For the Teaching of Mathematics Volume 2

For the Teaching of Mathematics Volume 2 Psychological Studies on Films

Caleb Gattegno

Educational Solutions Worldwide Inc.

First published in the United States of America in 1987. Reprinted in 2011.

Copyright © 1987 – 2011 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-336-4 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, NY 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 Part III Psychological Discussions ............................... 7 1 Remarks on Mental Structures .................................9 Particular Analyses. .............................................................14 1 The Crawl. ....................................................................14 2 Diving. .........................................................................16 3 The Russian Language. ...............................................18 4 The African Language. ............................................... 23 5 The Study of the Guitar .............................................. 25 Conclusion........................................................................... 29 2 A New Theory of the Image..................................... 31 3 Mathematical and Mental Structures ..................... 51 Introduction .........................................................................51 1 Mathematics and Psychology ..................................... 54 2 Evolution of the Mind in the Apprenticeship to Mathematics and Other Sciences............................. 59 3 Experimentation In Progress..................................... 66 Conclusion............................................................................ 71 4 Perception and Action as Bases of Mathematical Thought .......................................... 73 5 Mathematical Thinking and the Use of the Senses....................................................89 6 Adolescent Thought and its Bearing on Mathematics Learning ..........................................95

A Method of Investigation in Schools ........................... 95 7 Investigation Through Teaching ...........................109 8 Three-Dimensional Vision and its Psychological Application to the Teaching of Mathematics......................................................... 121 9 Pupils’ Reactions to Geometrical Classifications ...................................................... 133 1 Cube and Parallelepipeds.......................................... 134 2 Trapezia and Parallelograms.................................... 136 Conclusions ....................................................................... 138 10 Mathematics and the Deaf................................... 141 Part IV On Films ...................................................... 151 1 The Idea of Dynamic Patterns in Geometry ...... 153 2 Geometrical Intuition and Mathematical Films 159 3 Teaching Through Mathematical Films ............ 163 1 Lessons With the Nicolet Film Dealing With A Circle Passing Through Three Points. ................ 164 2 Lessons With The Nicolet Film on the Locus of Points in the Plane From Which Two Given Circles in the Plane Are Seen From the Same Angle..............................................174 3 Lessons Suggested By Three of Nicolet’s Films on Conics................................... 178 4 Conclusion................................................................ 182 The Sources of this Volume ...................................... 185 Index........................................................................ 187

Preface

In this volume I have gathered together a number of articles written between 1948 and 1962 which consider the teaching of mathematics as the forming in everyone’s mind of structures that are somehow called mathematical. The articles are selfexplanatory but it will perhaps be less obvious in some that I was attempting to develop a classroom technique for studying the content of the mind of our pupils, related to Piaget’s but which seemed to me, at the time, likely to offer greater yields insofar as the number of people studied would automatically be multiplied by 30 or so. Though I did not continue to publish reports on the many experiments I was carrying out, I went on lecturing about my findings, the techniques becoming more and more my own. The understanding of what I was encountering forced me away from the preconceived ideas I had gleaned in my reading, and from the acceptance of some conclusions offered by writers. In particular, I understood that in psychological studies such as those I was engaged in, one never studies more than the content of one’s own mind; in other words, in psychology as in nuclear physics, the observer alters reality by simply observing it.

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This awareness made me look for a relativistic approach for the description of events that would be acceptable to other observers related to me by the knowledge that we were looking at the same reality. It was in the transformation of mental structures into mathematical structures that I found the key that satisfied my requirements. So I started by showing how one can develop techniques for the study of structuration upon one’s own mind. Mathematics was not mentioned then for it is only a special area of activity. When I considered the specific problem of the formation of mathematical mental structures, I was aware that a huge field was opening up in front of me and that I should only be able to get a glimpse of it. The notions introduced in two or three of the articles following the first in this volume I believe to be powerful tools whose effect may be, if they are properly used, a dramatic increase in our knowledge of the psychology of mathematics, and not merely an a priori description of what one wishes to see or find.

More and more often since most of these articles were written (mainly around 1950) I have found, (1) that pupils at school operate far below their true level as a result of the way they are taught, (2) that our tests usually describe as much the impact of society on the minds of children as the children’s own mind, (3) that things are very different indeed if we mold our teaching to whatever learning is taking place. This clearly indicated to me that my early psychological studies were laboratory experiments with diminished individuals and that if I changed the procedure I should learn more and contribute more to true knowledge.

Two or three years ago I tried to present my findings on the psychology of learning in a sizeable book; this has not yet been published, but the score or so of my texts on elementary 2

Preface

mathematics both for pupils and for teachers and which are extensively used all over the world confirm that my psychological thoughts and findings now embodied in day-today activities with learners are akin to what children really do.

Because of this, children today can learn modern mathematics in primary schools. What no one had tried before became easy, and is now commonplace. Each lesson actually undertakes the transformation of mental structures into mathematical structures by a special awareness that is the aim of that lesson.

The arrangement of the chapters that appear in this volume does not follow their chronological order of publication. It appeared to me that the material available fell into two trends of research, experimental and theoretical. The first trend concerned itself with the production of the notions and the techniques that can be related to structuration of the mind through practice; the second with the classroom technique of investigation which I still believe could be most helpful and produce quick and impressive returns.

The first of these articles does not seem to have anything to do with mathematics but is concerned with the study of the self engaged in various structurations. After almost 15 years, with some trepidation I re-read it, and found in it a number of insights which had matured and become cardinal notions in my latest work.

This is followed by a study of images which led me to a new conception no-one besides myself has found of any use but which, I am convinced, contains germs that would greatly assist

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any student of learning or of the mind in its functioning. Though not couched here in the clearest of terms, this conception has served me so well that I dare draw it to the special attention of the reader.

Between the time of writing the two articles mentioned above (1949) and the ones that follow, I published in French in collaboration with A. Gay a report on studies on a new psychosomatic phenomenon* which, on the one hand, seemed to justify my way of thinking of dynamic mental structures, and on the other, provided new tools for the study of the mind in general. Two of the articles that follow are influenced by these psychosomatic studies. In them I tried to formulate themes for research, but today we still await their investigators. I would urge some of my most enterprising readers to consider them seriously if they are looking for a worthwhile field of study that would yield results for decades of work and reflection.

These first four articles are all sustained by the same inspiration.

The following four are also united by a common preoccupation: how to develop a technique of investigation that teachers could use while teaching their classes. As said above, I expect much from it where a sufficient number of colleagues use it systematically and for some time.

*

Un Nouveau Phénomène Psychosomatique (C. Gattegno & A. Gay 1952, Delachaux et Niestlé Neuchâtel and Paris.

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Preface

The last two articles belong to the series but in a looser way.

I included the first section of my book Teaching Mathematics to Deaf Children as the last article partly because I let this book go out of print, partly because I believe that in it I hit on at least one point of importance, that is, the possibility of by-passing verbalization, passing instead directly from perception and action to notation. For deaf children I believe this has been a short cut making life much easier and mathematics a much livelier subject.

A group of three articles on mathematical films constitutes Part IV of this collection. For years now I have been hoping to see teachers of mathematics recognize the great value of teaching through films of the type J. L. Nicolet and T. J. Fletcher have made and which attracted me from the very first moment I saw them. I know of a handful or so of people who have understood what such films have to offer. It is a great pity that the number of users remains so small. My faith in their ultimate acceptance by the profession is as strong as ever, particularly since I saw that adult education through television towards a way of thinking more adequate to the needs of our time is not only feasible but is the only way possible, in view of the challenge presented by our complex society and the numbers involved. I have been intensively engaged in the consideration not only of this challenge but also of the kind of solution to the demands of today that can be found in the mass media. Mathematical films have helped me a great deal and I believe they will help everyone.

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Once again I should like to thank my young colleague Jeremy Steele for his valuable help in making my texts more readable and for his good translations of some of my most difficult articles.

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1 Remarks on Mental Structures

This article first appeared in France in Enfance, May–June, 1951, No. 3.

It would seem of interest, for the analysis of learning, to study in detail a number of learning apprenticeships. Two courses are available to the researcher. On the one hand, he can examine what is accessible to him in subjects involved in an act of apprenticeship and confirm by means of appropriate tests that the matter is very much as he had understood it. On the other hand, he can act as a guinea pig himself so that he may, over a long period, study in himself a certain number of these acquisitions.

In the present article I shall limit myself to the second method and my remarks will be largely concerned with an exceptional experimental material since what is put forward can only be checked if other psychologists have undergone the same experiments or collected analogous statements about subjects they have examined.

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I voluntarily subjected myself to five different apprenticeships in order to check the conclusions reached one by one. In 1940, I learnt how the crawl in swimming is done; in 1941, how to dive; in 1942, I learnt Russian; in 1948, an African language, and since September 1949 I have been learning how to play the guitar.

These five apprenticeships were intended solely to help in the study of the mechanism of learning; they were dropped as soon as they ceased to provide new information about the apprenticeship in question, which occurred sometimes when the technique had been mastered, and at other times at a more rudimentary stage.

The choice of the fields was on the one hand arbitrary, and, on the other, particularly fortunate. The techniques involved in the five apprenticeships are fundamental and primitive in the sense of personal history and the observations made touch on a basis common to a host of other and more elaborate techniques.

From the method point of view, the relentless way in which the first three techniques were pursued, must be mentioned. Daily sessions lasting an hour and a half were for a fortnight devoted to the apprenticeships to the crawl and to diving. For six weeks I worked up to 12 hours a day studying Russian. Reading and observation brought their complements; on one occasion a diving champion showed me how to do a difficult dive. For the African dialect I had a teacher, and for studying the guitar, for a week I followed a “method� found in a bookshop.

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Let us note first of all that the techniques I was bent on acquiring between the ages of 30 and 40 are easily acquired at about 10 years of age, as my own children showed me at this time by learning with no apparent difficulty, how to swim, to dive, to play the piano and so on. I knew how to swim, but not to dive; I knew several languages, but none of the Slavonic or African families. I had no musical knowledge which might allow me to broach the study of the guitar. I had a natural tendency towards abstraction and a great difficulty in making myself conform. The problem which arose each time was how to attain consciously or construct consciously a mechanism which must satisfy certain conditions perfectly grasped by the intellect.

Thus, in the crawl, I knew how my arms, legs and head had to be moved, when to breathe in and when to breathe out, what rhythm of leg movement had to be combined with arm and breathing rhythms. But, once in the water, chaos ensued. Diving is such a simple act, but, once on the spring-board, not only did all grace disappear, but the resulting splash was so considerable that people stopped what they were doing to look on and pity me.

Clearly, an understanding of the mechanism was of no use whatsoever; I can even state that rational understanding is an obstacle in all apprenticeships analogous to those studied here.

This is my first general observation. There is a vast gap between mentally understanding a physical or functional structure, and actually making it exist as a conscious structure which must become automatic. The gap is not always due to emotion as might be deduced from the observation of adults learning to ride a bicycle, for example. It is due rather to the nature of the mental structures in question. Intellectual structures are those 11

Part III Psychological Discussions

which are as far removed as possible from physical and psychic structures and which make use of those with a rhythm and a speed almost reaching infinity. Physical mental structures have a limited functioning speed which allows each partial mechanism to take its place in turn.

In the intellectual conception of swimming, all the mechanisms are perceived as being simultaneous and organized in a theoretical being: the swimmer. From there chaos follows once the theory is put into practice. The mind can only concentrate on one of the movements, the others either stopping or being carried out in the way that the limbs are accustomed to move, resulting in the stiff and ungainly nature of the individual movements and of the whole action. The same also applies to diving.

In the case of the apprenticeship to a tongue totally foreign insofar as its alphabet, writing, syntax, vocabulary, sound and intonation are concerned, a complete view of the whole field of enquiry, as with swimming and diving, is not possible, and the above analysis is no longer applicable. The remarks to be made here and which seem extremely interesting are of a completely different character again.

In the case of the apprenticeship to the guitar, the role of analytical intelligence occasionally reappears in its paralyzing form though in a more subtle way. Here verbal and intelligible instructions (but written, however, for an anonymous pupil) must be translated into movements which correspond to sounds, the ultimate judge of the accuracy of the imitation. If the

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ability to make this judgment is lacking, as it was in my case, only an indirect means remains, still intellectual, of knowing if the correspondence is correct: arithmetic. Through this the instruction is checked and accepted before moving ahead with the study.

The notion of structure used here is perhaps familiar in only one or two of its meanings. It indicates any organization whatsoever at no matter what stage of development it is considered. The structures I am studying are mental even when they are somatic or psychic, but on condition that they may be put into action by a deliberate act of the self at any moment in one’s life in-utero as well as ex-utero. The difference between somatic and intellectual mental structures for example is that the former claim a muscular and nervous support while the latter are selfsupporting or supported by images, or operations whatever the origin of these may have been.

The first observation above was that intellectual structures are too detached from somatic structures to allow an immediate translation from the first into the second. A second general observation is that the free play of mental structures cannot take place except at their own level: the somatic structures evolving an organization between themselves, the intellectual structures between themselves. If there is a relationship between the levels, it is because the intellectual structures have abandoned their own laws in order to accept those of the somatic structures; the inverse is not possible.

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Particular Analyses 1 The Crawl After five days of relentless toil the structure had been acquired: the movements had been integrated, efficiency built up and a feeling of liberty acquired. The method to start with consisted in making just one of the movements automatic, being forever conscious at first of each of the factors intervening in this movement: foot, ankle, knee, thigh, hip. Hours of this same movement constituted a complex somatic structure which got progressively better and more efficient as it got more unconscious. Later, while it was in action, the consciousness moved on to a second movement which was studied in the same way, without trying, at this point, to co-ordinate it with the preceding movement; it seemed possible, if not necessary, to coordinate two organized and more or less automatic totalities, rather than integrate the new structure into the preceding organization, little by little. Co-ordination is an interesting phase in itself. Both structures appear to search for a point at which they can link themselves together which will make them function like a new and greater structure but will leave to each one a great deal of autonomy. The moment of discovery of this linking point is accompanied by joy and encourages a feeling of new power which facilitates the later progress.

In the case of the crawl, the greatest difficulty encountered was in breathing. A considerable emotional element is attached to breathing and the crawl is never well done if this is not carried out perfectly and unconsciously. It is necessary to breathe in quickly through the mouth and breathe out slowly into the water 14

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through the nose, the face being out of the water for only a fraction of a second. Air brings not only oxygen to burn in the effort, but also contact with life and the external environment. A whole set of emotional prejudices must therefore be overcome and a particular rhythm of breathing acquired different from the normal rhythm, and inserted into the complete action of the legs and arms which have been separately coordinated. A week after starting I was able to do the width of the swimming pool for there was no need to breathe except at the beginning and the end. It took a week to solve the problem of breathing, in principle much simpler than the other problems and one which it was possible to work out straight away, separately. The difficulty seemed once again to lie in the interference by the mind. Since everything that had to be done was proven and since the term “co-ordination� came to mind spontaneously, all I had to do was to introduce breathing into the movements already acquired and with luck this could have been successful, but as soon as I leapt into the water to do a length of the pool the rhythm was lost and it became necessary to start all over again. I exhausted myself by doing a length of the pool without breathing in order to be assured that the emotive factors had been overcome; but it transpired that it was only all of a sudden that the waiting was over and the whole action pieced smoothly together.

Children do not learn in this way. They may do these things badly, but they do them all at once, and perfect their style through practice and through increasing the number of obstacles in their games. All the adults who undertook this apprenticeship at the same time as I, experienced the same difficulties, but, as they had no urge to discover the secret of 15

Part III Psychological Discussions

these difficulties, they took three years to acquire a mediocre style, never doubting that it could, in time, be perfect.

2 Diving Here the analysis is more difficult since the fact of plunging suddenly into the water several dozen times in a few minutes is considerably tiring, and can be costly insofar as colds are easily caught. Thus it was in my case. My study was interrupted for a year after the first attempt which lasted a few days. When taking it up again it transpired that I was much better equipped than the previous results had allowed me to hope. An unconscious apprenticeship out of the water had taken place, the knowledge of which is precious.

The movements in diving are different from those of jumping and, according to the height of the diving board, they call on different muscles. My study was limited to a springboard one meter from the water and a fixed board two meters from the water.

Diving involves the co-ordination of the balancing movements of the arms, the tension in the knees, and the angle of penetration into the water for which the muscles of the back are used. This all seems so easy when a diver springs into the air and makes a neat entry into the water. But as soon as one wants to coordinate these movements voluntarily it seems that the muscles are seized with panic and everything goes haywire. The powerful slap the water can deliver is a sure means of making one take into account one’s manner of entering the water, which

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observers find so amusing. Everything is possible. There is total chaos, therefore lack of co-ordination at the somatic level even though one might have been convinced that the muscles had been given the correct instructions in accordance with the judgments made of what other divers do.

Here, unlike the case of swimming, the movements cannot be separately analyzed. More skill is required in making the necessary movements in order to stay on the diving board than in diving correctly. There is therefore only the concomitant analysis and the analysis during the action. If a benevolent observer is present, he can assist in the analysis of the result but not in the tensing of the leg muscles, or in the balancing movements of the arms, or in the jerking of the body before entering the water, and so on. On dry ground I could move my arms very well indeed, my feet and legs were in the correct position and everything seemed perfectly harmonized but the same movements on the diving board only led to a fall on my back or a somersault and a colossal displacement of water.

The following year my dives from the edge of the pool were correct from as early as the first day. Those from the diving board showed considerable progress. An accident in the vertical jump however obliged me to discontinue my investigation.

These two studies of physical apprenticeships necessitating the formation of structures far removed from my normal preoccupations which had been for years specialized at the intellectual and social levels, have a triple interest for the psychologist and for the educationalist.

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First of all they show us in detail activities which escape observation because the sportsman is rarely a psychologist and also because the psychologist generally speaking looks at external phenomena and tries to describe them in terms of behaviors rather than as structural syntheses. Second, they show us that the apprenticeship to the physical activity becomes all the more difficult as the self preoccupies itself with other questions and builds for itself the tools which correspond to them; that it is necessary to become aware that an optimal moment exists for physical structures to be acquired and that this moment is the one where the self in general organizes its hold on the world of direct sensory-motor relationships; that it is necessary to become aware that once this moment is past, the price of the apprenticeship will be all the higher since we shall have shifted our interest and our habits onto relationships with a rigid and removed rhythm; that here intelligence is a hindrance. Finally, they teach us in what way the organization of structures occurs in an adult in the light of his intellectual and relational awareness, in a child under the influence of the direct interest for the situation and for its integration in the general activity. The springboard is, for the child, an opportunity, but for the adult, a constraint.

3 The Russian Language Here we are faced with an extremely complex question whose various aspects I cannot pretend to have disentangled. I can only offer a few observations which appear to be of some use.

When undertaking the study of a language as difficult as Russian with such help as is afforded by a teach yourself book, it is 18

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obvious that the nature of the observations made depends on the line of attack chosen by the author of the method.

Nevertheless it was possible to observe the learning of words, sentences, syntactic forms, rules, and above all, to observe in myself the emotive and mental reactions to the situations thus created.

I wondered if I was making use of a knowledge of Greek in reading; if I was making intellectual and perceptive use of the rules of gender; if I was using mnemonics; or repetition in order to consolidate what had been acquired; if it were easier to go from Russian to English or German, or the inverse. Of the two books I was using, one was written in English, the other in German; I wondered what particular advantages each of these auxiliary languages offered to help me penetrate better into the spirit of the new language and so on.

What is certain is that at the end of six weeks I could read Russian and understand conversations overheard but was very wary about speaking. The principal reason for this was that I had learnt the pronunciation from a book without having it corrected by someone who knew Russian, and I believed that it must be faulty largely because of the pronunciation of the l which I never managed to understand from the written suggestions offered.

I have noticed that the more the material is scanty and inadequate, the more the slavery is total and the future of the apprenticeship precarious. The whole of the mind is tied to books and to exercises done and a feeling of unreality is

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constantly present, paralyzing the least effort towards the utilization of the acquired knowledge. Confidence only reigns in the solitude of one’s own studying. But this feeling of helplessness is lost as soon as what has been learnt is sufficient to tackle complex situations and where the principal factor lacking is vocabulary. The learner then knows that with a dictionary and practice these difficulties are surmountable.

In the study of the language I made use of my analytical intelligence in many ways. One way was to prevent habits from taking the upper hand and to watch that the words were pronounced according to the rules in the first pages of the books. I was thus obliged to repeat a word over and over again until the complete satisfaction of an intensely critical inner judge had been achieved—it watched over, and took note of, the linguistic interferences that occurred when I was trying to say something, when instead of the Russian word coming to mind a word from another language offered itself. It asked itself why that word and that language. It found logical groups to connect the knowledge acquired in one day and make it depend on the previously acquired knowledge, noting in which way the structures complete themselves, what the gaps are, what is needed in order to progress more surely and more rapidly, and what type of revision provides the best results and the most thorough consolidation.

Vocabulary and the beginnings of grammar were easily mastered and already a possibility was developing of creating in my mind an entire unconscious and automatic structure for the Russian language, separating itself from the means used up to this point for the acquisition of these elements. It seemed that 20

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the language took hold of me internally by its own power, entirely independent of how I did it. The very nature of this living entity, this structure of the Russian language became identical to my nature, the Russian words seemed the only ones spontaneously supplied by objects, and Russian sentences were coming to me to express my thoughts. The language substitutes for experience seemed to exist in me by the same right as all the other languages learnt spontaneously in my childhood and early youth.

This state of identification was not long in collapsing under the totally artificial circumstances in which I was studying. As soon as the list of verbs was tackled despair set in. There was nothing to catch hold of in this maze of verbs with their strikingly difficult future forms. Several attempts were made. There was some progress. The task appeared possible, but at what a price in those circumstances. The environment had not the smallest help to offer. Everything had to be retained by sheer willpower and as there was neither logic nor pressing need, the apprenticeship stopped short. All the badly assimilated parts which, in the first flights of success and progress had been accepted with ease and with the confidence of improving, became vast gaps and the knowledge which had seemed so full of promise collapsed into a jumble of words and formulas linked by a studied text. The structure of the language as a whole had escaped me—I did not know Russian.

This experience helped me to understand that a language can be learnt for specific needs, for example, in order to read scientific works without grasping the structure of the language, which is only learnt either in living it through literature or in living it 21

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with those who use it. In the first case there is constant recourse to a familiar linguistic structure which brings with it the feeling that the meaning is grasped even if the structure (even grammatical) of the language read escapes us. This auxiliary language makes for understanding, the one read provides the signs or signals.

Subsequently several opportunities arose for the completion of this analysis of linguistic acquisitions in terms of structures. I shall not enumerate here all the lessons learnt but shall mention just two of them.

I have seen the inverse phenomenon of the apprenticeship to a language, in observing how, under certain circumstances, people can forget their mother tongue or other of the languages they knew. French children speaking French in the home but living in England, learn French at school just like any other pupil, in an artificial way. At home, the structures which spontaneously present themselves in French are correct, but in the classroom the structures are those of the environment. The conflict lasts for some time until finally confidence is so undermined that the child refuses to speak French in his home, learning a distorted French at school and abandoning an assured acquisition which he should without doubt regain, but how? Soon the only linguistic structure present is that of the surrounding environment not only for English, but also for the languages used.

Another interesting phenomenon is that of the construction of a universal language. This problem tackled only in an empirical

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manner poses a number of very difficult questions about mental structures, particularly when concerned with a language both spoken and written. The solution of C. K. Bliss, the Australian, for a written language pronounceable in all languages, however remarkable, poses still further questions to which the solution is not easily seen, above all, if the structure of complicated ideographic sentences claims a word order borrowed from one culture rather than from another.

4 The African Language What was learnt in this apprenticeship was that there is no impossible obstacle to the assimilation of a language as foreign as this one, on condition, of course, that the pupil could hear and listen. Unlike the apprenticeship to Russian, I was here, from the very beginning, helped by a native who corrected my pronunciation and who produced all the examples necessary to illustrate both his remarks and his answers to my questions. The language in question was a tone language, its first difficulties being entirely auditive and musical. One had to be able to utter the same sound in three different tones and retain meaning not according to form, but according to tone. The newly introduced spelling of this language was still uncertain, which made the use of analytic intelligence more difficult. Here also intelligence appeared to be a hindrance rather than a help.

These two linguistic experiences, although very different and the two languages themselves being both extremely difficult, have taught me some similar things.

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The role of our normal means of understanding is to intervene at the outset and as a whole, to require that the linguistic material take a form which can be grasped, and not to bend under the existing conditions except after the event. Only when our structures are found to be inadequate do we force ourselves to create new ones. Most people pronounce foreign languages similarly to their own language because they allow themselves to think that the habitual structures are truly somatic and that a throat accustomed to pronouncing French can only emit determinate sounds; that an eye reading a word in which the letters of the reader’s language are used can only see it in the way that he learnt the letters, and so on.

Those who have learnt several languages can assure others that all this is so much preconception and carelessness. “Somatic” and “psychic” structures are functional and must be understood along with their functions. Every language requires a certain rearranging of the vocal organs if it is to be pronounced correctly, but the fact that there are men who speak dozens of languages, even as many as a hundred, assures me that different functional structures can be imposed on the anatomic structure, each one conforming to the stratified somatic structure that prevails in a certain culture. This is a problem for both the psychologist and the educationalist and an urgent and important problem in a world which searches for unity and which it can perhaps find by disentangling itself from false mental structures which it takes as being fixed and definitive.

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5 The Study of the Guitar This last study has been going on for eighteen months and I have decided to describe it just at the moment when it seems that I am reaching a crucial stage; in fact, I am just beginning to hear the sounds produced by the guitar. Let me explain.

I undertook the apprenticeship to the guitar knowing how to sing, knowing how to learn a song by ear, but never having tried to learn to read musical notes, to produce their equivalent vocally or on an instrument of any kind. I had no idea what time was. I was as ignorant of music as could be wished for, and therefore an excellent subject for the experiment. Moreover, having studied over ten years the apprenticeships outlined above, I was very interested in this opportunity to check my conclusions.

To start from scratch, as I did, is dangerous because it cannot be foreseen which direction progress will take. So it was that I very soon discovered that I was not hearing the piece I was playing, and that I could not play on the guitar any set of notes I could sing. I could recognize arias played by others and could learn to sing these arias from the guitar. To put this differently, there was co-ordination between ear and voice, but nothing else. To the question: “Is this note higher than that?� I hesitated; I was not able with any degree of certainty to produce two notes equal whenever I so desired, although I could sing them in imitation without knowing that they were equal. Here, then, my knowledge of singing was extra-intellectual, my intelligence had no grasp at all of musical matters.

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At the present moment I am just beginning to hear and I risk losing contact with my understanding of what the experiment has shown up to now. The problem could thus become the problem common to every musician and my contribution would lose its value of particularity even though it gained a value of generality. It is up to the readers to judge which analysis would bring them the more valuable information.

Never having touched an instrument, never having tried to read music, I was obliged to invent both the positions and the method. My way of holding the instrument proved to be false, and having been corrected three times as a result of fortuitous observations, it became necessary to begin the apprenticeship of the hands again at a stage where the incorrect position was more or less established. My method, which permits me to be assured that progress could be made, is of the type I called above “arithmetical�.

The book I had initially contained a pattern parallel to the stave on which were marked the strings and frets of the guitar and it was enough to read which finger had to be placed on which string and do this on the instrument with the left hand, while the right hand plucked the appropriate string. Thus a visual and a motor pattern had to be coordinated and the rule of translation was enough. I had a third person tune the instrument although this was scarcely important to me.

Moreover, I was often told that it needed tuning again.

Having overcome this difficulty, the question of rhythm arose. I had no idea at all about this and the numbers written at the

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beginning of the lines meant nothing to me since I had never learnt the scales. I learnt that a white note is worth two blacks; a black is worth two quavers and so on. This was visual arithmetic and within my reach. This was entirely different from producing notes on the instrument without wanting to count. But after a great many attempts the technique had been established. It was not, however, the end of the question of rhythm, but the battle had been won in principle and it was necessary only to gain dexterity before being finally able to resolve this difficulty.

I was faced with three problems of structures. Firstly a visual structure formed by signs called notes, which one had to learn how to read in terms of finger position on the guitar; there was no question of any other method for me since I had no desire to memorize the names and the sounds of the notes produced by the guitar (as is done when one studies with a musician with the intention of being able to play the instrument). The second visual structure, translated onto the guitar, had to be occupied by the fingers, and therefore corresponded to a third interior and motor structure. This new motor structure whose constitution it was the most difficult to foresee owing to the absence of any check on it, formed itself spontaneously and constantly perfected itself despite the incorrect positioning of the hand.

After two months my first simple piece of classical music was learnt in this fashion. Every day the piece was repeated, and though the notes were not always heard, the whole of the piece was produced as a motor translation of a visual structure comprising correct notes and a correct rhythm. Despite this apparent success there was absolutely no musical progress and 27

Part III Psychological Discussions

it was still impossible for me to play by ear even the simplest of pieces. At the end of six months some pieces had been learnt by heart, but were always played as a motor structure linked to a visual structure which by this time was virtual. Auditive tests systematically proved abortive.

At the end of a year a new exercise was undertaken, that of using the guitar as an accompaniment to singing. It was an entirely new technique. The only progress was registered when a piece bearing the chords of accompaniment was produced, then the old mechanism could play and all that had to be done was to learn how to reproduce a visual structure on the guitar. This was easy, but did not indicate in any way a technical progress.

After several weeks the visual structures were spontaneously transformed into auditive structures by playing the guitar; that is to say the sounds were no longer filtered as before and the third structure was slowly merging with the two others. It naturally still remains to be seen how this fusion is going to change itself spontaneously into motor and visual structures which could not have been found earlier since the sounds and the signs, or the sounds and the notes of the instrument, had not been consciously associated.

Clearly a host of other problems is contained in this same situation, but for our present needs the above will suffice. In another publication I shall try to give more elaborate details of the analysis of the visual, motor and musical structures I have acquired.

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1 Remarks on Mental Structures

Conclusion The above observations are of a purely experimental character and made me realize that very little has been written on mental structures. However they are interesting insofar as they show that mental structures can be systematically used in order to describe, from the psychological point of view, a certain number of mental phenomena on which we have little data. Having been led by the same research into widely different fields and over a fairly long period of time, mental structures will bring, I hope, a series of confirmations for those readers whose work has had a bearing on one or more of the fields of somatic, linguistic or musical structures.

For the educationalist, my study can provide a new reason for tackling problems of teaching from the angle of the acquisition of mental structures.

In another work I shall present a group of experiments made in order to bring into the open fundamental mental structures which intervene in the questions of mathematics at different levels of teaching. Here I can only say that the mental structure as a descriptive tool is equally appropriate to the analysis of abstract thought as to activity in general, and that there is no discontinuity in the apprenticeship to mathematical notions whether purely or partially abstract. The phenomenon studied in myself with the guitar and the structures it entails is repeated thousands of times in all the schools of the world where notions are learnt too early and where we do not expect, as in my case, consciously to see the merging of scattered structures effected.

29

Part III Psychological Discussions

We let our pupils advance blindly accusing them of incapacity or of lack of skill when the battery of structures has not been organized as a whole to allow of mastery and creativity.

It would perhaps be wiser to take a second look here.

The help the psychologist gives to the educationalist could well be that he will indicate to him not only the structures present in the mind of the pupil at the level at which he considers the pupil, but also the evolution of the structures. In this connection, I have brought together considerably important material particularly in relation to mathematical structures. Going through this will raise new problems of research and it is principally with the intention of attracting attention to these questions that these short monographs are published which I consider rather as probes than as expositions of a doctrine.

30

2 A New Theory of the Image

This article first appeared as Appendix 3 in Introduction à la Psychologie de l’Affectivité et à l’Education à l’Amour, Delachaux et Niestlé, Neuchâtel and Paris, 1952.

1 In order to provide an explanation of the progressive development of the drawings of a child I had recourse to various known theories of the image but found that each of these was lacking in some point or another. I was thus obliged to consider a new theory which I shall outline in the following pages.

In an article which appeared in Enfance,† I made a summary of facts relevant to this topic collected over a number of years. I shall not return to these facts here, but it was in attempting to take them into account that in the first instance this theory was elaborated.

In order to clarify my position right from the start I make the point that, both in my own collection of children’s drawings and

†

Psychologie du Dessin Enfantin, Enfance, No. 5, Nov.–Dec. 1948.

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Part III Psychological Discussions

in those of various art galleries, I found individual factors dependent on age and ability, and social and historic factors dependent on the social environment and the particular historic time. I have endeavored to incorporate these two elements into my theory and have also taken account of what I shall term the universal development of the view of reality. In my opinion, in fact, it is impossible truly to understand the progressive development of the drawings of a single child if the whole of “Reality” and what we understand by “external world” are not taken into consideration.

2 We know that the theory of the eye as a photographic instrument is a trivial one providing no explanation either of optical illusions, or of the vision achieved in daydreams, or of child art. Something more than the merely optic must be introduced and thus one has often had recourse to a perceptive activity or other dynamic entity.

Most modern psychologists dislike the concept of energy and avoid speaking about it; it has, however, always appeared to me that, at the crucial points in the various psychological theories proposed, this notion had been tacitly accepted, and moreover that, as if by a kind of conjuring trick, it even served to straighten out the situation. But I am sufficiently ingenuous to bring my fundamental hypothesis out into the open.

I shall start with the idea that all sense organs are extensions of the self* by means of which the self is objectified and creates for

*

Cf. J. E. Marcault and T. Brosse, L’Education de Demain, Alcan, Paris, 1939.

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2 A New Theory of the Image

itself substitutes for “reality”.† The substitutes are images which are therefore nothing but structured energy, whereas the self is pure energy. Child art, or anything else of an evolving nature produced by the child, is a sign of the progressive structuration of images described below.

3 At the moment when the self projects a fraction of itself into any one of the sense organs, it is simply temporarily delegating energy to this organ which proceeds to activate it; when this occurs, the energy becomes structured at the same time as it acts on the organ to form functional structures of it.

There are, in fact, two kinds of structures, which must not be confused. From the biological point of view, structure of an organ signified specific cellular arrangement, spatial and geometric so to speak. But it is clear that this dead structure cannot suffice for the psychologist or the physiologist.

Structurally, the eye is the same both in animals and in man; however, it serves a different purpose in each, it is selective in various ways. Superimposed on the somatic structure there is a functional structure whose nature is, for example, such that the sight of a prairie does not evoke the same reactions in a carnivore as in a herbivore, which obliges one to conceive of human sense organs, provided by heredity, as not functionally structured prior to the production by experience of the desired effect. It will therefore be permissible in this sense to say—as I shall say without further explanation—that the energy of the self forms structures within these organs.

†

For the moment I use this word in its most vague and direct sense.

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The fact that we distinguish the image from the self as a whole by its structural character alone and not by its nature and that we maintain the association between the image and the self as a whole through its possession of energy enables me to take account of many facts not usually considered together. In particular, we shall see more clearly how emotions and obsessions affect images.

The self, having projected a fraction of itself into the organ can retract it and either ensure that it retains the structuration acquired (the energy becomes image), or recast this energy in the whole (the image disappears), or project the image into another sense organ.

It is this constant relationship between the various organs which explains the generally recognized links between perception and action, between movement of the eyes and sight, between the socalled oral, visual and auditory areas and so on, through the fact that, firstly, it is the same energy which activates them all and secondly, that the structured energy in the one will proceed to form structures in the other.

I do not conceive of any sense organ as absolute, but rather as an objectivated part of the self, kept in operation and nourished by the energy of images, and structured for an increasingly complex, living and definite function. For the psychologist, in fact, there is no difference in nature between images and sense organs; they both create themselves and form structures within themselves. Insofar as they are specialized departments of the self, the various organs are so structured as to mark the images

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2 A New Theory of the Image

in a certain way, but a pure auditory image (energy structured by the ear alone) cannot be conceived except with a “virgin” ear, or alternatively by a blind person deprived of the senses of taste, touch and smell.

The old adage which holds that “the function creates the organ” here acquires a new meaning. The function to make images devolves on an organ which is constructed and maintained in function by these images. No sensations at all are experienced before the self cedes a fraction of energy to the organs and therefore before it produces images.

4 Dimensional theory. Let us now consider a fraction of the self a and let nt be the number of times a has been projected into any sense organ at all up to the time t; this I shall call dimensions of a. We shall consider the pair (a, nt) which shall be called image at time t. This is clearly a function of public, or clock, time, and one can say that an image changes in time if for t´ > t, nt nt´; development occurs if nt´ > nt and regression if nt´ < nt. If nt = nt´ for all t´ > t, it can be said that the image has attained its final form.

What is important here is that the development of the whole image can be described in a uniform manner and that at the same time a can be assigned an individual meaning, n a social meaning and (a, n) a meaning both individual and social.

In fact, the self is the sole judge, which, on the basis both of individual experiences within its own physical structure and of pain, can decide the limit to assign to a; in addition, it is the

35

Part III Psychological Discussions

relative intensity of a with reference to the self as a whole on the one hand, and to the objectivated part in the actual body on the other hand which will render the images obsessional, distressing or insignificant. In my view, in fact, emotion is simply the quantity of energy contained in the images, or rather in every objectivated element, which may be a. Individual experience is, at first, within the edifice of the self* and the determination of the initial aâ&#x20AC;&#x2122;s in the first images is related to the capacity of the organs, measured by the threshold of pain in the organ. This gives internal and specialized sensitivity the means to provide the self with a field of projection for its activity of objectivation and would explain at the same time (1) why the first global image we have may be that of our own body and (2) why the images are usually in perfect harmony with these means and are not experienced as distinct from our self. In fact, we live in our images and in this sense there is no reality that is not human.

Two types of research I have undertaken are closely related to this concept of the body image. The one has a bearing on the often noted similarity between drawings from the imagination of human beings and the artist who produced the work, as these human beings appear to us â&#x20AC;&#x153;objectivelyâ&#x20AC;?; and the other bears on the relationship between the image proper on the one hand and the physical sympathies and the bases of taste in the plastic arts on the other. Although the results are still fragmentary, it appears certain that the body image plays a decisive part, before any education, in the formation of taste, and likewise in affections. In particular, it alone can explain the fact described

*

This is proved by the fact that myelinisation of the nerve fibres occurs at a certain stage in every life.

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2 A New Theory of the Image

in the article mentioned above of the two little four-year-old girls making drawings of themselves, for such knowledge of oneself would presuppose an intellectual skill of analysis inconceivable at that age.

5 n has a social meaning, in the sense that the image is not adequate for action except when n is composed of a number of units which is a function not only of the individual but also of other individuals.

Let us take an example. In French, the sound vin has at least four meanings accepted by society. If this sound corresponds to vingt, I cannot act as if it were concerned with vain or vint. Or in English, the sound or also has at least four meanings accepted by society. When this sound corresponds to ore, I cannot act as if it were concerned with oar or awe. It is therefore necessary to make a new structuration of a as often as necessary for the energy a to induce images adequate for the action about which the individual has nothing to say since he does not invent language himself. Social error will be the signal for the necessity for restructuration, in order that the new image may take into account the objectivation by other people. An individual will have to substitute (a,n´) for (a,n) so that his action may correspond to the content of (b,m) for another individual.

The choice of a was made on the plane of the self and of the body proper, and the choice of n on the plane of social action.

We have accepted as a postulate that the self objectivated itself in a body which limited its projection; we shall also accept that

37

Part III Psychological Discussions

the recognition by the self of limits to its activity is to be considered identical to giving oneself primitive data forming a primitive reality. The structuration of images in accordance with this primitive reality forms structures of this reality. Social structuration occurs at the same time as the structuration of natural reality occurs; as far as objectivation is concerned, both are, for the self, one and the same thing. This is why “reality” is grasped in an evolving and selective manner. It is the “reality” of a moment and a group: it is a kind of sum of images objectivated and structured by a group of “selfs”. Their adequacy is an evolving phenomenon and, in certain respects, conventional, in order to allow the survival of the species. The role of education in its widest sense is to create this alleged uniformity of convention which bears on n but never on a. The self can be made to recapture an experience but it cannot be forced to objectify what others have put in their images. (a,n) is both individual and social at the same time, it is the essential element of reality. Through its component a it introduces the individual, mental, energic and dynamic factor.

Through its component n the social, hereditary and static elements come in. In fact, it is not so simple as this, for the hereditary is made up of the individual socialized, and the individual, in acting on n, can force society to develop.

Nevertheless, this description allows us to tackle widely different problems and to construct from them the dimensional theory.

We have seen how the dimensions of an auditory experience are acquired in social action. In the same way various meanings of the same word form the framework of a language; these

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2 A New Theory of the Image

meanings, acquired by individual experience, are socialized and retransmitted to the individual through education.

These ideas could be expressed in neutral language in the following way: Let G be a group of individuals and A a collection of elements which by any given convention can be conceived of as belonging to all the members of G. To each of the elements a is associated a number Na which shall be called the number of social dimensions of a at time t, and the individuals of a group G´ G classed in the following way: 1

Those for whom (a,n) is such that n < Na; these individuals will be the ones who have not reached the level of the group G; minors, for example;

2 Those for whom (a,n) is such that n = Na; they are part of G; citizens, for example; 3 Those for whom (a,n) is such that n > Na; these are the innovators, the individuals who represent the future.

G´ does not necessarily comprise these three classes with regard to G. In the case of child art, A can be the class of elements occurring in a given object and their arrangement. For example, one’s face is composed of features, of relationships between these features, of positions, of expressions, of colorings and of various suggestions. If a is an individual’s face, Na can today be defined as the number of features, etc. . . . which a normal adult would associate with a face. Picasso would be classified in the

39

Part III Psychological Discussions

third category whilst children would be in the first. The development of child art would then be described in the following way: â&#x20AC;&#x153;Associated to each element a in art, a number of dimensions Na exists which the sequence of works of art successively explores, art, generally speaking, ceasing to develop once n = Naâ&#x20AC;?.

6 The notation would appear to be ambiguous since a first signified a quantity of energy and now signifies an element.

Many would be inclined to accept a dimensional theory of art without having recourse to the image but art nonetheless remains a production of the self and represents images. If we analyze the difference between my theory of the image and the theory of art I have just outlined, we can see that we socialized a, and instead of leaving it individual, we acknowledged the existence of conventions allowing A to be defined; in other words, a was made to lose its dynamic and mental character and for these a concept was substituted. In my opinion, certain psychological theories fall down in trying to socialize the whole of reality and thus are not able to attain the ultimate and dynamic entity, which I have succeeded, I hope, in preserving in my theory of the image. n contains the social aspect of the image which can also be found hidden in (a,n). The reality of art cannot be explained by assuming the uniformity of a, for the whole infinite range of quantities of a can be seen in the art of traumatic children and the whole series of values of n from O to Na and beyond in the drawings of normal or abnormal children and in ancient and

40

2 A New Theory of the Image

modern art. It is a pseudo-problem to require the identity of the aâ&#x20AC;&#x2122;s and the sole validity of my convention is that it can accept a vague principle of approximation as a criterion of identity. For example, one would have: (a,n) = (b,m) if the first individual recognized that his image allowed him an action adequate to that of the second, and vice versa. To claim the absolute identity of a and b in my opinion oversteps the bounds of science and imposes pseudo-problems; to claim the identity of n and m is possible experimentally. To define Na is only possible when a is a concept, that is to say, when a is equally socialized. This was the step taken in order to transform my theory of the image into an explanation of child art.

7 We must, however, return to our initial position if we want to take account of the role of emotion in images on the one hand, and the role of images in abstract thought on the other.

We have already insisted on the double individual and social process in the formation of images, and said that the role of the body proper is to define the scope of a whilst that of social action is to define the scope of n. Naturally, even in the case of social action, the projection of a is an individual phenomenon.

Let us take as an example the case of the newly born baby wanting to feed at its motherâ&#x20AC;&#x2122;s breast. It is limited by its means and its habitual movements, but the environment could be introduced by even the least traumatism, forcing a reconsideration on the babyâ&#x20AC;&#x2122;s part of an already strongly

41

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structured a. Now no longer is it n which expresses the influence of the environment but a which has ceased to be adequate for the present action. The traumatism acts centrally on the self and its objectivations. The quantity of energy to be objectivated is greater or less than the amount necessary in the image and through this the whole series already objectivated and structured in the form of affectivity.

Not all traumatisms have the same effect or the same importance. The most serious is that which produces a modification of the threshold of pain which as a consequence then acts on the totality of objectivated images and also induces a group of images (a,n) which can neither insert themselves into the self nor into the group. A complete variety of nervous behaviors, of disturbances of the central function, which are volitional, as well as the superficial disturbances of simple neuroses, enter here. It can be seen that the role of images is a double one: they are the symptomatic residue of a disordered objectivation-function and, since the accent is put on a instead of on n, they are an indication of the de-socialization which follows from the failure of the normal socialization function.

The drawings of difficult, abnormal and frail children show clearly this double interference of disturbance first on a, and then, as a result of this, on n. It could even be said that if a is not affected, n develops normally and consequently (a,n) is individually harmonious and socially adequate.

The images, being substitutes for reality and, in the case of traumatism, being inadequate for action, it follows that the

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2 A New Theory of the Image

individual attains a â&#x20AC;&#x153;view of realityâ&#x20AC;? accentuated in a certain way, where the image of the body extends itself to develop into affective radical egocentricity. Since we are all egocentric and the body-image is the first image, it is necessary to imagine the body as the equivalent, the social prolongation of the image, and to see in alienation and hallucinatory disturbances extreme examples of life within the image and the image becoming the entire reality of people who live it.* The role of alienation is to reduce n to its simplest expression and to act on a. The differences between normal vision and hallucinations are in the role of the usual objectivation function, the one acting on n with an adequate a, the other on a with an inadequate n.

The sleepwalker identifies reality absolutely with the group of his images which are perfectly adequate; this would be the extreme example, were one necessary to justify my idea of the image as a substitute for reality.* The role of emotion in the image is to represent there the central function and all its history. If the self is capable of controlled objectivations, its system of images will be characterized by the fact that each image will possess the greatest number of dimensions possible, and that, given a proper economy in the distribution of aâ&#x20AC;&#x2122;s, a much greater number of images will result, not one of which will be obsessional. On the contrary, in the case

* In my opinion, the works of Tzanck on biological awareness seem similar to the ideas expressed here.

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Part III Psychological Discussions

of the traumatized self, the one or the other disordered function will induce a multitude of possibilities of alienation.

Unity of images occurs in two ways: firstly, through the self, the aâ&#x20AC;&#x2122;s delegated by the self remaining an integral part of it, and secondly, through the social, according to the value of n, images then being classifiable as sensory, active, intellectual, and so on.

8 This point leads us to the last section of this work: the role of images in abstract thought.

An analogy would be very useful at this point. The richness of three-dimensional geometry compared to plane geometry and to that of the straight line is well known. There is little to be said about the straight line, while the plane is so rich. Symmetry about a straight line on one plane produces figures which are not super-imposable but which can be seen to be congruent, however, if one turns to three-dimensional space. The same thing occurs for images in our theory. If we accept that the images (a,n) are such that, for a same individual and for his various objectivations, the values of n are equal in a great many cases, then we shall be able to speak of levels. It is not necessary that all images should have the same number of dimensions, but those which have the same number are at the same level. Once a certain number of dimensions is reached, then only through an abstraction can the former image be found again. It is possible that the power of an image at a given level may not be comparable to the one it had when on the preceding level.

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2 A New Theory of the Image

What interests us here is the fact that when a new dimension is attained, the image becomes radically differentâ&#x20AC;&#x201D;and behaves in a different way in connection with the other images; some previously impossible identifications with a number of dimensions become possible when n has become n´.

The sensory and active image is the parent of the intellectual image, the latter being merely the former with the newly acquired dimensions. It is on the sensory-motor image that the intellectual image develops, but, whilst they may be very closely related, it nonetheless remains that the latter has characteristics not presented by the other, the specific result of its lesser number of dimensions.

Abstract thought has its origin in sensory-motor thought and must be supported by this. If it is desired that the image should lose its character of partial substitute for reality in order to become a more complete substitute, it is necessary to make it undergo a development which consists in the acquiring of dimensions appropriate to abstract thought.

The problem has two aspects. On the one hand, a question of psychology is involved, and on the other, a method of teaching.

The acquisition by the image of new dimensions renders previously impossible operations easy, and brings about the formation of images, in their new state, according to new laws.

The whole of Piagetâ&#x20AC;&#x2122;s experimental research confirms this thesis, even though it may be expressed in different terms. The development of thought and intelligence is also the development of images, and once these have acquired a number of

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dimensions socially recognized as adequate, Piaget speaks of a final equilibrium since this number, supposed equal for all images, allows their automatic composition. In my view the composition is always automatic, whatever the number of dimensions of the images, but that an image is socially adequate today only when it has attained a certain number of dimensions.

The composition of images, it is true, is more easily accomplished and longer lasting when images have the same number of dimensions; nevertheless it is not impossible, however difficult, among images having different dimensions.

This leads us to the second aspect of the question. In education, one has to deal with selfs which are in the process of giving to the aâ&#x20AC;&#x2122;s the structures necessary for action in the world; in other words, given images (a,n) having numbers of dimensions which vary with the aâ&#x20AC;&#x2122;s, we are concerned with helping the selfs acquire the dimensions necessary for free action in the adult environment.

As far as abstract thought is concerned, such as is met with, for example, in mathematics, the educational problem consists in starting from the dimensions actually present in the images and in introducing the new dimensions necessary for the freeing of the self for action. We shall develop this point elsewhere, but here we can give an idea of the method.

A child of eleven years of age, whose activity has not brought about the knowledge of abstract properties of geometric figures, has images corresponding to active reality and would much more readily be able to understand a dynamic pattern than the

46

2 A New Theory of the Image

conceptual dynamism of a theorem. We therefore introduce him to a class of figures having the same property, rather than to a figure representing that class. Instead of one circle, the family of concentric circles; instead of a tangent to a circle, straight lines moving in the plane and touching the circle at any one point on its circumference; instead of a parallelogram, all those which pivot on one side, and so on. At this age, perception is already structured at the active level, and by beginning with a dynamic pattern we enter into the number of controlled dimensions. One obvious consequence of our definition of the image is that, once the organs are functionally structured, every new image which is objectivated has a number of dimensions determined as a result of this fact.

A new conscious experience thus leads to a structured image which must be restructured with a greater number of dimensions. In the case of education for abstract thinking, it becomes necessary to avoid beginning with images having a greater number of dimensions, as is the case in the image of the adult as compared to that of pupils, but instead to increase the number of dimensions of active images. The whole of my technique designed to lead to a mastery of the abstract concept can be reduced to beginning with a dynamic family of active concepts and to discovering, in the dynamic image, the invariant element of displacement or of transformation, which creates a dynamic and abstract concept arising from the sensory-motor concept, embracing it, but irrevocably larger. The invariant element is the abstract concept; the conscious operation, which allows the isolation of this element from the dynamic pattern, is the objectivation of the intellectual image possessing all the dimensions of the active image and more besides. (a,n) is always 47

Part III Psychological Discussions

individual and social, but instead of beginning with a alone, one begins with (a,n´), n´ < n.

In fact, my theory holds that a does not intervene alone except in the first experiences of one’s life, those experiences which proceed to form functional structures of the organs on which, until then, no demands had been made; subsequently only energy already structured intervenes in the images.

Abstract thought makes use of images, but rather than the pure sensory images claimed by the contemporaries of Binet, these are intellectual images. Increasing the number of dimensions of an image radically transforms it, in the same way as a circle on a plane surface can belong equally to a cone, a cylinder, a sphere, or to any surface of revolution in three-dimensional space, the passage from two to three dimensions making the circle disappear in the surface.

Given its number of dimensions, the intellectual image follows different laws from those of the active image, but those of the latter can be found again in the former, though less flexible and less articulated, as Piaget has shown.

9 Conclusion. It has been my intention in these pages to outline a theory of the image as a substitute for social and historical reality and one which takes account of the facts, and, in particular, of the development of child art and the history of art as a double adaptive and creative process, both aspects of which result from the same basic reality: the existence of many selfs capable of objectivation.

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2 A New Theory of the Image

The few applications I have given illustrate the vast field that can be examined with this tool, and which I hope to return to later.

49

3 Mathematical and Mental Structures

This article first appeared in the Bulletin de lâ&#x20AC;&#x2122;Association des Professeurs de MathĂŠmatiques de lâ&#x20AC;&#x2122;Enseignement Public, Paris, March 1958.

Introduction In this article I propose to explore a field of research rather than to arrive at any final conclusions. What I am submitting here for the attention of readers will demand a great deal of investigation before it can attain a degree of certainty sufficient for it to serve as a basis for rational action in the sphere of education.

Nevertheless, it would appear possible that with the help of the method studied here more satisfactory results might be obtained and more light thrown on a path along which we have been guided until now by tradition and mere groping in the dark.

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As a first step, I note that the attitude towards mathematics has varied considerably from the very first. At a time when the principal preoccupation of men was their relation to one or several supernatural beings, mathematics was a divine invention, a mystery to be understood. In mathematics, excellent examples of pure ideas were to be found.

At a time when the attention of men was turned more towards nature and their relation to it, mathematics became the vehicle of expression of the content of nature under the form of its laws.

Mathematics was held to constitute an adequate language for the exploration of terrestrial and cosmic nature, in physics and astronomy at first, later in other natural sciences. Mathematics was the tributary of the natural sciences, despite its parallel development with these, helping them solve their difficulties and express their results. While geometry for Descartes may have been the model which all the other sciences should have striven to resemble, mechanics was in fact the source of the decisive progress of the 17th and 18th centuries. Newton and Leibnitz led the field with the differential and integral calculus: mathematics expressed the content of the universe, and its language was universal.

Once menâ&#x20AC;&#x2122;s attention concentrated on their mutual relationshipsâ&#x20AC;&#x201D;political, social and economicâ&#x20AC;&#x201D;mathematics abandoned the Euclidian absolute in order to found new geometries; the global point of view was dropped in order to adopt the Riemannian concept of the local; the idea of the definite in favor of that of the probable and the approximate; the idea of nature became a set of relationships propounded by the investigators and whose technical competence alone would 52

3 Mathematical and Mental Structures

permit to bring to a certain degree of completeness. Physical atomism required its own methods: intuition and rigor defined their scope.

But once man discovered himself at the center of the universe that he had made for himself, a crisis of foundations occurred in the field we are studying, that is to say in mathematics. The work of Cantor on infinity and sets, the attacks of the logicians, the formalization of mathematics, abstract spaces, and so on, led one more and more to the conviction that mathematics was a mental activity governed by its own laws, and that many spurious problems had been posed owing to unconscious prejudices, and out-and-out incomprehension.

This rapid sketch leads us to the fringe of the problem. If mathematics is no longer a contemplation of the absolute or of the divine, if it is not the language of physics, if it is not the means of communication between men (as social beings at a certain level of civilization) when they are dealing with their perception, their inventions, their mechanical, electrical and industrial organizations, if, on the contrary, it is rather a mental activity of men, busying themselves with it and being interested in it, then psychology has something to say on the subject which could perhaps be interesting.

Thus I am going to consider in turn the consequences of this awareness, its significance with regard to the relationship between mathematics and the other branches of human activity, and the direction in which an education aware of these data can serve the cause of better understanding of the present trend.

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1 Mathematics and Psychology Generally speaking psychologists are not mathematicians, although many psychologists learn a certain amount of mathematics and apply it to their research. As a result, rarely does a psychologist allow himself to regard mathematics as an activity. It is a tool which more often than not is beyond him, and on which he does not dare pass judgment.

Since the crisis of its foundations, mathematics itself has carried out an introspection such as no psychologist could do. This introspection is expressed in an esoteric language in the thousands of works published all over the world in the last sixty years or more.

This long crisis in mathematics has not prevented the less scrupulous (or even the more scrupulous in their less watchful moments) from pursuing their creative work. But in fact, among mathematicians, not one really believed mathematics had developed a malignant growth. The question was rather of detaching oneself from preconceived ideas, and that was as painful in mathematics as in any other walk of life. The crisis was a kind of collective neurosis brought about by the obsession of seeing clearly in this immense field with its diverse branches.

In particular, the repercussions of the formulation of paradoxes and of some new ideas of physics made it necessary to revise the content of mathematics which seemed specifically based on experience, and that content on which experience seemed to be based.

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The crisis (which some still believe is far from being resolved) can be summarized by describing it as the shock of the discovery, in terms that become increasingly precise, that mathematics is a creative activity of the human mind.

For the mathematician who is not a psychologist, yet who is entirely engrossed in his work, the feeling that there is a crisis of the foundations of his activity must be particularly disturbing.

Only the mathematicians capable of philosophizing have taken part in the study of the solution of this crisis. The history of this is too long to be adequately summarized here. It must suffice to say that the vigor of the tree makes the likelihood of its roots being rotten most improbable; the two principal tendencies have been, first, the search for firmer foundations for the whole edifice in one of its branches which was then to be examined in great detail (after analysis, the set of integers was, in fact, chosen); second, the seeking of help from disciplines of wider scope than mathematics. Logic was tried, but this was slowly absorbed into mathematics, changing its name to logistic, combinatory logic, theory of foundations, etc.

Psychology was also called on. But as no-one today knows what should be understood by the word ‘psychology’, whether a science in the laboratory, a non-exact science, a biological science, a science of the unconscious, of the emotions, and so on, the assistance expected varied with the particular ‘psychology’ solicited.

Here I am undertaking work of a pioneering nature, making use of a personal attitude towards the problem of human activities. I

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have elsewhere developed my concept of Psychology as a science of Time*. I shall not return to this point here. The main characteristic of mental activity is that it is temporal and that time is required for objectivation; time, under certain circumstances and through certain transformations can eventually become public time, or clock time, and mathematical thinking in particular, is a temporal activity. The feeling that thought really does structure itself once the subject is intellectually active, cannot be escaped, and it is on the certainty that my thought is both structured and structuring that the certainty of the correct foundation of mathematics rests.

The novelty of this attitude must be stressed. Despite the Cogito of Descartes, the mathematician arrives at certainty through his participation in constituted mathematics, to which he accords the firmness and vigor acquired in the social act of apprenticeship without there being any question (or even possibility of doubt) of the soundness of the foundation. Many mathematicians today, having tried all the â&#x20AC;&#x2DC;horizontalâ&#x20AC;&#x2122; outlets, have achieved certainty by recognizing as the sole foundations of their activity the very activity of their mathematical thought defined for them in its own terms.

Leibnitz had already begun the substitution of relationships for objects in metaphysical contemplation, but the task of mathematics of the nineteenth century was to familiarize the mind with the consideration of new abstract beings which were nothing more than relationships and for which algebras were

* Conscience de la Conscience, Paris 1954.

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created. Relationships appear more readily than objects as the creations of the mind. The works of Dedekind, Frege, Hilbert, and others at the turn of the century progressed even further in the direction of the abstract being and of a priori relationships, the source of understanding of structures of the chapters studied. This great leap forward cannot be emphasized too much. This was the step which was to provide the opportunity for understanding the true object of mathematics and to reduce the crisis of the foundations to a psychological problem.

If parallel chapters of mathematics exist, which stay separated only because the mathematician decided to call the abstract being one thing here, another thing there, the very nature of the mathematical activity required that these chapters be fused together and that the abstract structures they imply be studied.

Despite the fact that such a movement from the particular to the general only occurs under pressure, it is nevertheless not only a matter of economy but also of human necessity. Human communication in fact would be impossible without the substitution of the general for the particular, and were the particular only to be used.

Once one is brought to consider structures, it can be clearly seen that mathematical activity has never been anything but a study of specific relationships between specific beings and that these definitions are internally justified in the eyes of the mathematician through the nature of the activity itself. The preconceived notions of the investigators and their lack of interest in examining this kind of question has made them take what they were doing to be something else. Mathematics studies structures which it creates obeying the laws it imposes as well.

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That is to say that a mathematician regards as mathematics the activity where certain mental structures defined by him come into play relating to other mental structures which he borrows from other mathematicians.

To describe mathematics as an activity bearing on particular mental structures is to stress two things. On the one hand, it is using psychological language and consequently introducing a new preoccupation and, on the other, it is posing the problem in exclusively human terms. We have no other justification for doing this than the fertility of this new point of view.

When the mathematician goes right back to the point where he can pose his axioms of structures, he almost stops. It is well to attempt to put the structures in some sort of order, but his main function is to create something new, to objectivate unknown structures and to bring their properties into the open.

But the psychologist finds his task much simplified as a result of the contribution of this act of comprehension on the part of the mathematician. In fact, he sees that he can describe the whole of human reality in terms of mental structures, mathematics becoming a particular integrated activity and being substituted for this reality. He sees that human creativity cannot escape him any more than it can remain outside his scientific description.

He can see that the problem of the various sciences and of the natural and human universe is no new problem and also that he does not run the risk of being exclusively psychological in his analysis.

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The notion of structure in mathematics has provided the opportunity for an internal clarification of mathematics itself, a purification from which pseudo-problems were eliminated. It has similarly served to base mathematics on itself thus reducing it to an autonomous and organized mental activity obeying its own laws at all levels. It has been useful in giving a tool to psychology which, given the fact of its origin, is aware that this can be satisfactorily applied to mathematics (both existing, and in the process of developing) and can be used in the study of other problems. The notion of mental structure has therefore served the double purpose of understanding a fundamental intellectual activity of humanity, and of enriching science which must attempt to explain all human activities.

In psychology, the notion of mental structures is to be considered a primitive one.

2 Evolution of the Mind in the Apprenticeship to Mathematics and Other Sciences Now that man and his activity is to be considered rather than man in his dependence on a supernatural being (theology), or man in nature as an object of nature (physics, chemistry, biology), or in his relationship with the State (politics) or with his fellow men (sociology, anthropology), one must explain, on the one hand, the proliferation of his activities and, on the other, what happens to each individual in his mental evolution as it has been described by the various sciences.

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In the preceding section I considered the notion of mathematical structure as a mental structure. Have mental structures any equivalent in other human activities? It would seem easier for mathematics to be reduced to a mental activity than physics.

Let us note first of all that the question of reduction to a mental activity can be put in two ways. Are we concerned with reducing science or rather the whole of reality to a mental activity? The first question is simple and would not appear to raise the same objections as the second; science means knowledge and we cannot imagine this without a human activity of acquiring it, digesting it and communicating it. However, there is no denying that reality, given once and for all, is an idea difficult to accept.

It would seem rather that reality might unveil itself to us consequent on knowledge, on the activity which substitutes mental structures for reality, which come into play mentally. But I shall not insist on thisâ&#x20AC;&#x201D;indeed, I shall not develop the point any further here.

The real problem, it would seem, is to know whether the notion of mental structures can be used in order to explain the appearance of specialized structures about which the various sciences are concerned. If this were possible, the other question would surely be answered at the same time.

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In my article, Some Remarks on Mental Structures*, I spoke of somatic mental structures, indicating therein that I was considering any organizing of our organs with respect to any activity whatsoever as a mental structure, even if the somatic element predominated. After this article had been written, I found experimentally that the most evolved mental structures have a somatic support found all over the body of the subjects studied and invariant from one subject to another, account having been taken of alterations due to personality†. For my present purpose I need only ask the reader to accept the view that mental structures are somatic in the first instance, or rather, that the soma, in its embryological evolution, acquires all the functional organizations we call somatic mental structures and on which are grafted, or from which spring, all other mental structures.

In the accumulation of experience, man creates mental structures to replace past experience and to permit future experience increasingly detached from the somatic support. This organic development by stages, integrated in the self, is the mental experience the psychologist must describe.

Life at the stage of the embryo, indeed very early infancy and the rest of one’s life, must be taken into consideration. In another of my writings, Psychologie de la Pensée ou Psychologie de

* First appeared in Enfance, Paris, No. 3, May 1951, reproduced here as the first chapter in this part. † See Un Nouveau Phénomène Psychosomatique, C. Gattegno, A. Gay, Delachaux et Niestlé, 1952, Neuchâtel.

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l’Intelligence† I shall develop the details of this point. Here I accept that human experience follows a definite direction from the mental point of view (Piaget has demonstrated it from his point of view, and for intelligence) beginning with the initial somatic structures, becoming more complicated as mental structures embodying the primitive structures which are rendered more flexible and more comprehensive. It is from the characteristics of mental structures that the categories of reality will be formed, with the older structures acting as common support for the differentiated structures. It will be through the intermediary of the first structures that individual experience appears as a coherent entity despite the different relative development of the more recent structures. There is an element of the somatic structure in all subsequent structuration, but with the more highly developed structures (insofar as they are a function of time) it is possible that they will lead to only one type of structure which develops on itself, mathematical structures for example. Thus a child’s drawing reveals a sui generis evolution illustrating the manner in which images develop; in children’s games there can be found at first an active structuration which tends to become more abstract later in order to include mental structures of the mathematical type (crossword puzzles, chess, bridge, etc.) or of the type in physics or chemistry (playing with meccano sets, boats or experimenting with colors, etc.), or of the type of the biological sciences (various collections, classifications, etc.). It is quite clear therefore that specialization of mental structures is attached to

† This book has not yet appeared in print (1963).

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an interest of the individual, who finds in his environment, owing to the differing ages of the members of the group, experience objectivated as accessible structures. The multitude of possible combinations of accessible specializations takes the multitude of individual mental behaviors into account. Each individual can develop such and such a mental structure beyond its representation in the group and can add to the accessible structures. Accessibility, in fact, results from the nature of mental structures whose process of objectivation starts with other accessible structures.

This double movement of objectivation in the individual and of accessibility in the group acts in such a way that the life of the mind is simultaneously an intellectual and collective activity.

From the individual point of view the activity is creative, even when concerned with structures existing in the experience of other members of the group. From the collective point of view both codified structures and structures to be codified exist. The former comprise existing knowledge, the latter, science in the process of being developed. Clearly, it is an abuse of language to call them collective structures. They are present in the minds of a certain number of members of the group and we consider this as equivalent to being in the group.

Children today find accessible mental structures in the group which they integrate, or rather, whose equivalents they objectify.

The use of somatic structures in order to objectify certain corresponding mental structures takes place in the groupâ&#x20AC;&#x2122;s games, in the introduction to language, in the adoption of rites and usages and in the adaptation to particular geographic conditions. But apart from corresponding structures, there is the 63

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possibility of combining behaviors and of producing new constellations at variance with the environment. It is this variation which introduces the personal and temporal element.

Its pressure is creative in the group, by contrast with permanent individual creativity, but it can give up its place to a total conformist stratification.

Categories are formed in the constitution of structures supporting one another. If the individual who introduces the variation is imitated, the variation enters into the corresponding structures whose subsequent variations, proceeding from the imitators, constitute structured branches. Thus it is with music.

Each instrument has its own domain, each theme its variations, each combination its requirements and its possibilities.

Similarly with physics; each sense organ has its qualifications, every combination of senses its limitations and its contributions in the form of apparatus, and every instrumental conquest its relative field.

It is by eventually exhausting the capacity of the structures in question for extension and combination that the mind is able to grasp the mechanism of objectivation when it is in action. So long as objectivation is simple, there is a straddling between the various new objectivations and the automatic use of earlier structures; whence co-existence is derived, in all minds, of categories of all kinds, not necessarily at the same level, nor even necessarily coordinated.

To summarize, the fact that the same initial structures give rise to different categories is in no way self-contradictory if the point

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of view of creative objectivation, or of the variation on collective structures, is accepted. The categories of structures, by their differences the one from the other, form the branches of knowledge, but by the internal co-ordination of the individual, either in his spontaneous mental evolution or with his capacity for particular objectivation, make personal mental life an integrated entity and provide the link between the branches of knowledge. There is absolutely no need to appeal to a Reality, single and permanent.

Just as the human mind can produce mathematics, so it produces all the other categories of structures; its linking them together is sufficient for the applicability of mathematics to the physical, biological or social universes to be taken into account.

No transcendent reason is necessary for the explanation of the adequacy of mathematics to describe reality, but simply that our awareness of reality is achieved through the intermediary of mental structures having the same nature as mathematical structures. Psychology here retrieves from metaphysics yet another problem and provides an immediate and rational answer. It is sufficient to see in what way the introduction to the life of the mind effectively takes place in time, both for the individual and for the group, in order to grasp that the specialization of mental structures is one of the processes by which they are constituted and expanded.

In the next section I shall examine in what way certain mental structures among our pupils are transformed into mathematical structures. The same study for the other categories would enrich our comprehension of education in action.

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3 Experimentation in Progress The problem I have posed myself can be expressed in the following way. Pupils in our schools already make use of a number of mental structures: they can use their bodies and their sense organs to form functional structures and images. They have learnt how to speak, and can play a number of games. Later on, they will have a more or less profound social experience, they will know how to evoke situations and follow them in their imagination, assuring themselves of the plausibility of certain statements through a judgment involving their experience. It is with these facts that our work of specialization must begin.

It appears obvious that the basis of all teaching, of mathematics or other disciplines, must be formed upon existing structures.

Commencing with secondary school pupils, I supposed that the only acquired knowledge present was of perception and action, and I established a series of situations of increasing abstraction, in which later situations were nevertheless related to the earlier ones. The teachers had only one task: to assure themselves that the pupils switched their interest from one situation to another only after they had mastered the content of the first.

Here I summarize just one item in my pedagogical research concerned with geometry, which will suffice to give an idea of the method employed. It must be added that the method used is much more flexible than the example would suggest, and that it has been extended to the whole of the school syllabus and to a part of university mathematics teaching besides. Another important preliminary remark concerns my belief that the

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teaching of geometry to pupils need not follow a logical order similar to that of Euclid or of Legendre. In any case the results of the traditional teaching of geometry are hybrid: there are two opposing tendencies operatingâ&#x20AC;&#x201D;first we want to convey a sense of rigor, and secondly, we teach that propositions true under one particular set of circumstances are equally true in the whole of space. This conclusion could indeed only result from a treatment very different from that used at present for the subject matter of the syllabus.

The pupils each have a pair of compasses and draw a circle which encloses space; inside this they produce figures whose properties they can discover by simple inspection. The exploration of the circle takes place spontaneously in the first instance with the aid of the compasses; the figures obtained are then colored and are of a remarkable variety. The use of compasses alone provides the opportunity to establish familiarity with circular figuresâ&#x20AC;&#x201D;in particular: with symmetries with reference to the center; with concentric circles; circular lunes; with the fact that a certain number of divisions of the circle can be superimposed in some rotations, and so on. A remarkable fact, and a turning point in this spontaneous study, is the realization by the child that he can cross the boundary of the initial circle which had previously served as the whole of space on which to look at. We are thereby informed that he is ready to deepen his study, and we take advantage of this.

I then ask the pupils to describe on a sheet of paper two families of concentric circles, whose centers are not too far apart. If the radii increase in length, it can eventually be seen that, by taking pairs of equal circles, the points of intersection lie on a line. This 67

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is a striking experience and all the children studied (many hundreds over a period of five years), have been able to see the fact, to draw the straight line and make more or less correct statements about it. They can, with comparative ease, be made to discover that all points on this straight line are equidistant from the two centers and that every pair of new circles with the same radii in the family will provide points of intersection on it.

The ruler (the materialization of a straight line) is introduced at this point. It is now a useful instrument. It can be used not only in the case above, but also in joining the two centers.

Examination of the figure thus completed leads to new discoveries. The circles intersect if the radius of each is greater than half the distance between their centers. The straight line joining their centers is an axis of symmetry. The instance above, by construction, divides the straight line joining the centers into two equal parts; symmetry shows that the angles are equal at the point of intersection, folding, if necessary, will assure their superimposition; the words “right angle” and “perpendicular” are accepted.

From this beginning I return to the single circle and to the inscribed hexagon which was not possible to draw previously with the aid of the compasses alone, but whose vertices had, in fact, been found. The new figure and its diagonals provide a host of propositions and the introduction to various polygons (equilateral triangles, rhombi, isosceles trapezia, right-angled triangles) and to their respective equality which results from the group of rotations about the center. The abstraction in this set of each of the figures creates corresponding mental structures which gain their degree of freedom when the pupil experiments 68

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to discover whether his conclusions are independent of the initial circle, of the position of the first vertex of the hexagon on the circumference, of the rotation of the circle about its center, of the transposition of the circle on the sheet of paper, and of the displacement of the paper in space.

The next phase makes use of another circle centered at one vertex of the hexagon, and of the same radius. In this case one of the diagonals is the common chord. The problem is linked to the instance above, and an obvious extension is to fix one of the circles and to diminish or increase the radius of the other.

Results are obtained on the chords and the diameter (bisection, orthogonality, relative length as an increasing or decreasing function of the distance from the center, etc.). Moreover, the construction of a triangle and of its symmetrical triangles with reference to the sides, knowing the three sides, follows.

For the moment, vision and action have been all that was necessary in order to generate the knowledge thus thrown into relief. Its truth stems from the mental obviousness which perception and action provide. Leaving aside all the possible extensions of this method to a host of other geometrical facts, I give here two further extensions of existing mental structures in the direction of a deeper mathematical awareness, even though its rigor may appear, to certain readers, inadequate.

Pupils, even at eleven and twelve years of age, easily recognize that a multitude of curves can be drawn on a blackboard or on a sheet of paper. They recognize equally that both closed and open curves exist. A curve of the first type has the obvious property of

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enclosing a part of the paper within it. Having made sure that the curves used have been shaped in order to be sufficiently simple and to avoid their possessing embarrassing properties, pupils easily see that, having obtained a simple closed curve (C), any number of open curves can be drawn on the paper to form three classes according as they are all outside (C), or have points inside (C), or touch (C). This classification is purely topological, and the pupils, once they have overcome their initial surprise, can very soon state this and find it obvious. If the closed curve is replaced by a circle, and the open curves by straight lines, then by this consideration of a special case, still more can be said about the figure. In particular, the tangents to the circle and their metric properties follow.

The second extension is a variation on the first. All our illustrations make use of the intuitive content; here I make use of Dedekind cuts. Having been given a semi-circle and a halfline from the center, then by taking a point on the half-line relatively close to the center and joining it to the two ends of the diameter, we form an obtuse angle (almost flat); in choosing a point very far away on the half-line the corresponding angle obtained is practically equal to zero, hence visibly acute. All points beyond the former and on this side of the latter also possess their respective properties and we are thus able to form two classes of points on the half-line: those which provide an acute angle and whose vertex is to be found outside the semicircle, and those which provide an obtuse angle and are inside it.

The point on the circumference is neither inside nor outside and the angle formed is neither obtuse nor acute, but a right angle.

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Whilst remaining convinced that I shall have many mathematics teachers opposing these views, it seems essential that I should lay myself open to their criticism in the hope that certain of my colleagues in secondary schools will find a solution to their difficulties in my suggestions without feeling themselves ridiculed because they use childish methods. In fact, it seems to me that the sacrifice of rigor on our part is nothing, when I think that our pupils in any case do not appreciate it as we do, and that in our own thought, perception, action and intuition play a constant and creative part. In my treatment of these points of the syllabus I have retained the attitude of the mathematician who, by intellectual examination of a situation, deduces or abstracts from it what to him appears significant, and only then adds order, or what we call rigor, to it. The pupils molded by my methods are much more alert than those who study in the traditional way and are less inclined to feel that mathematics is an empirical science where everything is established by measurements.

Conclusion The three sections of this article are conspicuously different, the last being very elementary, the other two exceedingly general in character. The contrast is, above all, due to the fact that the field with which we have been concerned is enormous and the vocabulary used is still in the process of being developed. I consider it necessary to say that the psychologist can learn from the modern mathematician, and the mathematician from the modern psychologist. The final synthesis lies in the field of education where the evolution of thought is a necessary condition for success and where mathematics can acquire its full 71

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sense of mental activity both for the master and for the pupil.

Moreover there is here a new field to be explored for teachers who have not managed to dedicate themselves to mathematical creativity and who know that their first task is to get their pupils to surpass them, and to enter into the future well equipped and, if possible, not merely without neurosis, but even positively happy.

Teachers alone, in their daily work, can carry out this synthesis forced on us today. In examining the composition of their own work and of that of their pupils they will discover the truth about the psychology of thought.

This short analysis should be considered as a modest contribution to this immense task full of promise.

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This article first appeared as Chapter 1 in Le Matériel pour l’Enseignement des Mathématiques, Delachaux et Niestlé, Paris, 1958.

1 In every study of perception, questions of a specialized and difficult technical nature are soon encountered which, while certainly helping to clarify the fundamental concept, risk making us lose sight of its applications. Nevertheless, once we begin to study the problem of thought seriously, we recognize that we must know more about the subject of perception.

Confronted with these two opposing tendencies, I have chosen to present a synthesis which, though provisional, is in accordance with the facts, so that readers who are teachers of mathematics may find the assistance they expect from the psychologist.

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2 The first important observation to make concerns the conceptual difference between a static perception, which the philosopher claims to be the basis of thought, and perceptive activity, a view which, on the other hand, is gaining an increasing number of adherents among psychologists.

Vision is the most studied of all the various kinds of perception.

Numerous volumes have been written on this subject, but I am not concerned here with presenting the various conclusions reached by following a particular technique or various problematics. We nevertheless must note that, beginning with the eye, as a photographic chamber having optical functions, it has slowly been admitted first that a connection existed between the nervous system and the act of seeing, and later that the formation of images is a complex phenomenon making demands on all ocular muscles, on the affectivity of the subject, on the lessons learnt in the experiences of life, and on other mental exercises and sensory activities. The study of the education of vision among blind people operated on for removal of the cataract has attracted the attention of psychologists and physiopsychologists to certain aspects of this problem which they had apparently preferred to ignore previously. Evolution of vision (or of any other sensory function) is strictly associated with the mental development of the subject and leads different individuals to different stages in the collection of visual possibilities. We do not need other proof of this than the examples, first of perspective vision discovered so late in the history of mankind, and secondly of three-dimensional vision which so few adults even today manage to acquire in or out of school.

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Reflecting on what sensory experience provides us with, I have necessarily been led to ask myself whether “pure” experience on the sensory plane is ever possible (cf. A New Theory of the Image*). The answer to this is obvious: very soon after birth such pure experiences are no longer possible. Nevertheless some experiences do come very close to perception pure, although the vast majority evolve towards a complexus where it is very difficult to recognize the sensory components. Illusions which are optical, tactile, thermic, etc. present constancies which are altered neither by the intellectual knowledge that they are in fact illusions, nor by age, i.e. by one’s general experience.

On the whole in everyday life, there is in the case of each individual and according to his actual aptitudes, a series of experiences inseparable from each act of comprehension on the part of the senses, which transforms the message to be perceived. Here evidently, we are dealing with a complex perceptive activity, of which examples can be found at all levels of development and in every moment of one’s life, and which cause the mental image to be very far removed from what is called an optical image in instruments. The interference in the image of all activity of the mind gives rise to a transformation of the element perceived into a multidimensional picture.

As soon as the sensory organ is structured for a certain vision, it does not lose it again. Perspective, once it has been perceived, remains an attribute of vision. Spatial vision of plane figures, once understood, is always available. Once grasped, the

*

Reproduced in this part.

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structure of anaglyphs can be found again, the perception of relief requiring only a brief moment of re-adaption in order to be put into effect.

The aptitude of the sensory organs to perceive what is accessible only after experience, but which henceforth accompanies their usage, I have called â&#x20AC;&#x153;functional structureâ&#x20AC;?. It is possible for anatomical structure not to evolve; functional structure is essentially dependent on time and can be transformed through use and exercise.

From the fact that experience makes each image acquire new qualities which manifest themselves through a capacity to perceive what was not perceived before, I proposed to consider each image as a quantity of energy structured in the sensory organs and muscles and which, in its turn, acted on the organs to give them this functional structure, consonant with its own structure. Structured energy, but retained in the whole of the mind by many functional bonds, the image possesses a dynamism which makes it mobile in the life of the mind, and capable of being evoked at a simple call. It has acquired a total self-sufficiency, from the fact that it was structured by physiopsychological activity and therefore it can substitute for reality.

We work on images before we work on reality by means of an internal action, which though virtual, is adequate, since it can be transformed into effective action.

3 The act of comprehension of perceptive dynamism shows that sight is the result of optical as much as muscular actions,

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and the synthesis demands that the object seen must be scanned, taken in, and the work both of the optical muscles and those of the body be integrated in this unity which is visual perception. But the reverse also applies; all mobilization of energy in the muscles occasioned by an action is estimated by some sensory organ. The various movements of mental energy taking place “inside the mind”, actions and perceptions, are all intimately linked and naturally develop to a high degree of perfection. No longer is there action without perception, nor perception without action, although it may be possible to recognize internally that the structured nature of energy comprises more alterations in any given sensory organ, which is a specialized extension of the self.

On every occasion when the musician reads music, plays his instrument, and produces sounds which he controls and uses as a criterion of correctness, he participates in the audio-visualactive synthesis. But even these three words thus joined together are inadequate to convey the idea of the true synthesis which forms complex musical images, where sentiments, the intellect, memories, and hopes or ambitions are merged to produce this highly structured whole which the hearer perceives.

In a book written in collaboration with A. Gay., Un Nouveau Phénomène Psychosomatique (Delachaux et Niestlé, 1952), I showed how it was possible, with the aid of a gayograph, to follow the vicissitudes in time of mathematical thought, in collecting certain somatic components. It appears to me that this laboratory work has definitely proven that thought has extensions in the whole of our soma, and the specific nature of mental registration assures me that the evolution of thought 77

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corresponds to the reduction of action and perception to automatic processes, yet without making these inaccessible to the self investigating its awareness. Every mathematical thought clearly makes use of images (and thus of perception and action), and each mathematical operation retains in itself the traces of its active origin.

But if this link between action and thought is obvious, it is nonetheless obvious that actions require much more energy than thought and, in addition, require much more time. What has happened between the action which made demands on all our muscles, and the intuitive thought of the mathematician which allows him to understand in a moment a complicated theory with which he was previously unacquainted?

As I have dealt with this question in detail elsewhere*, I can only give a brief answer to it here.

4 The whole of mental activity is internal, hence images and thoughts are not distinguished by virtue of their internal character. It well seems that action calls in an external world on which it is to act, that thought has only virtual recourse to this external world, and that there is a fundamental difference between thought and action only to the degree of oneâ&#x20AC;&#x2122;s interest in this aspect of the problem. But if what we are examining is the triad action-image-thought, it is evident that the mind experiences them simultaneously and distinguishes between them, but passes from the one to the other, recognizing their

*

Cf. Mathematical and Mental Structures, in this part.

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equivalence in that respect in all the judgments it has to make.

This occurs at all ages, even right from the start.

At the same time as actions are carried out, there are acts of comprehension on the virtual level, and formations in the interior of the whole being, of co-ordinations which we call images. From the fact that these images work in symbols in order to transform future actions, and allow economy of action, they are therefore, effectively thoughts. Intellectual activity is not detached from all other activity except in the case of the schizophrenic (and again one can note that he is adult, or at least of a certain age). It is not therefore only at a given stage that it can be said that thought takes place.

Many experiments first with children from 4 to 6 years old, and later with still younger ones, have shown me that if we do not intervene with a definition, in the study of the processes, children must be accorded a strongly intellectual and symbolic multivalent thought.

In fact, as far as requirements are concerned, that is to say the different demands made by the environment (here including the experimenter), the child develops a kind of symbolism of symbolism or acceptance of making use of a substitute for substitutes of reality and of drawing various conclusions from this about the properties of reality. This, moreover, does not stop at the second stage. There is occasion to recognize that abstract thought can be developed well beyond what we have taken for the norm once we have accepted that the child does not think in a certain way before a certain age unless a prodigy.

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It is the use of the symbol and its hierarchies which open up for the child the increasingly vast domains of intellectual life.

This I repeat: the preconceived idea that there are definite stages in the evolution of thought could be a cause of a totally erroneous view of the true nature of intellectual activity of the individual demanded by, and appropriate to, the environment.

The historical development of culture, if it has something new to bring to our understanding of the present moment, may be entirely foreign to what can, or could be, done by the mind, on which demands of a new kind, unforeseen by the prior experience of the group, have been made. An over-rigid determinism burdened by a rather sentimental historicism risk our being kept unaware of entire continents potentially present in our mental universe. For example, the acquisition of mathematical notions in the historical order is a strait-jacket which can prevent the mind from showing us how it freely functions. All, or nearly all, we know about the child embarking on the study of mathematics is of this nature, and perhaps, given the fact of the projection of our own minds, we are assuming that a deformed version of what is effectively possible is in fact the reality.

5 What then are we to call mathematical thought if we do not consider it in its historic context?

On a number of occasions already I have identified mathematical activity as that attitude of mind which, in the situation in which it participates, allows the mind to become particularly aware of relationships per se. This mathematical

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activity is such that, once related to social communication, historical mathematics is produced. Mathematical activity is more general than its results, retained, expressed and formalized in accordance with the codes that rule in the congregation. It is as part of a group that mathematicians see themselves, although they may not always be able to understand all the formalized results. What happens in the group, since it is conscious of itself, is clearly and internally distinguished from the other codified human activities.

But if mathematical activity is used as the criterion, it is possible to see it at work well before its advanced formalization, in its abstraction from life situations which present themselves in number to every child. That it may not be formalized does not detract at all from the matter. My favorite example is that of the structure of a staircase. To climb it is a pure action: it leads where one wants to go. But one can become conscious of what is involved in the act of climbing it without actually going through the motions. The muscular contractions become symbolic, merely outlined. The possibility will be envisaged of taking bigger steps involving two or more stairs at a time; or of going backwards; or of turning round and coming down again. Each project will make use of virtual actions. But these enter into another much experienced and general pattern: the repetition of movements. A stroke of imagination is all that is required for the staircase to continue indefinitely both upwards and downwards such that the symbols gain in extension and freedom with reference to the situation being experienced. The more hypothetical the concept of the staircase becomes, the more will its symbolism be of a higher degree, and what can be extracted from this situation will be of a pure relational type. If emotions 81

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and confusion do not become dominant in this virtual activity, the mind will then be clearly less engaged in a pre-mathematical dialogue. Instead of a real staircase, to see a staircase whose component steps have the consistency of images, whose height can be varied and whose number can be indefinitely multiplied; to be able to vary the number of steps taken per jump: is not such a staircase a preview of all arithmetical progressions?

The virtual actions extend the real actions and remain linked to these in such a way that mathematical activity can still be seen to be really involved in the situation, acts of comprehension then taking place concerning the relationships to be found there. This situation is rendered symbolic by a process of the same nature as that used in the formation of images and is therefore fundamentally biological.

Mathematical activity appears essentially primitive. But less primitive however, is its substitute in the social context because formalization is a second stage where another dialogue makes its appearance. There two individuals must compare their personal activities and make them compatible with each other for collective action to be carried out. From this arises the added demand of communication which, today, is found in the formalist guise of mathematics, completely hiding the original naked reality.

6 To perceive and to act are characteristics of all animals.

To become conscious of this activity is the attribute of man, and when the child becomes aware of action being quantitatively reduced, he recognizes in this its symbolic nature. Once he has

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discovered that he can use resultants of actions, it is no longer necessary for him to repeat acts which are combined. There is therefore a symbolism, both active and sensory, entering into the intellectual activity, of which it is one of the elements.

Formalization will also be active. Games for very small children are designed for the transcendence of action and eventually for their own replacement by symbols both readily recognizable and mentally useable. As these games complete their symbolizing function, the child discards them in turn.

For example, if an infant of two, or even less, classifies counters or buttons according to a standard it has chosen, then their equivalence, once it has been recognized, can be made use of (which is a mental structure) in order to solve problems it has posed itself, such as pushing counters through a slot or giving them to an adult, who only accepts one criterion of equivalence, in exchange for an object the child finds attractive, and so on.

Once the very young are presented with the Cuisenaire, given the fact of the relational wealth of these, they very soon become aware of a symbolism of the second degree, which is already strongly mathematized. For example, a number of four year old children, after a few weeks of playing with the rods, were able to answer the following question: A light green rod (3 cm.) was placed across an orange rod (10 cm.) and a pink one (4 cm.) across a blue one (9 cm.), and the children were asked to arrange orange rods side by side in order to have enough such that the light green would be neither too long nor too short, and the same with the blue rods, to have a width equal to the length of the pink rod; (3 orange and 4 blue rods were necessary); the rods were then placed in such a way that they formed two 83

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trains—3 orange rods end to end and 4 blue ones likewise. The second line (36 cm.) was visibly longer than the first (30 cm.).

Returning to the two original arrangements, counting the rods and replacing the result by rods crossed, first, the light green across the orange and secondly, the pink across the blue. Then the question was put verbally: “If we ‘fill up’ the light green rod with orange rods, and the pink one with blue ones, which will have the longer line?” The children, (not all of them, naturally, but as every day passed the number increased) replied “that one”, indicating the blue-pink cross. If the light green and the pink were changed over and the question asked again, the same children would make no mistake, giving the pink-orange cross.

Similarly, if the light green were replaced by the pink, and the pink by the yellow (5 cm.), there would again be no mistakes.

The symbol of the cross was being employed symbolically.

Here perception was evidently integrated into an intellectual pattern, mastered and sufficiently articulated to solve a complicated problem without the use of numbers. The child thought like an adult. Even if he had not at this stage acquired the words used for verbal formalization, there was nevertheless complete formalization at the perceptive and active level. The child had been a mathematician insofar as he had made perfect use of a series of qualitative mathematical structures and relational criteria.

Can we ask for more? In repeating the same exercises, changing the different rods, the children perceived relationships at two subjacent stages and, from their dynamism, drew the conclusion that such and such a result would proceed from the action if in fact it were to be undertaken. After some time it ceased to be 84

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necessary to carry out the actions; their virtual substitute had been made dynamic, which is what we call thinking.

Children who have been introduced too early to the verbalization of situations not explored at the perceptive and active level will not have at their disposition these two dimensions which make intellectual dialogue possible. They will lack the realism which upholds the symbol and gives it the necessary dynamism to replace one act of comprehension by another.

7 To act and to perceive are not luxuries for the mathematician. Without them there is no experience, neither mathematical nor of any other kind. With them, there is some hope that the dialogue one is searching for will become available, and become available at the most common level, thereby offering the maximum chance of success.

It is only when understanding is no longer achieved or alternatively when nothing is understood about mathematical activity itself, that at the outset a dialogue with ready-made rules or abstractions is proposed. A practice a very great number of the members of the teaching profession continue to adopt is to propose a verbal and authoritarian apprenticeship, thereby confessing a total lack of understanding of the act of learning.

We only need to think of the older pupils of 16 and 17 who have not managed to acquire three-dimensional vision and who are asked to produce three-dimensional theorems of Euclidian geometry for their baccalaureate! or think of 6 and 7 year old

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children who are made to learn their multiplication tables by heart! or think of the 12 and 13 year-olds who have not managed to understand their ‘fundamental identities’ in algebra and trigonometry but must recite them all the same! Every one of our readers will recognize that in a certain way these remarks are addressed to him.

Mathematical activity, which must be experienced in order to be relived and which is the sole activity which can be the instigator of development since it transforms each person into a mathematician, has as one of its aspects the participation in situations. In these situations are to be found the problems, the theorems, in fact the relationships worthy of being brought out and communicated. Once recognized and stated, they can be established, or, in other words, then and only then the mind struggles to prove to others that it has not been deluded and that what it claims to have found in the situation is, in fact, truly there. Therein lies the source of the need for proof. After some time this need becomes a natural requirement; even before communicating with a concrete “somebody else”, one addresses oneself to a virtual “somebody else” who acts as interlocutor.

These two aspects of mathematical activity (acts of comprehension and their communication) become the attributes of the mathematician socially recognized as such. To insist on the one and not on the other is to see only one part of what takes place in reality in the field of mathematical production. Any article which contains neither a new result nor a clearer proof is not publishable; the most highly rated papers are those where fertile situations are to be found and where the author displays insight (an act of comprehension of relationships) and shows that he has effectively perceived what he is stating.

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8 Any pedagogy cannot be realistic if it does not take into account the education of the natural aptitude to enter into a dialogue with situations in order to extract relationships from them. For that, pupils must exercise their relational sense both from the very beginning and also continuously thereafter.

Abstraction must correspond to the hierarchy of substitutes, of symbols, and of symbols of symbols. If well conducted, this abstraction does not constitute a great obstacle to the pupils. All that it requires is that the teacher should recognize that it is necessary to act and perceive in order to become at all apt at making statements.

In the following chapters* the reader will come across sufficient data for him to avoid any trouble whatsoever in providing concrete examples for the propositions which, in this chapter, may have appeared too abstract for him.

In meditating on his own difficulties experienced in the study of a psychological chapter, he will perhaps become more tolerant towards his pupils who study a field no less arduous.

* Cf. Le Matériel pour l’Enseignement des Mathématiques, Delachaux et Niestlé, Neuchâtel et Paris, 1958.

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5 Mathematical Thinking and the Use of the Senses

This article, written by the author when Chairman and Director of Studies of the Association for Teaching Aids in Mathematics, first appeared in the first Bulletin, January 1953.

It is easily assumed that mathematics is concerned with abstract beings and that mathematical thinking is unrelated to reality; it is, however, also generally assumed that all that is in the mind comes through the senses. How are these two views to be reconciled?

Frequently it is those things that are the most obvious that escape our attention, and it seems to be so in this case. We are concerned here with a manifold reality, yet we use only one way of approach to it. The reality is formed of a variety of minds at different levels of experience (children and adults), of a set of mathematical structures interwoven in the various branches of mathematics, and of experiences which have evolved in a variety of ways. If then our approach is only from the angle of what

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constitutes the content of the adult mathematicianâ&#x20AC;&#x2122;s intellect, there is little wonder that we find ourselves in an impasse.

What is required is an understanding of the mind, both of the child and of the adult, at work on its experiences in such a way that it produces mathematics. The fact that mathematics has developed historically should make us realize that it is in time that the mind creates mathematical structures and that our perspective must be a temporal one.

The basis of our knowledge is the organization of our experience at its various successive levels. There are as many levels of abstraction as there are levels of experience. A pin, for instance, picked up from a surface on which there are a number of things, has to be abstracted; an angle has to be abstracted from a geometrical figure having many features; the operation to be used in the solution of an algebraic situation, or the precise word to express our exact meaning must be abstracted in our minds.

It is too commonly supposed that there is only one level of abstraction, that of the mathematical thinker. Even among mathematicians it is possible to find a whole series of graded abstractions such that papers written by some cannot be read by others as they present them with insurmountable obstacles.

If this is the case, it would seem natural that once we know the degree of abstractness attained by a given mind we can present it with situations in which abstractions at that level are in operation. Statements can then be made which, though to some minds may seem to be hardly abstract and therefore hardly

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mathematical, are in fact as truly mathematical as those at other levels.

In this perspective of the co-existence of minds at varying levels of mental development it becomes evident that the teacherâ&#x20AC;&#x2122;s awareness of what can be considered mathematical at this or that level is of supreme importance. We must, as teachers, discard the unfounded belief that a certain given level is solely and truly mathematical. It would not be difficult to find mathematical papers, whose difficulty is not of a technical nature, which would be quite beyond the grasp of many who hold this belief.

In the light of this awareness, the significance of teaching aids, and the part they can play, can be better understood.

Models, filmstrips and films are obviously made for a definite purpose which is, in general, to demonstrate a proposition or an organization or a sequence of changing facts. First of all therefore, there must be a choice of the point to be demonstrated, and this in itself is an abstraction. The manipulation of the model, the projection of the filmstrip or film introduces a being with which it is proposed to make the student familiar. The model embodies the thought of its inventor, of which it is an abstraction. The film selects from among a multitude of happenings those that will lead to the conviction that a given abstract fact about certain abstract figures has an abstract existence, through recognition of its existence in the situation contained in the filmâ&#x20AC;&#x201D;as, for example, in Nicoletâ&#x20AC;&#x2122;s film showing that a right-angled triangle results from the association

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of one side of a regular pentagon with one side of a regular hexagon and one side of a regular decagon.

Sensory aids are the molding of material by ideas so that the reverse process, the discovery of the idea through manipulation of the material, may take place. Thinking is essentially a mental process, and if the material is an aid to thinking, as experience has shown that it can be, then it is in certain respects a real substitute for discussion or suggestion in initiating a thought process.

It is important, however, not to conclude from this that any aid will be the most adequate for a given purpose. The making of suitable aids is possible only when the constructor has clarified his ideas and discovered how much of his own thought is susceptible of being embodied in a given material and thus producing the required idea. If, for instance, a sheet of paper is the material, by being rolled it can produce a cylinder or a conic surface, and by being so thoroughly crumpled that no fraction of a line remains on it, it can suggest a non-analytical developable surface. This last example was suggested by Lebesgueâ&#x20AC;&#x2122;s degree of abstraction. An action on a given material can produce an aid, but the action must be purposeful, and its purpose seen in relation to an idea exemplified in the material. Soap bubbles, for example, might be used, but would not suggest themselves as a permanent aid owing to their labile nature. Nor would ripples on the surface of water, whereas a photograph of them might do so.

Films are suitable for the discovery of a theorem involving a dynamic pattern whose parts can be photographed, but are not adequate to enable an idea whose basis is not pictorial to be grasped intuitively. Although algebra is as much concerned as 92

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geometry with the dynamics of the mind, no film has as yet been produced as an aid to its teaching. Filmstrips satisfy a different need and suggest themselves as the proper medium when the ideas in question require discussion based on a finite number of instances. Models of various kinds provide manipulation that will lead to a concept or an evaluation.

The Association for Teaching Aids in Mathematics* has as its aim the investigation of the part that can be played by each of the various types of aid in helping the mind to reach its conclusions, and the means by which, in each particular case, fuller understanding and greater readiness to apply the ideas involved can best be achieved.

1 Existing material will be analyzed in relation to what it claims to achieve, and the means by which it does so. Then the ideas it deals with will be examined in relation to existing aids in order to discover whether the present material is the best available. Experiments will be undertaken in various schools with a view to assessing any proposed modifications. When general agreement has been reached, the Committee of the Association will consider the question of sales and distribution.

* Now (1963) known as Association of Teachers of Mathematics (A.T.M.)

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2 New materials will be produced on behalf of the Association on co-operative terms, each producer or team of producers keeping the authorâ&#x20AC;&#x2122;s rights. The new material will be examined in the same way as existing material, preparatory to production and distribution.

The Committee is prepared to consider suggestions concerning both new and existing aids from individual members of the Association.

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6 Adolescent Thought and its Bearing on Mathematics Learning

This article first appeared in Brussels in Revue Belge de Psychologie et de PĂŠdagogie, 1951, Vol. 13, Nos. 55â&#x20AC;&#x201C;56.

A Method of Investigation in Schools In my professional life, it has been my duty constantly to consider the various aspects of adolescent thought as it manifests itself at school. In the following pages I submit for examination by psychologists and teachers a method of enquiry which has been tested over a period of four years in at least a hundred London secondary classes. This method recommends itself to me (1) insofar as it eliminates at least one of the factors which make methods of testing and individual interviewing so artificial, (2) because it provides material which is immediately useable and comprehensible, and which can be interpreted at once both correctly and without any doubt whatsoever.

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Experienced teachers having an interest in their pupils could easily accumulate through this method a mass of evidence such that decisive progress could be made. Furthermore, since I am concerned with the improvement of teaching (of mathematics in particular) by making it benefit from the contributions psychologists can make, I now submit to the teaching public part of the research I am at present undertaking.

Although we all agree that teaching in classes of 30 to 40 pupils may be far from the best means of disseminating existing culture, we cannot escape the fact that this practice is the one most commonly found today. Pupils come to consider the classroom as a real and normal situation and create for themselves a series of habits where a degree of spontaneity is possible, and also perhaps, where a true state of the adolescent (or of the child in general) is revealed. To me it seemed possible to benefit from this opportunity by carrying out research on adolescent thought, which is a conscious object of concern for teachers but who still leave to one side its affective and social aspects. Thus we see that from the very beginning pupils are found in a situation which does not worry them unduly, and which would worry them even less should their own teachers make use of the approach here advocated. The way of studying schoolchildren used here is not unlike the method a psychologist-teacher would adopt if he were using for his testing material the subject matter itself.

It is obviously not merely a case of observing how the children face up to the difficulties in the school syllabus, but rather a conscious experimentation, where the collection of questions asked and the manner of their asking have been varied and 96

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compared, and where the reactions of the pupils to the questions have been studied.

Here, I am not concerned with the statistical aspect of the research, which, whilst admittedly of great interest, does not seem as instructive to us at this moment. Nevertheless, this statistical work is now in progress.

To start with, I give below the protocol of an experiment to illustrate the method.

On this occasion I was dealing with a class of first year pupils in an east London girls secondary school. There were 35 pupils all about 12 years old, coming from a number of primary schools in the neighborhood after the 11-plus examination which includes an I.Q. assessment. This, in every case in the class in question, was over 105. It was a typical classroom and each pupil sat in her usual place. The sole change was that for 40 minutes I replaced their mistress. It was to be a mathematics lesson, making use of what the pupils already knew, in this case, a little arithmetic.

It would perhaps be best to add that it was a very cold day (â&#x20AC;&#x201D; 4°C), that it was freezing, and that the pupils had just finished their school dinner; also perhaps that the English I speak occasionally differs from current usage.

Having described the circumstances, I now outline the substance of the research. A triangle was drawn on the blackboard and the mid-point of each side was marked. As the 97

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pupils knew no geometry, it was merely a matter of making them exercise visual perception, but since I knew that all the facts of which we were to make use were valid in Euclidian geometry, their conviction was to be accepted as if it had been a certainty. By joining the mid-points of the sides we obtained new triangles, all equal. The class was asked how many triangles there were in the diagram. The uniform reply was 5 (one large one containing four smaller ones). Then, the median triangle was traced over with green chalk and the question asked: “What fraction of the white triangle is the green triangle?” All the girls, except for three, said , the three said . I first asked for an explanation from the ones who had said . They said that there were 5 triangles, therefore the one in the middle was a fifth. One for their answer explained that of the three who had given there were four triangles inside the big one, therefore the green triangle was a quarter of that one. After this explanation I asked: “How many think that it is a quarter?” There were several converts, but a good many still said .

The green triangle was subdivided into four by the same process and the inner triangle drawn in red chalk. The pupils laughed, and were perfectly confident, relaxed and even joyful.

I asked again: “What fraction of the white triangle is the red triangle?” The answers were , , , , and one of the three exceptions above said . To the question: “How many triangles are there in the diagram?”, the reply 8 was given by one pupil; the majority, forgetting the initial triangle, said 7. From this, a majority then decided on , and one on . What was remarkable was that the two other exceptions this time said . . The one who had decided on gave her One said 98

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explanation first, but not a single one changed her opinion. The one who had given said perfectly logically: â&#x20AC;&#x153;The red is a quarter of the green, and the green a quarter of the white, of the white.â&#x20AC;? Two girls changed their therefore the red is minds on hearing this reasoning, but nobody else; those who had said maintained their opinion. I subdivided in red the three other triangles to the great joy of the pupils, and numbered the triangles 1, 2, 3, . . . 16, thus forming the sixteen triangles, with the pupils chanting the numbers as they were written down. After this operation, and when the diagram was before them, completed, I asked the initial question once again, concerning the central triangle in relation to the white triangle, the answers remaining the same as before, the only exception who now accepted for her being the girl who had given answer.

The red triangle at the center was divided into four in the same way, the median triangle being marked in yellow chalk. The same questions were asked. This time the replies were as follows: the majority , some , , and two, one of whom was the girl who had consistently given the correct answers, said . to The same analysis followed and some changed over from once our little mathematician had again explained that the yellow triangle was a quarter of the red one, and the red one of the white one, so the yellow one must be of the white, obtained by multiplying 16 by 4.

The yellow triangle was subdivided into four, and the median triangle drawn in blue chalk. This time there were only two answers: and with the single exception of the little . Everybody understood the blue mathematician who said 99

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triangle was a quarter of the yellow, and the yellow of the white, but even so, 64 was doubled in many cases, and our little logician experienced the humiliation of having made a mistake in carrying over which, moreover, she did not correct straight away, under the impression that she was always going to be right, not accepting her error until 4 64 was carried out aloud by the class, and I had written the result on the blackboard.

One interesting incident was that when so many said , I proposed that the result be decided by a majority vote. This was unanimously rejected, the reason given being that their general agreement did not mean in the least that the answer was correct, and that it could well be that the girl who alone had thought it was was in fact right.

To end the lesson, once everyone seemed to have arrived at the right answer, I drew a series of triangles on the board next to the diagram used up to this point. These were the same triangles as before but arranged successively in their respective positions, all drawn side by side in white chalk. The pupils laughed a great deal to see them alternating their peaks and becoming smaller and smaller. By referring to the original diagram they immediately discovered that there were four of one size to the one of the preceding size, and realized that by inverting the order, one triangle was a quarter of the preceding triangle, that there were 16 of the third size in the first and that one of these was consequently of it.

It became apparent, at the end of the customary duration of the lesson, that the answers permitted me to believe that the

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difficulties had been overcome spontaneously, without other intervention on my part than the stimulation of their thought.

I could cite a great number of experiments like this, each successively more edifying than the previous one, all making use of the material of the school syllabus alone, with classes of boys, girls, or mixed, and of ages varying between 11 and 17 years.

But for our present aim it will perhaps suffice summarily to analyze the illustration above asking the reader to study it, if possible with a similar group which may be accessible to him, and to publish his findings, different though they may well be from my own.

In the first place, the fact that we were intimately involved, and that our activity was always directed towards minds struggling with difficulties, evidently created an atmosphere which could be judged negatively for research. But does this method not make good the means lost in ordinary tests? Furthermore, is it not a well-known fact that for man no test is passively endured, and that there is always an apprenticeship in every adaptation to a test. Here, at last, the presence of an experimentor in an ordinary place (or even in a place of inspiration) and his affective-intellectual participation are immediately recognized in the experiment. Thought is dynamic. The child both learns and corrects himself during the test, but he surrenders himself entirely, and what he gives us seems to be very precious.

In fact, though he may have eventually corrected himself, he has never refused to tell us his true and spontaneous thoughts. He

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has handed his conviction over and has told us on what it was based. When I did not question the answers (which in many cases the reader will have already observed) it became possible to make a hypothesis which the outcome of the test either confirmed or rejected. Thus, I was able (remembering what L. Johannot said in his book on ‘Le raisonnement mathématique de l’adolescent’, Neuchâtel, 1947) to ask myself whether the child confused subtraction and division as arithmetic operations, and if this does not explain the fractions (for 1/(4 + 2), instead of the other error 1/(4 2) which would give ) (for 1/(7 2), instead of the other error 1/(7 4), and supposing that the initial illusion of , which I have not tried to eliminate, is combined with the correct operation which requires division by 4).

When I questioned a girl she showed a triple lever in her thought process.

First, with the single and constant exception of the girl who had mastered all that a given situation demanded of her, there was in each case a close combination of arithmetic and perceptive logic.

Then, with one exception (the girl who had said in the second case), having initially noticed the external triangle, and isolated it from the triangles contained within, the children were able to include it in their calculations on the first occasion, but thereafter no longer took it into consideration as a separate thought. If the child lets himself accept for a moment that he is concerned with making a comparison between the part and the

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whole, the whole is not placed on the same level as the part, even though they may be entities of the same nature. The diagram as such, except in the case of the one girl, inhibits the powers of reasoning from becoming general all at once because the attention is divided; but in the field of representation, we are conscious of the fact that the problem is not impossible, provided that the operations are clearly understood.

This, in fact, shows us the third lever, that of pure arithmetic.

Once the method had become formal, at the end of four insertions of triangles, the pupils were happy about their mental analysis, and the diagrams did not trouble them any more; they no longer seemed funny, except insofar as they were detailed and retained their initial nature, and moreover, their essence here being strictly arithmetical should have had the effect of making them lose their detailed aspect.

Naturally, we do not suppose that the children have completed their mental evolution. All we claim is that the treatment of the situation has brought them to rid themselves of thoughts tied to a purely perceptive situation, and led them to a position of compatibility, where the equal and the unequal are added together as homogenous units, and made them capable of grasping that the activity appropriate to the situation is of an arithmetic character, or, as Piaget has said, operational. Here the proof is formal. Not once was any girl told what she had to do. Not once was her thought conditioned. At most, it was freed by pushing the analysis beyond what the apparent obviousness imposed at the start, leading to the knowledge that the diagram, though cumbersome, was indispensable, and that the arithmetic of the situation was obscure.

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The presence of the little 11 year old mathematician was a great help in enabling me to judge whether the questions were of the appropriate level for pupils of that age, and for language to be used which got through to them. She was able to express, in a manner satisfactory to us both and to her comrades too, and in pure and abstract terms, what the situation involved, and once her classmates had heard her there was no doubt that they had understood. I therefore had recourse to her on every occasion when a proposition had to be explained which she understood just as well as I did, reducing by this means the pressure exerted on the childrenâ&#x20AC;&#x2122;s minds by the mathematical omniscience they granted me.

I shall now outline briefly some experiments whose results I consider to be of interest, and which, in part, confirm certain results obtained by Piaget concerning the study of space and spontaneous geometry.

1 When the above technique is used, the notions of direction, angle, area and volume give rise to the most unexpected results. For example, 35 boys of 11 to 12 years of age claimed that two of their comrades, walking along the two diagonals in the rectangle of the space between the front desks and the blackboard in their classroom, changed direction when they crossed, and that even if one of them moved out of the way, and the other walked along the same path he would still have changed his direction at the point where he would have met his comrade. This same group told me that if one turned round oneâ&#x20AC;&#x2122;s axis on oneâ&#x20AC;&#x2122;s path and then moved a very little distance in the new direction (while the initial movement was quite large), no angle was formed, whereas if the movement in the new 104

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direction was reasonably great, the angle was formed. Some girls of the same age, when tracing the route they take on going to and from school in the bus, provided angular diagrams, but denied that there were any angles on the journey: â&#x20AC;&#x153;There are only bendsâ&#x20AC;?. One of these even drew the route going from left to right, when the others had drawn it from right to left without anybody being unduly troubled by their so doing. Children are not conscious of the universality of representation but rather regard it as individual.

2 The study of difficulties met by adolescents in the case of geometrical figures where various parts overlap, with threedimensional vision, with the algebraic aspects of geometry and the geometrical aspects of algebra, give rise to a remarkable collection of data but with which we are not here concerned.

3 In a series of particularly fertile experiments, the notions of reversibility (of which Piaget has made a psychological study) and of algebraic operation were consciously introduced right from the beginning of the teaching of algebra. For example, I have often seen whole classes achieve comprehension and mastery in one hour of all that is implied by the solution of such literal expressions as:

=k

their initial equipment being only the arithmetic learnt at an English primary school. This particularly important section of 105

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my research will be the object of a detailed exposition to be published later.

4 In conclusion, I shall briefly mention the idea of the â&#x20AC;&#x153;dynamic patternâ&#x20AC;? in algebra and geometry, which is an excellent tool in the analysis of the formal intelligence of the adolescent and adult. It is concerned with the transformation of the formula accepted by the child as a unique and static form into the formula known to be dynamically generated by reversible operations combined in such a manner so as to arrive at this pattern. There are several problems which deserve the attention of the research worker, and whose pedagogical significance is, in my opinion, very great. We must recognize that there is a sizeable perceptive element in the act of thinking and that, for the adolescent trying to learn mathematics, the concepts have not yet acquired the infinity of the class of the component notions they represent. Thus, in the case of the circle, for example, the child has a concept of one particular circle, and for a long time, when attempting to understand mathematical situations, he is completely at a loss since the circle under consideration is not his own particular circle.

Whence comes our proposition of making use of a potential infinity of concentric circles in order to lead him to one circle representing a class, and from there to the idea studied. This illustrates the great utility (which has been widely tested) of the films introduced by Nicolet, which, were their use made more general, would considerably improve the teaching of mathematics to adolescents.

These references to my research in progress, vague as they will surely appear to many of the readers, do not presume to offer a 106

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solution to all the questions raised; rather they are designed to attract the attention of a good many people to the opportunity of contributing to the analysis of adolescent thought.

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7 Investigation Through Teaching

This article first appeared in Amsterdam in Nederlansch Tijdschrift voor de Psychologie, 1952, Vol. 7, No. 3.

1 The purpose of the following pages is to draw the attention of teachers of mathematics and psychologists to problems that can be investigated in the school. I shall in fact discuss means by which the ordinary teaching period can be used for the investigation of questions which may prove to be of supreme importance for the adaptation of our teaching to our pupilsâ&#x20AC;&#x2122; real experience and ability. As it is my intention to be brief and merely to start the ball rolling, I shall confine myself to indicating what can be done, without giving an exhaustive description of various experiments.

2 It is generally considered that the role of the teacher is to teach, i.e. to take his pupils through a given syllabus up to a certain standard which is usually concretized in an examination.

It occurred to me that the fact of being in a classroom with a

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group of pupils was an opportunity for discovering how they reacted spontaneously to mathematical situations, and for noting these reactions verbatim (one or two adults being present to note down questions and answers, or the pupilsâ&#x20AC;&#x2122; written answers being collected afterwards). I soon found that such situations could be extremely productive, not only for the pupils but for myself as well, and that I had in fact discovered a new way of teaching mathematics and of investigating the content of the pupilsâ&#x20AC;&#x2122; minds. After making numerous experiments in London schools I feel confident in passing on my findings to other teachers and in proposing a large-scale enquiry into the true basis of mathematics teaching.

3 Before proceeding to a consideration of the draft scheme I propose, I shall give one instance of the method used. The experiment was made with a class of 35 girls, aged 11 or 12, in a grammar school in south-west London. A student and the senior mistress were present. The girls had been tested at about the age of 11 and had been found to be of an average I.Q. of over 105.

They had been having lessons in geometry for about a term and a half and I suggested that they were merely being conditioned and that they lacked the capacity necessary for the geometrical structuration of reality, a capacity which is taken for granted by all teachers dealing with this syllabus.

In order to prove my point, one girl was asked to stand in front of the class, who were then told to draw two lines representing her eyes. They all did so correctly, producing either or (the latter spontaneously). The girl was then asked to bend her head to the right and the class was told to draw her eyes in

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that position. Some of the drawings were correct were either

but several

When she bent her head to the left, the same girls produced

thus indicating that they were satisfied that their earlier representation had been correct. The girl herself was then asked to draw on the blackboard what she thought the class had drawn when they were looking at her. She bent her head, hesitated, then drew her eyes in the opposite position, left when her head was to the left and right when it was to the right, as if she were in front of a mirror. When the class was asked if her drawing was correct there was general hesitancy, with approximately the same number of answers for and against.

As a further test, I explained that I was going to give an order which I wanted them to obey immediately, without reflection.

This being understood, I said: â&#x20AC;&#x153;Raise your left handâ&#x20AC;?. Ten of the girls raised their right hand, a few quickly changed it to the left, and one could not make up her mind.

Another girl then came to the front of the class and was given two pencils and two rulers to represent knives and forks respectively. She was asked to lay the table for two people sitting opposite one another, without moving from where she stood.

She put the rulers opposite one another and the pencils opposite 111

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one another. She asserted, when questioned, that they were right, and only discovered her mistake when she moved to the other side of the table. Some of the other members of the class had raised their hands to suggest that she was wrong.

It had already become clear that for many of these girls there was genuine difficulty in mentally representing positions from various points of view and that their structuration of space was quite inadequate for any formal geometrical study. I did not stop there, however. I wanted to discover their ideas of direction and angle, both of which were already implied in the previous experiments. On enquiry, I found that four of the girls went home from school by the same bus, and I asked each of them in turn to draw on the blackboard the route taken. The results were approximately as follows (the arrow indicating the direction of the actual drawing being added as an aid to the reader):

The girls watched one another draw and with the four drawings on the board the class unanimously agreed that all of them could represent the route followed. The only explanation given was that they must be right since the girls who went by the bus had

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drawn them like that. There was no question of checking one drawing against another using a set of abstract and logical relationships. When the girls were asked whether there were any angles on their routes they said that there were not, there were only “bends”.

To conclude the experiment, I asked the class to make a plan of the room, using a long rectangle to represent each row of desks.

When this had been done I asked them to watch what I was going to do and to indicate my movements on the plan by a line, with an arrow showing the direction taken. I then moved as follows:

The girls’ plans were then collected and the lesson came to an end. Less than half the class had been able to indicate my path correctly. Difficulties of various kinds had been met, some of memory, some of placing the gap, some of putting the arrows correctly in opposite directions within the same gap, some due entirely to the abstractness of the task.

4 On the whole, the period made it apparent that there is no justification for the assumption that the notions of direction and angle, at least as we understand them, are present in the child’s mind at this stage. It also taught me that the usual method of introducing the idea of angle has no solid foundation and is mere conditioning, serving only for the type of question to be found in textbook exercises. Finally, it was a clear indication 113

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that space representation is not a simple matter, that it has not been mastered at the age of 11 or 12, and that it requires special methods of education.

5 Numerous experiments carried out with boys, girls and mixed classes at all stages in many grammar-schools in the London area (which contains as many pupils as Belgium and Holland together), all lead to the same conclusion: we must make a systematic study of our pupils as we teach them, beginning without any preconceived ideas and checking our findings by comparison with those of others. The task has been too long delayed and we suggest that others who appreciate the necessity for it should join us in an investigation of these questions. Suggestions from any readers who consider that in the scheme that follows problems have been omitted or difficulties neglected will be welcomed, and discussion can take place either through correspondence or in the international seminars I have organized for the purpose of stimulating research into the teaching of mathematics.

6

There are three main fields that require investigation: 1

The actual content of the minds of the pupils;

2 The course of development of the awareness of mathematical situations;

childâ&#x20AC;&#x2122;s

3 The methods by which we can use our skills as teachers to bridge the gap between mathematical and mental structures.

7 With regard to the first of these, the examples given in Section 3 are sufficient evidence that a single lesson period can

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provide material of the highest value both in quantity and quality. A large-scale investigation of similar questions, first in parallel forms and then gradually further up the school, would enable us to discover at what level this or that notion is present, or to what extent it is foreshadowed, thus providing us with the answer to (2).

Notions are geometrical or algebraic or, more generally, are mathematical structures, and we must discover how they make their appearance, how they correlate, and how they merge one into the other.

Geometry evolves from awareness of space relationships, grasped according to our mental tools and circumstances. In the case of angles, for instance, our eyes, our heads, the whole of our bodies give us the real basis for addition of angles and provide a complete rotation as a unit. But we cannot project these potentialities into objective space until we are capable of representational thought. Our analysis must therefore first be that of the awareness of angle in our own movements, then of angle in the motion of something outside ourselves and the translation of the one into the other. It also involves the analysis of representation in general and of the relation between images and thought.

Our space is structured, as is now well established, according to topological and metric properties. For the mathematician, there is no difficulty in distinguishing them, but are our pupils capable of non-metric topological thinking? Is “bend” instead of “angle”

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an example of it? The one is something “felt”, the other is a measure; the one is topological, the other essentially metric.

With respect to each of the ideas of our geometrical syllabus we must investigate the extent of “metricity” implied, the degree to which the metric element is a simplification, as for instance when we think of a circle rather than of a closed simple curve.

We must discover whether notions that we assume to be primitive are not in fact elaborate and variously interpreted by our pupils. It is assumed, for example, that we can use the idea of direction to lead to that of angle, yet direction presupposes considerable experience and develops hand in hand with that of angle; it is even in some sense more abstract than angle.

Oriented angle, in which the two notions merge, is so complex that it is not really grasped before the age of 15 or 16, and then only by those who make real efforts to do so.

Equality, other relationships of equivalence, areas, volumes, similarity, and proportionality are all difficult, complex notions, and require thorough psychological elaboration. We must investigate their underlying structures and discover at what point in our pupils’ experience they are firmly established.

8 Algebra presents problems of a different type. It involves, in fact, operation on operations, and once this is grasped the only remaining difficulties are those met in the construction of the new combinations that result from specialization. We must, however, investigate what is the content of our pupils’ minds in terms of operations and of the structures underlying these operations.

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In the primary school, operations are on numbers and on quantities, which are numbers structured according to certain laws into a system related to a different experience of the universe. In the secondary school the universe becomes more intellectual and quantities vanish into equations, which are abstract operations upon other equally or less abstract operations. Algebra is a closed system, open inside itself and developing by means of specialization, inversions, combinations, the new beings being new only by virtue of our specialized awareness of them. Thus ab becomes a2; am an = am+n, and so on.

Here our investigation is easier but more subtle. The discoveries to be made are few, but they are not obvious. It is evident that the translation of a problem into an equation implies difficulties, that systems of equations are complex, that graphical study illustrates difficulties, but what evidence can we find of the content of the child’s mind and his experience?

In my view the clue lies in the fact that mental activity does not necessarily tend towards ever-greater interiorization and abstraction. It is possible for the mind to be absorbed in its activity and to extend “horizontally” rather than “vertically”, as for example when we become absorbed in woodwork and make more and more new objects. In order to discover how to help our pupils in their mastery of algebra we must know to what extent their minds are familiar with operations upon numbers and quantities, and we must be able to shift their interest from the number or quantity to the operation itself, which will enable us to see whether they can replace a given quantity by a potential quantity which is obviously the outcome of operations.

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Our investigation must therefore be related to three distinct levels: 1

Numbers and quantities, their relation through operations and their generation through operations: arithmetic.

2 Potential numbers and operations: generalized arithmetic.

3 Operations upon operations: algebra.

At which of these levels has a given pupil arrived? Is he still at the first, or has he reached the second and perhaps the third?

These are the questions we have to answer.

9 The second problem of Section 6, that of discovering how the child’s awareness of mathematical situations develops, will be automatically solved if we deal satisfactorily with the first throughout the school. If we are aware of the content of our pupils’ minds over several consecutive years, we shall ipso facto be aware of how their minds develop.

It should however be pointed out that neither the content nor the development of the child’s mind is independent of our activity with respect to him, and that both will be affected if we discover a method of adapting his spontaneous mental growth to the apprehension of mathematical structures. It has been the task of education to effect this change, and the fact that we consider that at the primary school stage the child is capable of performing operations upon numbers means that we already hold certain views as to the content of his mind. It remains for us to examine by what means we can ensure that at the 118

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secondary school level this content can be made definite instead of indefinite. Hitherto the content of the secondary school pupilâ&#x20AC;&#x2122;s mind has been examined only at the end of his studies, when he enters the university, and even then only in cursory fashion. It is legitimate to enquire whether students of mathematics who have not studied geometry in three dimensions are capable of solving problems requiring spatial representation. My experience with British students leads me to reply emphatically that they are not capable of doing so, but preconceived ideas ignore the evidence and prevent acceptance of this view.

10 The field I propose to teachers and psychologists for investigation is rich and largely unexplored and will yield an impressive and valuable harvest. Detailed investigations through teamwork can easily be set on foot. A number, which I have already undertaken personally myself or with students in English schools, require only corroboration or critical examination and can serve as a basis for study of the method used. Summaries of them are available and will shortly be published as brief monographs.*

* This did not in effect take place; articles were written instead, some of which appear in this part.

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8 Three-Dimensional Vision and Its Psychological Application to the Teaching of Mathematics

This article first appeared in Gymnasium Helveticum, October 1952, Vol. 6, No. 5.

The following few pages have been written with the intention of sparking off an exchange of views between members of the teaching profession and those who experiment in the field of the psychology of intelligence and its pedagogical applications. My own personal position is that of a professional mathematician who has dedicated a number of years both to the teaching of mathematics at all levels, from the kindergarten to the research seminar, and to experimental psychology, understood not so much as a laboratory study performed on animals, but rather as the empirical science of the life of the mind and spirit. I have therefore had the good fortune of having had direct experience of mathematics and psychology, and from a blend of the two have developed a method for the clarification of the problems of

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teaching. This technique is within the reach of all teachers; and I shall explain it by an example, chosen from a wide selection of possibilities, any of which would have served the purpose equally well. It will suffice to apply it once oneself to be assured that it is both easy and fruitful. Readers who would like to communicate their reactions to me may be assured of my interest in their results and of my desire to correct the mistakes which I must most certainly have made, since my field of experimentation has been restricted and particular.

In some countries where three-dimensional geometry is not studied until pupils have reached the age of 15 or 16, I have had occasion to notice both in myself and in secondary school pupils that “spatial vision” is acquired one day all of a sudden, and that once there, it is always at our disposal in the same way as the ability to speak or write. I define this “spatial vision” as a characteristic of the way we look at things such that our representation of objects can be recognized as being threedimensional, and that shapes drawn on plane surfaces gain a relief, permitting us to consider them as if they were truly in three dimensions. I wished to submit this observation to the test of experience.

As in all other questions I have studied, I replace a mathematics lesson with a class retaining the pupil-teacher relationship but the teacher merely asking the questions, supplying none of the answers, and leaving the pupils to express themselves as they wish. Someone takes notes of the questions and of the answers, which are studied later at leisure. The questions are put in such a way that a complete sentence is not required for the answer;

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consequently the problem of technical language does not arise and difficulties of expression are eliminated.

In the case described below, it was not necessary to test all the classes. On the strength of the observation above, I argued that if 15 to 16 year old pupils and groups of students reacted as if the particular attribute of vision in question did not exist, a fortiori it would be lacking in still younger pupils.

In British schools, the teaching of three-dimensional geometry is not part of the mathematics course, even when the curriculum for the first two years at school includes a practical study of solid bodies. As a result, I was able to meet minds completely new to this subject. I found the ability to be as little developed when examining adults with a number of years’ experience in the field of science, as when examining graduates who were to become mathematics teachers, and ordinary schoolchildren.

The following are the questions that were asked: 1

“Draw a skew quadrilateral.”

2 Once this had been done, either spontaneously or with my help, I asked that another be drawn, and then: “If A, B, C, D are the four vertices, choose the positions of these four points with reference to your sheet of paper, and mark them ‘on’, ‘in front’ and ‘behind’.”

3 This choice having been made, I asked: “Mark the midpoints of the four sides M, P, Q, R, and say what position is obtained by the lines joining these 123

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points with reference to your paper, according to your choice made in Question 2.”

4 “Can you say anything about the figure MPQR? If so, can you prove it?”

5 “Draw another skew quadrilateral and mark in the diagonals. The figure having six sides, join the midpoints of the opposite sides.”

6 “Can anything be said about these straight lines as regards their intersection? Can you prove your statement?”

This test, to which answers can immediately be supplied by some pupils in Continental secondary schools at the age of 15 or 16, seemed to recommend itself to me for its simplicity, for the possibility it provides of being able to put questions without any ambiguity and in terms such that the response is not influenced in advance, and for the light it sheds on the problem under consideration.

The answers given by pupils are on pieces of paper which can be collected, and which represent material permanently available for analysis. It would seem essential that the experimenter himself should analyze the results in order to be able to add to the study of the replies such details not noted as hesitation, wavering, bewilderment, exclamations of joy and so on. Absence of a reply on one of the sheets does not necessarily mean that the pupil is completely at a loss about that particular question.

All the pupils tested in this way experienced difficulty with Question 1. They did not know what a skew quadrilateral was.

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Once it had been defined as a quadrilateral whose four vertices do not belong to the same plane, they had no idea how to draw it. When an ordinary quadrilateral was drawn on the blackboard, general protest followed. The following explanation, however, was accepted: â&#x20AC;&#x153;Is it possible to draw anything other than plane figures on a flat surface? No. The quadrilateral drawn will necessarily be a plane figure, but I shall consider it to be skew.â&#x20AC;?

With students from 22 to 30 years of age, though they may not have known immediately what a skew quadrilateral was, two or three were able, by delving into their respective vocabularies, to provide the definition. When it came to drawing the figure, one person had no idea at all what he should do, the best among them thought of dotting in one or both diagonals, while the remainder hesitated. Not a single one spontaneously drew an ordinary quadrilateral.

Question 2 was not understood at all, despite its simplicity.

Quite a few readily made their choice, and all possibilities were represented.

The replies to Question 3 were very interesting, and prove the good foundation for the choice of the example. The difficulties were all of representation in which the drawing itself cannot help unless oneâ&#x20AC;&#x2122;s representation of it is correct. Many children of 15 and 16 years of age and even older students could not, with any degree of certainty, work out the relative positions of the straight lines in question by reference to the purely arbitrarily chosen positions of the vertices. Some of them could, however,

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which goes to show that once spatial representation is operative there is no difficulty whatsoever in describing what one clearly sees. To this question, some were already replying that MPQR is a parallelogram, and were fixing it correctly in the quadrilateral.

Only one could show that MPQR was truly a parallelogram, by using the diagonals and a formal proof, written in telegraphic style.

All the answers to Questions 5 and 6 showed that the functioning of representation had not been mastered, even for those who had said that MPQR was a parallelogram. But the answers confirmed the hypothesis that this functioning can be made apparent and educated.

The detailed analysis of the answers was interesting both statistically and psychologically, but for our purpose it will be sufficient to draw the following conclusions.

The series of experiments has proved firstly that threedimensional vision required by geometrical thought is a very different thing from the spatial view which results from the use of our two eyes; secondly that it does not result from the knowledge and integration of perspective in our visual consideration of objects; and thirdly that it is not the product of mathematical instruction at school and university, since such instruction does not provide the necessary exercises to bring it out and educate it. My experiments have shown rather that this three-dimensional vision can only be a consequence of a

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particular kind of apprenticeship which has, to my knowledge, escaped the notice of secondary school teachers.

In fact, there still are pupils in the secondary schools of France, Belgium, Holland, Switzerland, Spain and Germany who study three-dimensional geometry and yet do not understand it since they have not acquired the vision which to such a large extent helps in the success of their fellow pupils who have managed to develop it. When I taught it myself, I devoted many hours to making pupils aware of the variations in representation when the viewpoint was changed. All the pupils eventually saw the figure of Desargues’ theorem as a three-dimensional one, the test of their mastery being for me its proof (so simple) in three dimensions.

It seems astonishing that the concept of the maturing of faculties with age may here be at fault. A priori psychologists to whom the above facts were submitted were skeptical, understanding by ‘spatial vision’ the organization of our activity in order to meet certain complex spatial situations, such as climbing stairs, climbing onto a moving vehicle and so on. The mistake lies in this confusion. The experiments are clear on this point. There is no cause and effect connection between these two visions since the one can exist without the other. When both exist, they are coordinated, as is the case for architects, cabinetmakers and others. But even here it remains to be proved that the co-ordination goes beyond actual needs as is the case of the geometer. Maturity permits apprenticeship to a function—it does not cause it. Thus I merely state that everyone is potentially capable of ‘spatial vision’, but that no one can see things in this

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way until the appropriate exercises have been placed within his reach, permitting mastery of the technique.

Let us not forget that it is not necessary to make use of this function in order to be a mathematician and that there are ways of making good its lack by precise rules of action as in the case of descriptive geometry, where spatial vision can even be a hindrance.

I have dwelt at some length on this example, which was chosen from a wide selection, but I believe that this has not been in vain. I have thus been able to stress the interest of this method of inquiry which brings teachers face to face with the content of the minds of their pupils and adds clarity to some aspects of their activity. In fact, if we are better acquainted with what our pupils need in terms of mental structures, we shall avoid many disappointments for them and shall spare ourselves considerable trouble. If in our teaching we could distinguish between what is psychological and what is mathematical, our art would be so much more sound and effective. If we knew which mental structures permit the assimilation of mathematical structures, then we would be in a position to plan a program which would transform our pupils into beings who would think about the problems with joy and spontaneity, and broaden themselves as a result; we would know how to avoid having so many failures in our classrooms, pupils whom we unjustly accuse of being lazy or stupid, because we do not appreciate their difficulties.

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By my technique of investigation I have gathered enough material to prove to all those whose minds are not closed to the point of refusing to consider the evidence that the setbacks encountered with certain pupils are due to a lack of understanding of the mentality of the adolescent; this arises from the fact that we are first and foremost teachers of mathematics and that this subject is our primary concern. This comprehension, although far from complete, is now possible.

The principal factors involved in understanding are known to us, or at least I believe them to be. A greater number of our colleagues should study them. Teams should be formed to exchange ideas and results so that a body of knowledge might be amassed adequate to the task and serving as a basis for the development of an education which takes into consideration the learner (who is no less important than the matter taught).

I have studied through my method the psychological problems that the ideas of angle and direction pose for the adolescent; coordination of representation from various viewpoints* and the concepts involved in the notion of a solid body; the difficulties which result from the conflict between day-to-day experience and abstract thought (for example between the physical axiom of space being occupied by only one object, and, in geometry, the fact that parts of figures belong to several wholes);â&#x20AC; the

* That is to say the fusing of different perceptions of the same object viewed from different angles into one abstract entity. For example, in the section of a cone, the geometer forgets he is looking at the cone from above, from below, or from one side; this is outside the problem he is setting himself. But such is not the case with a child. â&#x20AC; On this point see my article The Uses of Mistakes in the Teaching of Mathematics, reproduced in Volume I, part I of this work.

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separation of component structures from more complex ones (for example in discarding the assumption that a metric exists in problems where this seems to have a natural place but where, in fact, it merely renders the geometric situation under consideration more special); the role of the continuum in the creation of a feeling of certainty in geometry and how it helps in obtaining intuitive results; the role of the dynamic pattern in the organization of oneâ&#x20AC;&#x2122;s consciousness.â&#x20AC; In the field of the teaching of algebra, I have been able to establish that if the reversibility of operations is systematically observed in lessons at school, and if algebra is considered as a system of operations upon operations, the pupils can very quickly acquire a complete mastery of large parts of the present syllabus and experience great delight. Algebraic thought becomes spontaneous and the pupils do not distinguish it from their habitual intellectual activity.

My experiments have established that the difficulties experienced by pupils can be eliminated not by repeating the theory and multiplying the exercises proposed, but rather by a progressive development of the system of operations, making the learner conscious each time of the abstract elements in new

â&#x20AC; The notion of the dynamic pattern has been introduced by me into the teaching of mathematics and is applied in many British schools. A more detailed exposition of this topic is to be found in my paper written in 1949, entitled A New Theory of the Image, reproduced in this part.

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situations in order to supplement what is acquired and automatic* In trigonometry likewise I have discovered the weak points and, in elaborating a psychologically true means of approach, I have made pupils competent who, insofar as questions of this nature are concerned, had been given up as hopeless by their teachers as a result of an apparent lack of interest in mathematics in general, and for those chapters considered hard and boring in particular.

I have already described, in many brief papers, a number of these experiments and have drawn some conclusions which I consider to be of use. But I must insist on the fact that I am merely beginning the study of mathematics teaching and that there could not be too much collaboration between all the better teachers of our generation in order to get it under way. We must systematically explore the field of our own activity by means of the indispensable tool of psychology.

My suggestions can be summed up by saying that, in the exercise of these functions, the teacher will find his own classroom his laboratory, and the material all there within his reach. It is up to him to pick it up.

* In geometry there are situations to be examined mentally; to abstract from a situation consists in interpreting a figure as the symbol of a general structure rather than as the particular arrangement of its constituent elements, to read in it one thing rather than another.

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9 Pupils’ Reactions to Geometrical Classifications

This article first appeared in Mathematica & Paedagogia, No. 9, 1955–56.

The following notes are a study of pupils’ reactions to two situations which may be said to contain a conflict between a perfectly understood concept of classification and an element which, although belonging to classification, is not always recognized as such, because of an extreme specialization of that element.

Let me first of all give the point of view of the mathematician. If we recognize the fact that plane figures of a certain type are called trapezia when they have two sides parallel, it follows that a parallelogram is a trapezium. In the same way, a rectangle would be a parallelogram, a square would be a rectangle, and so on. If solid bodies are rectangular prisms (like the Cuisenaire rods), the cube too would be a rectangular prism.

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Now let us pass on to the reality of the classroom. My investigations took place with a class at West Bromwich (children of 11 to 12 years of age and considered to be of low I.Q.), a class at Antwerp (children of the same age, whose I.Q. I do not know), a class at Neuch창tel (13 to 14 years old), a class at London (14 year-olds, but thought by their teachers to be ineducable), and a class at Madrid (pupils of 10, but already at a Grammar-School or Instituto). Other similar experiments will not be studied here.

For the plane figures, the geo-boards are used; for threedimensional figures, the Cuisenaire material.

1 Cube and Parallelepipeds At West Bromwich, the class had 35 pupils considered unintelligent; a group of teachers was observing. Having familiarized them with the rods, I asked them questions about the number of faces, edges and vertices of the cube and the other rods. All the children found the invariants with ease. Before this lesson, these pupils used a non-mathematical vocabulary. But they accepted without any difficulty a geometrical language and used it correctly. The problem which interested us then was to find out what line must be taken for the properties of the cube to be recognized as belonging to the cube in its capacity as a parallelepiped. The discovery of the invariants above does not allow this. Passing on to a classification of the rods, the children were unanimous in saying that the orange, the blue, the tan, and so on down to the red rod were parallelepipeds, but whenever the cube appeared, the only word put forward was cube. If I

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asked: “Is the cube a parallelepiped?”, the reply was always: “No, it’s a cube”.

After two or three attempts, repeating the same actions and the same questions, there were one or two pupils who suggested the cube was a parallelepiped, but the shouts of the others forced them to retract this. As I continued to put the rods in decreasing order, drawing their attention to the fact that, while the length diminished, the term they put forward stayed the same, they saw clearly that I was leading up to the inclusion of the cube in the series. A few pupils hesitated and replied that the cube was a parallelepiped. But when called upon to justify their opinion, they rejected it. We had to abandon the attempt for lack of time.

But it was clear that the conflict lay in the field of the concepts which are covered by words, rather than between two obviously distinct experiences.

Since the word parallelepiped had been introduced to cover the group of those objects having one dimension greater than the other two equal ones, the pupils replied correctly so long as they could see this. But to accept the cube as belonging to that group would have meant abandoning that criterion and coming to the realization that dimensions do not affect the fact of belonging or not belonging to a class. With ten rods and a difference in length of one centimeter, there were not enough elements to force the change in perspective without interfering.

The Antwerp experiment was quite similar to the one at West Bromwich; the result there too was that the children were not willing to recognize the cube as belonging to the class of

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rectangular prisms. But at Antwerp, my lack of command of the Flemish scientific language gave rise to a belief among the many teachers observing that the fault lay with the experimenter rather than with a thought structure organized by certain conditionings. It was enough to propose to all skeptics to repeat the experiment in the same terms.

It is to be noted that I was surprised by the pupilsâ&#x20AC;&#x2122; resistance to the assimilation of the cube into the relevant classification, especially as these children were fairly old and not without experience in handling such simple objects.

My explanation that mathematical language covers several points of view between which the mathematician moves easily, while the studentâ&#x20AC;&#x2122;s language is rigid, seems sufficient for the facts mentioned here. It should be noticed that at West Bromwich some pupils arrived at the threshold of those regions where two points of view are transformed one into the other.

2 Trapezia and Parallelograms On a rectangular, 25-pin geo-board, rubber bands can be stretched to form polygons. If the pupils do not know the names of the figures thus formed, these can be decided on, and it can be shown that neither rotation nor translation affects the number of sides or certain properties of the figures.

In particular, one can set out to study quadrilaterals. Having recognized that a quadrilateral with two sides parallel is called a trapezium, a large number of these can be produced, taking care 136

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to keep the parallel sides unequal. In general children know the parallelogram, which they call by different names according to their country.

At Madrid, a square was a square as long as the diagonals were not one vertical and the other horizontal: in this case it was a “rhombus”. If one moved the geo-board slowly in such a way that the children had the impression of continuity, the result remained the same. The direction of the rotation made no difference. The spontaneous verdict was unanimous: one name for one position of the figure, another for all the other positions together.

In this experiment at Madrid the parallelogram could not be thought of as being a trapezium even when I moved the elastic band and showed that I obtained it as the limit of two classes of trapezia respectively larger and smaller than the parallelogram.

The children were perfectly at ease, and named with conviction the trapezia and the parallelogram, but never said that the limiting case was a trapezium. Even when I told them that I was going to show them that this was the case, they refused to make this step, and maintained their position.

At Neuchâtel, with older children, the same thing happened. But with the “backward” children at London, they began at once, and of their own accord, to consider the parallelogram, in two ways, as a trapezium, and said these most satisfying words: “A parallelogram is twice a trapezium, this pair of sides being parallel and this pair too”. To check their understanding, I began with a right-angled trapezium and finished with a rectangle.

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This was also, in two ways, a trapezium. What is more, the rectangle was recognized to be a parallelogram by way of this double belonging to the class of the trapezia, which was the property isolated in each case. I did not have time to find out whether the right angle was conceived of as being a particular angle, nor to consider the rectangle in this form as a particular case of the parallelogram.

Conclusions Leaving aside the examples given here, it is evident that we have millions of opportunities in our teaching to study what our pupils make of our definitions. Without any doubt the verbal rigidity to which these examples bear witness is the result of uncertainty in the face of the entities we put forward to them. To conceive the entities of mathematics without bearing in mind their future vicissitudes is one of the faults of the teacher, and which can be eliminated by a dynamic approach. If entities are a snapshot in their classes, it will be easier to see that a name is an abbreviation for a group of relationships rather than a ticket for a unique and invariable product. Our â&#x20AC;&#x153;backwardâ&#x20AC;? children were virgin soil, and, given a dynamic form of teaching, they threw themselves wholeheartedly into the game. The children destined for various forms of advanced education are excellent imitators, and in the world of speech consider words as solid realities to be associated with equally rigid figures.

It would be interesting to hold over a period of time a discussion backed by experiments to consider that aspect of the teaching of mathematics which calls on the relationships of speech and

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thought to the structuration of ideas of classification and to the property of the belonging to various classes.

This ought to interest both mathematicians and teachers.

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10 Mathematics and the Deaf

This article first appeared as an introduction to Teaching Mathematics to Deaf Children, Educational Explorers, Reading, 1956.

It is commonly believed that thorough knowledge of a field is necessary before it is possible to make any significant contribution to it. But it is also well known that the history of science shows that a number of important impacts have been the result of a penetrating question put by a fresh mind, and it is for such a mind to recognize that it can serve the cause of the study it embarks on.

Although teaching has been my special field of study for over twenty-five years, I cannot claim long experience of teaching deaf children, but I can describe my experiences in schools for the deaf and of my immediate success in teaching mathematics there.

My experience began in Birmingham, when I was invited to introduce the Cuisenaire material for the teaching of arithmetic 141

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and chose to do so by using it with classes instead of talking about it to the staff. On the first day I met five forms, children whose ages ranged from 5 to 16. I had no knowledge whatever of the problems of teaching the deaf, and no idea of the lines of attack most favored by teachers. Children were before me who were to be taught some arithmetic, and as far as I was concerned that was the whole situation.

Five minutes with the first group accustomed me to their behavior and noises, and I then gathered three or four of the nine children round a table and tried a game with the colored rods. One of the boys at once grasped the idea and showed that he could master a complex set of qualitative relationships. The others were slower and less secure, but did not take long to see that by trial and error they could succeed in finding answers to the questions put to them in terms of the material.

I was told by the Head and the class mistress that these children were intelligent and would probably succeed by any method, though perhaps not so quickly or so well. I then took four other children round a second table, one of these being deemed ineducable and unlikely to be kept in the school. This child did indeed look a miserable little creature, but I did not assume that she would be unable to perform the simple exercise I had in store for her which had always worked with normal children of her age. I took three rods, small enough to be held together in the hands of a little child, one white, one red and one green. I showed them to her, let her feel them one by one, then all together, and put them in her hands behind her back. I then showed her another white one equal to hers, and indicated, by doing it myself, that I wanted her to bring it out of the three that 142

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were held behind her back. She looked dismayed and made signs for help to her class teacher who had obviously been her support in all her previous attempts at learning. I insisted however and she brought out the right one. I asked for the red one and she brought that out, then the green, then the red one again, then the white, and every time she was successful. Her eyes were sparkling, and she had no difficulty in repeating the whole process when I increased the number of rods to four and five. The other children joined in the game and all of them were able to decide by its feel that the rod they were bringing out was the same length and color as the one I showed them.

My own confidence had been established in that lesson and I felt ready to go on to other classes and teach other topics using the rods. I did so, my only difficulty arising from the fact that, as the whole trend of training is verbal, I could not reach a satisfactory level of communication with the older pupils. But with both these and the others it was obvious that thanks to the Cuisenaire material I could achieve much more than the teachers who taught them regularly. It was generally agreed by the staff that the lessons had shown that the colored rods were of use, though not many of the teachers considered themselves ready to use them. The discussion that followed dealt with the use of the material in teaching this or that topic.

For me, the principal fact which had emerged from these successful lessons was that, with no experience of teaching the deaf, I had been able to hold the childrenâ&#x20AC;&#x2122;s attention for at least three times as long as their usual teachers, and had secured an understanding which was shown to be spontaneous by the reactions of the pupils to my questions. I had certainly not 143

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attempted to communicate with the children by speech or manual gesture. I merely submitted a situation to them in which they could easily perceive a certain relationship; either by selecting their own rods, or in writing or in words, they indicated that they had grasped it.

The behavior of the children and the remarks of their teachers stimulated me to think again about all that had happened. A few monthsâ&#x20AC;&#x2122; reflection and a few more days of experimenting in schools in London, Birmingham, Zagreb and Belgrade assured me that I was on a fruitful track. In experiments, in lectures and discussions, I have tested my ideas and have been encouraged to write this booklet by the many teachers who have found that my lessons solved problems for them which were hitherto difficult and often baffling. *** For a teacher of children who can speak, a first contact with groups of deaf children obviously provides a shock that can set off a train of thought. In my case there were two points on which to base my reflection. Firstly, it was clear that I was capable of meeting the challenge offered by the pupils, and secondly, I was absolutely ignorant of any special traditional approach to the instruction of these children. From my success I derived confidence, from my ignorance the freedom to see what was happening rather than what had to be done in order to put across this or that point of the syllabus. What did actually happen, broadly speaking, was that we had self-teaching methods which worked equally well with deaf children as with

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others and which suggest that perhaps the main obstacle met by teachers results from conceiving of the lack of speech as a handicap, instead of attempting to see the reality of a mind without speech in relation to its own universe.

As I did not think of my pupils as being handicapped, but as being what they were, I did not think of teaching them as if they were myself minus some of my qualities, but rather as persons who have a universe of experience of their own into which I might enter if I were sufficiently sensitive. It also occurred to me that much of my own experience is unrelated to verbal material and that communication is neither wholly conscious nor wholly social. For several months as a baby I lived without either knowing any language or even wanting to know one. What kind of experience does one have before speech occurs? Perhaps deaf children have an evolution of that universe rather than ours where words increase in importance with intellectual capacity for communication.

Then came the question: are we really understanding deaf children if we try so hard to condition them to speech, and consider that we are â&#x20AC;&#x153;correcting their handicap by making them lip read?â&#x20AC;? In a school in London where this conditioning is considered by all the teachers to be the most important thing for their pupils, I had a class of older girls (16-18). Instead of teaching them some arithmetic, I created resentment both in them and in their teachers, while with girls who were less intellectually endowed I obtained spectacular results.

Remarkable results were also obtained in Zagreb with classes that had little speech: the pupils could solve mentally problems that the teachers present found difficult to solve on paper.

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It seems likely that if we reconsider the way in which thought is actually formed, we shall find that words occupy the later stages, and that they should only appear as labels for lived experience clearly separated from the rest of life for the purpose of being labeled. What occupies the first stages is, to my mind, at least as vivid to deaf children as to others.

Though this is not the place to develop a psychology of thinking, it can be said that as a background to future designation of objects and relations, perception and action are fundamental. It is not the word that is meaningful, but the perception of the meaning of the word that both allows of its retention and gives it its value in communication. Only when experiences are meaningful can words appear as convenient short signs for them.* By giving precedence to the elements that are really basic to thinking we can allow the deaf children to elaborate their intellectual powers just like other children who indicate what they are doing through the use of words.

In fact, we are very far from understanding ordinary childrenâ&#x20AC;&#x2122;s ways of thinking and we have to rely mainly on their speech, as they rely on ours for the understanding of what we say. The main cause of our incomplete relationship with deaf children is our failure to conceive imaginatively of a non-verbal universe. If we could do so, then our methods of teaching would be related to the real child in his own real universe of experience. What I

* Note: In my attempts at teaching language to a group of deaf girls, the situation was made so clear to them that in a very short time four difficult spatial relationships were mastered: between, in front of, behind, next to.

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claim here is that I now have some insight into that universe, and that this is proved by the fact that my lessons are planned in such a way that the deaf pupilsâ&#x20AC;&#x2122; minds are stimulated and show understanding and creativeness.

To describe the mind of a deaf child in its elaboration of a nonverbal thought is not difficult, but limitations of space here only permit a general sketch.

When a mind is confronted with a situation, it must perceive in it those elements that can be singled out. The contrast required for this is a result of perception in which the feeling of tension in the muscles plays a considerable role. Seeing is both an optical experience and the experience of what happens to the ocular muscles that scan or delineate the field of vision. There is no perception without a concomitant experience of variation in the tone of some associated muscles, so the true basis of thinking consists of perception and action. The weeks and months when the baby has no speech are full of this kind of structuration of experience, perceptive and active categories being formed and recognized internally precisely by the amount of feeling involved. These categories, being fundamental, are also basic for speech production and hearing. The deaf child can form a complete universe of his own experiences, structured according to the existing means; the hearing child forms a different universe precisely because other dimensions of experience are present. The two universes cannot be compared in terms of extension, quality, etc.; they are essentially different to the extent that the stresses and emphases lead to results sui generis which do not conflict if looked at in their relativity. It is not by adding or subtracting speech that one could pass from one to the 147

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other. Only in the abstract is such a passage possible. Those who live the one experience do not live the other, and generally fail to have the imagination necessary for entry into the other. It is not easier for those endowed with speech to imagine a life without it than the converse. But if we who can speak think of ourselves forming our universe of experience before we decided to learn speech, we can see that there exists a reality, with all its richness, vividness and promise, which has in fact become our life, and we can begin to give the deaf child a universe of his own as valid to him as ours is to us. Because the deaf are relatively so few, we make the mistake of conceiving of our life as normal, and instead of trying to understand the mystery of a life without words but with its own data, we attempt to give it a speech at all costs. By this view we see the deaf child as being handicapped, but we might equally well say that it is we who are handicapped because being word-conscious we live in a verbal universe and make communication the most important thing in life.

Since words are not the primary, but the secondary elements in true experience, we have, when we can reach our own minds at work, the means of reaching the mind of the deaf in its dynamics. It is remarkable that we can form for ourselves a universe of experience resembling that of the deaf, and when we do so we seem to have created for ourselves another world, a world in which perception of relationships is direct and translated into the language of relationships without the intermediary of words. There is no neutrality as in the case of concepts, but a kind of permanent significance of the image that occupies the mind. The individual experience is true, as for the artist and the poet, and the generality follows from the memory of what is common to singular experiences. It is a memory 148

10 Mathematics and the Deaf

which is not verbal and which is similar to a film having the additional property of not requiring to be unwound to bring back a particular image.

Because the thought processes of the deaf are still at the image level, they appear to us as less labile and less articulate than ours, but when we learn to form thoughts from our images without passing through words, we find that for most of their qualities they can compete with ours. Thought is the result of a special awareness: that of the mental dynamics behind our images, and mastery in thinking results from the bringing to clear consciousness of the dynamics to be used for its own ends.

I can illustrate the formation of mathematical mental structures without assuming in any way that translation of perception and action requires verbal material. I shall use notation and written signs to establish the sequence of thought. If, after they have mastered the processes and the notations, some children choose to use sound production, they will be much freer in the use of their own mental energy, and this, I forecast, will make for much greater success than the normal process which requires that the deaf child be first versed in sound production and then meet the experience he is to know.

The statements in this introduction which may seem somewhat vague will be replaced by ones that are much more definite and more qualified when we come to show how the deaf child can in a very short time master sufficient mathematics to allow him to compete with any other child.

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If the teaching of mathematics to the deaf can be improved in this way, it is hoped that teachers and authorities will consider whether the arguments developed here may not be relevant to other educational themes.

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1 The Idea of Dynamic Patterns in Geometry

This article was first published in the Mathematical Gazette, London, September 1954, Vol. 38. No. 325.

The purpose of this article is to present an idea which is being successfully put into practice in a number of schools by teachers who were formerly our students. It is related to geometry, but it is based upon the understanding of what constitutes the foundations of geometrical thinking.

Every teacher of mathematics is aware that his work involves a two-sided process. On the one hand, there are his pupils, and on the other his subject. The latter has to a certain extent been mastered in his college studies, and he sees mathematics as being a set of concepts and relationships linked together logically. It seems to him that any presentation of his subject must to some degree involve these notions and this way of thinking, and when he observes that his pupils do not use these tools as he does, he attempts to alter either the order of

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succession of the various topics, or his method. The whole problem of teaching mathematics seems to revolve round this difficulty of relating childrenâ&#x20AC;&#x2122;s spontaneous thought to mathematical rigor and abstractness.

Our solution to this difficulty has been not to water down the syllabus but to ask ourselves how oneâ&#x20AC;&#x2122;s experience becomes mathematical. It is obvious that those who become mathematicians were at one time children, and that they had had to construct a mathematical type of mental activity to which they later adhered. We shall be concerned here only with the question of geometry at the school level, and more particularly with the relation of geometry to spatial experience.

In our view, the whole of school geometry is within the grasp of every adolescent capable of concentration upon his perceptual and active experience, provided that this is presented to him in an organized form which yields geometrical facts. This is the basis of the method of dynamic patterns.

As children, we learn in our games to organize perception for the purpose of action, and to organize perception and action with a view to more complicated action and perception. The perception that concerns us here is that of geometrical facts. These, for the adult, are contained in geometrical situations, but from the point of view of the teacher they are hidden in the situation and have to be extracted, abstracted. Our idea is to animate any given figures so as to form a pattern containing an infinite number of figures, but to do so in such a way that only the fact that we want to abstract shall be singled out. If we begin with a

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circle, for instance, we can arrive at a variety of situations, according as we make the radius vary, the center move on a line, and so on.

The first point, then, is to see the concept or the relationship in question as the invariant of a dynamic pattern. The second is to organize the pattern in such a way that one invariant only can be discovered in it. The third is to enable our pupils to perceive it with their senses, supported by their minds, and to express what they had perceived in their own language. When this has been improved on, the result may prove to be a statement significant for the geometrical education of our pupils. It then remains to give the proof, i.e. to relate the newly acquired experience to earlier experiences in orderly fashion, although this will not necessarily increase the conviction of our pupils.

One or two examples will serve to illustrate how the method can be carried out in practice, firstly with the use of the blackboard and colored chalk, and then with a film.

Let us suppose that our aim is to produce a dynamic pattern which will bring home the property of the angle in a semi-circle.

Having drawn a semi-circle of diameter AB, we shall then draw any line L from the center, meeting the circumference at M, but going beyond. We then choose two points, P, very near the center O, and Q very far away, on the line L. Angles APB and AQB are obviously respectively almost a straight angle and a very acute angle. Points between P and O yield angles approaching 2 right angles, while points beyond Q yield angles approaching zero. It is a matter of a few minutes, in questioning

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the class, to obtain the certainty that the points on L inside the semi-circle yield obtuse angles, and those outside, acute angles.

The point M of intersection is neither inside nor outside, and yields a right angle. The whole of the awareness of the value of the angle AMB is now linked with that of many others, by virtue of the dynamism of the pattern. The requirement of the syllabus has been met, but also this particular angle has been placed among an infinity of cases as the only one that is definite. For the teacher, two things remain to be done. One is to vary the semi-circle, and the other to vary the line L, so that the fact discovered is seen as an invariant of all these situations. Only then, when the result has been clearly stated by the pupils, can we know that it has become part of their experience and constitutes a geometrical fact for them. The formal proof is then the easy part of the work.

Our second example will be presented in the form of a film, one of those made by J. L. Nicolet and called Animated Geometry.

The film is silent, only delineated figures are used, and there are no letters. The problem is to find the locus of points whose property is that the angles between the two tangents to each of two circles are equal. This is an interesting rider, and one not easily solved by pupils at the lower sixth form level. The film is animated in the following way. First of all a point is chosen near one of the circles, one angle then being obtuse and the other acute, and the same is then done with the other circle. The angles are seen to be unequal, but by comparing the two cases we find that the relationship changes, and the idea begins to dawn that there may be an intermediary position in which the angles are equal. When one has been found, the question arises as to whether there are others, and if so, where. The locus is a 156

1 The Idea of Dynamic Patterns in Geometry

circle containing the smaller of the given circles. The dynamic pattern then follows: the two circles remain centered where they were, but one increases and the other decreases, the locus gaining its full significance from the infinite number of cases of figures. The lessons contained in this film are obviously many, and we leave the reader to imagine their uses.

We have given these examples at length to illustrate the value and range of the idea of dynamic pattern. If more teachers of experience are prepared to put it into practice, it may well be that we shall find ourselves in possession of a truly exciting set of geometry lessons, lessons no longer divorced from our pupilsâ&#x20AC;&#x2122; interests.

Nicoletâ&#x20AC;&#x2122;s films are obviously much finer tools than the patterns we ourselves can draw, and they are also to be highly recommended for the aesthetic quality with which the genius of the Swiss pioneer had endowed them.

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2 Geometrical Intuition and Mathematical Films

There are two ways in which one can think of the relationship between mathematics and visual aids. On the one hand, one can investigate how one can improve on the existing blackboard technique by using good diagrams ready made in strips, and introduce illustrations not accessible to most teachers as visual material. On the other hand, one can investigate whether the learning process in mathematics has components of the nature of dynamic patterns and relate them to a technique of film production. We shall not discuss the respective merits of these approaches but rather shall concentrate on a fact little known to mathematics teachers, though a mere reflection on their experience might give them ground for assessing its deep reality.

We want to talk of the dual-level approach to geometrical reality.

We are frequently told that geometry is the study of space relationships, and we also find very often that our teaching of geometry encounters both boredom and a lack of insight into its

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reality. Is it that geometry is not concerned with space relationships, or is it that we, as teachers, rarely distinguish in our syllabus between mere conditioning and awareness of space relationships? I think the truth is that, as mathematicians, we are capable of equating our thought about reality and our thought about geometry and that we begin with our pupils at this level without remembering how we arrived there.

Let us take a few examples to bring into relief this problem of levels of awareness in space relationships. First consider perspective. Is it founded on reality? Yes and no. Yes, for when we become aware of it we can no longer not see it and not use it.

No, because it was only discovered very late in the history of art and representation, and is not spontaneously understood by children under a certain age in certain civilizations. Hence perspective is a geometrical treatment of an appearance of reality and is not a direct grasp of the organization of realityâ&#x20AC;&#x201D; this remark is meaningful when we discuss whether congruence or similarity is the more primitive notion in geometry. Second, let us imagine a figure in a textbook. Do pupils look at it as a representative of a class of figures as we do? Certainly not, at least not for quite some time. Why do they not? The reason seems to be that with regard to reality we investigate properties of particular objects, the relationship we put in our theorems are true of the objects involved in the figures and, unless we extend them to the class of those objects in all positions and sizes, we are demanding only an awareness of the visual type. Moreover, we have ourselves made a profound assimilation of relationships and objects and require our pupils to regard these as if they were interchangeable.

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In these few lines, we have suggested that, if we want to be successful with our pupils, it would perhaps be wiser to return to the way we form our own concepts and thinking process in geometry. We have also suggested that the task may be made easier if we act on the experience of the pupils in order to direct it in the path of our own and to make them use it.

Generally speaking, images are essential for thought. But thinking is effective to the extent to which it is not prevented by the image from achieving its full range and from functioning with speed. This dictates two ways in which our work may proceed. First, we must use images and second, we must go beyond them by converting them into concepts and relationships. Naturally, both must be abstracted from the situations. They acquire their intellectual life in the mind of the pupil through our intellectual grasp of their internal connections and necessary bonds.

Films are an obvious answer to these requirements. If their dynamic nature is properly used, they may display in front of the pupils the whole family of situations which in a textbook are illustrated by perhaps only one example, and may produce a theorem which expresses the invariant property imbedded in the dynamic pattern. For instance, any locus is of this nature: for example, all points equidistant from a given point, or all points in a given relationship to two given points, and so on, involve an infinite number of representations possessing the same property which is the invariant; but every geometrical fact can be viewed in this way.

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The Swiss mathematics teacher, J. L. Nicolet, has produced a considerable number of films which use a technique of animation fulfilling these requirements. The idea is that the pupils should gain from the dynamic pattern awareness both of the whole range of facts involved in the geometrical situation, and also of the invariant that they themselves formulate in language satisfactory to them. The task of the teacher is then to improve on this acquired awareness and to relate the first to a desire for a more abstract proof.

The genius of Nicolet allowed him to discover not only the value of intuition in our formation of mathematical ideas, but also the right technique of animation which makes use of patterns of such beauty and simplicity that no one can see the films without a great sense of excitement. The simplicity appeals to all minds and the aesthetic component opens the way for appreciation and easy recollection. A description of any one of his films would require a great deal of space, though the film itself might only last for just a few, and never more than five, minutes.

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3 Teaching Through Mathematical Films

This article first appeared as chapter 7 in Le Matériel pour l’Enseignement des Mathématiques, Delachaux et Niestlé, Paris, 1958.

The number of mathematics teachers who use films in their teaching can at present be counted on the fingers of one hand.

There is, therefore, at the moment, only a small body of experience on which to draw in this field and the evidence is necessarily limited.

It was J. L. Nicolet who first had the idea, in 1940, of using animated geometrical drawings for teaching classes of pupils from 10 to 18 years of age in part of French Switzerland. Since 1949 I have had the opportunity of using his films as teaching aids in a number of countries. In these notes I shall confine myself to an account of my own experience with the films.

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I shall first give examples of lessons with various groups of pupils, then discuss the value of the film as a tool in these particular lessons, and finally examine what seems to me to be the methodology of teaching through films.

1 Lessons with the Nicolet Film Dealing with a Circle Passing Through Three Points This film lasts for two minutes. First there appear on the screen three non co-linear points, followed at once by a circle which moves in from the left, without changing its dimensions, and passes through one of the fixed points. The circle begins to sway, and then expands until it reaches one of the other two points, and for a moment the figure is fixed. The movement then begins again, the third point is taken in, and the circle remains fixed in this position. Then, one after the other, there appear other circles which have the same fate; whether to begin with they are exterior or interior, centered on one of the given points, or come from the right or the left, they develop so as to reach first one, then another of the given points, finally merging with the first circle.

Nicoletâ&#x20AC;&#x2122;s films are silent, and there is no accompanying text. The length of the film as a whole, and of each image, has been calculated in accordance with Nicoletâ&#x20AC;&#x2122;s idea of how the film might be used with a class.

I have followed a variety of paths in using this film. Sometimes it has been shown once, twice, three times without comment and the pupils have then been asked to give an account of it. On

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other occasions I have shown the film and put questions to the pupils from the beginning, questions varying from: “What do you see?” in relation to a set of images, to: “And now where is the center of the variable circle?” These various lessons with different groups of pupils produced a variety of results, all of them of interest for the study of films as a teaching aid.

It is not yet the moment to proceed to an evaluation of teaching through films. The fact that Nicolet’s films are silent, and that no valid norm has been established for the use of films leaves freedom for considerable variety of method. Whereas the author had envisaged one atom or molecule of the syllabus of elementary geometry, my use of his films has taken a different path. Having seen that they offer the means of creating dynamic mathematical situations, I have taken the liberty of exploiting in this sense what they offer.

The pupils with whom I have used this film have usually been newly introduced to geometry; they may even not have studied circles. Sometimes, therefore, the scenario has been discussed in geometrical language, but sometimes in other terms which have revealed the affective dimensions linked with this series of images.

I have noted the spontaneous statements made by pupils in certain of the classes. When a child’s reaction is to see the film as a story in which the three points are hands that catch the poor circles, which dance, then struggle to escape, and are finally strangled when the three points are fixed, it is obvious that his whole personality is involved in the story, through which he is

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again showing that mathematical thought depends on perception and action. Although the main theme remains the same, these nice little stories present considerable variety.

When the pupils are able to use truly geometrical language, we find another set of possibilities. Merely by watching the film and living the vicissitudes of the circles, the children produce suggestions which are very significant. On one occasion, when the film was shown to a mixed class of forty pupils of 12 who had failed to gain entrance to a grammar-school, one of the girls, who was considered to be by no means an apt scholar, described what happened to the circles by saying that each time one of them met a point it lost one third of its freedom. This image pleased her fellow-pupils as much as it amazed the adult spectatorsâ&#x20AC;&#x201D;and it was adopted.

On another occasion a pupil said that a circle that passes through one point can swing without changing and can expand if it wishes, but when it passes through two points it can only slide.

In another class, a child saw the points as obstacles, which could not be avoided, placed on the path of the expanding circles.

(Here, as might be expected, the film was in command for the moment.)

But if we begin with more precise questions, children who have not yet studied circles show striking ability to answer complicated questions. Without worrying about whether they know by heart the definition of a circle, we can ask where the

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center is and what its locus is—without of course using these terms. In relation to the first images, and when the film is again shown, this last question produces only vague answers, because in fact there is nothing necessary in the movement of this circle.

But as soon as the circle oscillates round the first fixed point, the correct reply comes without hesitation: “The center is on another circle centered on the fixed point”. If the film is then stopped and the children are asked to draw on the board the locus of the center, they suggest an arc of a circumference centered on the fixed point, and, somewhat doubtfully, that the radius of the circle is equal to that of the swaying circle. (With older pupils the proof of this can be required.)

When the circle begins to expand, the younger pupils cannot answer this same question, whereas older ones, who see that passing through the fixed point there is an invisible straight line to which the various positions of the variable circle are tangent, can clearly indicate the locus. It is remarkable how definitely the answer is suggested by the film.

After the film has been shown a number of times and the children’s attention has been drawn to what happens when the circle is “caught” between two of the fixed points, even the youngest pupils see that the centers describe a straight line perpendicular to the line joining the two points, and, after some discussion, that it passes through the center of the segment they define.

Finally, the children are asked to summarize the constant fact in the film. Sometimes, but not always, the answer that Nicolet

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wants is forthcoming: “There is one circle and only one that passes through three non co-linear points”. Usually the answer given is an equivalent proposition, but one that is nearer to the child’s imagination, e.g. “If on a circle there are three fixed points, all the circles passing through these three points merge into it” or, “Passing through three points there are a lot of merged circles, whatever circle we begin with” and so on.

Nicolet created this film in order to get this result, but in my experience it is one of the least of the many facts that children can find in the film. For me, the real lesson begins after the film has been seen a number of times and discussed. It has served to provide experience, to present a set of visual images and to suggest a line for the integration of these images. Now the task of the mathematics teacher begins.

My procedure on any given occasion depends on the time available and on the age of the children. Blackboard, paper and compasses make their contributions to the study.

Starting with three points, as in the film, we draw a circle and mark the center. At once the question arises: “Why place it there? Where else can we place it?” There follows a study of the position of the center and we reach agreement as to the meaning of the words: “The center has two degrees of freedom”. This discussion, which can be of considerable importance for the future development of the pupils, can take a variety of directions, very interesting in themselves and related to the two dimensions of the plane, the possibility of locating a point in the plane with the aid of two coordinates, the fact that a vector in

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the plane can be decomposed into two vectors relatively to two fixed directions, etc. Common sense must of course be used in guiding the discussion, but my experience shows that even children of 11 can go a long way in these directions which have become concrete as a result of the original problem.

The next step is to show that once the circle has been chosen it can be displaced in the plane; we find either its various positions or the circles equal to it in the plane, and we see that these are two ways of looking at the same reality. We then play the following game. Having drawn a circle on the board, I ask the children to imagine all the circles equal to it in the plane. Some children keep this circle within their field of vision and duplicate it indefinitely (reversing the process in the film), sending off circles in all directions. Others displace the circle and imagine it occupying all the positions. What they say is fascinating, and through it they come to understand the equivalence of certain virtual actions, while the teacher gains insight into the varied content of the minds of his pupils.

When the center has been thoroughly studied, I ask whether there was any reason to choose this particular circle rather than a smaller or bigger one. It is at once agreed that the size is irrelevant. Then, fixing the center, I ask the children to imagine all the circles having this point as center. They do not all see the same thing. Some begin by imagining smaller and smaller circles, but without reaching the end of the operation, and ignore bigger circles, while others take the opposite course and never come back to the interior without however finishing with the exterior. This situation provides the opportunity for a very rewarding dynamic lesson.

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Through further study of the family of concentric circles we can gain valuable information as to the geometrical thought of our pupils. As long as the visual field is under his control, the child says things that are similar to what we ourselves know, but if he is asked to make his circles expand indefinitely he does not let them go outside the field, although he maintains that they are expanding. When he is asked to reduce the size of the circles, he reacts in the same way and refuses to reach the center; there is always a space between the circumference and the center.

For this stage to be passed, the images must be replaced by awareness of the dynamics underlying their evocation, and of the fact that if we cease to look at the circles and see instead whither they are tending, we shall have on the one hand the center, and on the other the â&#x20AC;&#x2DC;indefiniteâ&#x20AC;&#x2122; in the plane. The images and their dynamism are easily reconstituted by using numbers.

If R is the radius, can we double it? Yes, and we get 2R. And can we still see the circle? Yes. Can we double this one? Yes; and its radius will be? 4R . . . What is the radius of the biggest circle possible? Is there one?

If R is the radius, can we halve it? Yes, and we get Can we still see the circle? Yes. Can we halve the radius of this one? Yes, and the radius will then be?

. . . What is the radius

of the smallest circle possible? Is there one?

By patiently letting the children express themselves, we discover what are the mental obstacles to this mental activity and also

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how the basic difficulties, which often remain even after several years of teaching, can be overcome.

We now start with a circle of radius zero or point and produce from it the family of concentric circles. The radius increases from O to infinity. Starting from a circle which we call infinite, we bring it into the field of vision and make it tend towards its center. The radius diminishes from infinity to O. We summarize this by saying that a family of concentric circles has one degree of freedom: the value of the radius which varies from O to infinity.

By fusing the two experiences, we see that any point in the plane can be the center of a family of concentric circles and that we can displace a family and generate it at another point. Once again this exercise gives rise to a game similar to the one described above.

The film is shown again and we now see that the variable circles are members of families. We describe them, in particular the ones we have just studied, and prepare the ground for pencils of circles.

We study loss of one degree of freedom, i.e., families of circles passing through a given point. They comprise an infinity of infinities of circles, because on every straight line passing through this point there are centers of circles that are within our terms of reference. The pupils draw some of the circles in their exercise books and discover that they can find sub-families formed of equal circles having their centers on a circumference

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whose center is the fixed point, and others formed of circles which touch, at the fixed point, a straight line passing through this point. This is only intuitively grasped and I do not attempt to establish the proof except with more advanced pupils.

The pencils are, however, within the range even of beginners and prove to be an excellent field for geometrical experience.

The actual drawings present problems which need to be resolved. The children have seen in the film that the circles â&#x20AC;&#x153;slideâ&#x20AC;? between the two points. Starting from two points, placed either vertically or horizontally, we see that the straight line passing through them is a symmetry axis of the situation since right and left, top and bottom are irrelevant. So the circles will go in pairs. Moreover, the symmetry of circles strongly suggests that the centers of the circles drawn are in equilibrium with respect to the same variables, and being neither higher nor lower, nor more to the right than to the left, they are on the other symmetry axis of the situation: the perpendicular bisector of the above segment. We imagine it drawn, and by looking at it we see where the right angles and the equal segments are. By using the radii of the equal circles, we get a rhombus, and we give it all possible positions. With each rhombus we associate two equal circles which pass through the two points, and we can consider the figure from three aspects: the centers on the perpendicular bisector and their distance from the initial straight line; the rhombuses which give the radii, and finally the circles themselves. A little practice with this figure allows complete articulation of the situation and the transition from one perception to another.

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We can now ask the children to find the smallest circle in the pencil. This presents no difficulty, and even children of 11 have always succeeded in finding it. It is however another matter when it comes to finding the largest circle, and it becomes necessary to work on the figure with its pairs of ever growing circles. Usually the position has to be forced so as to arrive at the circle-straight line, and the class is left with the certainty that there is no largest circle since a straight line is not a circle.

Having reached this point, we consider the three pencils on the three sides of the triangle formed by the three points in the film.

We find that they have in common the circle passing through the three points. We find that this is also the case if we take only two of the pencils.

We now proceed to make this experience of families of circles specific by coordinating it with the drawings. What we want to do is to construct the perpendicular bisector of a segment and the center of the circle passing through three non co-linear points, these being the elements met in our previous qualitative analysis.

We want, with ruler and compasses, to construct the rhombus we had earlier. This is the first time that an intuitive question has been replaced by a precise question whose answer must be arrived at, not through thought or perception, but through action. All that we know is that we want to draw a rhombus, which means that the four sides will be equal although their value is irrelevant.

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We begin, then, by marking two points on a sheet of paper; we open our compasses to any width, and try to find the two other vertices of the rhombus. In this way we discover: (1) that a solution can be found only if the width of the compasses is greater than half the distance between the given points; (2) that to obtain the required points we must use intersecting arcs.

This is all that is necessary, and we can now construct as many rhombuses as we wish. Any one of them gives us the required perpendicular bisector. If we take two segments having a common extremity, we solve the problem of the center passing through three points.

Before leaving these questions which arise out of the film, we also study the particular case of three co-linear points.

2 Lessons with the Nicolet Film on the Locus of Points in the Plane From Which Two Given Circles in the Plane Are Seen from the Same Angle I have used this film with pupils of 13 and 14 years of age who have not yet studied tangents to the circle.

The film shows two circles in the same place. Then two half-lines or rays are drawn from a point and become tangent to one of the circles, while two others become tangent to the other circle. By rotation of the set of two of these rays, we see that the angles they include are not equal. When the point is displaced in the plane, we see that the inequality can be reversed, which suggests that there may be points from which the two angles are equal.

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The film ends by showing the locus of the points in the plane where the two angles are equal.

This film is such that the answer cannot be found through intuition alone, and I have therefore chosen to use it in a different way, at least in secondary schools.

I begin with an intuitive study on the blackboard, and I have frequently found that in a single 45 minute lesson we have covered ground that would seem to require several hours. The lessons have always been given to very ordinary children and have been observed by a number of teachers.

It should be remembered that the pupils have not yet studied tangents to a circle.

I draw a circle on the board and ask whether we can classify the straight lines in the plane with respect to it from the point of view of intersection. The children discover, quite quickly, that there are three classes: secants, non-secants and intermediaries or tangents. We choose a point on the circumference and draw the corresponding radius, and when a half-line is made to turn round this point the children quickly realize that as long as the angle formed with the radius is acute, on either side, the halfline is secant, and that the case of the right-angle gives the tangent, since the half-line is then neither secant nor nonsecant.

We draw this tangent in yellow. Then, taking any point P on it, we draw a half-line which can pivot round this point, and we 175

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examine when it is tangent. Since it can be either secant or nonsecant, there is a moment at which it changes from one to the other. We draw this position in yellow, join the point of contact to the center, and examine the resulting quadrilateral. We quickly discover that one of the diagonals is the symmetry axis and that the other is perpendicular to it. We find that the tangents are equal, as are also the angles at the center and at point P. We go on to see what happens if P moves along the symmetry axis. We find that when P moves away from the circle the angle at P diminishes, and that when P moves closer to the circle the angle increases. When it is very close, the angle is almost 180째, and when it is very far away the angle is almost nothing.

We now put the symmetry axis in the plane by making it pass through the center. Since the position of this axis is irrelevant, we know that what we said earlier is valid for any position. In particular, the locus of the points from which a given circle can be seen from a given angle is a circle concentric to the given circle, increasing in size as the angle diminishes.

We now put two circles in the plane instead of one, and repeat the construction, starting from any point P, so that we have two pairs of tangents from P to the two circles.

The children are now ready for the showing of the film. I stop it before the end so that the locus is not shown, and I ask them what they think is the locus of these points in the plane. The two original circles were deliberately drawn so as to be clearly seen as being different in size. Often the reply does not come at once.

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The children need more time to look at the problem. The film is shown again and various questions are put during the projectionâ&#x20AC;&#x201D;questions for instance about the displacement of the point with respect to the two circles: â&#x20AC;&#x153;Must we bring the point nearer to the big circle or the little circle if we want to reduce the inequality? Is there a symmetry axis in the situation? Do these facts tell us something about the locus? Are there points such that the half-lines carrying the tangents are continued or coincide? What can we say about the angles in these cases?â&#x20AC;?

If the questions are too numerous, the film is interrupted and we have more discussion, using the blackboard to settle some points, or the film is shown again.

Experience shows that under this pressure, after the preliminary study, and when the film has been well assimilated, the children succeed in finding the locus. And they prove that they have understood by stating that since the figure is symmetrical and the points on the symmetry axis from which the tangents common to the two circles start are part of the locus, the locus is a circle having these points as extremities of the diameter. When the end of the film is shown, they are delighted to find that they have guessed correctly.

In another lesson, we study what happens when the variable elements of the film undergo change. What happens, for instance:

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1

if the smaller circle expands and the other remains unchanged, or conversely? And in particular if they are equal?

2 if one is reduced to a point, the other remaining fixed?

3 if one expands indefinitely, the other remaining fixed?

4 if one expands and the other is reduced to a point?

All this is extremely interesting and very useful, and the children take great pleasure in exploiting the situation to the full.

3 Lessons Suggested By Three of Nicoletâ&#x20AC;&#x2122;s Films on Conics These lessons have not been given with films, but as Nicolet has made films dealing with these topics, the use of the films in the way described above would obviously prove profitable. As I was giving the lessons, I saw how they could be made into films.

Nicolet had already made certain parts and his synthesis appeared in 1956 in the form of a very attractive film.

Let (C) be a fixed circle, Po a point in its plane, and (K) a variable circle passing through Po and tangent to (C). We first study all the possibilities of figures depending on â&#x20AC;˘

the position of Po with respect to (C);

â&#x20AC;˘

the fact that (C) may have a finite radius o, or an infinite radius or one equal to O. 178

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Then comes the question: â&#x20AC;&#x153;What is the locus of the centers of(K)?â&#x20AC;?

1 (C) has radius nought and Po coincides with this point. The locus is the whole plane.

2 (C) has radius nought and Po is different from this point. The locus is the perpendicular bisector of the segment PoC. 3 (C) is a straight line and Po is on it. The locus is the perpendicular to (C) at Po. 4 (C) is a straight line and Po is not on it. The locus is the parabola having Po as focus and (C) as directrix.

5 (C) is a circle with finite radius and Po is on the circumference. The locus is the straight line passing through Po and the center of (C).

6 (C) is a circle with finite radius and Po is interior to (C). The locus is the ellipse of which one focus is Po and the other the center of (C), its major axis being the radius of (C).

If P is at the center of (C) the ellipse is reduced to a circle concentric to (C) with diameter equal to the radius of (C).

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7 (C) is a circle with finite radius and Po is exterior to (C). The locus is a hyperbola with foci Po and the center of (C) and its major axis is equal to the radius of (C).

Having determined all these loci, we must show their mutual transformations one into the other by using the constantly changing relationships between Po and (C). We begin with Po coincident with the center of (C) and (C) having a positive finite radius; the circles (K) are all equal and their centers are on the concentric circle with diameter equal to radius (C). If we displace Po towards the right, the locus is distorted and becomes an ellipse having the same major axis and which gradually collapses as Po tends towards the circumference while its major axis tends towards the radius of (C) that is on the right half of the horizontal diameter. But as soon as Po reaches the circumference the locus becomes the whole straight line carrying this radius. When Po passes beyond (C), two branches of a hyperbola make their appearance. When we see how these branches open as Po moves further and further away, giving for Po at infinity a straight line, we get an idea of the transformations of these loci one into the other with (C) remaining fixed.

The second phase, the displacement of Po to the left, reproduces the same series of figures; a rotation round the center of (C) will complete the awareness of the whole range of these patterns.

We get a different picture by transforming (C) into a circle with ever-increasing radius, moving the center further and further to

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the left. The locus of the centers of (K) becomes a parabola instead of an ellipse. The parabola is on the left when the center moves to the left, and on the right when the center moves to the right. But if, at the same time, Po tends towards the limiting straight line of the circles, the parabola opens and becomes the straight line itself when Po reaches it. On the other hand, the parabola collapses on to the half-line perpendicular when Po disappears towards the right to infinity. If we had begun with Po exterior to (C), the parabola would be the limit of the hyperbola, one of its branches being imaginary. Straight lines, circles and conics proper are thus fused into one definition and its dynamic production.

It will be obvious to the reader that these last pages contain the outline of a film which could be used for teaching conics in the secondary school. Nicoletâ&#x20AC;&#x2122;s last film does not cover all the cases, nor the continuous transformation of one into the other.

Through their dynamic reality, the transformations of the curves one into the other become obvious and the cases of degeneration are thoroughly integrated as particular cases of particular cases.

If the film were studied in conjunction with models showing the conic sections, a different type of movement and a different relationship between these curves would emerge.

It has always been my experience, in dealing with the content of this section, that pupils are truly happy at seeing things from this angle, which makes them easy to remember and gives reality to the classical statements.

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4 Conclusion Since as yet there are few teachers who have experience of teaching through films, the above account of one personâ&#x20AC;&#x2122;s experience with respect to only a few of the possible uses of this tool may carry little weight.

My conception of the function of films as being the one we have just discussed: the creation of pregnant situations which the teacher can exploit and so reorganize his classical syllabus, is not shared by all the authors of mathematical films. Some want the film to illustrate a difficult point of the syllabus or to serve for revision of a section of it. Others feel that there is a place for a mathematical documentary, i.e., a film which attempts to show every aspect of a question, and in particular creates through this synthesis an opportunity for re-thinking a series of theorems in terms of geometrical images. This point of view has sometimes led to the discovery of new theorems (T. J. Fletcher).

I have chosen here to illustrate some aspects of mathematics teaching through films because I am convinced that if teaching is to be dynamic this tool which has served me so well has its part to play. It leads to real progress in the understanding of geometry by all pupils, the bright as well as the less bright, and also by their teachers. But it also represents, for the re-thinking of the school syllabus, a new principle which the traditional verbal habits of work would never have revealed.

The examples I have given are an illustration of what is meant by the study and improvement of mathematics teaching.

182

The Sources of this Volume

“Enfance”, Paris.

“Revue Belge de Psychologie et de Pédagogie”, Brussels.

“Nederlansch Tijdschrift voor de Psychologie”, Journal of the Psychological Society of the Netherlands, Amsterdam.

“Introduction à la Psychologie de l’Affectivité et à l’Education a l’Amour”, Delachaux et Niestlé, Neuchâtel & Paris, 1952.

“Gymnasium Helveticum”, Journal of the Swiss A.T.M., Basle, Switzerland.

“Teaching Mathematics to Explorers, Reading, 195 6

Deaf

“Bulletin of the A.T.A.M.”, U.K.

183

Children”,

Educational

For the Teaching of Mathematics – Volume 2

“Mathematica & Paedagogia”, Journal of the Belgian A.T.M., Morlanwelz, Belgium.

“Le Matériel pour l’Enseignement des Mathématiques”, Delachaux et Niestlé, Neuchâtel & Paris, 1958, published by the I.C.S.I.T.M. “Bulletin de l’A.P.M.”, Journal of the French A.T.M., Paris.

“The Mathematical Gazette”, Journal of M.A. of U.K.

184

Index

Abstraction, 11, 44, 66, 68, 81, 85, 87, 90, 91, 92, 117 Action, 5, 11, 12, 13, 14, 15, 17, 19, 20, 33, 34, 35, 37, 39, 41, 42, 43, 44, 46, 51, 64, 65, 66, 68, 69, 71, 73, 76, 77, 78, 79, 81, 82, 83, 85, 87, 90, 91, 92, 97, 98, 102, 110, 117, 122, 128, 133, 135, 143, 146, 147, 149, 154, 165, 166, 169, 173 virtual, 28, 76, 78, 79, 81, 82, 85, 86, 169 Anaglyphs, 76 Analytical intelligence, 12, 20 Awareness, 2, 3, 18, 53, 65, 69, 78, 91, 114, 115, 117, 118, 149, 156, 160, 162, 170, 181

Binet, 48, 127 Bliss, C. K., 23 Cantor, 53 Categories, 62, 64, 65, 147 Cuisenaire, 83, 133, 134, 141, 143 Dedekind, 57, 70 Descartes, 52, 56 Dimensional theory, 35, 38, 40 Dynamic pattern, 46, 47, 92, 106, 130, 153, 154, 155, 157, 159, 161, 162 Emotion, 11, 14, 15, 34, 36, 41, 43, 55, 82 Energy, 32, 33, 34, 35, 36, 37, 40, 42, 48, 76, 77, 78, 149 structured, 33, 34, 35, 38, 42, 47, 48, 56, 64,

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For the Teaching of Mathematics â&#x20AC;&#x201C; Volume 2

75, 76, 77, 116, 117, 147 Environment, 15, 21, 22, 32, 42, 46, 63, 64, 79, 80 Euclid, 52, 67, 86, 98 External world, 32, 78

101, 117, 130, 145, 146, 161 musical, 11, 23, 25, 27, 28, 29, 77 sensory-motor, 18, 45, 47 Investigation, technique of, 3, 4, 129

Fletcher , T. J., 5, 182 Formalization, 53, 81, 82, 83, 84 Foundations, 53, 54, 55, 56, 57, 153 crisis of, 53, 54, 55, 57 Frege, 57

Johannot , L., 102 Learning, 2, 3, 4, 9, 10, 11, 12, 19, 22, 85, 95, 143, 159 Lebesgue, 92 Legendre, 67 Leibnitz, 52, 56

Gay, A., 4, 77 Geometry, threedimensional, 44, 122, 123, 127

Modern mathematics, 3 Marcault , J. E., 32 Mass media, 5 Mathematical activity, 57, 81, 82, 85, 86 films, 5, 91, 92, 107, 151, 157, 159, 161, 162, 163, 164, 165, 178, 182, 183

Hilbert, 57 Images, 4, 13, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 62, 66, 74, 76, 77, 78, 79, 82, 115, 149, 161, 165, 167, 168, 170, 182 active, 44, 45, 46, 47, 48, 56, 62, 77, 78, 83, 84, 85, 147, 154, 179 body-, 36, 43 intellectual, 12, 13, 17, 18, 19, 25, 37, 44, 45, 47, 48, 56, 59, 63, 71, 75, 79, 80, 83, 84, 85,

Newton, 52 Nicolet, J. L., 5, 156, 162, 163 Notation, 5, 40, 149 Objectivation, 36, 37, 38, 42, 43, 44, 47, 48, 56, 63, 64, 65

186

Index

Operations, 13, 45, 102, 103, 106, 116, 117, 118, 119, 130 on operations, 116, 118, 130

Relationships, 18, 39, 52, 56, 57, 81, 82, 85, 86, 87, 113, 115, 116, 138, 139, 142, 148, 153, 159, 160, 161, 180 space, 44, 48, 53, 67, 69, 104, 112, 114, 115, 116, 129, 147, 159, 160, 162, 170 Riemann, 52 Rigor, 53, 67, 69, 71, 154 sense of, 10, 67, 72, 162

Perception, 5, 34, 47, 53, 66, 69, 71, 73, 74, 75, 76, 77, 78, 84, 98, 146, 147, 148, 149, 154, 166, 173 Perceptive activity, 32, 74, 75 dynamism, 47, 76, 77, 85, 156, 170 Perspective, 74, 75, 90, 91, 126, 135, 160 Piaget, 1, 45, 46, 48, 62, 103, 104, 105 Psychology of learning, 3 of mathematics, 1, 2, 5, 29, 53, 54, 56, 57, 59, 65, 66, 73, 80, 82, 89, 96, 107, 109, 110, 114, 119, 121, 122, 129, 131, 138, 139, 150, 153, 163, 183 Psychosomatic phenomenon, 4

Self, 1, 3, 4, 9, 10, 11, 13, 14, 15, 18, 19, 20, 24, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 46, 48, 53, 54, 56, 57, 59, 60, 61, 62, 64, 66, 71, 75, 76, 77, 78, 81, 83, 85, 86, 91, 95, 96, 101, 102, 103, 109, 110, 111, 117, 118, 119, 122, 124, 125, 127, 142, 144, 145, 162, 163, 181 Sense organs, 32, 33, 34, 66 Situation, 18, 19, 20, 28, 32, 66, 71, 81, 82, 85, 86, 87, 90, 91, 96, 102, 103, 104, 106, 110, 114, 118, 127, 130, 131, 133, 142, 144, 147, 154, 155, 156, 161, 162, 165, 170, 172, 173, 177, 178, 182

Reality, 2, 19, 32, 33, 36, 38, 40, 43, 45, 46, 48, 58, 60, 62, 65, 76, 79, 80, 82, 86, 89, 110, 134, 145, 148, 159, 160, 169, 181, 182

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For the Teaching of Mathematics â&#x20AC;&#x201C; Volume 2

mathematical, 1, 2, 3, 5, 29, 30, 51, 56, 57, 60, 62, 65, 69, 72, 73, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 104, 106, 110, 114, 115, 118, 119, 126, 128, 134, 136, 149, 154, 159, 162, 163, 165, 166, 182, 186 Structuration, 2, 3, 33, 34, 37, 38, 62, 110, 112, 139, 147 Structures, 1, 2, 3, 4, 9, 12, 13, 14, 17, 18, 20, 22, 23, 24, 27, 28, 29, 30, 33, 34, 38, 46, 48, 51, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 84, 89, 90, 115, 116, 117, 119, 128, 130, 149 abstract, 11, 29, 41, 44, 45, 46, 47, 48, 53, 56, 57, 62, 66, 68, 71, 80, 81, 85, 87, 89, 90, 91, 92, 104, 113, 116, 117, 129, 130, 148, 154, 155, 161, 162 anatomic, 24, 76 auditive, 23, 28 collective, 54, 63, 65, 82 functional, 11, 24, 33, 47, 48, 61, 66, 76 intellectual, 12, 13, 17, 18, 19, 25, 37, 44, 45, 47, 48, 56, 59, 63, 71,

75, 79, 80, 83, 84, 85, 101, 117, 130, 145, 146, 161 intellectual mental, 13 linguistic, 20, 22, 23, 24, 29 mathematical, 1, 2, 3, 5, 29, 30, 51, 56, 57, 60, 62, 65, 69, 72, 73, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 104, 106, 110, 114, 115, 118, 119, 126, 128, 134, 136, 149, 154, 159, 162, 163, 165, 166, 182, 186 mathematical mental, 2, 149 mental, 2, 3, 4, 9, 10, 11, 12, 13, 19, 23, 24, 29, 32, 38, 40, 41, 45, 51, 53, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 75, 77, 78, 80, 82, 83, 86, 91, 92, 103, 112, 115, 117, 119, 121, 128, 129, 145, 146, 147, 149, 154, 171 motor, 18, 26, 27, 28, 45, 47 musical, 11, 23, 25, 27, 28, 29, 77 physical, 11, 12, 17, 18, 35, 36, 53, 56, 65, 129 physical mental, 12

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Index

psychic, 12, 13, 24 somatic, 4, 13, 14, 17, 24, 29, 33, 61, 62, 63, 77 somatic mental, 61 visual, 26, 27, 28, 34, 74, 77, 98, 126, 159, 160, 168, 170 Symbol, 79, 80, 81, 82, 83, 84, 85, 87 of symbols, 87 Symbolism, 79, 82, 83 of Symbolism, 79

Unconscious apprenticeship, 16 Verbalization, 5, 85 Vision, 5, 20, 32, 43, 67, 69, 73, 74, 75, 86, 102, 105, 121, 122, 123, 126, 127, 128, 147, 169, 171, 182 perspective, 74, 75, 90, 91, 126, 135, 160 spatial, 33, 75, 119, 122, 126, 127, 128, 154 three-dimensional, 44, 48, 74, 86, 105, 121, 122, 123, 126, 127, 134

Tests, 2, 9, 28, 101 Thought, 3, 21, 29, 41, 44, 45, 46, 48, 56, 71, 72, 73, 74, 77, 78, 79, 80, 84, 91, 92, 95, 96, 100, 101, 102, 103, 107, 111, 115, 125, 126, 129, 130, 134, 136, 137, 139, 144, 146, 147, 149, 154, 160, 161, 166, 170, 173 abstract, 11, 29, 41, 44, 45, 46, 47, 48, 53, 56, 57, 62, 66, 68, 71, 80, 81, 85, 87, 89, 90, 91, 92, 104, 113, 116, 117, 129, 130, 148, 154, 155, 161, 162 representational, 115 symbolic multivalent, 79 Tzanck, 43

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