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A. REVUZ FURTHER TRAINING OF MATHEMATICS TEACHERS OF SECONDARY LEVEL* Programmes, Methods, Means The following reflections arise from more than 10 years experience with 'further training' of mathematics teachers; they aim at showing some of the conditions of efficiency in this work. Let us first point out two mistakes which I am afraid have been committed, at least in the beginning, by all university professors who took an active part in 'further training'. The first mistake is to believe that the average mathematical level of active teachers and their susceptibility to modern thinking are from the start equivalent to those of good university students. Such a supposition is not entirely wrong since most of the teachers, indeed, after an adaptation period reach the level of good students. Yet this mistake may be a harmful one as a large number of teachers can be discouraged from the beginning if the university professor is not aware of the burdensome working conditions which prevent teachers from meeting new ideas in their field and engaging in genuine creative activity. This means that there is the problem of an adaptation period, which has to be faced: new ideas should be presented in order to persuade the listeners that change is possible and desirable, but this should be done so slowly and moderately that they do not get the irrevocable feeling of being overtaken and left behind. The second mistake is to believe that the transposition of a question from one level of teaching to a more elementary one would be easy for the ma- jority of the teachers. Sometimes, indeed, such a transposition can be per- formed by mere rewording and amplifying the exposition, but mostly it requires much more work, which is often underestimated though never with impunity. At the university level one admits, though perhaps wrongly, that the students have got a sound idea of what mathematical activity means, and that it is possible if not even useful to offer them rather ready-made theories in their strictly logical development. If such a procedure is contro- versial even at the university level, it is strictly impossible at the secondary level where the first task is to make the students understand what mathe- matics means and how it serves to master the reality; this task cannot be performed by confronting them with ready-made mathematics. Mathe- matization is as important as mathematics in secondary education. * Report to a Unesco Institute Conference in Hamburg, 21-26 October 1968. 493 Educational Studies in Mathematics 1 (1968--69) 493--498; 9 D. Reidel, Dordrecht-Holland

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Page 1: Further training of mathematics teachers of secondary level

A. R E V U Z

F U R T H E R T R A I N I N G O F M A T H E M A T I C S T E A C H E R S

O F S E C O N D A R Y L E V E L *

Programmes, Methods, Means

The following reflections arise from more than 10 years experience with 'further training' of mathematics teachers; they aim at showing some of the conditions of efficiency in this work.

Let us first point out two mistakes which I am afraid have been committed, at least in the beginning, by all university professors who took an active part in 'further training'.

The first mistake is to believe that the average mathematical level of active teachers and their susceptibility to modern thinking are from the start equivalent to those of good university students. Such a supposition is not entirely wrong since most of the teachers, indeed, after an adaptation period reach the level of good students. Yet this mistake may be a harmful one as a large number of teachers can be discouraged from the beginning if the university professor is not aware of the burdensome working conditions which prevent teachers from meeting new ideas in their field and engaging in genuine creative activity. This means that there is the problem of an adaptation period, which has to be faced: new ideas should be presented in order to persuade the listeners that change is possible and desirable, but this should be done so slowly and moderately that they do not get the irrevocable feeling of being overtaken and left behind.

The second mistake is to believe that the transposition of a question from one level of teaching to a more elementary one would be easy for the ma- jority of the teachers. Sometimes, indeed, such a transposition can be per- formed by mere rewording and amplifying the exposition, but mostly it requires much more work, which is often underestimated though never with impunity. At the university level one admits, though perhaps wrongly, that the students have got a sound idea of what mathematical activity means, and that it is possible if not even useful to offer them rather ready-made theories in their strictly logical development. If such a procedure is contro- versial even at the university level, it is strictly impossible at the secondary level where the first task is to make the students understand what mathe- matics means and how it serves to master the reality; this task cannot be performed by confronting them with ready-made mathematics. Mathe- matization is as important as mathematics in secondary education.

* Report to a Unesco Institute Conference in Hamburg, 21-26 October 1968.

493

Educational Studies in Mathematics 1 (1968--69) 493--498; �9 D. Reidel, Dordrecht-Holland

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If insufficient regard is paid to the importance of the scientific and peda- gogic analysis which has to precede the introduction of mathematical notions at an elementary stage, a complementary mistake is possible: to assign to every notion a well-defined, and often quite high, level, and to claim that below this level this notion cannot be dealt with honestly. There have been university professors who declared that the use of groups at the secondary level is an absurdity. Certainly this is so if it is undertaken in the way it is done at the university. Yet experience has shown that at very elementary stages it is possible to unravel this notion from a variety of exam- ples, and that finally those young students who had acquired it by their own activity, had understood it more profoundly and were better prepared to recognize and use it than the older students which had been confronted with this notion from the beginning in its most abstract purity.

These considerations lead us to formulate the central problems of further training. What are its goals in the present state of science and education?

The modernization of mathematical instruction was started by mathe- maticians who were alarmed by the gap between a static traditional instruc- tion and a dynamic 'Science'. However, two more, and not less serious, reproaches can be made against traditional instruction: its dogmatic methods, and its wrong attitude towards applications. The applications, if not com- pletely ignored in favour of a uselessness which was identified with pure scholarship, were reduced to ridiculous, old-fashioned examples or con- signed to schools with a narrow-minded utilitarian programme.

The mistaken contrast between pure and applied mathematics, the one, beautiful but haughty and barren, the other, shaky and less attractive, is one of the most serious defects of traditional instruction. It is hardly an exaggeration to say that the students were divided into a majority who could not handle any mathematics, and a minority who could not go outside the field of mathematics. Many efficient theories are ignored by people who would be happy if they mastered them and who instead use routines based upon inconsistent reasoning.

The purpose of modernization of instruction, and consequently of further training, cannot be to introduce 'modern' concepts into primary and second- ary education and meanwhile to stick to dogmatic teaching methods and to keep aloof from applications. Nor is it to stress so-called applied mathematics while still neglecting the forceful tools of modern mathematics. Nor is it to change teaching methods while conserving the old contents. The true goal unites all these three aspects, which may be distinguished by analysis but may not be separated in teaching practice.

The outstanding means to reach this goal is the 'situation-model' didac- tics. The natural start of any mathematical activity is to place somebody

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F U R T H E R T R A I N I N G OF T E A C H E R S OF S E C O N D A R Y LEVEL

in a situation which he has to analyze in order to act with the highest degree of efficiency. Mathematics participates in this analysis by creating models which represent certain aspects of the situation in a precise but schematic way. Two fundamental problems then arise:

(a) To find an adequate model among the variety which may be proposed and which have to be discus~sed according to their advantages and dis- advantages.

(b) To study the model as such, that is, to develop a mathematical theory of small or large extent, with all desirable ingenuity and rigor. The properties conserved in the model then appear as the axioms of a corresponding mathe- matical theory; the adequacy of the model can be tested at the level of the axioms or by important results of the theory which allow an easy comparison with the reality of the situation.

In any case, for the efficiency of teaching it is essential that these two aspects are not disconnected too early, and that the students take an active part in the study of the situation, in the elaboration of the model, in the development of the theory, in the confrontation of the theory with the original situation.

Since the goals of further training are determined by those of school and university teaching, it is not too improbable that teaching methods are based on the same principles in all cases. It is a strange and sad experience to state that expositions on pedagogics are often deprived of all pedagogical qualities as though what is recommended in dealing with children does not hold true when dealing with adults. The old contrast between the adult who knows and sets the norms, and the child who learns and has to accept the norms, remains unconsciously in the spirit of those who denounce it. A human being would not progress if he is not strongly motivated and if he does not make a great personal effort, whatever his age or his level of education may be.

The most direct and strongest motivation for most of the teachers is the desire to improve their teaching. This is one more reason why the presenta- tion of some concept should as much as possible go together with the consequences which it allows on different levels of teaching.

Unfortunately painful motivations cannot be avoided if teachers are to be persuaded that concepts they have taught in good faith for years are actually useless, or ambiguous, or wrongly viewed. One should not forget that being persuaded or starting to be persuaded of such facts, is a psycho- logical discomfort; it is highly embarrassing to have taught in a way which now appears to be imperfect, and perhaps not to know how to change it. Therefore they have to learn as clearly as possible, and as soon as possible, the solution which should end their uncertainty; maybe one should even

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organize the work in such a way that the solution of change is at least partially known in advance. In any case, whoever starts retraining teachers should have purged himself of any dogmatism, lest he expels an obsolete dogmatism by means of one that at the moment looks more enlightened. He should never forget that his work is worthless if, instead of persuading, he imposes some authority and that the result of his efforts will be nought or negative if he does not subordinate his work to the collective intellectual progress of humanity, which is the basic motivation of all scientific and pedagogic work.

The spirit of fraternal collaboration, the susceptibility to open discussion are even more needed if the retrainer addresses himself to teachers who initially have only been motivated by administrative pressure or by the an- nouncement of a new programme. Though such circumstances may stimulate indifferent or recalcitrant teachers, it must be stressed that any such motiva- tion should soon be replaced by a more profound one.

The best way to activate participants is to form small t e a m s (e.g. of teachers f rom the same institution or neighboring ones) who collaborate not only in giving but also in receiving instruction. Each group should have a leader with a more profound mathematical background though any hierarchy should be avoided. Such a team should not work in an air-tight compart- ment. I t should be exposed to periodic confrontations with other teams and participate at more general meetings, where a highly qualified person would deal with some theme which has arisen f rom the teamwork and has been judged too difficult, or perhaps with a new theme which has not been tackled before and is now handed down to their discussion.

All ranks of the educational system should participate in these activities; administrative or social obstructions to close and frequent contact should be removed.

This work may be organized in various ways. It is not useful to dwell upon details which depend heavily on local conditions. Attention should be paid to a not too formal organization. The more free and responsible people feel and the more initiatives are welcomed, the more successful the enterprise will turn out.

Television may be a good aid. It allows information to be transmitted to teachers who are too far from a university centre to visit it frequently. In France it has been used for six years. Collective viewing by a team followed by a discussion would increase its impact.

The themes should be carefully chosen depending on the knowledge and the mentality of the teachers in question. I f it is a group who have never been touched by modernization efforts, one should start f rom traditional instruction to show them:

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(a) that classical algebra becomes more lucid as soon as the structures of the underlying sets of numbers are made explicit. Too often traditional expositions obscure essentials and unduly stress special mathematical entities instead of sets and structures,

(b) that most of the traditional expositions of geometry are far below that degree of perfection which is usually attributed to them, that they contain numerous concepts which are governed by confusion (such as angles), that they often use petty artifices with no further bearing, that an exposition based on vector space and scalar product is easier to master and to use and that it provides students with tools which are indispensable in all fields, pure and applied,

(c) that the concepts of calculus can be introduced much earlier than is the case now, by means of concrete examples of step and piecewise affine functions which lend themselves to numerous operations, and which lead in a quite natural fashion to discovering differentiation and integration. This could also be an opportunity to show that computing need not be a repelling and oldfashioned business. The use of computers has the double advantage of initiating the students into the use of a material which becomes more and more popular and giving them a better understanding of the operations they perform,

(d) that very simple logical notions provide a more conscious under- standing of the structure of reasoning,

(e) that the traditional treatment of measuring magnitudes is inconsistent and should rather be replaced by an elementary theory of measure which, among others, has the advantage of including an elementary theory of probability.

If the teachers are already partially acquainted with these questions, the goal should be to go into some greater depth. But this should be done simul- taneously in two directions: on the one hand, to promote the study of the mathematical subject-matter so far that the teachers get reasonably acquaint- ed with the mathematics they are supposed to teach and with its more general perspectives; on the other hand, and in agreement with these perspectives, to study the best ways of teaching the notions of the subject-matter, to find out how to make the students discover the utility and efficiency of these notions, that is, in particular, how to get them familiar with a large number of situations where these notions play a role, by preference situations close to reality. It is true that artificial situations are often simpler than real ones, and this may sometimes be a reason to prefer them, and it is also true that sometimes questions can successfully be presented in the form of games. But finally the elucidation of a relatively complex situation which needs more time will have more effect, and on the other hand, any abuse of artificial

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situations would condemn teaching of mathematics to the same kind of sterility as I mentioned earlier.

For many years one of the major goals of further training will be to provide the teacher with a large collection of examples of situations studied in other disciplines and capable of mathematization.

To finish, I would insist upon further training being a long-term task which to be fruitful has to be pursued continuously in every individual case. Short and intensive stages do not pay; the minimal formula for reasonable efficiency is one and a half hours a week for one year (expositions, discussion, exercises . . . . ). This work should be closely connected to the teacher's classroom work. In this respect the constitution o f teams o f teachers who collaborate in learning as well as in teaching, discussing freely and sharing their experiences, is highly promising: in the course of the experiments in 'sixi~me' classes (12-year-old pupils) in France in 1968-69 numerous teams were formed; their efficiency in pedagogical research as well with regard to information has been one of the clearest lessons of these experiments.

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