

Did I ever take into account the needs of the pupils and their

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For the Teaching of Mathematics

way of thinking? Did I believe that one and the same approach was valid for all ages, all moments and all topics?

From that time onwards I began to revise one by one all the ideas and attitudes I had held about education, teaching, learning, mathematics, the evolution of ideas and so forth. I was really fortunate in having let only ten years of my professional life pass before becoming aware of what was confronting me in my mathematics lessons. Indeed, I had even made profitable use of these ten years, concentrating on mathematics and allied subjects, but it was a shock to find that I could live in a situation, be so intimately involved in it, and yet not notice the essential factors that make that situation what it is. How could I avoid falling again and again into the traps which once caught me?

Today I can say that constant watchfulness is required and that one dies before one has fully protected oneself. The articles of this book are a testimony of the progress made by constant watchfulness, but they also show that constant revision is needed if one is to meet changing circumstances adequately and be prepared for the increasingly broad groups of people one teaches (the subnormal, the handicapped, the super-gifted, and so on). These two dimensions of progress are why the title begins with For instead of On, or About.

I am quite convinced that mathematics teachers (and perhaps teachers of other subjects too) need to improve themselves in a number of respects if they are to be able to cope with the challenge of teaching their subject at the end of this century. For the teaching of mathematics to be done properly, we need to

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Introduction

become more concerned with complex and multivalent thinking rather than with simple classifications, univalent proofs and systems of axioms; we need to develop the concepts that describe the realities we are working on and with; and I believe that the work I have done, described in part in these chapters, is a contribution towards meeting these needs which I know to be the needs of all teachers and not of myself alone.

What I did after 1937 was For the teaching of mathematics; it was done through teachers, in national and international gatherings, in seminars lasting from a day up to a fortnight, where more and more people came to realize what was needed, who was working on what, and how to take advantage of the discoveries of mathematicians, logicians, historians of mathematics, epistemologists, psychologists and experimental teachers. As I forged ahead bringing people more and more into contact with one another, stirring up the ranks of the professions with proposals that called upon everyone to participate in the joint effort, and reducing the role of the “authorities� For the improvement of mathematics teaching (since teaching, after all, takes place in classrooms, not in closed laboratories), I found that very few people indeed were really interested in working For the teaching of mathematics. Let me give an example. From 1945 until 1951 I believed Piaget had something to offer for the improvement of our teaching. I devoted my energies to propagating his thinking in lectures, translations and experiments. Piaget had not asked me to do this, but he was pleased nevertheless to have me as an aide.

When I discovered how limited his conceptions were, and how he refused to consider views that conflicted with his own (though he would concede privately that one might be right) I 5

For the Teaching of Mathematics

was obliged to decide that, For the sake of the needs of mathematics teaching, it was not helpful to propagate the work of Piaget, particularly when improved viewpoints could be proposed. Today, after twelve years of constant examination of the material used to support Piaget’s views, I can only say that, For the teaching of mathematics, his work has little contribution to offer. His concept of number is out of date and too schematic, only satisfactory in an a priori system; his “stages” are more convenient than real, and certain of his conceptions of children’s learning are disproved daily when more adaptable techniques are used.

Nevertheless, most people engaged today in educating the young acknowledge Piaget, and accept his conclusions as if they were irrevocable premises. This situation, I feared, could easily lead to the disappearance of empirical attitudes, more experimental moods and constant watchfulness, to be replaced by the mediaeval criteria of Authority and Opinions. Indeed, this is what I see happening more and more, and not only in journals on the Continent; the empirically minded Anglo-Saxons are being attracted by the tidy set of stages and see in them the germ of the psychological equivalent of their syllabus. They also see how easy it would be to prepare tests on such bases.

For the preservation of the right to reject all statements not in accordance with reality, people will hear my voice at least; and it must be noted that I was once an enthusiastic exponent of Piaget and his school, only changing my mind under the pressures of reality and as a result of a better understanding of what he writes. For the sake of more correct mathematics teaching we must return to the source of everyone’s knowledge (including 6

Introduction

Piaget’s), to the set of actual learners engaged in actual learning, and watch what they actually do rather than classify their responses to the experiment-centered challenges. What can be learnt from these observations is becoming more and more accessible as more and more teachers watch their pupils at work with the new materials that I, among others, have made available first to mathematics teachers and am now presenting to language teachers. In other publications to appear shortly these lessons learnt from the classroom will be presented for discussion.

I should at all costs like to avoid the belief that my last word about anything human could be the last word on the subject.

The study of man is in its infancy and may remain so, as the field will forever expand as awareness deepens and more of us burst the boundaries of observation with the appearance of new behaviors in some beings not previously taken into account.

This book, containing so many contradictory statements, is evidence that what is clear one day loses its clarity when further elements are taken into consideration. The writer of each of the articles though the same individual (?) is on each occasion a different one as a result of what he has learnt from the time of writing one article to the next. I am neither ashamed of the contradictions that will be noted, nor inclined to give an explanation for them. They are a testimony of a changing vision, affected by what is seen and by the reflection on what was lived.

Since the articles extend over many years of work and reflection, progress of the writer in his own studies is revealed; one can feel that with deepening awareness came a simplification of the simplifiable, a better articulation of what seemed to remain 7

For the Teaching of Mathematics

isolated, an elimination of the irrelevant, and a greater insistence on the important. The large number of ideas discussed and the many details retained become fewer and more naked as the insights become knowledge both analytic and synthetic. By developing its own language (but not a jargon), and assembling the most urgent problems of the teaching of mathematics, I hope that this book will contribute somehow to the education of the new recruits to the profession. It is being offered to the old hands as well as to the newcomers.

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Part I Mathematics and the Child

1 Logics

This section of this chapter appeared in the Mathematical Gazette, October 1947, Vol. 31, No. 296.

When teachers consider the problems of mathematics in the school they recognize that they have to make a definite effort to simplify the study of these difficult questions, but, it seems to me, they work in a frame which is imposed by the curriculum, and their temptation is to think very little of the child. The problem, however, has to be faced as a whole and the aim of this article is to discuss the teaching of mathematics with special reference to a child-centered point of view.

Syllabuses are produced by adults who tend to think that mathematical knowledge can be divided into parts which can be allocated to children of such-and-such an age. If this view does not correspond to reality, then a new committee decides to replace one question by another seemingly more suitable to the children concerned. A priori, one could say that if there are so many changes in the syllabuses, some deeper mistake must lie behind.

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Part I Mathematics and the Child

What is this mistake? It seems to me to be merely this: the child is ultimately the agent of his own apprehension and his growth of knowledge. Without considering the child as an active unit, as someone to be known before thinking of subjects to be taught to him, we are bound to make mistakes and to undertake continuous amendments of the curriculum, of books and of methods. In fact the art of teaching mathematics is measured today by the ability of the teacher to disclose new methods outside the field of mathematics to convey the truths of mathematics. Is this not an implicit admission of failure? Is there not, therefore, an urgent need to consider the reality as it stands and to think of the teaching of mathematics as the problem concerning mathematics and the child?

In advocating a complete reconsideration of the situation I am quite aware that I shall meet many objections. But one of the qualities of mathematicians (not always shared by other men) is that they know that when there is a mistake somewhere there is no use hiding it. It will only appear somewhere else in the form of a contradiction. This time the contradiction can be felt when we consider how many individuals dislike mathematics or have a fear of it. This forces mathematicians to write popular books to recapture adult interest.

Historically the development of schools was from schools for grown-ups to schools for all children, and in consequence children were taught without being known, and with the implicit assumption that they were prospective men and only that.

Moreover, the needs of industry demanded technicians first of all and it was tacitly agreed that fifteen years of school, spent in acquiring some technical ability, was a good price to pay for a 12

1 Logics

specialist. The curriculum was therefore devised to give at the end of these years the required ability. It was the task of the teacher to get the result, no matter what the method and the effort required: for example, solve fifty exercises on removing brackets, two hundred on simplifying fractions, do six years of drill on multiplication tables, etc. As only the result counted, there was no need to consider the method, there was no room for reform, and still less for consideration of the reality of the child.

But as machines were doing so much work, the need for technicians was both reduced in relation to the school population and increased in quality. More workers of a higher efficiency were required and more had to enter careers where other subjects were more useful. A competitive society needs more lawyers to settle its disputes, more trade travellers to place the goods, etc. Syllabuses were revised so as to introduce more advanced material and also to give room for other training. But the methods remained the same. Every now and then a crisis forced the educational authorities to undertake a slight change here and there; sometimes because a better method was found to expound such-and-such a theorem, or even because the opinion of some authority in a Committee prevailed. But there was not a real feeling for reform.

The reform in spirit was imposed from outside. First, it was found that the child is not a small man but an individuality with its own ways of thinking and its own purposes. He, in the end, will be an adult, but this will be later and, what is very important, he will be an adult in the world of the future which nowadays cannot be described with great certainty. Secondly, 13

Part I Mathematics and the Child

some educational pioneers have let the child develop, observed it and given it little teaching in the ordinary sense. In these schools sometimes there was no change of syllabuses but there was an attempt to change the method. It was still believed that the aim of education was the transmission of culture and that the test for the success of this is the official examination. But it was also believed that much depends on the way in which the subject is treated and the child’s natural curiosity used. The project method, or the Dalton plan, prepare candidates for the same examinations as traditional methods, but convey the material in a new way. These methods were successful. In America, where everything can be attempted, and finds followers, hundreds of methods were tried. The result is that now almost everybody who teaches mathematics wants to appeal to some external help in performing his task.

Today there is an intense interest in methods and an experimental era has started. Discussion on how to improve the teaching of logarithms is common today among the teachers. In front of such a situation it seems to me to be time for a thorough reconsideration of the whole problem of the child and mathematics.

Today we find people who ask whether it is not a pity to waste so many years in the learning of mathematics when nothing remains. For us mathematicians, it is impossible to agree with the implication. We feel so strongly that mathematics is one of the greatest achievements of the human mind, and enjoy its beauties so much, that we cannot conceive a school curriculum without mathematics. But the facts are there. Huge efforts at school do not leave any trace. It is not true that a mathematician 14

1 Logics

is an abler man or a better man, or even a pleasanter man. His place in society is often that of a crank. But society cannot progress without mathematicians, and needs many of them, more and more as the world becomes more mechanized.

We shall find a way out of this dilemma if we consider the problem as a dual one. There is the problem of the necessity of having mathematicians among the scientists and there is the problem of securing that mathematics comes into the school in the natural development of the child. The first problem will be solved if the blind law, which has provided the world with mathematicians since the remotest times and without special care, is not opposed. Let people whose taste for abstractions is great have a chance to meet mathematics at the proper moment.

Therefore the first problem will be solved as a corollary to the solution of the second problem, which is the really difficult one.

We shall attempt to offer a solution of this problem, at least a theoretical solution. It is not true that there is only one way to obtaining the truth, and that this is the method by which geometrical truths are obtained. In fact, if one studies logic, one finds that there are several possibilities, the axioms of each being themselves not logical, and it is ultimately a question of taste which decides the choice. There are several geometries, contradicting each other, one of which is taught as true in the schools, the others being simply ignored. If we call logic the set of basic “tools” which allow us to obtain “truths”, let us see if what we have found above is not the case for the logic of the child. Perhaps it may be decided that our plain adult logic is the only one, and the others may be deliberately ignored. Perhaps

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Part I Mathematics and the Child

all our system of education may have been built on that assumption.

Since the development of theories of relativity we have learned to consider different observers and not to blame anyone if results are different unless a suitable transformation is found to decide the results in any system of reference. Have we undertaken such a study of the different observers in the school case: ourselves and the children? Have we thought that our successive views about the universe, about ourselves, about logics, . . . imply the necessity of a relativistic view of the problems? A relativistic view is possible. We shall undertake it on the logical lines mentioned above, not as a problem of abstract logic, but of plain philosophy: logic will be what makes us believe that we have obtained the truth.

If I see my daughter in the street, I recognize her and can go to her, talk with her about things we know. There is no doubt for me that this girl is my daughter. I have used a judgment which is instantaneous and due to perception. I have used logic of perception. The same happens if I look at a right-angled triangle or an equilateral triangle. I do not need to think. I recognize each of them immediately. Obviously, this logic of perception has its limits. If the angle A is 890, I shall still say that the triangle is right-angled at A. In similar fashion, I may make a mistake if I think I see a Mr. Smith whom I once met in a crowd.

The logic of perception exists even though it may not be secure.

If we want to know definitely whether ABC is a right-angled triangle, there is only one way, that of measurement. To agree upon the result is then logic of perception based upon logic of action; similarly for Mr. Smith, we can ask him his name. Logic 16

1 Logics

of action exists in every one of us. If I tell anybody that I can walk on my hands from Liverpool to London in ten minutes, he does not need to think, he will state immediately that it is impossible. Although he has not seen me trying it, his experience lets him judge it impossible. He uses logic of action.

In the well-known problem which leads to the conclusion that a right angle is equal to an acute angle, when we want to prove that there is a mistake, there is no other way of proving the falsity of the result than by drawing a figure using circle and ruler. It is not mere reasoning which is used, but action (the construction). Those who think that geometry is the development of abstract notions in the mind, must stop when they meet the previous result, for if no action is used, any following theorem is indifferently right or wrong.

The Middle Ages are known for the use of faith as a tool in the search for truth. What is in the Bible was true and everything which can be deduced syllogistically from the contents of the Bible was also true. Angles are as real as men.

For the Greeks, beauty was a criterion of truth. For adults of today, neither faith nor beauty is used as a criterion in geometry, but in many cases emotional judgments are used. There is a logic of feeling (or of emotion) which contains faith and aesthetic feeling as special cases. If we walk in a street and see a driver of a car deliberately driving over a dog which he could avoid, we throw ourselves on the driver to punish him for his villainous action without asking ourselves any question. The response is immediate. Our moral feelings are hurt and do not require any deliberation. The logic used to find this truth is merely logic of feeling.

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Part I Mathematics and the Child

Can we find examples in mathematics in which this logic of feeling is in action? It is a daily experience of the mathematician that an awkward way usually leads to a mistake and a “fine” method of approach leads to truth. Hundreds of theorems in mathematics are given on assumption. Their proofs are sought for, sometimes for centuries (Fermat’s last theorem, the fourcolor problem, . . .). Mathematicians believe that these results are right, or that there must be a way of proving them. It is mere faith. The fact that every continuous curve has a tangent, that there is only one infinity, are mere beliefs. Zermelo’s axiom still divides mathematicians; those who do not want to use it call the others “mystics”. Logic of feeling is also used when we approach any problem in which the content gives us joy and happiness.

The logic used in every book of mathematics is usually called the logic of the obvious (or, evidence) based on some simple ideas such as: the whole is greater than the part; it is impossible to be and not to be at the same time; the same part of space cannot be occupied by two different bodies simultaneously. It is only when we can reduce our problems to obvious ideas that our mind is satisfied. All our books of mathematics proceed by it. We do not need to give any example. But on the contrary we want to show its limits. Zermelo’s axiom is an example which shows that what is obvious to one may not be obvious to another. If we have more than one person involved, the obvious must be defined.

That is what the positivists did when they decided to use for finding truth in society the logic of mental convergence: What we agree upon is true. Has this logic any representatives in the field of mathematics? Obviously all the statistical results are of this type; they draw their acceptance from this kind of logic. To choose a law to agree with the majority of the results but not 18

1 Logics

with all is a definite new step in the search for truth. Graphical and statistical methods are representatives of the logic of mental convergence and were considered simultaneously in history.

Clearly, if some fact is obvious to everyone, it is obvious to the majority, and so logic of the obvious is contained in logic of mental convergence, but not conversely.

We can find other types of logic: pragmatism, which is the method of finding truth by the validity of the result; logic of invariance, the relativistic substitute for relative truths.

We have discussed only one point: logic. But obviously it has great importance in relation to the teaching of mathematics. It shows that if a child goes through these different logics the syllabuses must be molded on this evolution. It is unreasonable to use graphs at an early stage, for they are abstractions and not concrete things. They satisfy a man trained in social life and social thinking. Their meaning is acceptable only there.

There must be a worldwide investigation: 1 of what are the ways used by children to get their truths; 2 of what are the substitutes for the truths we want to hand to the child in the field of mathematics, substitutes which will use the logic present at the stage in which the child is, which will build the next logic as a tool.

By this process we can still keep the teaching of mathematics in the school syllabus, give joy to the child as he does what he 19 21

Part I Mathematics and the Child

understands, not interfere with his growth, build his knowledge in all fields at the same time, obtain his spontaneous adhesion to abstract mathematics when at the age of 14 or 16 his logic of the obvious in his tool of judgment. Normal children, if not disgusted earlier, like algebra and geometry in post-puberty. The knowledge gathered outside the school in their early childhood is the only support for their logic of perception, logic of action, logic of feeling, and they use these elements to support their logic of the obvious. It is because the school does not provide this support that we find boys of 15 who think that a brick can weigh 1 o grams or a hundredweight. If the school provides mathematics corresponding to the existing logic not only will it save the future of mathematics in the schools but also the future of the child in society.

In the space of such a short article it is impossible to discuss thoroughly all the points raised and we ask the reader to consider this article merely as an invitation to consider the problem from this angle. Another article will give some personal attempts to find substitutes for the problems we meet in teaching practice.

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2 Substitutes and Examples

This section of this chapter appeared in the Mathematical Gazette, May 1949, Vol. 33, No. 304.

In the first part of this article we studied the relation between mathematics and the child and tried to suggest that a greater place could be given to mathematics in the schools by taking into account how the child learns. We tried to show that, to give the child a knowledge of mathematics, various approaches can be used, each of which can serve as a basis for the following one if we follow the development of logic, using the set of tools mastered at each stage in order to progress from one level to the next.

Before we embark on our second part, two points require clarification.

1 We spoke of substitutes for mathematical facts at various levels without perhaps giving an adequate explanation of what we mean by a substitute.

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Part I Mathematics and the Child

To show pupils that in a triangle A + B + C = 180째 the method most commonly used is to ask them to add up the measurements of the three angles. For us, A + B + C = 1790 is not a substitute for the theorem. It is an approximate result if the true result is known, otherwise it is a particular finding resulting from certain actions of the child.

Substitutes are mathematical truths obtained otherwise than by reasoning. In spite of being temporary expedients, they are quite as efficient for the child as a mathematical understanding is for us. A + B + C = 1790 being wrong mathematically is not a substitute. But logarithm understanding either of substitute. If in order to fractions and obtain

tables correctly used, though without their origin or construction, are a solve I suggest reversing the

I am giving a substitute, an action as substitute for the solution making full use of the rules.

These examples will, we hope, make clear what is meant by a substitute.

2 The second point concerns the idea of abstraction, a word used very loosely by many psychologists and whose meaning is assumed to be quite clear. Obviously there are hierarchies of and etc. It is also evident abstractions such as 2 and x and that animals and even plants have some capacity for achieving analytic concepts of the kind that we are continually manipulating. Are the selective powers of a tissue anything else than the ability to distinguish between like and unlike? The point we wish to make is that it is not only verbal possibilities of

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2 Substitutes and Examples

distinction which must be taken into account when there is some truth to be grasped. Analytic powers work very early in the mind of human beings, but in a way not usually appreciated or recognized by the adult mind. The reason for our failure to recognize this fact is to be found in an ordinary experience of the human mind, the construction of ever-widening classes of attributes, each new one containing and taking the place of the previous one. For instance, the class of red objects is contained in the class of colored objects. To grasp the first of these, we needed a certain amount of experience and some power of abstraction; to grasp the second, a conception of various classes each of a definite color was necessary. Red becomes a color only when it is dissociated from the objects which have this quality.

When it becomes a color it is an abstract red and can be used in combinations of color.

This somewhat detailed explanation has been necessary to make us aware of the value of substitutes in the teaching of mathematics. Red does not lose its value as vibrations of the ether when it is used and investigated by painters merely as a color acting on the retina and combined with other colors. The same can be said of what we are proposing here. We do not lose the mathematical value of any of the propositions we present to the child if we present them as containing only those “dimensions” which the child can grasp. The process of his learning will be gradually to add more and more “dimensions” to the idea already grasped.

(In the next paper, “The Use of Mistakes”, we develop this point.)

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2 Substitutes and Part I Examples Mathematics and the Child

The giving of geometrical toys to children has long been advocated, and many are used in Dr. Montessori’s method.

Though this idea has been offered to educationists for more than fifty years, very little has in practice been done with it, and there is great opportunity for improvement and extension. (We hope one day to show how much can be done in this respect by a single contributor, and to stimulate others to contribute.)

The well-known “insets� certainly provide a true mathematical acquaintance with figures and relationships of figures. One of my daughters at the age of three and a half was able in a Montessori class to distinguish clearly between the center of an area and the middle of a line, and would not allow me to confuse these two concepts in ordinary speech.

Apart from this material for early stages, very little exists that can be used by children for further development, partly because the only mathematics taught in primary schools is arithmetic, and partly because no one has been sufficiently interested to produce active material corresponding to the ages 7 to 12. Our ancestors used simple machines for counting which have been easily replaced by mental arithmetic, and it has seemed that there was no real need for new devices in this part of the mathematics syllabus. I cannot accept this view, but a description of some substitutes for arithmetic would take me too far.

After the age of 11 children are suddenly brought into contact with algebra and geometry, whose nature is quite alien to their previous experience and their present interests. What they really

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2 Substitutes and Examples

know and can do is connected with sensory and motor abstractions. For instance, at that age 2 + 3 = 5 is a motor operation; the sign + gives a dynamical order. Then suddenly they are faced with x + y, which is static, and they fail to understand it, since there is really nothing there which can be understood at their level. x + y is always and for everyone x + y.

What is new in it is quite outside their sensorial and active power of analysis. They cannot put x and y together unless they write xy, as they often do, remembering the fusion of 2 and 3 into 5, the symbol thus replacing the result. They cannot make the addition as they have been trained for years to do, they cannot conceive of x and y as anything but two fixed letters or objects. Very late in their school life they still react in this way, because algebra has been kept outside their real life, which is sensorial, active, emotional before being freely intellectual.

My approach to the creation of substitutes is to give a sensorial, active, emotional value to the intellectual process. I make small machines to perform mechanical processes and I accept a process as being understood if the right result is obtained following mechanical and not verbal-mental lines. For example, I ask pupils to factorize ax + bx with their eyes, and to show them how to do it I cover x with my hands, leaving visible the other factor, a + b. I do not remind the pupils that we multiply a and x, b and x, and add them, or add them and then multiply by x. For me what is involved is an action which the child can perform and which gives the required result. Later on in their school life the pupils will see the full meaning of the process, as they do that of so many others which we never explain.

Presented like this, factorization of expressions like ax + bx + ay + by follows immediately, and the existence of a minus sign 25

Part I Mathematics and the Child

does not introduce any new difficulty. The method has been completely successful.

What are the parts of algebra that can be dealt with by mechanical processes? Obviously factorization, simplification of fractions, L.C.M., H.C.F., arithmetical and geometrical progressions are of this type.

Let us take the example of arithmetical progressions. I make a staircase starting from any height a0 and call d the height of a step. I put, say, a rabbit on the floor, make it jump, and calculate the heights it reaches. If we start from any step instead of from the floor, we can use the same process and let the pupils discover that an = a0 + nd or aq = ap + (q - p)d. Then we look at the figure and see that the first height a0 may vary, as also may d and the step on which we stop. The final height reached will depend on these three data. A certain amount of play is possible here, and is indeed inevitable if the picture is to become familiar. We then make holes in the steps, produce the floor to a wall, and have two people to shout and listen to the echo at various stages, and we find that the time for a return journey of the sound is such that at two steps equidistant from the ends it adds up to the same. an – p + ap is independent of p and therefore = an + ao.

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2 Substitutes and Examples

an = ao + nd, aq = ap + (q – p)d, an – p + ap = an + ao, 2S = (n + 1) (ao + an) If we now send one rabbit up from the floor and another down from the top (an), both jumping at the same speed, and make them stop on each step to discover the height they have reached, we find that both have obtained the same numbers, but in reverse order, and that the numbers add up to the same result.

The formula we obtain is to some extent artificial for the pupil, but each operation can be visualized and the difference between an – 1 and an appreciated, a rather difficult achievement by some other methods. We have tried this method with success several times. The gestures necessary to complete this description are left for the reader to supply. Many other methods are of course possible, and from what we have said it is obvious that a film could easily be made to show what an arithmetical progression is. For geometrical progressions the staircase would be replaced by a sequence of wheels magnifying the numbers.

27

Part I Mathematics and the Child

I would ask the reader not to criticize these methods on logical grounds, for I do not defend here my ideas on these grounds. I merely suggest for the attention of teachers of mathematics devices to simplify their work. When we teach by drill that (–) (–) = (+) we are using a similar method. It is not understanding that we aim to achieve, but a perceptive and active habit of replacing an even number of minus signs related by by a “ + ”.

Let us now consider the identity A2 – B2 = (A – B)(A + B). The difficulty experienced by pupils lies in the recognition of the generality of the formula and of how much can be done with it.

The method I use is first to get from the pupils the fact that A and B are not essentials, since they can be replaced by X and Y or by any other letters we care to choose. What then remains is the pattern, 2 – 2 = ( – ) ( + ), which can also be reversed, ( – ) ( + ) = 2 – 2.

We then see what is identical on both sides. It becomes a recognition of perceptions. All that is then required is to see that a perception can be as wide as we choose to make it and that there is no limit to the complications we can reach. This obviously does not cover the case A4 – B4 or A8 – B8, since here another kind of perception is involved, that of A4 = (A2)2, which must be grasped before we can concentrate on the second step of the process.

Let us pass on to quadratics: ax + bx + c = 0. Literal quadratics and formulae for solutions are sometimes taught to third forms,

28

2 Substitutes and Examples

and it is work at such a level that I have in mind here. We have all experienced the kind of difficulty met by pupils; for instance, the presence of x on both sides of the solution. Without going into details it may be said that it is usual to make pupils thoroughly familiar with the equation before attempting to solve the literal equation. I differ from most teachers in that I do not begin by solving numerical examples. My approach is to separate the visual impressions one gets from the equation in which all the letters seem of equal value. I write a, b, c, x, in different colors and also the exponent2, using say red, yellow, green, white, blue, and white again for the signs +, –, =, 0. I give to a, b, c, different names, calling a Sheila’s number, for instance, b Robert’s, c John’s. With their help I build up sets of equations and show that the arbitrariness of the coefficients is that of given numbers chosen at random, x remains untouched and is shown to be the outcome of the equation. This first acquaintance separates two planes, that of a, b, c, from that of x, then three planes, that of a from that of b and that of c. When we have agreed that it is more economical to work the literal equation than the oo3 of equations with numerical coefficients, we begin to see that there are various types of equations. In order to have quadratics a must differ from 0, but b and c can be 0. Also a, b, or c can be equal in various equations. We then write down the four types in color: I. ax2 + bx + c = o, II. ax2 + bx

= o,

III. ax2

+ c = o,

IV. ax2

= o,

29

Part I Mathematics and the Child

and we see that all quadratic equations must fall into one of these classes. This full investigation of the meaning of the various terms of the equation frees the pupils’ minds from all doubt as to what they are looking at. It seems to me essential, and I spend one or two periods with the class on this work which is usually considered to be unnecessary. It is only because my psychology of mathematical learning uses sensorial, active or emotional approaches to each question that I find it necessary to give such foundations to my teaching. So far I have found it profitable in that it saves much unnecessary repetition.

For the solution of the literal equations I begin by solving a x 2 = 0. As a 0, the only possible answer is x = 0. This case acquaints the pupils with what we call solution of equations, finding the value of x, which by going through various operations indicated in the equation must reduce the left hand side to nought. a x 2 = 0 requires but little thought, and may not show that the pupils have really understood what we are aiming at. But ax2 + c = 0 is a much more profitable test of understanding. As only one operation is visible, the addition shown by +, we can direct the pupils’ thoughts towards the idea that the addition of two similar quantities can never produce nothing. The lesson is two-fold: (a) a quadratic equation may not always have a solution, and (b) ax2 and c must be of opposite signs, or ac < 0, from which we get that –

> 0. This

step is rather difficult to obtain from the pupils, since the minus sign is overwhelming and makes ac < 0 count almost for nothing in one’s awareness. But after some skilled questioning and

30

2 Substitutes and Examples

direction we can familiarize them with x2 = –

and with the

simultaneous fact that ac < 0. The persistence of the sign – is really troublesome, and the trouble is extreme when we come to deal with x = ± (–

), with its new element ±. Most careful

preparation of the ground is necessary here, to make clear its relation to what has already been mastered. This situation shows, by the way, how little of the algebra so far learnt has really been assimilated.

Another type of difficulty is met in ax2 + bx = 0, for although it is easy to factorize, x(ax + b) = 0 has to be dealt with as a product to be equated to 0, the pupils still think of x as one thing and we want them to discover that x could be either 0 or –

,

which to them appear to be simultaneous incompatible values of one unknown. Our task consists in letting them see that what we are seeking is a possible value for x and not to fix x in two ways.

In my experience of literal quadratics these difficulties are great, for they represent the understanding of what is abstract in all this. It should be noted that the solution of quadratics by factors is purely mechanical work and acceptable as a substitute only in the case of easy factors. The solution of (I) contains new difficulties, which can easily be described by the reduction of (I) to (III).

Quadratics have been dealt with in so much detail in order to enable my readers to have a clearer view of the way in which I

31

Part I Mathematics and the Child

undertake the analysis of the teaching of mathematics, seeking every possible device to ensure a thorough acquaintance with all that is implied in a formula, but never forgetting the elements in the childâ&#x20AC;&#x2122;s mind which must be the bond between his experiences and what we offer for his attention.

In geometry the same process is followed, but space will not allow me to go further. In trigonometry at a higher level I have obtained results of the same kind, since in fact trigonometry is an algebra of a special kind using definite relations between symbols.

32

3 The Use of Mistakes in the Teaching of Mathematics

This section of this Part, submitted and accepted in 1948, was published in the Mathematical Gazette, February 1954, Vol. 37, No. 323. under the impression that only their more fortunate fellows who are â&#x20AC;&#x153;mathematicalâ&#x20AC;? can hope to avoid the mistakes which to them are inevitable?

To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that is all-important.

In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.

There is no disguising the fact that, generally speaking, little or no use is made of the opportunity presented by our pupilsâ&#x20AC;&#x2122;

33

Part I Mathematics and the Child

capacity for making mistakes, and yet it is only these mistakes that can throw a light on the significance of our activity as teachers of mathematics. Though obviously we ultimately come to the point at which the mistakes must be corrected and the possibility of their recurrence eliminated, I would suggest that we should do well to curb our tendency to correct, and develop the habit of incorporating into our lessons the observations that we cannot fail to make in marking homework or in using an oral approach with our classes.

It is manâ&#x20AC;&#x2122;s privilege to make mistakes; only through experience, experience that is often painful, does man learn and acquire some degree of wisdom. In the teaching of mathematics, the opportunity for gaining true understanding through experience is too often reduced to the minimum. There is always someone who knows, who can produce the right answer, which is imposed upon those who cannot. But how often must the teacher make the same correction, and how many children reach the end of their school career?

I would suggest that a different approach to this situation is possible. With the help of my students, I made and analyzed a collection of mistakes made by children learning mathematics, with a view to discovering what could be learnt from them. It would not be appropriate here to give the list, and every teacher could certainly draw up his own, but I shall indicate a certain number. They are dismal evidence of the defects of the work we are doing, but they can be put to useful purpose if we are prepared to learn from them what they have to teach.

34

3 The Use of Mistakes in the Teaching of Mathematics

1 Mistakes in elements of algebra: •

failure to understand the meaning of signs,

•

incorrect use of brackets, both with and without (a),

•

incorrect squaring of a + b and a – b,

•

substitution of ab for a + b and of 2x for x x,

•

substitution of division for square root; e.g. x2 = 16, x = 8,

•

substitution of subtraction for division; e.g. 3x = 15, x = 12; ab = c, a=c – b,

•

solution of quadratics with the unknown on both sides of the root e.g.

•

incorrect simplification 0f fractions: e.g. ,

or •

wrong use of sign =; e.g. 3x + 5 = 14 = 14 – 5 = 9 = ,

•

inadequate understanding of rules for operations upon powers; e.g. although x° = 1 is considered true, 90 is thought to be 9 or o,

•

various combinations of the above.

35

Part I Mathematics and the Child

2 Mistakes in geometry related to: •

confusion between what is proved and what is to be proved, and between hypothesis and conclusion,

•

lack of understanding of the notion of “converse proposition”,

•

confusion of data and theorems,

•

inability to suggest constructions,

•

difficulties in syllogistic reasoning,

•

difficulties in preserving the data throughout the proof,

•

lack of sense of generality,

•

lack of trained intuition,

•

confusion of names and notions,

•

inability to separate purely geometrical from numerical reasoning,

•

inability to use parts of a figure separately and to argue about their relation to the whole,

•

inability to make a mental representation of the situation in question, particularly in three dimensions.

36

3 The Use of Mistakes in the Teaching of Mathematics

Arithmetic, calculus, trigonometry and mechanics can all provide us with similar lists. Those given above are sufficiently impressive for our purpose here.

All teachers in secondary schools are familiar with such mistakes, but by no means all see in them an opportunity. For the most part, there is merely an attempt to instill right habits by repetition and drill. Too often we console ourselves, and shirk our responsibility in the matter, by deciding that those pupils who persistently make the same mistakes are dull. In my view, such an explanation is entirely unacceptable. Mistakes are mainly due to mishandling of mental situations on the part of the teacher. How otherwise are we to explain the fact that mistakes are so universal and so similar? Does the explanation not lie in the fact that behind the uniformity of the mistakes there is a uniformity of mental structures of which we are still largely unaware?

Why is it, for instance, that when there is confusion between two operations, the confusion is between addition and multiplication, and between division and subtraction? Why is it that in the proof that parallelograms on the same base and between the same parallels are equal in area, the same obstacle is met by the same teacher every year with different groups of pupils, and by other teachers in other schools, and that the difficulty is overcome in a similar way?

To my mind, there is here a clear indication that such mistakes can, if we so choose, constitute a guiding light into our pupilsâ&#x20AC;&#x2122; ways of thinking. If we are prepared to learn from them, our

37

Part I Mathematics and the Child

teaching will no longer be that of mathematics, but of adolescents engaged in the process of acquiring mental structures that are akin to mathematical structures.

In physics, it is axiomatic that the same part of space cannot simultaneously be occupied by two objects, and this accords with our everyday experience. But in geometry, when we have two overlapping figures—a situation only possible insofar as the common element is seen as belonging first to one and then to the other of the two figures—a different principle is involved.

This simple fact is, however, overlooked, and we fail to see that the resulting lack of understanding of theorems and proofs must inevitably lead to mistakes. We are prepared to concede that mistakes are related to lack of understanding, but we are not prepared to secure understanding so that mistakes may be avoided. Understanding results from integration in existing structures, or from the creation of adequate new structures. It must mean that we stand “over” the problem in question, whereas the presence of mistakes means that we are in a position of inferiority, and it is the task of the teacher to reverse this situation.

The example given above illustrates the difficulties created by a contradiction between everyday experience and a mathematical situation. For many children, such a contradiction presents an insoluble problem; they abandon the effort to understand what is involved and accept the view that mathematics is for the gifted few. The wise teacher, however, will help his pupils to overcome the difficulty by showing them that the figure represents a mental image, and that it is possible to imagine a situation in which actual figures overlap, only because we focus on parts of 38

3 The Use of Mistakes in the Teaching of Mathematics

the figure first in this way and then in that. The wise teacher will remove the contradiction by making it plain that in actual fact there is no overlapping (in the sense that the same space is occupied by two different objects), but that the figures can be envisaged as if there were. He will thereby create a new awareness, a new structure that goes beyond the empirical experience, a mental experience essential in mathematics. It is this mental structure that intervenes in the childâ&#x20AC;&#x2122;s mathematical thinking, not the idealized object called triangle or circle. Since it is a mental structure, it can be acted upon mentally, and when manipulated mentally it can give rise to other mental situations consistent with the whole of reality, with physics and with everyday experience.

And what of algebraic mistakes, so common and so much more obvious than those in the field of geometry? Here again the remedy is understanding, not repetition and drill, and the role of the teacher is again that of discovering the mental structure that intervenes in the algebraic thinking required.

Few teachers teach addition and subtraction together, and division and multiplication together, as different aspects of the same operation. In my view, it is precisely this fact that accounts for all mistakes in the four operations, apart from numerical mistakes. The fact that in our textbooks different chapters are devoted to multiplication and factorization, etc., shows that the separation is deliberate on the part of most teachers. Such an arrangement is presumably intended to isolate difficulties, whereas in reality it merely created them.

39

3 The Use of Mistakes inPart the Teaching of Mathematics I Mathematics and the Child

In algebra all operations are reversible. By their very nature, operations constitute pairs. To become aware of the meaning of algebra is to be able to perform both an operation and its inverse with the same ease. Addition and subtraction are two aspects of the same algebraic situation, and are reducible one to the other.

It is therefore essential that the pupil who can add algebraically shall at the same time be able to subtract algebraically, thus avoiding mistakes in transferring terms from one side of an algebraic equilibrium to the other. The same holds for multiplication and division.

The mistakes made through confusion of addition with multiplication point to another source of difficulty. In arithmetic, multiplication of integers is repeated addition, while multiplication of fractions is a much more complex algorism.

The same word is used in both cases because there are connections between the two processes. In algebra also, multiplication yields addition when some of the potential numbers become actual, e.g. when ab becomes 2a, but in general it is an operation defined, like addition, by its own laws, and distinct from addition in that the identity operation is 1 instead of o. This structural quasi-identity of addition and multiplication in algebra constitutes a stumbling-block for beginners when numbers vanish into letters, and more attention needs to be given, not to drill, but to the actual formation of the abstract operation, its properties being abstracted from arithmetical situations by the use of problems of the â&#x20AC;&#x153;think of a numberâ&#x20AC;? type, as a first step to equations.

Algebra is operations upon operations, and can be grasped without difficulty if it is constructed in this way during the first 40

3 The Use of Mistakes in the Teaching of Mathematics

two years of the secondary school. Operations are reversible by definition, and gain their individuality through specialization, generalization then being the result of the combination of the new operation with those already existing. All the mistakes quoted are due to lack of awareness of the operations involved, and this lack is in turn due to insufficient emphasis on operations by teachers who have a different conception of what constitutes algebra.

Operations in arithmetic are part of the mental structure of our secondary school pupils, and can be performed, at least in the grammar schools, with some skill. The problem of teaching algebra is, to my mind, that of passing first from operations upon numbers and quantities (the result being numbers and quantities) to awareness, not of the result, but of the operations themselves, and then to the substitution of operations for quantities. The process might be as follows: from 7 4 + 12 = 40 to 7 p + 12 = 40, 12 = 40,

p=

or r =

p=

to r p +

; to r p + k =

40, and finally to r p + k = h and its various solutions. The numerical aspect vanishes slowly and is replaced by a set of operations that are regarded as equivalent since they can be reduced one to the other directly either or through their inverse. (a + b)2 is the square of a sum, and (a â&#x20AC;&#x201C; b)2 the square of a difference; each of them is also an operation upon another operation.

41

3 The Use of Mistakes inPart the Teaching of Mathematics I Mathematics and the Child

The teaching of algebra is the teaching of the dynamics of operations, and this is a self-checking mental activity that requires no other authority than its own smooth functioning. It is for the teacher to create awareness of the mental processes, to bring the structures into existence and let them function according to their own laws. The mistakes which then occur will either be slips, or will be due to insufficient awareness, and will serve to indicate which structures have failed to integrate with previous systems to form new wholes. Once mental structures are present, they impose themselves on the awareness with an inescapable rigor, and determine the framework within which what is true is immediately recognized. Mistakes then no longer have a place; they are spontaneously eliminated through further efforts towards mastery, which is biologically necessary for mental health.

Our responsibility to our pupils demands that we shall not attempt to give norms and right answers, but that we shall use all our skill and imagination to discover the mental structures required for dealing with our mathematical problems, which originated in some mind endowed with those structures.

42

Part II Pedagogical Discussions

1 Note on Pythagorasâ&#x20AC;&#x2122; Theorem

This article, written and accepted for publication in 1948, first appeared in January 1952 in the U.S.A. in The Mathematics Teacher, Vol. 45, No. 1.

In the following pages we propose to discuss some of our thoughts on the beautiful theorem which still fascinates millions of people all over the world.

Our suggestions are those of a teacher facing youngsters whose mathematical insight is greatly hampered by the fact that they have to learn things chosen by adults at moments chosen by adults for adult reasons. The teacher who is aware of this difficulty will try to introduce as much variation as he can into the treatment of any question.

In teaching a theorem like that of Pythagoras it is usual to suggest a variety of intuitive proofs until the formal classical proof can be developed. In this note we are undertaking a study of the whole question rather than suggesting how it should be developed if used in a classroom. Although what we say is the 45

Part II Pedagogical Discussions

outcome of real class teaching, the stress here is on the logical content of the lessons we would give.

In Figure 1, let ABC be the right-angled triangle, A = 90째; and a, b, c, the name and length of the sides BC, CA, AB, respectively.

There are two ways of dealing with the concepts contained in the Pythagorean theorem: areas and similarity. Either we assume areas and assume that we do not know what similarity is, or we start with the concept of similar figures and then apply similarity to problems of areas.

In addition, the possibility clearly exists of starting simultaneously with areas and similarity, but that is somewhat contrary to the deductive and axiomatic tradition prevailing in geometry.

Here we shall separate the two approaches and see what can be done in each case that makes one distinct from the other. It is obvious that experience in teaching will help in the choice of material introduced into actual lessons with secondary school students.

Case I. We assume areas but not similarity.

46

1 Note on Pythagoras’ Theorem

Let us consider the right triangle ABC and the altitude AH drawn to the hypotenuse (Figure 2). Then Area ABH

+Area AHC=Area ABC. If we unfold the three triangles ABH, AHC, ABC by rotating around AB, AC, BC respectively,

we obtain a figure very comparable to the one in Pythagoras’ theorem except that right-angled triangles replace squares. Here we do not need a proof, but we fail to see the relation between this fact and that contained in Pythagoras’ theorem.

From this simple result we can deduce an interesting extension of Pythagoras’ theorem which appeals to us because it frees us from the static original pattern and allows for the consideration of an infinity of patterns in the same figure.

47

Part II Pedagogical Discussions

In Figure 3, let us draw the parallel to BC at A’ and the parallels to AB and AC at H’ and H” respectively (see Figure 4). If P, Q, R are any three points chosen respectively on YZ, ZX, XY we have Area PBC = Area QCA + Area RAB.

And in the case of Pythagoras’ theorem we get:

48

1 Note on Pythagoras’ Theorem

“If 1

triangle ABC is such that angle A = 90°,

2

the lines x, y, and z are respectively parallel to a, b and c and their distances are equal to a, b and c respectively,

3

P and Q are two points on x so that PQ = a, R and T are two points on y so that RT = b and U and V are two points on z so that UV=c; then Area PQBC = Area RTCA + Area UVBA.”

How far can one replace right-angled triangles and squares in the proposition in order to obtain extensions, not making use of similarity?

If it is proved that the area of a circle is R2, we can give the following figures and results: In Figure 6, Area C1 = Area C2 + Area C3 since multiplying both sides of a2 = b2 + c2 by gives the result.

49

Part II Pedagogical Discussions

In Figure 7, Area D1 = Area D2 + Area D3; the factor being

.

In Figure 8 Area Z1 = Area Z2 + Area Z3; the factor here being ( ) as can be seen in Figure 9.

These extensions are immediate and could be multiplied, but the limitations we encounter are due to the fact that we cannot use any other means of calculation than elementary algebra, and that we cannot make use of similarity.

50

1 Note on Pythagoras’ Theorem Part II Pedagogical Discussions

The following pattern is in the same field of causality and is useful in suggesting some of the important extensions of Pythagoras’ theorem.

Let Q0 be a fixed square, Po a point on side BC, Pox the perpendicular to BC at Po. Let P be a moving point on Pox. In each position we draw the squares Q and Q on PB and PC. Quite clearly when P is at Po, Q, + Q’ < Q0 and when P is at P’, Q+ Q’ > Q0. We can look at the moving pattern and see that by moving P continuously on P0x, we pass from a sum of two squares less than Q0 to a sum of two squares greater than Q0. In the first case the angle at P will be obtuse, in the second case will be acute, and the limit case of the right angle leads to the sum of the squares being equal to Q0.

51

Part II Pedagogical Discussions

The consideration of Figure 10 leads therefore to the proposition: “The sum of the squares on two sides of a triangle is less, equal or greater than the square on the third side according as the angle between these two sides is obtuse, right or acute.”

Naturally it is a further step to change the inequalities into equalities by adding or subtracting twice a suitable rectangle, but this pattern has the advantage of putting Pythagoras’ theorem in the natural sequence of patterns contained in Figure 10.

Case II. We know similarity and can prove that similar figures have areas in the ratio equal to the square of the ratio of similarity.

From Figure 2 we can show by similar triangles that a2 = b2 + c2, where a2, b2, c2 mean the squares of the lengths of the sides.

“If similar figures are drawn on the three sides of a right-angled triangle then the areas of the figures on the two sides add up to the area of the figure on the hypotenuse.”

In fact we have a2 = b2 + c2

52

1 Note on Pythagoras’ Theorem

and if F1 F2, F3 are the similar figures respectively drawn on a, b, c then

and

Hence Area F1 = Area F2 + Area F3.

Pythagoras’ theorem, as well as the results of Figures 2, 6, 7, and 8, are obviously special cases of this theorem and we see the deep reason why the case of the right-angled triangle (Figure 2) is so like that of Pythagoras, a fact that escaped us in the first instance.

Underlying Pythagoras’ theorem are therefore the idea of similarity and the obvious result of Figure 2.

Similarity opens the field for an infinity of figures replacing the squares of Pythagoras, removing the need for actually finding the factor by which to multiply the equality a2 = b2 + c2, as we had to do previously and this last equality is itself the application of similarity to Figure 2. The fact that Area ABC = Area ABH + Area. ACH in Figure 2 is obvious. It does not

53

Part II Pedagogical Discussions

require any proof and represents the first of a sequence of results of which that of Pythagoras is only one, and indeed no longer such an outstanding one, since squares are not such striking figures when we compare them with other analytic figures.

But we must not forget that the results of Figures 4 and 5 are true even for non-similar figures, and this is an extension obtained within the logical field of the concept of area.

If we add results like those to Figure 10, we see that the two treatments of the question, though they overlap in some respects, are definitely complementary and can bring to our teaching additional sources of light and of understanding.

Our short analysis shows some possibilities of dealing with Pythagorasâ&#x20AC;&#x2122; theorem in a simple yet quite deep way.

54

2 A Note on the Teaching of Mathematics

This article first appeared in the U.S.A. in the Journal of General Education, Vol. VI, No. 4, July, 1952 The teaching of mathematics is to be regarded, first and foremost, as a specialized creative activity of certain adults. This specialized creative activity involves a synthesis of three fields of study which takes place, not as a philosophical exercise, but in the actual teaching situation and its planning. The three fields are those concerned with the working of the mind; the mathematical activity that results in mathematical statements; and the needs of society, science, and technology.

It is a specialized job that can be done only by teachers of mathematics. No-one else is concerned with the actual linking of the pupilâ&#x20AC;&#x2122;s mental activity with mathematical knowledge, and it is only the teacher who needs to know how best to achieve this particular task and who recognizes the value of years of effort

55 57

Part II Pedagogical Discussions

spent in the study of this or that part of the huge field of mathematics.

Once the problem of teaching is seen as the creation of an actual synthesis of mind, mathematics, and the needs of the future, it remains to discover how this synthesis is to be achieved. Here there are at least two fruitful lines of attack.

The first concerns mathematics itself. After sixty years of study of the foundations of mathematics, it is now recognized that mathematics is a specialized activity of certain minds, giving rise to statements which we call “mathematical” and which are most easily communicable in the esoteric language to be found in treatises and journals. That this is so is easily seen in the attitude of the working mathematician of today. He deals with structures and relationships between structures, rather than with so-called “mathematical objects”. In other words, mathematicians have become aware that the “objects” of their science are particular mental constructs to which they apply their mental dynamism in order to make explicit the content involved in them by virtue of the imbedded structures. Not all constructs are susceptible of further development: some are too quickly exhausted, and some are too difficult, in the sense either that they do not show their component structures or that little experience exists of such types of construct.

Mathematics being a mental activity of a specialized type, it is obvious that it takes place in minds and that these minds have had to build their mathematics out of their mental experience.

And what do we know about this experience? We know that

56

2 A Note on the Teaching of Mathematics

mathematicians were once children and that mathematical structures did not exist in their minds, but merely the possibility of forming them. It is this formation of the mathematical structures that needs to be understood before the first step toward the synthesis can be taken. Mathematicians have helped by showing how certain evolved mathematical structures can be reduced to simpler ones and how re-combinations produce the fundamental theories.* The second line of attack concerns the mind itself. Let us consider the possibilities open to the psychologist who is aware that our mathematical knowledge has to be constructed out of our mental experience.

He can investigate at what stage of mental development the child is capable of answering this or that question in the field of mathematics. For instance, when can the child use metric properties in order to ascertain a geometrical fact? When is he capable of a generalized statement, such as that of the transitivity of equivalence relations? This line of attack is particularly exemplified in Piagetâ&#x20AC;&#x2122;s work. For him, mathematical notions are constructs, but they are necessary constructs, and his concern is to discover how they reach their final stage, which is identical with what he himself finds in the notion.

Another possibility is to assume the existence of mathematics as seen in the school curriculum, and to investigate at what stage

* Cf. N. Bourbaki, ElĂŠments des mathĂŠmatiques (Paris: Hermann & Co., 1938- ).

57

Part II Pedagogical Discussions

this or that difficulty is at a minimum and the subject capable of taking it in his stride. This is the approach of many schools of psychology in Germany, Holland, and the United States.

For the writer the situation is entirely different. Accepting the lesson given us by mathematicians, we attempt to discover an explanation of the mental activity that is valid both for the child and for the creator of mathematical notions and statements. A temporary approach which has already proved fruitful is by way of what we shall call mental structures. It is to be considered as temporary, since in these pages little is said of the emotional elements involved in thinking, and it is likely that affectivity plays a considerable part in our thought-processes.

A concise definition of “mental structures” is not easy, but the empirical awareness of our mental activity provides a useful starting point. Everyone knows how habits are acquired through repeated drill and increasingly frequent check to make certain that the aim set is achieved. It is the interaction between check and drill that determines whether we continue the exercise or pass on to another activity, as can easily be seen from observation of children playing marbles, hopscotch, or leapfrog or of the pianist or the scientist at work. It is in the formed set of mental patterns in which perception, action, and representation are intermingled that we find the mental structure. It is this that underlies the symbol and is the basis of our social intercourse. It is this that is the starting point for elaboration into a more developed structure. Thus, to remain in the field of mathematics, the mental structures which we call the “first integers” will be formed out of the perception that sets of objects, each with a specific use (such as pears, nails, coins, 58

2 A Note on the Teaching of Mathematics

cups) can also be spoken of in terms of their cardinal numbers.

When a few integers enter our awareness and are regarded in relation to one another and not to the objects, we are on the way toward forming the mental structures that we shall call â&#x20AC;&#x153;integersâ&#x20AC;?. These mental structures are now new elements of reality and form part of another system of communication, verbal and notational, which will develop into arithmetic.

Each moment of a mental structure is a mental structure, and there is, in general, differentiation in time between them even if the new structure is recognized as containing a previous one.

Triangles are not always seen as a special case of polygons or as intersections of some solid figures by some planes.

The mental equipment common to all men is formed of perceptions and actions. Early childhood is used for the organization of experience through these two types of structuration, which are intermingled from the start. The mental structures thus obtained will be the material to be used in the formation of mathematical structures discovered by mathematicians. But not all mental structures of a mathematical character are singled out by mathematicians, since for them there are sets of primitive notions. For instance, the criteria for recognition of membership of a set are assumed, although for the psychologist it is an elaborate problem to decide whether a A because of this or that attribute. This distinction between the working mathematician and the psychologist has its bearing upon teaching. Though today mathematicians use sets as primitive notions, it took generations of thinkers to arrive at the point at which a set could be conceived as a mathematical notion. Mental structures are richer than mathematical 59

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structures, but the latter are â&#x20AC;&#x153;freerâ&#x20AC;?, in that they attain wider fields of existence. For instance, each integer is a mathematical structure which, when it is involved in the life of its users, is a mental structure that intervenes in many situations but loses its power of being replaced by all its equivalent definitions in terms of the other integers. It does not appear as being linked with these but with a different kind of reality; its mathematical character is blurred. In the case of the smaller numbers, these two aspects are evident when, on a shopping expedition, we pay for what we buy and add up our expenses; in the first case money is a medium of exchange, in the second it is numbers.

The evolution of certain mental structures into mathematical structures is not from the simple to the complex, but from mental structures involving perception and action to mental structures in which these are symbolically preserved. A useful example is that of the transition from measuring the angles of a drawn triangle to the statement of the constancy of their sum in any Euclidean triangle.

To draw is clearly to combine perception and action in a specific way leading to the awareness of a mental structure as their combined substitute. To measure angles is also to combine perception and action in another specific way, and it is only when we are struck by the astonishing fact that the result of our actions always falls between numbers very close to 18o° that we begin to wonder whether this is a property of the mathematical structure which is not any one of the drawn triangles but a substitute for all triangles. But what is meant by the statement that the sum of the angles of a triangle that is not drawn is 1800?

This question has no answer unless we see that mathematical 60

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structures are specialized mental structures, those whose link with perception and action has become potential instead of actual. We shall be able to substitute for the actual drawing a potential drawing by producing a triangle as an organized mental structure; for the angles, the set of all the equal angles that can be obtained by displacements preserving their magnitude; for their sum, the angle obtained by eliminating the common sides of the adjacent angles we can form and inquiring whether the exterior sides are on one line. The mental activity thus displayed is called a “formal” proof, as distinct from the other, which is empirical. In our view it is quite otherwise, and the two proofs are concerned with two levels of awareness of what our mathematical mental structures are. The formal proof takes place with mental substitutes which are those mental structures in which perception and action are felt to be remote but are, in fact, potentially contained. Thus the formal proof is neither more convincing nor more generalized than the empirical. It concerns material extracted by an ordinary mental process from the set of substitutes for combined perception and action and seems different only until we question our conviction and discover that it is a matter of agreement that makes mathematical structures of a different essence. If we look closer, as the psychologist does, we find our thought not only can “abstract” but can also return that to its more primitive mental structures.

The word “abstraction” is adequate to express the formation of mental structures that apparently rest upon themselves rather than upon the actual material of perception, but abstraction is not a process that can be dated in life. It is active all the time that the mind is actively experiencing. It is, in fact, the process 61

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of producing mental structures. Language learning, symbolic play, and dreams use substitutes abstracted from the process of objectivation or creation of structures. From the start, human life puts the child in front of a man-made universe, and both the natural world and the social world in which he lives require him to be able to make judgments involving perception and action combined in a multiplicity of ways.

When children recognize that their life consists in some measure in this process of forming mental structures that become increasingly abstract in order to meet ever wider challenges, they are aware, without its actually being expressed, that they are potentially mathematical creators. For, indeed, those among us who specialize in the formation of mathematical structures, more or less particular, more or less involving known structures, more or less productive, are as truly creators as the artists who form particular plastic structures with special properties.

Owing to the multiplicity of the elements involved, the relationship between mental and mathematical activity may still be somewhat obscure, but this discussion may have simplified the situation sufficiently for us to make further dogmatic statements.

In contrast to other psychologists and teachers of mathematics, we consider mathematics as a creation of the mind of mathematicians, and the learning of mathematics as the creation, by each individual student, of mental structures akin to those of the mathematician. It therefore becomes possible to define the first step of the synthesis we are seeking as the

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organization of the child’s experience in such a way that he becomes aware that his mental activity yields mathematical structures having their own organization, which organization is called “mathematics”.

It is the teacher himself who must first make this synthesis in his own mind. For him, however, the process is rather the reverse of the child’s. The teacher must see that mathematical structures are mental structures and must be able so to disentangle them that he can see at what stage of his pupils’ experience he can make them aware of the components involved. It is possible to develop such a study in the form of a “functional syllabus,” and the writer has done so in relation to the English grammar-school syllabus. One or two illustrations will suffice here.

Let us consider the teaching of simultaneous equations to grammar-school pupils of thirteen to fourteen years of age. We begin with a game: A thinks of (but does not announce) a number, and B announces a number. Then C asks A to increase (or diminish or multiply) his number by B’s, and to announce the result. The answer—A’s original “unknown” number—is obtained by all the pupils when they “undo” what has been done.

This, repeated and complicated, leads to the solution of equations in one unknown (either numerical or literal).

Moreover, the equations need not be linear. Quadratics can also be solved, first in the set of non-negative numbers and later in the set of all numbers. The game is then extended, with two people thinking up numbers, and the teacher adding them. This indeterminate equation can be solved by the two pupils in question, while the others can give only the two equalities 63

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x = A – y or y = A – x. If we all want to find the two numbers, we need another equation; and we begin by studying, e.g., the pattern x + y = 3, x – y = 1. The solution is immediate by inspection. It is almost as immediate if the right-hand sides are, respectively, 30 and 10, or 300 and 100. But if they are, respectively, 30.1 and 10.1, the solution becomes possible only through an awareness that the answers to the previous pairs of equations are obtained by adding and subtracting the two equations.

When someone in the class discovers and expresses this fact, several pupils see it immediately. Then a whole lesson is spent in establishing the method, keeping the left-hand sides x + y and x – y, respectively, and changing the right-hand sides. The numbers are replaced by letters, then by polynomials in different letters. When this has been mastered— and no more than one period is required for this in any form in a grammar school—the work that follows consists in discovering all the permissible changes of pattern, i.e., that the choice of x and y is immaterial; that, if powers of x and y are used instead of x and y, a further step is involved (extracting the required root at the end); and that the numerical values of the coefficients of x and y do not need to be 1 and 1 but must be the same in the two equations (though not the same for the two letters). Thus, after two lessons the class can solve equations of the following type by inspection:

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The operations can be correctly carried out by all pupils, and the answers,

can be checked by reversing the operations.

The next step is to establish that the order of the equations and the relative arrangements of the right- and left-hand sides are also immaterial. Thereafter the pupils solve

and all the other permissible variants either by explicitly reducing to previous forms or by doing so mentally. This having been mastered (in one lesson), fractional coefficients are introduced in all these types of equation. In five lessons the class is ready to solve a wide class of pairs of simultaneous equations (which they themselves have constructed from the start), expanding the initial pattern to its extreme degree by all permissible operations at their disposal. We are still dealing, it is true, with the same pattern of co-efficients, but all changes within this restriction are experienced as the dynamics of the situation. Mental structures are expanded to the full with mathematics that is related to the childâ&#x20AC;&#x2122;s previous experience.

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The step forward to the general solution of simultaneous equations involves the extension of the number set to that of all numbers, and of the pattern to that involving arbitrary co-efficients. The difficulties here are smaller than previously.

We only need combine the existing dynamics with two new operations: to multiply equations by suitable coefficients, thus producing patterns comparable to the original one, and to note that negative expressions are susceptible of the same treatment as positive ones.

The dynamics of the original case can be described as the passage from

to

and that of the general one, as follows:

to

to

to

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to

to

and

Each system is deduced from the preceding one by multiplying by appropriate coefficients or by forming a linear combination of the two equations. This analysis shows where careful planning of the lessons is required. There are two delicate points in the construction of the mechanism: first, the transition from two pairs of numerically equal coefficients to one pair; and, second, the forming of linear combinations of equations. A comparison of systems such as the following shows that equations which are merely proportional lead to identical answers and that any one equation can be the representative of all:

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The comparison of the answers and of the new equations obtained by multiplication does not supply sufficient awareness.

The main point to be established is that there are infinitely many equivalent pairs of equations. It is awareness of this fact that is essential for the freeing of the children’s minds.

The true mathematical value of the study of simultaneous equations does not lie in the pupils’ capacity to solve automatically any pair of such equations. It lies rather in their understanding of the compatibility of data, of the distinction between unicity of solutions and indeterminacy, and of the possible extension of the method to any system of linear equations. It is in the translation of this awareness of the mathematician into a system of lessons involving mental structures and the mental activity of our pupils that our functional syllabus can make its contribution to the improvement of the teaching of mathematics.

Our second illustration concerns geometry. We do not wish to teach Euclidean geometry as a deductive system. We want to give our pupils geometrical experience, a capacity for making statements about properties discovered in the context of spatial experience. Our pupils can see that a pair of compasses used in a certain way enables us to draw a figure we may call a “circle”.

They see that we can use one center and various radii and produce a pattern of concentric circles; or varying centers and the same radius; or varying centers and varying radii. This gives

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them experience of the three degrees of freedom of the circle in the plane. Combining these procedures in various ways indicates how the drawing of a circle is affected by various conditions. For example, the circle must pass through a given point, or through two given points, or through three, etc. It is a striking experience to discover that four points are not, in general, on any one circle and that three points not on a line fix a circle. All this can be obtained as part of the organization of perception and action. By selecting pairs of circles from two systems of concentric circles, it is further possible to provide the excitement of discovering that, if corresponding radii are equal, the points of intersection seem to be on a straight line; that, if the sum of the radii is constant, the points are on an ellipse; etc.

Loci are dynamic patterns denned by a property and defining a property. It is possible to give our pupils experience of a geometrical nature when in their actions they perceive invariant properties. It is an amazing fact that the radius of the circle can be taken six times on the circumference and produces a hexagon; that the figure can be rotated in six different ways and superimpose the hexagon, while the circle is invariant for any rotation round the center. Group properties thus emerge quite early and may play a part in future work. We have to introduce â&#x20AC;&#x153;abstractedâ&#x20AC;? figures contained in patterns and their properties contained in given situations. For example, the circle and the hexagon bring in equilateral triangles, rhombuses, trapezia, and rectangles, when vertices are joined in different ways. These figures are duplicated under rotation around the center, and six different positions are experienced. Metric properties for sides and angles can be abstracted and experience in statements obtained.

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These few examples show that greater freedom in our work as teachers in no way diminishes the possibility of our pupils becoming mathematicians if they so choose. On the contrary, through the functional syllabus they enjoy their creation of mathematical structures and take it as far as time and interest allow. The results are as securely imbedded in their experience, in their minds, as is their capacity to walk and talk. After all, it is our attitude toward our mental structures that produces mathematical structures. Once the latter exist in our awareness and are recognized as being of the same nature as our other mental equipment, there is no shyness in considering them, talking about them, expanding them. This is in no way idealistic; it takes place in all the classes we teach in London.

Mental structures are the lot of everyone. Some of them are mathematical because they are looked at by the mind in a particular way. It is our task as teachers to eliminate preconceived ideas and to make our pupils aware of their capacity to evolve the attitude that produces mathematics out of their mental activity. Our testimony is that this is not only possible but remarkably easy when we ourselves believe the child and ourselves to be creative minds.

So far we have made the link between mental and mathematical structures. There is a third term in our synthesis: the needs of the future. We have to discover means of linking the mental activity that is mathematical with a world possibly different in content from that in which the adult generation of today was born.

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Politically, the world has changed in the last forty years from a situation of European supremacy to one in which new and powerful countries are developing new ways of life—ways that to our generation are apparently conflicting. But it is also true that science and technology are offering new challenges that are not necessarily different in the various countries. For many people the future is obscure, and it seems absurd and unreal to plan for it. However, it is obviously equally absurd and unreal to imagine the world as continuing as it was forty years ago.

The teacher’s function is clearly to equip his pupils for a life which will take place in the future. He must make assumptions about this future and plan for it. Otherwise he must accept a special responsibility for any shortcomings in the next generation’s ability to adapt to the new conditions.

The Commission* considers that it is possible to begin a study of the actual demands of the future and sees it as a duty to undertake this part of the synthesis earnestly and with all competence. In 1953 a seminar will be held in Holland to examine those aspects of the challenge of the future that are already apparent and their relationship to the functional syllabus.

* The International Commission for the Study and Improvement of the Teaching of Mathematics was founded by the author in 1951 and he was its secretary until 1960.

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In conclusion we would say that human endeavors require human actions and that the specialized function of preparing the young for life in an industrialized world puts a particular responsibility on the science and mathematics teachers who are their guides and to whom society entrusts the task of that preparation. If these teachers do not do it, it will not be done.

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This article first appeared in Belgium in Mathematica & Paedagogia, No. 3, 1953-1954.

In addressing myself above all to men and women who teach mathematics, I intend to broach the question of intuition from the angle of the activity in the classroom. What is discussed can be equally applied to matters other than the introduction to schoolworkâ&#x20AC;&#x201D;to research, for example.

The first remark I should like to make is that mental activity cannot be defined except in terms of situations. A whole series of possible situations exists making demands on individuals, and in every case the being mobilizes its capacities, its experiences, its will, etc. in a certain way. This is not simply a reaction to a stimulus but rather a complex preparation on the perceptive, affective, active and intellectual levels, an opening of internal floodgates to allow a restructuration appropriate to the

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situation. A restructuration, not only of automatisms, but also of initiatives which make selective use of the means.

Thus, for the teacher, a true lesson assumes the character of a multiple challenge where the topics to be considered, the weather, the mood of the pupils, their previous preparation, his own mood and his limitations at the time will all play a part.

Every teacher knows that affective elements have an important place in the act of learning.

If therefore we begin with the knowledge that at every moment of our lives we are in relational situations, we shall then be ready to consider the question of learning and teaching in a more realistic light. After all, every teacher should know what his personal position is in relation to mathematics. Does he understand it? Is he sufficiently acquainted with it for his feeling of inferiority induced by the vastness of mathematics to be counteracted? Has he considered the role of logic and its place in the whole? Does he know in detail the history of any single important discovery and has he seen what made this possible?

Does he realize the part played by the temperament of the individual mathematician, or what the dominant trend is, or what difficulties arise in choosing which fields of study to undertake, at any given time? Has he taken into account the gap between social requirements or the needs of other sciences, and the state of mathematics as studied or taught in various contemporary institutions?

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There are many other questions which it would seem necessary to ask before one can adequately place oneself in the situation arising from the teaching of mathematics. Up to this point we have considered the situation solely from the point of view of the teacher. But here pupils, syllabi, methods, examinations and social habits must also be taken into account.

I do not intend to discuss in detail all the component elements in the situation influencing each other; a fat book would be necessary were I to do so. But I shall try to bring real pupils into the situation, created by the syllabus and the teacher, of a lesson at school.

The syllabus, to me, appears only to offer some signposts in the world of the mathematics the teacher must make accessible to his pupils so that they may find themselves, at the end of their study, in a better position in relation to the questions to be tackled later on. The syllabus can do no more than provide a series of headings; it is up to the teacher to make himself understood in the reality of the classroom. He should be able to work out the methods necessary in order to overcome the particular difficulties introduced into the schoolroom situation by the personal characteristics of the individuals with whom he is concerned, the pupils. It must be understood that, even as seen by inspectors, methods to be used cannot but fall within the province of the teachers. If they fail to avail themselves of this privilege then the blame obviously falls fully on them. The classroom situation has thus been clarified a little, even if, in the real world, owing to the temperament of certain teachers or inspectors, a much greater influence is accorded to a number of factors whose importance may here appear minimized. It seems 75 77

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important to begin an analysis of a teaching method with these considerations which place the professional and social responsibility squarely on the shoulders of the teacher. In the last resort we are concerned with relationships between individuals; rules or recommendations have no effect except insofar as they are applied by individuals.

In a real mathematics teaching situation there is a more particular situation still than that described above, a rather artificial one which proposes to make demands in a certain direction on the mental activity of pupils. It is with this situation that we shall now be concerned.

The syllabus â&#x20AC;&#x153;atomizesâ&#x20AC;? mathematics, that is to say it isolates its component points in order to schematize it. Textbooks atomize in this way in order to overcome difficulties, teachers atomize in order to adapt themselves to the abilities of their pupils. This seems perfectly legitimate in the contexts in which it occurs and is accepted by the great majority of teachers as the only satisfactory course to adopt.

However, if it is true that the dialogue with the universe at every level occurs within meaningful situations which mobilize the whole being, this systematic atomism cannot be justified since it eliminates the true spur to action, the affective challenge. The difficulties seem greater rather than smaller if the mind tackles one point in isolation, at a time when it is attracted by the whole.

Every teacher knows that the technique of subtraction with borrowing is hard to teach; that theorems not associated with other theorems are speedily forgotten; that teaching how to

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prove rigorously a theorem while not assuring oneself of the pupilsâ&#x20AC;&#x2122; understanding is to build on sand. What seems to result from this it that we must reconsider precisely the way in which we tackle these difficulties, allowing ourselves, if necessary, to be guided by other principles of work, principles we should discover were the classroom a true laboratory and not merely a place where theories are applied.

In my intensive experimental work over a period of years I have, with a considerable degree of success, I believe, employed methods of teaching mathematics based on the following observations.

To do mathematics is to adopt a particular attitude of mind in which what we term relationships per se are of interest. One can be considered a mathematician when one can isolate relationships from real and complex situations and later on when relationships can be used to create new situations in order to discover further relationships.

Teaching mathematics means helping oneâ&#x20AC;&#x2122;s pupils to become aware of their relational thought, of the freedom of the mind in its creation of relationships; it means encouraging them to develop a liking for such an attitude and to consider it as a human richness increasing the power of the intellect in its dialogue with the universe.

It is apparent that we are here far removed from trying to convey a mechanical process and stereotyped reasonings.

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The cult of rigor is, in fact, a weakness. Before we demand the adherence of our pupils to very strict standards, we should make sure that they have sufficient experience to justify the development of the rules of communication towards their formal aspect. Rigor demanded too early indicates a lack of understanding of the real situation. Our task is to make our pupils first acquire experience in mathematics, and as the awareness of relationships increases, we progressively bring in a spontaneous demand for more precision in communication, required because of the ambiguities of speech. Thus the increasing awareness in the discovery of relationships and the demands of communication together provide a natural place for verbal proof and formal rigor, and now this does not seem to be imposed from outside or only designed to torture the minds of the young.

It would seem that all this is only very vaguely associated with the title of the article; where does intuition come into it?

There are so many definitions of intuition that it is essential that we make clear what we mean when we use the word. For the psychologist, and it is as such that I am now speaking, the word intuition means an undifferentiated contact with situations. It means contact of the whole being, open to the demands contained in the situation, perceptive with all its senses, active with all its means, intelligent with all its experience, artist insofar as it is sensitive to the beautiful, to the organized, to the unusual, etc.

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To penetrate a complex situation, to allow oneself to be penetrated by it, and to extract from it what is appropriate is, for me, accomplished with the aid of intuition, which is more than reason, intelligence etc., but which is also less than these since it is less specialized and less adequate for the reaching of particular goals.

It is clear that this definition of intuition made our remarks about situations in general, and mathematical and pedagogical situations in particular, a necessary introduction to the view we hold about education.

No one can boast that he is able to dispense with intuition, but there is a tendency in mathematics to minimize it and be suspicious of it. I should like to suggest to readers that the educationalist should be the last to do without it, and indeed should learn to make a wise and more circumspect use of it.

Here briefly are a few examples by way of illustration. Any teacher, however little experienced, could produce others.

1 Let us assume that we are introducing our class to plane geometry. Giving every pupil a pair of compasses and asking them to draw two families of concentric circles, we are creating a situation from the outset. To meet this situation in any given way would require participation by gesture, perception, abstraction, verbal translation, organization of its component elements, polarization of the attention, and so on.

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For example, pairs of equal circles can be taken and their points of intersection (if they do, in fact, intersect) can all be seen to lie on a straight line.

From one of the families a circle can be chosen at random and the circles of the other family classified with reference to it according as they intersect, touch, envelop or are completely outside it.

Many other things besides may be done depending on the point of view taken. In particular, one can extract from the situation the construction of the bisector of a line; the construction of a triangle when the lengths of the three sides are given and the conditions for such a construction; the inequalities in the triangle, etc.

By encouraging the children to say what they are doing and how they are doing it and by making use of group work, we can introduce into the exercises taught all the rigor desired.

2 By animating a particular figure, a dynamic pattern can be created which brings about awareness of a number of points which are treated separately in the text-books. For example, let 0 be the center of a circle of diameter AB, and P a point somewhere on the circumference. It can be seen that OAP and OBP are two isosceles triangles, and that the angle APB is therefore a right angle; but in addition, the interesting relationship to be seen in this figure is that the angle APB is half the angle formed at 0 by the straight line AOB. This becomes obvious if P is fixed and B moved, and the same figure 80

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considered for the new theorem. The role of the diameter in the first instance was a double one: it was a support for the two radii AO and OB, and also for the chord AB. Various statements can therefore be made about the same situation depending on whether one underlines or not this double role. Moreover, as P is arbitrary and its position has no special significance, we can see at the same time the two supplementary arcs and the cyclic quadrilaterals that can be inscribed; all this can now be simultaneously considered.

All these theorems are different aspects of the same situation and provide an excellent illustration of the part played by intuition in geometry. The mathematical films made by J. L. Nicolet of Lausanne and of T. J. Fletcher of London are based on this idea of situation and are eloquent examples of it.* 3 Geometry is not alone in benefiting from this way of looking at things.

Arithmetic in our primary schools is today no longer abandoned to the cares of dutiful teachers, it is rather the object of concern of mathematicians and psychologists and primary school teachers of great ability and intelligence alike. In fact we know that today the concrete is not just what can be perceived by our senses alone but above all what is grasped by the mind, what is familiar. Whole numbers progressively become concrete as they are studied; and if we limit ourselves to situations that are really

Since this article was written I myself tried to animate diagrams and have produced a number of films.

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comprehensible to children, then their mathematical work will be sure, vigorous, spontaneously varied, and rich. Therefore the crux of the matter lies in seeing that the child structures sets, that the structures used have their origin in all the activity of the child, and that the sets which come into play are functions of these structures.

Atomization in the study of arithmetic by pupils can only lead to incomprehension, insecurity and incompetence. It is perfectly possible to present material to six-year-old children and have them operate on it in precisely the same way as university students do, provided that the sets used are related to their ages.

At the age of six, addition, subtraction, and multiplication can be carried out, fractions can be discovered, and division undertaken in the case of the set 1, 2, ... 20, for example. At the age of twenty, this structuration becomes afield, the set is indeterminate and the fractions clearly have acquired a status similar to that of whole numbers. At six, a mathematical attitude exists which is comparable in every respect to the attitude possessed by many at twenty, as much from the positive as from the negative point of view, that is, with respect to what is known and to the extensions still possible.

The teaching material of the Belgian Georges Cuisenaire has made this way of looking at things quite obvious.

4 When arithmetic operations are understood as such and not as resulting in numbers, the pupils become aware of the dynamics of algebra. The object of elementary algebra is to operate on operations. The double possibility therefore exists of presenting

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situations in the classroom in order to lead the pupils to a true understanding of elementary algebra. On the one hand, by suspending arithmetic operations, making them impossible yet maintaining the awareness that they could become possible, one moves from the position of carrying these out, which is arithmetic operation, to the relationship between potential numbers, and from there to the realization that operation per se can be envisaged. On the other hand, awareness develops that operations can be combined and even operated on, in turn.

This way of looking at things is very different from that shown by secondary school teachers in their lessons and text books, but it follows naturally from the all-important question facing one who wishes to see clearly into the difficulties of mathematics teaching. In what way do algebraic situations differ from those to be found in arithmetic? If arithmetic is concerned with numbers and known or unknown quantities, with what, then, is algebra concerned? The question deserves a great deal more attention from teachers than it in fact receives. My own answer is that algebra deals with operations per se, the result being operations, that it corresponds therefore to a higher degree of abstraction and does not require the same knowledge as is necessary in arithmetic. (This would explain, on the one hand, how a knowledge of algebra â&#x20AC;&#x153;blocksâ&#x20AC;? the ability to solve problems of arithmetic, and on the other, how some children who are weak in arithmetic find themselves completely at their ease with algebra.)

These examples are not intended to be anything more than illustrations and should not lead to discussions about their details. My main contention is that teaching through situations 83

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is more effective both for the teacher and for the pupil and that intuition allows one to grasp the whole of the set of relationships contained in the situation. From this arises a stimulation of the whole of the mind, a co-ordination of results facilitating memory and increasing experience and power, and the habit of not atomizing the intellectual function.

At this price and for these results our colleagues would, without doubt, be inclined to try out a form of intuitive teaching which might not be the farce which is altogether too often encountered.

Pythagoras’ Theorem is “proved” intuitively with the aid of paper, scissors and glue. This is neither intuitive, nor a proof.

For me, intuitive appreciation of the theorem occurs if, in the situation which consists in comparing areas organically contiguous to the sides of triangles, the problem of the rightangled triangle and squares on its sides is seen as a particularly simple, special case providing a useful relationship.* An intuition, in my view, cannot be atomized, yet this is the case for a theorem “proved” in isolation by some ad hoc method.

To conclude this article intended to stimulate discussion and not to be an ex cathedra expression of opinion, I shall say that to the extent that we free our function as teachers from the opinions of philosophers and make it a flow of experience based on the reality of the classroom (empirical, not but strict), and make it generous as a result of our becoming aware that we are dealing with people and dialogues of minds at various stages of

* See Chapter 2, Section 1, above.

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development, to this extent shall we penetrate even further into an era where the frustrations to be suffered as a result of countless set-backs will be replaced by the joys of our new discoveries and illuminations.

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First published 1956 in Nastava Matematike I Fizike, V, No. 1, Belgrade.

A month ago I left Yugoslavia after a fortnightâ&#x20AC;&#x2122;s stay; I had met a large number of primary and secondary school teachers, and had given some forty lessons to illustrate the method of teaching mathematics on which I have been working since 1937, constantly modifying it in the light of increasing experience.

As this is no a priori doctrine, contact with pupils is essential for its explanation, and I believe I am the only teacher who is willing to take a class with any group of children whatsoever, anywhere and at any time. That this should be possible seems to me the important thing. Someone exists then who knows that mathematics can be taught without worrying about questions of language, syllabus, what ground has already been covered, or the childrenâ&#x20AC;&#x2122;s abilities according to the official grading.

Similarly, children also exist who learn mathematics from a

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teacher who cannot talk to them, or who speaks incorrectly, or even perhaps very well, and who does not make them repeat over and over again what they have learntâ&#x20AC;&#x201D;children who, without his instructing them, re-invent as much mathematics as time allows, and who still know this months afterwards as well as if they had only just discovered it. These children are the pupils I met in Spain and Greece, in Scotland and Yugoslavia, in Germany and France, in fact, everywhere where I have taught.

Witnesses are legion. All are not equally impressed, but many are utterly convinced. Our pupils have the right to demand that we examine our attitudes in the light of these facts, and that we, as logical and honest people, should be prepared to let our conclusions dictate our conduct.

Unless I am taken for a magician (which has indeed happenedâ&#x20AC;&#x201D; I have even been accused of hypnotizing my pupils), and my students, who have understood and now apply these same principles are there to prove the contrary, behind this approach must lie facts which a man of reason and goodwill ought to be able to grasp and use.

In these few pages, I explain some of the principal features of my particular work with children, related to mathematics and such as I understand it today.

It should perhaps be mentioned that ever since my adolescence I have been an autodidact, and, not having received the benefits a tutor would have bestowed on me, I chose to be instructed by reality and to accept no other authority than that of empirical truth. I studied authors and consistently found that their works

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dated; no one had either seen or understood everything. There was room for my research and for my conclusions alongside theirs, just as there will be room for the contributions of others who see further and more truly than I do.

Mathematics, of which I have made a serious study for many years, likewise struck me as being strongly marked by time and changing fashion. It was not produced ready-made by the absolute mind, but rather was built up by individuals who developed it according to their acts of comprehension of the relationships within the grasp of their vision. Their competence did not stem from their technical achievement when speaking of the whole. Often they lacked one quality. They were not able to distinguish their prejudices and their personal point of view from all the possible ways of understanding the universe known to mathematicians.

Their interest should have been the reality of mathematical activity, rather than the collection of published theorems. They should have considered the mind in the act of learning and conceiving mathematics. That is to say, they should have been psychologists of mathematics. But even this is not enough.

One must be aware of mental processes in their complexity, not reduced to a verbal description or an unsupported introspective testimony; complex thinking involves the mobilization of perception, affectivity, of cultural habits and unconscious prejudices. All this is relevant to a realistic study of what is fundamental in teaching, or in other words in the mathematical dialogue between pupil and teacher.

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Whenever I go into a classroom to give a lesson, I am never concerned about my personal position; I am not preoccupied with my chances of success or failure or the difficulties that the particular circumstances may present. Each lesson is a complex experiment in a human laboratory, which should teach me something new about what is, in truth, the mathematical dialogue for the minds I am about to meet. The only failure I can possibly experience is my failure to learn something, which, happily, never seems to happen: the children invariably succeed in teaching me many a lesson in one of their lessons. I go into the classroom with a real feeling of freshness, never saying to myself: “I must get to such and such a point; the children will answer this or that”. I have no right to say to myself anything but: “Each child knows what he knows, and it is thus that he will come to me. I myself shall never be able to make an inventory of his knowledge, and if the class, being an artificial group, has no common level, this is nothing but the natural result of their different lives. All I must do is to present them with a situation so elementary that they all master it from the outset, and so fertile that they will all find a great deal to get out of it.

Moreover, it is possible that, liberating their energies and their perception, they will, in a few seconds, reach a point of awareness where they can survey the whole situation from afar, and even see more in it than I can myself”.

In fact, it must be admitted that, many a time, I have seen children do several months’ work in a few moments, even teaching me something I ought to have seen, had my vision been truly innocent. Experience has convinced me that our pupils, whatever they are, are a good deal more able than they can be shown to be by our torturing methods. There is, then, in my 90

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relationship with the pupils, an attitude rigorously different from that of the teacher who is bent on teaching, pursuing one end chosen either by himself or by the syllabus, and who ties the object of his lesson to the objects of the previous lessons, rather than to the actual knowledge and real capacities of the children at the moment of his contact with them. The reader will understand nothing of what follows if he does not imagine exactly what, in terms of classwork, this attitude of mind means.

If my desire is not to teach a theorem or particular group of theorems, how do I know that my contact with the children will not be reduced to useless and idle talk? To guarantee that this will not happen, it is enough to know that mathematics consists in the act of apprehension of relationships per se, and that, if the teacher is in earnest, this will take the form of a contemplation of the relationships contained in any given situation interesting both to all the children and to himself. We must now provide a little more explanation of what mathematical situations are, and put forward a number of examples to show that simplicity and fertility can go hand in hand.

Every mathematician who has conceived a theorem and considered what its proof involves knows that the mental processes are, first of all, the abstraction of a relationship from a complex context, and, only after this has been done, the association of this relationship with others recognized as belonging to the situation.

I shall take as an example of this a personal study which will adequately illustrate the analysis I suggest here; but I shall not

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endeavor to develop a full and rigorous argument. As one who knows several languages, learns new ones enthusiastically, and studies their particular structures and difficulties, I find it perfectly natural to consider the set of all the sentences I know. I shall adopt the attitude of the mathematician when I begin to formulate what I see in them in terms of structures and relationships, leaving arbitrary the words, and only retaining perhaps their function. The set of these sentences can be subdivided into numerous sub-sets having different properties, which can be considered from various angles. Let us take, for example, the transformations which allow us to pass from one sub-set to another, and which we call translations. These transformations create a proper correspondence between sentences and sentences, and between structures and structures.

On the set of sentences the set of translations forms a group.

Indeed, to every translation is associated its inverse, and an identical operation exists which keeps the sentences in the same set, and two translations combined form a third belonging to the group, the law of associativity being respected.

More fully developed mathematical studies of this question, which would give information on this subject in a new way, are possible. For our purpose it is enough to see that the initial complex situation is within everyoneâ&#x20AC;&#x2122;s grasp and that the mental process has consisted in isolating one or two relationships and in seeing their dynamism in the simplified situation. The fact that translations form a group adds nothing to our knowledge, but that we are able to deal with these questions as mathematicians and not as linguists gives ground for hopes of conclusions valid for all languages, which could be interesting indeed.

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Another example—an artificial one—is obtained with the help of the Geo-boards, which I have introduced precisely to show that the angle of approach to situations is worthy of the educator’s attention. On various boards protruding nails form a lattice on which polygons can be produced with the aid of colored rubber bands. When the rubber bands are stretched over a sub-set of the nails, situations are created which induce the children to make statements that are mathematical. Thus, if one takes the 25-nail geo-board (a rectangular lattice) various quadrilaterals can be produced (squares, rectangles, parallelograms, trapezia and any concave or convex quadrilaterals) and in addition to their recognition as individual figures, the following points can be studied: •

classification of quadrilaterals;

•

finding the area of quadrilaterals, taking a square (the smallest that can be made) as the basic unit;

•

comparing areas of rectangles or parallelograms having the same base and different heights (these particular cases of the classical theorems appear as general cases if the choice of the initial unit is recognized to be arbitrary, and if various squares of the lattice are in turn taken as a unit);

•

the properties of the diagonals of quadrilaterals are very easily seen by watching what happens when they are placed in the various quadrilaterals;

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•

the triangles that are half parallelograms give rise to similar propositions.

In addition to the above points peculiar to quadrilaterals, the following observations can be made: •

equality and similarity are immediately perceived;

•

the properties of segments of transversals cutting parallel straight lines; the case of a triangle and a line joining the mid-points of two sides; of a trapezium and a line joining the mid-points of the non-parallel

sides;

Desargues’

theorem

of

homologous triangles and Pappus’ theorem are all immediately seen; •

Pythagoras’ theorem and its extension to the case of an obtuse angle are easily demonstrated in the case of right-angled triangles with sides a and a, or a and 2a, or a and 3a; and in the further case with the angle 1350 and sides a and 2a;

•

finding the areas of complex concave and convex figures, introducing naturally the subtraction of areas;

•

enumerations of figures of the same type; this involves questions of combinatorial arithmetic;

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•

oriented figures and questions of symmetry, translation and rotation;

•

an elementary study of fractions can form an introduction to this subject;

•

problems

of

trigonometry,

construction,

of

measuring lengths using, in various degrees of difficulty, similarity, Pythagoras’ theorem and less obvious properties; •

properties of concurrent lines; medians, heights, bisectors and others easily demonstrable.

The interesting point here is that with seven different geoboards we can meet most of the properties of plane geometry that are taught in secondary schools as well as many problems that are more advanced.

Thus, in this way the atomistic point of view, which is the approach where the emphasis is placed on syllogism and which paralyses the children, is replaced by a method which stimulates inventiveness, and allows every child to use his perception of relationships and to work at his own speed.

Our role then is that of observers receiving the pupils’ propositions, of judges demanding that these should be really clearly enunciated, and of leaders of debates giving to each child the chance to express himself.

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The classroom becomes a laboratory where the mathematical propositions contained in a situation are discovered; a workshop where they are hammered into a presentable shape; a research seminar where one learns to choose pregnant situations, and to classify and criticize the conditions which have permitted the finding of only the propositions met.

The work of the teacher becomes much more interesting and he himself grows more, while only using the school time for which he is paid. His discoveries about what happens during the apprenticeship in mathematics, and which are accessible to him alone, since it is he who is living this experience with his pupils, will put him in a new position in the world of explorers. Whereas before it was rare for primary or secondary school teachers to have any very important contribution to make to science, now, if the teacher understands that his position is a unique one, and that he is the only person to meet what he is meeting, he will see that, for a certain time at least, he can make a very important contribution to our study of real mathematical activity as it manifests itself in the child and the adolescent.

In fact, professional psychologists can only see one stereotyped facet of this apprenticeship, and in order to make headway in their investigations they seek help from precisely those people who have the reality to be studied in front of them. It seems obvious to me to ask these people to make this enquiry themselves, for all it demands is an alert eye and an apprenticeship in observation that seems to me to be within reach of everyone. *

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What is not within everyone’s reach, at the moment at least, is another experience which forms the basis of my pedagogy and which could also transform the teaching of mathematics. This concerns what I shall call “modern mathematics”.

As I said at the beginning of this paper, mathematics is not one single entity, but a succession of acts of awareness of the relationships which give mathematics various forms. Galileo’s mathematics yielded to the mathematics of Newton and Leibnitz in the first instance; later this in turn yielded to that of Cauchy and Riemann; now we can speak of the mathematics of Cantor and of Dedekind. A mathematics based on numbers and calculation seemed so promising that the 17th century seems to have been overwhelmed by it, leaving aside the correspondences that Galileo had studied. It is still the natural ruler in the field of technology, to which it gave birth. But with Cauchy and Riemann, functional thought, first conceived by Leibnitz, became the heart of mathematical thought, proposing new and pregnant challenges. Cantor and Dedekind have made us turn our eyes towards sets and structures.

Even today, one can see these three tendencies side by side in mathematical publications. For us here, the important point is that, if we are to attempt to base our teaching on these three broad lines of attack on mathematical relationships, success is most easily obtained starting with sets and structures, and achieved only with the greatest difficulty if we stay in the field of numbers.

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It is well known that an immense difference in the teaching of arithmetic has resulted from the introduction of the Cuisenaire rods. Whereas before interminable hours of repetition were needed to ensure the remembering of the links between whole numbers, now children of five can work on fractions, and at seven can demonstrate that they have the mathematical experience of other pupils of twelve or more.

My prolonged and detailed study of the causes of this success prompts me to say quite unequivocally that it is due to the fact that these rods place within the grasp of the child the most primitive notions we know in mathematics, and thus which we only met at the end of our own process through an everdeepening abstraction.

A large number of lessons, often observed by groups of teachers, given to children of thirteen or more, who are, at their age, still ignorant of set theory, have shown that the algebra of sets and Bourbakiâ&#x20AC;&#x2122;s notation seem quite natural to them. In the same way, at the age of ten, the group of permutations can be studied.

At fifteen, pupils without any warning have proved that they can propound theorems on enumerable sets and prove properties of bounded infinite sets, finding quite easily theorems on points of accumulation, on the limit of monotonous series, etc. At fourteen, fairly large classes where the experiment has been tried were able to distinguish the axioms of measure, analyzing the stages which give the length or area or volume of ordinary continua, and fully understood the advantage of an abstract language.

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Every day my experience confirms the truth of this discovery: our pupils, much sooner than we believe, can study questions not usually considered except at university or even later. I have drawn up a whole catalogue of such questions, and there are many more which I have not had time to examine. Professors at the Ă&#x2030;cole Normale SupĂŠrieure of Paris assure me that their students are easily able to understand questions which they themselves find singularly abstract. They are of the opinion that, if the candidates for the Grandes Ă&#x2030;coles are taught according to the axiomatic method, there will soon be no further need for them to learn and unlearn masses of theorems that can be acquired all at once in a well-constructed modern theory.

It seems unreasonable to continue to teach a syllabus dating from another era and to give our pupils old-fashioned mental habits which cannot possibly help in the study of the modern challenges arising from multivalent situations. At an international seminar at Echternach (Luxembourg) in 1953 we tackled the problems of mathematics syllabi at universities and of the needs of technologists and, although our conclusions were by no means clear-cut, it was evident that univalent mathematics teaching, such as the majority of universities continue to give, is very far removed from the modes of thought needed by telecommunication engineers, nuclear physicists, economists, strategists, etc.

It would be impossible in this brief paper to do justice to the complex recasting we have in mind, but the above notes will emphasize the urgency of the need for an examination of a remodeled mathematics teaching program in which our pupils will play the role of guides in what they can understand and 99

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learn, and in which our role will be that of the inventors of methods. We shall then be able to create syllabi that do not confine the studentsâ&#x20AC;&#x2122; thoughts to preconceived patterns, and which allow them to tackle the problems their age poses them.

Numerical mathematics is too highly structured and would be an advanced rather than an introductory study. It is not a question of disguising traditional techniques and of starting from wholes to arrive at the point at which our forebears began, but rather to start from a different and realistic point of view. If the notions of equivalence, order, isomorphism, quotient-sets etc. are the primitive elements, then we must start with them, seeing how these multivalent structures reveal, through their individual differences, the natural chapters of modern mathematics. Children of all ages have constantly shown me that they have a keenly alert mathematical intelligence and that they are perfectly at ease with such questions, whilst the complicated structures give them trouble. As soon as I understand the mental processes of children, their progress takes the form of spectacular leaps in a very short time.

Experienced teachers of the deaf-and-dumb find the results I have achieved with their charges amazing both because the mathematical perception of these children is natural and also because I confront them only with situations easy to comprehend. The whole secret lies in moving from the general to the particular, while our forebears held that the mathematical process should be directed towards generalization. In fact, it is rather different: the multivalent precedes the univalent, and children facing an adult universe strongly marked by the history of mankind would not be able to adapt themselves to it had their mental processes not spontaneously contained that suspension

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of judgment which postpones the fixing of a word or of an opinion.

The dynamic mode of thought possessed by our pupils remains long unfixed and is closely akin to the mathematics which sets out to embrace all possibilities. The definite is of interest to this thought only insofar as examples are suggested from which a start can be made in the process of freeing it from all restrictions.

All this should encourage us to have a far-seeing and broad vision, and to serve as we ought the young generation in our charge.

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First published in the Times Educational Supplement in August, 1957.

Some 12 years ago I began to consider mathematics teaching in Britain, first at the university level and then in various types of schools. It was no use taking as a basis for comparison what is done in similar institutions on the Continent, for what I had seen there was no more reassuring that what was to be found here. Everywhere there was need for reform. There is an absence of enlightened opinion as to why mathematics is considered by so many otherwise intelligent people as being beyond their capacity. I shall not spend time here in criticizing the research done on the teaching of mathematical topics, but shall turn at once to a consideration of the vicious circle in which we find ourselves in this country and which has little to do with the question of research.

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Those who, after leaving the university, return to the grammar school as teachers, may be proficient in certain techniques, but they have no idea of their subject as a whole, nor understanding of the relation between their newly acquired knowledge and the preparation of their classes for official examinations. That this preparation is the main object of education in the grammar schools, 11 years of continuous contact with them have forced me to believe. In whatever way teachers envisage their work, grammar schools are devoted to the worthy social task of getting their pupils “through”.

Bits and Pieces Mathematics is an examination subject, and its syllabus consists mainly of bits and pieces of antiquated subjects. Among those who are least able at this kind of work are most of the students who hope to become teachers in infant, junior and modern schools. The few who specialize in mathematics at training colleges are given more of it as a preparation for the task of teaching youngsters, while the bulk, who will in any case have to teach arithmetic, get a few hints, often from an education lecturer whose mathematical horizon is limited. Since the sensation caused by the A.T.G.D.E. report of last year, training colleges have been appointing “well-qualified” mathematics lecturers—i.e., men and women with good degrees, able perhaps to offer more, but more only of the same kind of mathematics.

The young teachers thus go out to the schools with a narrow vision. They can only attempt to reproduce in their pupils the confusion that is in their own minds; in doing so they rely on the

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hints and tips given to them with assurance and authority. It is not long before the inevitable disappointments bring an increase of their insecurity. Often in my contacts with primary and modern school teachers I have heard it stated that for the first time in their lives they have understood what a fraction is, or a measurement, or an area.

Universal Error There is indeed a human mortgage on the profession. The universities are doubly at fault, in the training they give to future grammar school teachers and in their preparation of examination syllabuses. The grammar schools are doubly at fault in blindly accepting examinations as the only criteria and in encouraging their weaker pupils to take up teaching. The training colleges are at fault, in perpetuating methods which are rarely conducive to understanding and in deeming it ethically acceptable to send out to the schools teachers who are illequipped to take charge of the coming generation. And, of course, the teachers themselves cannot disclaim responsibility.

There are other causes for the present depressing state of affairs.

There is the widespread belief that mathematics is a difficult subject, accessible only to a minority having special qualities. In many experimental lessons I have been able to show, to the satisfaction of most critical audiences, that this is in fact not the case; children can do mathematics as naturally as they walk or speak, but teachers and authorities are unaware of the fact. This ignorance is the second cause of the present stagnation. People are too pre-occupied with other matters to find time to examine

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their preconceived ideas, and the few who have done some investigation into the position have preferred to perpetuate the prejudices.

It is of little use to know what is wrong without knowing how to put things right. The usual solutions proposed appear so naive that one is amazed that they should be suggested. What can be the contribution, to a task of this magnitude, of a third year at college, or the inclusion of more mathematics, or of alternative syllabuses offering history of mathematics or statistics? What has an increase in special allowances to do with improving the quality of teaching?

It will have been noticed that I have not mentioned the work of university departments of education. In the first place this is because I have been engaged for 11 years in the work of one such department; and secondly because such departments are concerned only with a small number of the teachers. But they have been ignored chiefly because I wish to consider them together with the greatest obstacle to reformâ&#x20AC;&#x201D;respect for tradition. This I have found strongest in relation to the role of the grammar school as seen from departments of education. It could, of course, be argued that, having lived for only 12 years in England, I am unable to understand the value of the tradition.

My work has been dismissed by many on those grounds. If we look closer, however, we find that traditions are often hard to defend. The history of education is a continuous history of patching, of haphazard decisions, of trials and errors, which become institutions, which succeed when their adversaries have died, because it is no longer known how they came into being.

They are then adhered to â&#x20AC;&#x153;by traditionâ&#x20AC;?.

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Mathematics teaching in this country is suffering from acute loyalty to a tradition that is doing tremendous harm. The reports of the Mathematical Association show that generally speaking the profession is blind to the real trouble. If there is no serious effort to examine the work done at all levels and no immediate attempt to do something drastic, those who claim to be the leaders of the profession must accept responsibility for the resulting state of affairs. *

*

*

To be effective, a reform must cover all the points at which the present organization is faulty. There is work to be done in the training colleges and with the practicing teachers to improve the foundations of mathematics learning, but it must not be forgotten that the efficiency of the grammar schools and universities can also be increased.

Within the framework of an article it is not possible to do more than sketch the solution I have in mind. For further details the reader is referred to more extensive texts already published or soon to be published.*

* L’Enseignement des Mathématiques, 1955. Le Matérial d’Enseignement des Mathématiques, 1957. Both published by Delachaux and Niestlé, Paris.

Teaching Mathematics in an Expanding Economy I, Reading, 1957.

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First Nucleus If a limited number of efficient people were brought together for an intensive course of study, each could become a nucleus round which a group could be formed and informed, and so the process would continue. A determined effort could in five years change the face of things. I have been able in some six or seven years to create a movement within the profession which augurs well for any more systematic and more deliberate effort by inspectors and teachers at all levels. My experience suggests that there is no reason to believe that the profession as a whole would not support such a scheme.

Once people have been brought together, what should the procedure be? Are they to listen to, and then discuss, the opinions of others? My view is that discussion should take place only after it has been made plain in lessons that the current opinions are often only preconceived ideas, and that new ideas do in fact work in the sense we suggest. It is my hope that more teachers will come forward who are ready to illustrate, with classes that can be observed, the ideas they have evolved in their own practice. If this is done there will be no question of believing what someone else claims; when what has been forecast actually happens it is open for all to see.

Schemes of Lessons It is in arenas such as these that a number of slogans for which I am responsible have been put into circulation. When I realized that mathematical activity is essentially the result of an

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awareness of the dynamics of relationships I developed schemes of lessons which could be put to the test by any observer, and which have in fact often been tested with classes of all kinds.

Such assurance is founded on evidence as rigorous as that in any other scientific field. I know that if pupils are confronted with situations that can yield information they will extract that information from them as surely as they discover the plot of a film.

The idea of mathematical situations is open to development.

Teachers must learn to consider such situations, to suggest them to their pupils, and to utilize them as a source of mathematics.

My geo-boards, on which the knots of particular lattices are marked by nails round which elastic bands can be placed, are designed to produce a vast number of mathematical situations which are mainly geometrical. The Cuisenaire material is another simple means of creating a great variety of situations appropriate for pupils at all levels, which lend themselves to systematic study of arithmetic and algebra. The Nicolet films are yet another way of suggesting situations that synthesize chapters of mathematics.

There are also, of course, the verbal situations which can be produced by marks on paper or on the black-board, and those contained in any piece of apparatus. What is being recommended here is a new organization of knowledge by means of complex situations through which relationships can be made explicit. This is advocated in place of the usual presentation which requires first the understanding of separate items and then their application to given situations. In the new approach theorems and problems are interchangeable: before 109

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enunciation all propositions are problems. All facts have to be ascertained: to do this is to prove them.

There is, of course, no difference in this respect between algebraic, arithmetical, and geometrical situations.

Underlying Notions As well as this background of situations there is the important idea of mathematical structures. When we consider the quantitative aspects of mathematics, we find that there are always underlying qualitative differences or similarities to be recognized first. These are, as it were, more primitive than the highly organized notions met in textbooks. By deliberately starting with the qualitative properties we can ensure a wider grasp and more general understanding by our pupils of what is suggested. It is obvious, though it has long escaped notice, that besides the â&#x20AC;&#x153;objectsâ&#x20AC;? of mathematics such as figures, numbers, relationships, there are actions, or ways of handing the objects: it is in becoming aware of these and formalizing them that we meet the algebraic aspect of mathematics.

Thus, awareness of the group structure, so important in the whole of mathematics, is only possible when we have grasped that some virtual actions are reversible, that they can be combined so that two of them can be replaced by a single one, that two can be found which neutralize each other. It is easy to see that many of our activities have this qualitative structure, as for example the permutations of a few pupils in their places in the classroom, or their displacements in the playground.

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It is interesting to find that although historically it is only recently that mathematicians have discovered how to work mathematically with qualitative relations, these are readily understood by young pupils and lead to more rapid development of mathematical ability.

To sum up, we can say that a re-education of our teachers in short intensive courses could take the form of giving them experience in producing pregnant situations and of working on them with pupils, so that they may recognize the hierarchy of structures and their architectural connections. A more qualitative approach would serve to distinguish sharply between understanding which comes with awareness of them and techniques which follow the exploration of their field of extension.

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6 Mathematics and the Needs of Society

The original text in French was a recording of a television lecture given on December 6th, 1960, for Radio Canada in Montreal in the series ConfĂŠrences.

I am here this evening to talk to you about certain problems which arise in the adaptation of mathematics teaching to the needs of a rapidly developing society like ours.

To all those who are concerned with education as an historical phenomenon, it is clear that there is a considerable gap today between the needs of our society and the school syllabi. People in all countries are trying to change these syllabi. In general, these changes are being made primarily under the influence of mathematicians. The influence of psychologists, on the other hand, is minimal, likewise is that exerted by the general needs of society and of industry in particular. But in my own work, I have tried to take into account the four requirements which have become essential today:

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1

The need for mathematical rigorâ&#x20AC;&#x201D;this is evident since we are concerned with the teaching of mathematics;

2 The psychological requirementsâ&#x20AC;&#x201D;since we have the problem of relating mathematics and the learner (for pupils ought not to be forgotten in this!); 3 The pedagogical requirementsâ&#x20AC;&#x201D;since these have a bearing on methods and means; 4 Finally, the social needs, which are reflected in the syllabi.

Naturally I shall not say anything about mathematical rigor: everyone knows what this is. But the psychological requirements should temper the demands of mathematical rigor. No one would dream of teaching a baby something for which it is still not ready. The content of the material to be taught must therefore always be subordinated to the capabilities of the child.

The child thinks in multivalent terms because the world is spread out before him and the unknown is ever present in his experience. He is open to all that comes and cannot restrict life.

He always expects that there will be a meaning in whatever comes his way.

Mathematicians have rediscovered this multivalent thinking; they have recognized it in mathematics. So-called modern mathematics constitutes a collection of structures which act upon each other. It is remarkable that such a connection should 114

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exist between the multivalent thinking of the young who are not yet formed or molded by social thought, and modern mathematics which has tried to rid itself of all of its restrictions that were merely the result of historical development.

Since the child is endowed with a multivalent mode of thought and also that mathematics presents a number of multivalent forms, it will perhaps be a simple task for modern psychologists and educators to relate the two.

Society, on the other hand, has become so complex, presents so many problems, and we are submitted to so many influences simultaneously that we cannot limit our thinking, as before, to ready-made patterns operating almost like a strait-jacket. We must learn to think in a complex manner about complex questions. And in order to do this it is necessary to examine what can be done in the face of that which, though still insufficiently denned, might be called successive realities which, one day, will become a metaphysical concept: Reality.

To confront a situation, to enter into a dialogue with it, is to establish a relationship situated at several levels simultaneously.

If I look at something, the situation is organized by perception.

One moment I see colors, at another, shapes, at another, size or other relationships. We are constantly changing our point of view in relation to the same situation; indeed, we are obliged to do so continually. The mathematician, however, is called upon to do this in a professional way all the time. Let us take the example of someone who wants to go from one place to another in a town he knows well. Generally speaking, several routes are

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available to him. He knows these routes because he has been over the roads and has a picture of them. With these pictures in mind, he can say: “To go from here to there, I can either take this road or that one. What are their respective advantages?”

Once the advantages have been determined, the study of relationships is at an end. One might reason thus: this road is shorter, but that one has fewer crossroads, fewer traffic lights, and is less congested; but that one, on the other hand, is cluttered up with road works, etc. The totality of these routes represents the reality. But to think of these routes per se is to do mathematics. Thinking of the roads independently of all they contain is to be a mathematician with respect to that reality. The interest of the mathematician could be said to lie in relationship, or relationships, or, better still, in the dynamics of relationships.

As soon as he comes into contact with this, every individual becomes a mathematician.

A life situation contains a great many factors. Bat I may alter it by taking away the properties I consider to be accessory. For example, if I consider a situation which includes color, and then leave the colors to one side, I shall have ignored color, abandoned it, I shall have altered the situation so that color no longer interferes. “Mathematisation” is precisely this activity which consists in the isolation of the relationships within a situation. To mathematise is to ignore everything apart from the relationships in the situation with which one is faced.

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I shall give you an example. We shall now take a situation composed of tangible objects.* Let us take this rectangle composed of small sticks, or rods. In fact, these rods are colored, and it can immediately be seen that the colors are a perceptual element in the situation. There are several rows in this rectangle. All these rows are equal in length, but they are all different in composition. Here is a situation which is immediately taken in by the eye; we look at it, we see several rows, and enter into a relationship with them via the eye. It could be that the situation had been arranged by oneself initially. In that case there is simultaneously perception and the memory of an action. But if, looking at it now, you have noticed that all the lengths are equal, you will have isolated one relationship, leaving aside a number of others. But you might also examine it for something else and observe: all these rows, though equal, are however, different, for they are composed of smaller parts added together (below the top row) to form all the others. To add is an operation. We are therefore now thinking in terms of addition and we note that this same rectangle contains various additions.

We therefore have a concrete situation in which we have isolated the relational elements. But these addition-relationships are not the sole possibilities, these same rods being capable of rearrangement in entirely different situations.

* Here the Cuisenaire rods appeared on the screen; the decomposition table for the yellow rod could be seen.

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Here is a staircase composed of ten rods.* The difference between two successive rods is equal to the length of the smallest rod. Looking at this group a staircase can be seen; but if the mathematical eye, which is concerned with relationships, intervenes, it can be observed that these rods might be described in various ways. If I call the smallest rod “one”, then the next, which is twice its size, could be called “two”, the next, “three”, and so on, up to the last, which would be called “ten”. We should thus have encountered ten whole numbers. But we are not obliged to ascend; we might just as well descend and say: ten, nine, eight, etc. down to one.

It might equally be asked: why choose the smallest rod as the unit? Would the same language be used whichever rod were chosen? Would we still be talking about the same things? New relationships would then be perceived. Thus, by choosing the orange rod we should have

,

,

,...

. If the second, the

red rod, were to be chosen, we should then have

, 1, 1

, . . . 5,

when all the rods are included. The language changes as soon as the viewpoint changes. The situation exists in the world of physical reality, but in the world of mathematical reality it is the point of view which counts. By changing the point of view, both the language and the awareness of relationships are changed.

* The staircase of the Cuisenaire rods appears at this point.

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We shall now see how it is possible to create much more complex situations at much higher levels.* Here is a mathematical film made by J. L. Nicolet. It demonstrates that families of circles satisfying certain conditions can, through the loci of their centers, describe curves such as were celebrated among the ancient Greeks and which are important today in the fields of astronomy and the launching of missiles: they are called conics.† Fletcher succeeded in presenting extremely complex theories through sets of images. Pictures have an advantage over words in that they very quickly engage our attention and can also be rapidly recalled. While memory associated with meaningless words is difficult (which is why our children have so much difficulty in their studies), pictures on the other hand are recalled unaided, in response to a simple signal.

The mathematical situations which are the heart of my pedagogy are situated at the perceptive and active levels. We make the learner (be he child or adult) establish a relationship with the situation by entering into a dialogue with it. He learns to

* At this point one of the films made by J. L. Nicolet, a Swiss schoolteacher, for the teaching of geometry at the secondary school level, is shown. † During the showing of this film, and an extract from another on conies made by the English mathematician T. J. Fletcher shown immediately afterwards, the lecturer comments on the pictures. These commentaries are omitted here and the interested reader is referred to the chapters devoted to the mathematical film in Le Matériel pour l’Enseignement des Mathématics, Delachaux et Niestlé, by Nicolet, Fletcher and Gattegno, of which one chapter is reproduced here, in Volume II, chapter 4.

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question a situation; he transforms it by his questions, polarizes it; he allows it to give him the answers in the desired direction.

But in continually changing oneâ&#x20AC;&#x2122;s point of view, one and the same situation is able to offer a considerable number of ideas so that we can restrict ourselves to a small number of extremely rich situations. This is what our pedagogy stresses. In fact, with the two examples of material which have been shown, the films and the Cuisenaire rods, a great number of lessons can be given.

The habit picked up by the pupils at a very early age of not limiting themselves to a single aspect, of always being ready to meet the unknown, means that mathematics, as a subject to be taught, becomes educative in the broad sense of the word. The child and his sensitivity are educated to be open to everything that happens, and surely, the big problem in education today is the preparation of children for a changing world. Is it not true that the reality we all know is a function of time? But not a known function of time, so no one can predict what it will be. It may be so different one day from what it is at the present moment that we cannot prepare our children for a static world without doing them harm.

Education of the sensitivity in mathematics is a novelty.

Education of the sensitivity in any subject at all would be a novelty. But this is a task we must undertake. Since the needs of society are variable, our aims must be that each of us be prepared to meet the unknown rather than ready-made situations. The example of mathematics, which has already effectively made a start in this direction, is very encouraging.

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Our pupils will have realized that a situation is always full of unknowns, that an opening of the mind will allow such and such a thing to be perceived and another opening of the mind the perception of something else, and they will thus, it seems to me, be prepared for a changing world.

Transformation is but one aspect, complexity is another. When the traditional educational process is observed, always giving the pupils one atom of knowledge followed by exercises having a bearing on that atom, then another atom, sometimes related to the preceding one, sometimes unconnected with it, likewise accompanied by a great number of similar exercises, one realizes that our pupils are not being prepared for contact with the worldâ&#x20AC;&#x2122;s complexity.

But if each of us has always been deliberately put in contact with complex situations, has always seen that what he knows is perhaps nothing beside what is yet to come, and if the situations chosen are only slightly structured and really promising, then the dialogue may last a long time.

Consider, for example, the fact that, generally speaking, addition is taught for the first months all by itself, that subtraction comes later, followed much later by multiplication and division, and later still by fractions, while in the study of the situations offered by the Cuisenaire materials, all this is met at one and the same time. Addition, subtraction, multiplication, division and fractions are all present in the situation. The only change which takes place lies in the degree of complication. The more experience the child or the adult has, the more complex the

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situations he is presented with become. The two films you have seen, one by Nicolet, and the other by Fletcher, are on very different levels. The first can be shown to pupils of twelve while mathematicians would have difficulty in following the second.

But both follow the same principles: in a very short space of time they show a large amount of material and dialogues can be established. In fact, possibilities have been opened up which had not been thought of up to now.

From a pedagogical point of view, the discipline we have introduced is relative to the method. From a psychological point of view, we have recognized that it is the pupil who must learn, that it is the pupil who must, as far as possible, experience this movement of assimilation which makes him master of his material, so that it exists in him and he is able to recall it. The number of rules is considerably reduced. Instead of rules we have an act of comprehension of the activity. To become aware of oneâ&#x20AC;&#x2122;s activity means different things at different levels. One can become aware of oneâ&#x20AC;&#x2122;s activity by learning to dominate the situation. For example, when one wants to ride a bicycle: when all that has to be done has been mastered, command is demonstrated by getting on the machine and pedaling away.

In a mathematical situation, one can be master of it once all it contains has been understood at the perceptive and active level.

It can then be verbalized and later written down. There is thus, a succession of phases in the dialogue. First of all, one enters into the situation, throws oneself into the water, and learns to swim.

When it has been proved that one can swim, then one can tell others about it, talk about it. To talk about it is to reach another level, that of the creation of patterns, structures and 122

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propositions which allow one to say that one has a grasp of the matter. But writing, when a certain notation is adopted, provides a substitute for all the previous work. Writing, notation and the mathematical text should therefore be regarded in terms of activity. In traditional teaching methods the mistake is made of starting with notation, and the pupil has great difficulty in understanding the meaning as a result. After a certain time he understands naturally, but on pedagogical and psychological grounds, we should in fact begin at the other end.

It is only when a text is full of the dynamism engendered by action and perception that it becomes as real for the reader as for the writer. Naturally our pupils who have worked with the materials and seen films are characterized by a considerable aptitude at the usual mathematical level, both notational and verbal. But for deaf and dumb children who are deprived of words, and also for blind children who cannot evoke an image, other means must be found. With the deaf and dumb one simply moves directly from perception and action to notation; with the blind one must go directly from perception and action (tactile perception of course) to verbalization. And one remains at these levels.

By introducing dynamism into the activity, seeing that each child has all the social elements contained in language and notation at his disposition along with his previous experiences, a harmonious unity between the individual and the group is created. In allowing him to enter into a dialogue with rich and possibly inexhaustible situations, he is led very close to the infinite and the unknown, and herein lies a very great difference

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between what we know and suggest today and what we did when these disciplines were unknown to us.

Mathematics is an example: today it has been proved that the subject hated most or feared most by pupils, the least popular subject of all, has been able to take its place in showing that education can adapt children to a changing world, to a world full of promiseâ&#x20AC;&#x201D;for the unknown is not necessarily frightening; on the contrary, when one is accustomed to meeting it always with joy, then one is prepared to be in harmony with what the future might bring.

Technology and science are complex and require that everyone attempt to form a synthesis, a fusion of ideas stemming from different sources. We believe today that the theory of mathematics teaching which takes into account the four requirements of which I have been speaking makes contact possible between the mind opened out towards the unknown and the infinite, or in other words, contact between the mind of the child and the needs of our changing society.

Technology today, with its specialists, is a succession of separate disciplines. We all deplore this specialization which appears to be essential in order to penetrate a little further in each domain.

But when it is known what children can do, and do so well in such a short time; when it is known that they can, for example, cover in five years a field of study which, until now, had required twelve or nearly as much; when it is seen with what perception of relationships and with what ease they grasp complex sets of ideas, then one may be allowed to hope that we shall lose

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nothing of what is demanded of the specialist today and that at the same time we shall replace this fragmentation of knowledge by broader syntheses which will be possible because each of us is adequately equipped, and has demonstrated this ability from early youth right up to the time of entering university.

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This article first appeared as a Presidential Address to A.T.A.M. in Mathematics Teaching No. 12, March 1960.

The main tasks confronting teachers of mathematics generally, and more especially, those in Great Britain, will be the subject of this, the first Presidential Address of the Association. Even if we did nothing to meet the challenges of the present and the future, no-one would blame us, but I shall assume here that quite a number of mathematics teachers, at all levels of instruction, have uneasy consciences in this regard and are prepared to examine their work critically.

Those who have decided to give their energy and their time to meeting these challenges, already know that they are engaged in a creative work of great importance. Teaching can be transformed, as can any other field of experience, into an area in which the creative qualities of the mind find their place and use.

When this happens teachers feel that the reality they are meeting is larger than themselves, is inspiring and rewarding. It is extraordinary to find that so few teachers know that the 129 125 127

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situation in which they live and operate is endowed with such qualities.

We are in need of as many inspired teachers as there are classes.

To these classes children are sent without their consent and are submitted to whatsoever we may be as persons and to what we do with them. It is only if we really care for them, and know what we are doing to them, that we can justify our relation to the future and to the education of the next generation.

On the whole are we, in the profession, to go on believing the many opinions current today which are neither founded in fact nor are revealed truths?

Are we always to leave to someone else the responsibility of telling us what to do and how to do it?

Are we to close our eyes to all realities that do not immediately fall into the patterns to which we are accustomed?

Are we to be content with a change in emphasis, or a new shade here and there, even if we feel that an overall transformation is needed?

Are we to continue to think in terms obviously inadequate to the challenges, or are we to attempt to gather the necessary skills and knowledge needed to illuminate our paths?

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At present, as a profession, we are dismally ill equipped. The Training Colleges and Departments of Education are doing what they can, but that is reduced to little because the necessary knowledge for meeting the challenges is either not yet available or is considered unimportant.

It is in order to answer some of the questions above, and to define some others needing consideration from the best among us, that I choose this theme for my address.

We must stop believing that the bright children are those who are verbally inclined and who manage to get into Public Schools (if they have the means) or into grammar schools. Brightness is a complex quality, one dimension of which is emotional. If children are given what corresponds to their make-up, they become bright. If they are denied it, they give up the struggle to make sense of so many words. Thus we have a share (and I think it is an important one) in the formation of the legions of the less bright pupils as a result of our preconceived ideas of how we should carry out our task of teaching. These need reexamination at once.

A.T.A.M.* has tried to concern itself with ways of doing things, with reaching pupils who usually shy away from words, and has questioned the entirely verbal approach to mathematics. In contrast to this attitude, as far as I can tell from my experience, the great majority of teachers still do in the classroom what was

* Association for Teaching Aids in Mathematics, now known as A.T.M. or Association of Teachers of Mathematics.

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done centuries ago, that is, unquestioningly follow the syllabus and the suggestions of someone else. We must stop thinking that there exists a person hidden somewhere who knows exactly what each one of us has to do in his class at such-and-such a moment. The responsibility for teaching would seem obviously vested in the teacher who is at that moment there. But, in fact, it does not work out in that way. There is the syllabus to cover, the many pages of the textbook to turn, the demands of several shadowy persons and institutions to satisfy before we consider the concrete situation of our class. The failures in the shadowy regions we interpret as failures of our pupils or of our own when, clearly, all that is at fault is an attitude: our loyalty. This we owe, as teachers, only to truth and to the future. High sounding words! —but without what they stand for we are nothing; or worse, we are frauds.

We must learn to take our share, fully, if possible, in the educational set-up. This is very important. We can more surely satisfy inspectors, parents and examiners by being part of reality, then by allowing all sorts of illusions to come in. (One such illusion can be singled out: the equating of ‘covering the syllabus’ with ‘our pupils have mastered its content’.)

Only when we know better how to teach can we expect to lead our pupils towards mastery; we shall only know this when we know how our pupils learn. It is there that our main task is to be found. We have to study learning as a matter of course and all the time: not only in books, but where it takes place: in front of us, in the classroom. Then we shall drop that other illusion of believing that each question has an answer. It may have; more likely, however, if it is a difficult question it will require study 130

7 What Matters Most

and sleepless nights, and will only lead to one of many possible approximate solutions. Life questions are not as carefully schematic as the ones we meet in textbooks—particularly of mathematics. Questions concerning teaching are complex and we have to learn to think in a complex way about complex questions. Any other way is illusion again.

In our reality, people are involved. People are moody and unpredictable, demanding that we meet them without preconceived ideas. They want to be interested and challenged, and dislike being taken for ‘things’, regimented, and their individuality forgotten. (Of course, some seem to like it. I describe such people as being anti-social since they have agreed to conform to what has emptied them of all their will to be themselves.) Because we are dealing with people, teaching is a great art and we should see it as such. It is a difficult art requiring us to work hard at it to be good at it. This is just the opposite of what teachers believe when they embark on their careers—and often when they retire, for they too can be made soulless as a result of ‘good’ behavior for so many years.

Teachers must be resigned to the fact that in their classes and at any given moment they alone must decide what to do when confronted with any given situation. Neither book nor lectures can tell them, once and for all, what to do in every case that may arise. If we are honest, we must admit that our ideas about teaching are not as articulate and as full as they should be. We have clichés and slogans and use them freely.

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We are frequently hostile to new ideas and methods. If someone comes up with a new proposal, and if it is not presented in its final form (for example, a syllabus with suggestions, or even textbooks, examination questions, etc.), we do not want to waste time looking at it. In this way we so often miss a very good opportunity of improving ourselves and of giving better service.

Closing our eyes to reality because it appears in an unfamiliar guise is very widespread. But can teachers afford to do this, since they do not work ultimately for themselves, but for children and for the future? Were teachers to become sensitive to the uniqueness of their function and of its importance, they would look for what could help them improve their work, and they would accept responsibility for using it in their own way.

Techniques are nothing if this background of sensitivity is lacking; they are everything if it is present. For then one can reach that level of awareness where one is at once able to translate into action what is known to be for the good of the studentsâ&#x20AC;&#x201D;both as individuals and for their future life.

The techniques I think of are those I personally know; my readers will think of others. But we must all be agreed that what matters most is that we reach a level of confidence based on knowledge, so that we can stand up for what we think and do, not because we have faith, but because it is true.

This is possible if we deliberately and with all the necessary energy pursue the knowledge needed in psychology, in mathematics, in methodology, and in experimental teaching.

Not all will pursue everything; not all will need to. As an

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Association we can pool our experience and share it. This was the reason for its being formed.

In psychology, we are primarily interested in knowing the process whereby mathematical experience is acquired; what makes it a specifically mathematical experience; the various ways in which the mind creates those dynamic mental structures that become a power when they are possessed. We also want to know what is creating obstacles, which factors can be dispensed with, and which are indispensable (for example, is sight necessary?). We want to know, too, if we can substitute one experience for another, and whether special techniques are needed to make the substitution efficient. (How far are words part of the mathematising process? Is there a mathematical thought that exists outside all the uses of senses, languages and signs? Can we attain it? Do we need to know of it to reach handicapped children properly? What problems do we meet in these people whose universe of experience differs from ours?)

We want to know how long is needed for such-and-such a mind to reach mastery of this or that field of mathematical experience, and whether personality, language, and the current environmental modes of thought have parts to play.

These are some of our psychological problems, and we can help each other by finding parts of the answers and by commissioning the specialists whose additional help is required.

In mathematics, our needs as teachers are not identical to those of scientists and technicians. What we need most is an understanding of the mathematical thinking processes (which

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are not reducible to deductive reasoning, as logic is only a section of mathematics and is only used in some stages), and the overall picture of the field, with examples of sufficiently varied mathematical behaviors; we need to be acquainted with the historic movements and their contributions. We need to learn to transfer to our pupils, not only proficiency in some skills, but also the power that mathematics gives over reality.

The curriculum for teachers in Universities and Colleges must be altered to fit present needs. Our duty to the future demands that teachers have an insight different from that of scientists and technologists. We educate through mathematics, and that means opening new vistas, taking the student to higher levels which can no longer be equated with algebra, geometry, trigonometry and calculus (all subjects of 300 years standing).

We must understand the way that mathematicians become aware of structures; which are the important ones; how they can be made to act upon each other to provide classes of relationships about which this or that can be said. The truth of any mathematical statement depends on the fact that everyone can ensure for himself that the statement is contained in (is deducible from) the set of relationships used to introduce the situation. As teachers, we need to know how to classify the mathematical species so that they can be recognized and the appropriate statements made immediately. We need to know when a situation requires that we use topological or algebraic arguments; how to disentangle the components of a complex entity and take steps to improve our grasp of it.

This sort of curriculum is being developed in some places, and it is possible to become acquainted with it by reading, for example, 134

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Professor G. Choquetâ&#x20AC;&#x2122;s course of analysis or the latest publications in the U.S.A. We in England have little to show.

What is needed is the continuous development of a sensible approach valid throughout the whole course of school education, replacing the present bottleneck resulting from all sorts of influences dating from various times.

We need to know how to present our work to those learning mathematics. In fact, we need a methodology of the methodology of mathematics teaching.

The teaching of mathematics has an aspect that relates to the classroom, which is its methodology; but each science has developed, besides its results, an interest in how these results are obtained, what the problems of the science are, how they differ from those in other sciences, etc. This aspect is the one that has, so far, been so utterly neglected in the methodology of teaching.

When we know why we do something in the classroom and what effect it has on our pupils, we shall be able to claim that we are contributing to the clarification of our activity as if it were a science. One aspect of this, at least, has been developed a little; I refer to the study of childrenâ&#x20AC;&#x2122;s mistakes and what we can learn from them.* This is one of the most fascinating fields of creative work for teachers, since they are at all times facing the teaching

* See The Use of Mistakes in the Teaching of Mathematics, reproduced in Part I of this work.

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reality and are, besides, the people most interested in seeing clearly in this reality.

Another aspect of our work could be the study of the mathematical possibilities of children. So much here is sheer prejudice at present. Indeed, it is now becoming known that children’s mathematical abilities are incomparably greater than has previously been suggested. Much time, energy and frustration could be avoided if teaching were related to children’s thinking powers.† My last point will be concerned with experimental teaching. In our Association this theme has often been touched upon. We do not want anybody to share our views until these have been proved right in the classroom. We do not think that we are entitled to anyone’s hearing unless we have assured ourselves in advance that we ask for attention on the grounds of our successful experience. If all teachers demanded this standard of behavior from all their advisers, it could contribute greatly to making the profession responsible, where words mean exactly what they say. Experimental teaching is the use of the classroom in order to learn something worth communicating about the activity of teaching, so that improvements can follow in the work of others who are facing similar circumstances.

In conclusion, I want to say that what matters most is that we stop being the tool of prejudice and cliché, and seriously set about our work as if it were the most precious activity we have.

† See in particular Volume 3 of this work.

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By becoming aware of the various components of our professional life, we shall at the same time give ourselves exciting and exhilarating moments, make a contribution to science, help the young generation to meet its future, and, above all, live on the side of truth.

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This article first appeared in the Washington Post Sunday Magazine, in the U.S.A. in September 1961 When I was a schoolboy several of my classmates could not make any sense of the purpose of studying mathematics and many neglected it as a result. Nevertheless, they themselves considered anyone who was good at it as worthy of their respect for such intelligence and ability. I was not in the least distinguished as a schoolboy, and even though I had some ability at mental arithmetic, I showed very little sign that one day mathematics could become my full-time occupation. One day, the teacher gave us a test involving a problem of geometry, and I saw that if I dropped a perpendicular from one point to one line everything became obvious. Not one of my classmates saw this, and so the test did not count. I had, in fact, proved to all that the problem was an easy one whose simplicity resulted from a vision rather than from memory. As most people believe that learning is memorization they refuse to agree that seeing is as easy as, or even easier than, memorization.

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For me (then aged 13) it was a turning point in my life. I understood that mathematics was something of the mind, resulting from the inner movement of the imagination coupled with experiences that became successively deeper and deeper.

Teachers and books facilitate the meeting of significant and interesting challenges but in fact, mathematics is all around us and within the reach of everyone, provided we develop the sort of sensitivity that corresponds to these challenges. Indeed, it is more a matter of sensitivity than of reasoning (though reason is a kind of super sensitivity applied to ideas and thoughts) and I have since then perceived in everyone this ability to sense the beauty of mathematics.

Consider a drop of water falling at the center of a bucketful of water. It creates a ripple which is a circle whose radius increases until the ripple reaches the side of the bucket. If drops are allowed to fall one after the other, a succession of concentric circles is created making such a fascinating design that one cannot stop looking at it. If the drops are allowed to fall closer to the side of the bucket the design becomes much more complicated for now the first circle formed is reflected by the curved side of the bucket and interferes with the next one.

The ordinary eye enjoys this game of waves while the mathematical eye asks: â&#x20AC;&#x153;What exactly am I seeing, and what should I see were I to move the drop closer and closer to the side of the bucket; what would happen were I to let the drops fall individually at various rates, or more than one drop at a time?â&#x20AC;?

In addition, since the mathematical eye operates within the mind, the experiments are not carried out in fact, but the imagination is left free to bring in its contributions. This is done 140

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on a piece of paper with such simple instruments as a pencil, ruler and a pair of compasses. To look at the problem in this way is much simpler, since so much less is required than in the experiments. One can try it for oneself, but the fascination resulting from observing the effect of the falling drops, which was the first impulse received by the mind, must not be forgotten. When drawing, we partly describe what we saw, but since we are now free to do whatever we wish with our instruments we can show ourselves how far we can go by using them.

Drawing two families of concentric circles is a simple enough task. To pair them up according to different criteria may lead us to another fascination, for we can develop a new vision seeing not only the actual but also the potential. For instance, there was no straight line in our drawing but we can see one if only we mark the points where the circles of equal radius, one in each family, cut each other. These points, in fact, lie on a straight line.

But what would happen if we were to pair together those circles whose sum of their radii was equal to a chosen length, rather than circles of equal radii? By simply marking their points of intersection we generate ellipses, and as many ellipses as we choose different lengths. If instead of pairing by sum we pair by difference, we find hyperbolae, the majestic lines running to infinity. Many more things besides jump out of these two simple families of concentric circles.

Indeed the reason why we should study mathematics is because it educates in everyone a third eye; an eye capable of scrutinizing relationships per se, capable of finding in even the most ordinary of situations something worthy of note, and capable of 141

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indefinite extension. The concept of infinity lies in the realm of mathematics, and mathematicians are those who among us have made it accessible to the minds of all. Once relationships are observed, a big forward stride has been taken, proof both of the power of the mind and also of an opening to the concept of infinity. One can reason that, for instance, if one thing is twice as big as another, then at the same time the second thing must be half the first. If something can be doubled, doubling per se can be observed, and it can be seen that it is possible to double double double . . . something, and conversely, to halve halve halve . . . it. Doubling indefinitely increases magnitude indefinitely while halving reduces magnitude indefinitely. The infinitely large and the infinitely small are reached at one and the same time.

Mathematics, like poetry, satisfies the deeper senses in us all; both maintain us in contact with the ever changing, the unformed yet formed and subtle. But mathematics stems from the cares and attentions of a mind receptive to experience, and our minds are engaged in all kinds of activities, financial, economic, practical, scientific and so on. Relationships are involved in all our activities; once we become aware of these relationships we are thinking mathematically. Mathematics is part and parcel of our lives since we are especially concerned with relationships. All men have need of mathematics in order to understand better what challenges face them, and to grasp better how to act on reality. To think of a better route to drive from one place to another is to act mathematically (though this may not, as yet, be taught at school as such) since a comparison is made in advance between the relative merits of each of a set of routes in accordance with the properties attributed to them.

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These properties have been abstracted and compared in the mind; this activity is mathematics.

When mathematics is really understood as the natural approach to the universe of experience in which things are simplified by the deliberate neglect of irrelevant attributes and in which the relationships between these things are stressed, one cannot fail to see that it is in oneâ&#x20AC;&#x2122;s own interest to develop this power of abstraction and comparison and to attempt to achieve oneâ&#x20AC;&#x2122;s aim of being better equipped in this way.

The value of mathematics cannot be exaggerated; as it is impossible to live without breathing, so the universe of relationships per se cannot be entered other than through mathematics because these are one and the same thing. By refusing to study mathematics, we refuse to give ourselves a power this study can develop, and which is necessary in order to enter more deeply, more safely and, in particular, in a more healthy way, into the world of today so highly organized, and thus working at a level one or two stages removed from the immediate. To be in a position to live in, and serve, our so complex and diversified modern communities, the presence of a right way of thinking which can consider complex entities in large numbers, seems indispensable. Until today no one has developed a system more adequate to achieve this than mathematics, as can be seen by its manifold uses in all walks of life.

Though mathematics may appear difficult to students it must, however, be stressed that it is not alone in its possession of this

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attribute. No one would deny that it is necessary to work hard in order to reach a high standard of performance in games, sport or business. Sport is open to all, but not everyone takes it up professionally. Mathematics is also available to all, but again, only a small percentage will make a career in it. Mathematics is so enjoyable and so enriching when well understood that one should ask oneself when repelled by it: â&#x20AC;&#x153;Why have I not yet been infected by it?â&#x20AC;?, rather than assume that it is hard, or boring, or useless, for it is none of these things.

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This article was first published in the U.S.A. in November 1961 in The Mathematics Teacher, Vol. LIV, No. 7.

Mathematicians are often mistaken for logicians. Among themselves, mathematicians know who is doing what, and often those who produce the meat of mathematicsâ&#x20AC;&#x201D;that is, the new theorems and theories, the promising notionsâ&#x20AC;&#x201D;look upon logicians as the cleaners of the premises, mainly preoccupied with cobwebs. Logicians, in their turn, look down on any creation that has any content; they are convinced that mathematics in essence is one type of game and all the rest is folklore and of lesser value.

This competition among professionals would be an interesting sight for onlookers did it not threaten to invade the field of mathematics education and lead to sterilization in not too distant a future.

If all writers were only grammarians, we can easily imagine how dull bookshops, libraries, and bookshelves in our homes would 145 147

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be. If grammarians believed that only grammar books or articles were literature, we would readily be convinced that illiteracy was around the corner because no one would like to read and no one would find it justified to suggest reading as a source of fun, growth in taste, and experience. In fact, grammar is a codification of what exists already. It provides the specialists with interesting problems that give them excitement and make their efforts worthwhile for themselves. But evolving languages (live, written, and spoken speech) are the justification of grammars and grammarians. Their raison dâ&#x20AC;&#x2122;ĂŞtre is the existence of linguistic challenges that come out of the creative activity of writers and orators.

Similarly for logic and logicians. They certainly have a place in the world of mathematical thinking, but not all the room. They come in, after the mathematician has done some creative work, to look at it and see what they can learn from it or what they can say about it. If mathematicians stopped producing in their way new ideas, soon the activity of the logicians would stop or become purely historical analysis. For the sake of logiciansâ&#x20AC;&#x2122; continued existence, mathematicians must go on thinking in their artistic way, i.e., not legislated in advance.

Indeed, mathematicians find their ideas where they can, as they can, and clarify them as far as they can. Their efforts at extracting theorems from the situations they are contemplating cannot be said to follow any one line or even any sequence of lines. Readers of mathematical papers find in them the polished final product of painful and often unsatisfactory struggle. These readers may see that there are still relations to be found in the

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same situations or in related ones, and, in their turn, produce additional theorems.

The activity of mathematicians is often very complex and has, so far, escaped description. It still remains to find someone who will write an adequate study of mathematical creativity and the actual histories of important contributions in adequate terms.

(Poincaré, Hadamard, Polya, and others have written most interestingly on the subject; still everything seems very mysterious and hidden in this field.) The reasons for such an absence of a comprehensive knowledge on this matter could be: (1) the multiplicity of the approaches of mathematicians to their mental content; (2) the difficulty of creating and at the same time watching how creation takes place; (3) naiveté mathematicians may experience when doing something other than mathematics, making it difficult to contribute something worthwhile outside the activity for which they trained themselves; (4) the word “mathematics” covers so many areas of experience that it is hopeless to attempt to reduce them to anything but themselves—in other words, mathematics is the expression of the activity and must be known from within to be known at all.

If all these reasons were true, to teach mathematics would mean, then, to give experience in as many aspects of mathematics as possible to let the learner find by doing, struggling, and selecting which sections of the wide fields correspond to his gifts, his powers of work, and his insights.

Then the acquaintance with real challenges will bear their fruit, and the learner will have the opportunity to become a

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mathematician who can trust himself to achieve something worthwhile, at least for himself if not for others.

But to teach formalized material is to select for the student; it is to extract all nourishing juice from the situation; it is to present skeletons that only challenge a power of dealing with signs and rules devoid of significance for temperaments different from that of the selectors. Indeed, we fall again into the mistake of teaching grammar instead of literature. Formalization is one of the stages of mathematisation, and no mathematician can afford to leave out that very important phase (called problem-solving by some) which is the transformation of a perceived situation into a mathematical oneâ&#x20AC;&#x201D; as he would not wish to neglect the presentation of what he has found in the most adequate formal manner current in his day.

To teach mathematics well seems to require that we 1

always stress mathematisation of situations either perceived or given by a set of statements, drawings, etc.;

2 relate the schematized new situation with other sets of relations already explored by similarity, analogy, transformations, specialization, or generalization; 3 isolate in the situation one or more relations and their bonds or dynamic links; 4 express most adequately at the level of actual experience of the learners all that has been reached; 148

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5 discuss among the learners, and sometimes with a more advanced student (the teacher), whether the expressions proposed cover the experience met, and check all that in an effort of formalization.

Formalization is thus the last step in the mathematical activity of learners as well as of creators. If we do not place it there, we are courting sterilization, since the meaning of the last stage will have to be guessed on uncertain grounds and lead to a feeling of magic, instead of an activity of knowing minds. It would also unnecessarily tax the imagination that may well be adequate if meaning is always present, but that does not accept guessing at random and without direction.

Teachers of mathematics, as much as any elder members of our community, carefully have to avoid impairing the mental health of the young generation. It is our view that anyone who forces children into a sterile dialogue with meaningless notions and a formal sequence of ideas presented at the beginning of experience, rather than at the final stages of analysis, is actually taking a very grave responsibility in the social sense.

Can we learn music without tunes? Can we learn languages without contexts? Can we learn to swim without being in the water? Why should we then learn mathematics without the very substance that makes it?

These are not emotional notes. They are common sense at a moment when scientific and mathematical education is suddenly beset by professors coming from ivory towers to tell

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what we should be doing with our children—whom they have perhaps never studied. Our own responsibility begins at the moment we accept the leadership of these men (clearly not resulting from their work in education and with learners, but in far-removed areas of activity). Our responsibility increases if we accept their opinions as true, particularly if they are not.

If the ideas of these professors are right, we shall be grateful.

But if these ideas are not correct, have we prepared ourselves to present our own alternatives, or do we only have recourse to another professor?

A man of experience may not be able to value experience until he reflects on how he reached the level of maturity that is his.

But teachers, who have looked at pupils learning, know that the children’s way of knowing mathematics is through deeper and deeper acquaintance with the labile entities they meet. Children have to know the classes of beings which are covered by names and how to classify into subclasses when attributes are added to the defining ones. As mathematics is, in part, an attitude toward reality in which relations are singled out in their dynamics—half involved in the situation where they are met and half free to be involved in other situations—an important part of mathematical experience is in the acquaintance with that psychological ambiance which gives different meanings to the same signs on paper. Just as notes are notes and not tunes, mathematical signs are signs and not mathematics; it is the activity behind them that matters. Every mathematician knows that a mental context goes hand in hand with a mathematical text, and he knows how important it is to be able to tune in when contemplating a situation. All this is indeed folklore, but how important! Without 150

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this there may not be any mathematics, as we can observe from the complete lack of any consequence of the axiomatic systems produced in large quantities forty or so years ago. Even when mathematicians say that they play a game with some rules, they put in the game images, meanings, and intuitive relations that guide them in their moves.

If imagery, initiative and sets of meanings are important for the performance of mathematical activity at all levels, we must take great care that our pupils experience them for what they bring while they spend their time at mathematics.

The substance of mathematics in education is its folklore, and the more varied it is, the keener our minds will be in finding behind the great variety of experience those large, unifying structures that are like the machine tools of mathematics. Once these structures are perceived, they give a new meaning to all previous experience; but we cannot likewise expect that the contemplation of machine tools may lead everyone, and with reasonable ease, to visualize more structured beings that are useful companions in life.

As teachers, we would like to be successful in the classroom and, if possible, all the time and with most children. That legitimate wish, based upon the knowledge that our pupils have keen minds ready to do and undo complex arrangements and see things happen as a result of their activity, can be reconciled with the fact that for logicians what is most important in mathematics education is to learn to reason according to certain rules. Indeed, we can well see that since there are several years

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reserved for mathematical education, we may be permitted to build up both a varied and interesting experience in studying the folklore as a source of challenging questions, and the means to find in them the germs of new attitudes that have developed historically to the present. This will also allow us to see that it is possible that the mathematics of tomorrow may be very different from the one fashionable today.

Since the intuitive content of mathematics is as much a reality for learners as for creators, we would be well advised to maintain it in all our reforms, at all levels of teaching. This is one of our jobs as teachers when we examine proposals coming from different horizons.

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The Sources 0f this Volume

“The Mathematical Gazette”, Journal of M.A. of U.K. “The Mathematics Teacher”, Journal of N.C.T.M., U.S.A. “The Journal of General Education”, Chicago University Press, U.S.A. “Mathematica & Paedagogia”, Journal of A.T.M., Belgium “Nastava Matematike i Fizike”, Journal of A.T.M. & P., Belgrade, Yugoslavia “Times Educational Supplement”, London Radio Canada, recorded television lecture, Montreal “Mathematics Teaching”, Journal of A.T.M., U.K. “Washington Post” (Sunday Magazine), Washington D.C.

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Index

Abstraction , 15, 19, 22, 23, 25, 61, 79, 83, 92, 98, 143, 155 Action 16, 17, 18, 19, 20, 22, 23, 25, 26, 58, 60, 61, 62, 69, 72, 76, 110, 117, 123, 132 Arithmetical progressions , 26, 155 Axiomatic method , 99 systems , 5, 42, 69, 71, 151

Curriculum , 11, 12, 13, 14, 57, 134, Dalton plan , 14, 156 Dedekind , 97 Dialogue , 76, 77, 84, 89, 90, 115, 119, 121, 122, 123, 149 Dynamic pattern , 69, 80, 156 Examination , 6, 14, 75, 99, 104, 105, 129, 132

Bourbaki , 98 Fermat , 18 Films , 1, 81, 109, 120, 122, 123 Fletcher, T. J. 81, 119, 122 Formal proof , 61 Functional syllabus , 68, 70, thought , 97

Cantor , 97 Cauchy , 97 Choquet , 133 Complex thinking , 91, 156 Cuisenaire , 84, 99, 111, 122, 123, 156

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Galileo , 97 Geo-board , 93 Geometrical progression , 24, 25 Grammar-school , 63 Group structure , 110

Mathematisation , 116, 148, Memory , 84, 117, 119, 139 Mental structures , 37, 38, 42, 58-65, 70, 133 Methods , 12, 13, 14, 19, 27, 28, 75, 78, 90, 100, 105, 114, 123, 132 Mistakes , 12, 23, 33-42, 135, Mode of thought , 101, 115, Modern Mathematics , 98, 102, 114, 115 Montessori , 24 Multivalent , 5, 99, 100, 114, 115 situations , 99 structures , 100 thinking , 5, 114, 115

Hadamard , 147 Intuition , 36, 75, 78, 79, 84 Leibnitz , 97 Loci , 69, 119 Logic , 1, 5, 6, 11, 15, 16, 17, 18, 19, 20, 21, 28, 42, 46, 56, 76, 90, 116, 124, 125, 133, 134, 145, 146, 150, 151 of action , 17, 20 of feeling , 17, 18, 20 of invariance , 19 of mental convergence , 18, 19 of the obvious , 18, 19, 20 of perception , 16, 17, 20, 28, 61, 63, 71, 91

Newton , 97 Nicolet, J. L. , 81, 109, 119, 122 Notation , 59, 98, 123 Operations , 30, 35, 37, 39, 40, 41, 65, 66, 83 upon operations , 40 Pattern , 28, 47, 51, 52, 58, 64, 65, 66, 68, 69, 80, 100, 115, 122, 128 Perception , 16, 20, 28, 5862, 79, 89, 90, 95, 100, 115, 117, 121, 123 Perceptive , 28, 73, 78, 119, 122

Mathematical films , 1, 81 situation , 38, 91, 109, 119, 122 structure , 38, 57, 59-63, 70, 110

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Index

Piaget , 5, 6, 7, 57 Polya , 147 Potential numbers , 40 Project method , 14

Substitutes , 19, 20, 21, 22, 23, 24, 25, 61, 62 Syllabus , 6, 11, 12, 13, 14, 19, 24, 63, 68, 70, 71, 75, 76, 87, 91, 99, 104, 105, 106, 130, 132

Quadratic equation , 30 Quadratics , 28, 29, 31, 35, 65

Univalent , 5, 99, 101

Relationships , 3, 24, 56, 76, 78, 84, 89, 91-93, 95, 97, 109, 110, 115-118, 124, 134, 141 Representation , 36, 58 Riemann , 97 Rigor , 42, 78, 114

Verbalization , 123 Zermelo's axiom , 18

Simultaneous equations , 65, 66, 68 Situation , 4, 6, 12, 14, 31, 34, 36, 37, 38, 39, 40, 57, 60, 62, 64, 67, 71, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 92, 93, 94, 95, 97, 101, 102, 111, 112, 113, 117, 118, 119, 120, 121, 122, 123, 124, 125, 128, 130, 131, 134, 141, 146, 147, 148, 150, Structure , 37, 38, 39, 41, 42, 58, 59, 60, 61, 62, 63, 64, 65, 67, 70, 72, 83, 84, 94, 99, 101, 102, 112, 113, 116, 123, 124, 133, 134, 151

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