9
46 A. Wittenherg 6.12 Lesquelles de ces recommandations vous suggCrez de mettre en pra- tique au fur et i mesure que des professeurs adCquats deviennent disponibles ? 6.131 Lesquelles de ces recommandations ne doivent, A votre sens, cons- tituer que des ahypoth6ses de travail)) pour des rCflexions et expbriences ult& rieures, Ctant bien entendu que vous ne suggeiez pas leur mise en application immCdiate ? 6.132 Pour ces derniQes recommandations, prCcisez la nature exacte des cdflexions et experiences))que vous envisagez. 6.2 Veuillez indiquer, au moins dans les grandes lignes, de quelle mani&re vous envisagez I'avenir du mouvement de rCforme de l'enseignement des rnathkmatiques. PrCcisez en particulier si vous considCrez la procCdure actuelle (propositions de rCforme ClaborCes par des confCrences d' cc experts)), choisis et rCunis par des organisations telles que YOCDE; manuels basks sur ces pro- positions Ccrits par des groupes de professeurs de l'emeignement secondaire) comme essentiellement satisfaisante ? Le cas CchCant, prCcisez les modifica- tions que vous aimeriez voir introduire dans cette prockdure. 6.3 Indiquez ici votre agrCrnent ou vos divergences d'avec les idkes ex- primCes dans l'kditorial. 6.4 Veuillez ajouter ici toutes autres considkrations non couvertes par ce questionnaire que vous aimeriez consigner pour le b6nCfice de cette enquCte. A. Questionnaire on the Teaching of Mathematics prepared by A. WITTENBERG t, Toronto To reduce the size of the problem somewhat, you are asked to consider only secondary school instruction in mathematics for pupils in your country who will be proceeding to university studies, i.e. in general, pupils at grammar schools, high schools, IycCes, gymnasia or athbnees in Europe, or college-bound high school students in the United States of America. It should be borne in mind that the situation in Europe and in the United States is not strictly comparable, since the first group includes about 5-10% of an entire age group where the second includes about 30%. In other words, the questions which follow concern a much less rigorously selected group of young people in the United States than in Europe. The group in question is referred to as the 'reference group'.

A. Questionnaire on the Teaching of Mathematics

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46 A. Wittenherg

6.12 Lesquelles de ces recommandations vous suggCrez de mettre en pra- tique au fur et i mesure que des professeurs adCquats deviennent disponibles ?

6.131 Lesquelles de ces recommandations ne doivent, A votre sens, cons- tituer que des ahypoth6ses de travail)) pour des rCflexions et expbriences ult& rieures, Ctant bien entendu que vous ne suggeiez pas leur mise en application immCdiate ?

6.132 Pour ces derniQes recommandations, prCcisez la nature exacte des cdflexions et experiences)) que vous envisagez.

6.2 Veuillez indiquer, au moins dans les grandes lignes, de quelle mani&re vous envisagez I'avenir du mouvement de rCforme de l'enseignement des rnathkmatiques. PrCcisez en particulier si vous considCrez la procCdure actuelle (propositions de rCforme ClaborCes par des confCrences d'cc experts)), choisis et rCunis par des organisations telles que YOCDE; manuels basks sur ces pro- positions Ccrits par des groupes de professeurs de l'emeignement secondaire) comme essentiellement satisfaisante ? Le cas CchCant, prCcisez les modifica- tions que vous aimeriez voir introduire dans cette prockdure.

6.3 Indiquez ici votre agrCrnent ou vos divergences d'avec les idkes ex- primCes dans l'kditorial.

6.4 Veuillez ajouter ici toutes autres considkrations non couvertes par ce questionnaire que vous aimeriez consigner pour le b6nCfice de cette enquCte.

A. Questionnaire on the Teaching of Mathematics

prepared by A. WITTENBERG t, Toronto

To reduce the size of the problem somewhat, you are asked to consider only secondary school instruction in mathematics for pupils in your country who will be proceeding to university studies, i.e. in general, pupils at grammar schools, high schools, IycCes, gymnasia or athbnees in Europe, or college-bound high school students in the United States of America. It should be borne in mind that the situation in Europe and in the United States is not strictly comparable, since the first group includes about 5-10% of an entire age group where the second includes about 30%. In other words, the questions which follow concern a much less rigorously selected group of young people in the United States than in Europe. The group in question is referred to as the 'reference group'.

Questionnaire on the Teaching of Mathematics 47

1. Edzlcation for whom?

1.1 Do you consider that the entire reference group should receive the same instruction ?

1.2 If the reference group is in practice subdivided into sub-groups receiving disparate kinds of mathematics instruction (whether you regard this procedure as justified or not), what are, in your opinion, the criteria which should govern this subdivision :

1.21 The pupils’ intended profession ? 1.22 The mathematical leanings and aptitudes displayed by the pupils ? 1.23 The general intelligence of pupils? 1.24 A combination of several of these criteria, leading to a subdivision

based on at least two of them? (If the latter solution is chosen, the result would be several sub-groups consisting, say, of future scientists at different levels of mathematical aptitude and leaning, or even perhaps several sub- groups consisting of future ‘art’ students more or less interested in mathe- matics.)

1.25 Other criteria? Please specify. 1.3 Should the subdivision be extensive, i.e. with a large number of

highly differentiated sub-groups (for instance, future mathematicians, future physicists, future chemists, future sociologists, etc.), or minimal, consisting of two or three sub-groups only (for instance, future ‘art’ students and ‘science’ students; the two or three categories into which Swiss and German gymnasium pupils are divided; or others ?).

Should the subdivision apply to all subjects studied by the pupil (the traditional subdivision into classes which are taught in commoii), or should it apply only to mathematics teaching (the English practice of ‘course setting’, or the American practice of regarding high school courses as separate units ?). Or should it apply to a group of subjects, for instance, mathematics and physics ?

If solution 1.21 or 1.24 is the one applied, do you consider that the subdivision should be final, so that a pupil cannot study science, for example, unless he had belonged to a ‘science’ sub-group at school (the English and French practice) ? Or do you consider that the subdivision should merely be a guide, leaving open the possibility of admitting the entire reference group to scientific or technical studies provided the pupils successfully complete their secondary studies (the Swiss and German practice, which is to admit t o pure and applied science faculties the holders of any type of matriculation certificate or ‘Abitur’) ?

At what age should the subdivision of the reference group take place ? If the subdivision applied is of a final nature, do you consider that

it should become final immediately after selection (with perhaps a very short

1.4

1.5

1.61 1.62

48 A. Wittenberg

period to correct obvious mistakes), or that it should be regarded merely as a guide for at least one year, and only then become final ?

1.63 1.71 1.72

In the latter case, when should the subdivision become final? On whose authority should the subdivision take place? Should the pupil be compulsorily allocated to a specific sub-group

by the school authorities ? Should the allocation depend entirely on the free choice of the pupil or of his parents ? Which of the parties should prevail, in case of dispute ?

1.81 Which authority should, in your opinion in the event of your having felt unable to reply to one or other of the preceding questions, be responsible for making the decision ?

1.82 On what criteria should this authority base its decisions, and on the basis of what criteria should it subsequently judge whether its decisions were the right ones ?

2. EdUcati0.n for what?

This section explores the aims of mathematics teaching and the correspond- ing choice of subject matter.

2.11 Do you consider that mathematics teaching should impart to the pupil the practical knowledge he will need in daily life ?

2.12 There seems to be general agreement that this knowledge should include at least the following subjects : simple arithmetic; a highly concrete grasp of function and of graphical representation; a highly concrete grasp of statistics that will enable the pupil to understand such concepts as ‘an aver- age’. Do you agree with this list? What other subjects should the pupil be taught, in your opinion, under category 2.1?

N.B. We should like you to be as specific as possible in your replies to this question and to the following ones. Please avoid generalizations (such as, ‘a grasp of finite mathematics’) and give explicit answers (such as, ‘elemen- tary matrix, algebra - addition, multiplication, inversion’).

Show how the subjects you have just listed can be useful in daily life? (Please be explicit: if you are unable to give explicit reply in the space available, please give specific examples that are as typical as possible. If for instance, you have included elementary matrix algebra in your list, do not give a reply such as ‘the growing r61e of automation’ but cite an example that you consider typical of a common situation in which the pupil will one day find it necessary to multiply or add matrices.)

2.21 Do you consider that the pupil should be taught subjects that are not listed under 2.1 but would contribute to his general culture? In other

2.13

Questionnaire on the Teaching of Mathematics 49

words, subjects justified on the following grounds : familiarizing the child with our cultural heritage; enabling him to understand the culture of our time, etc. ?

2.22 If your reply is yes, indicate the kind of subjects you have in mind. 2.23 Indicate the contribution made by each of the subjects on your list

to the aims set forth in 2.21. (Here again we should like you to be completely explicit. If, for instance, your list includes familiarizing the child with the language of the theory of sets, do not reply ‘familiarizing the child with the spirit of contemporary mathematics’, but specify what precise aspect of contemporary mathematics you are conveying to the child by teaching him this language.)

2.24 Co-ordinate your replies to 2.22 and 2.23 with those given in section. 1. In particular:

2.241 Should the subjects referred to in 2.2 be taught to the entire reference group ?

2.242 If your reply is no, please explain how each of the sub-groups resulting from the subdivision you envisage relates to the sub-groups of the list given in 2.22.

Should the pupil be taught subjects not listed under 2.1 and 2.2 as a preparation for his future studies or profession ?

If your reply is yes, state which subjects you have in mind. If necessary, confine your reply to the areas of higher studies on which you feel competent to pass judgement (for instance, higher studies in the mathematical and physical sciences).

Indicate the connexion between the subjects you have listed and uni- versity disciplines. (Avoid generalizations. Be specific; for instance : ‘a know- ledge of elementary group theory will be useful in such and such a way for a particular first-year university course’ .)

2.34 Co-ordinate replies in this sub-section point by point with those given in section 1. In particular:

(1.1/1.2/1.3) : Should pupils study subjects listed in 2.32 because even if they make no contribution to their own professional training, they will be useful to some of their classmates?

(1.5): If you are in favour of a subdivision that serves as a guide only, are your replies under this sub-section compatible with that of subdivision ? In other words, will pupils who have studied the subjects listed in 2.32 be in a position to undertake scientific or technical studies without being at a serious disadvantage ?

(1.6) : If you are in favour of a subdivision of a final value, state whether the subjects listed in 2.32 should be taught from the point where the subdivi- sion becomes final or before then - and in the latter case, for how Iong before ?

2.31

2.32

2.33

4 Dialectica 1 4 / 67

50 A. Wittenberg

2.41 Should pupils be taught subject not yet listed under 2.1,Z.Z and 2.3. because learning these subjects is a good academic exercise or helps to ‘sharpen their wits’ (intellectual gymnastics, training of the mind).

If your reply is yes, please indicate the scope of such subjects in relation to the pupil’s overall mathematical training, and give examples of the subjects you have in mind.

2.42

2.43 2.5

Co-ordinate your replies to 2.42 with those given in section 1. Should the pupil be taught subjects not yet listed, for reasons other

than those considered in 2.11, 2.21, 2.31 and 2.41 ? If your reply is yes, please state the reasons and subjects you have in mind. Once again, please co-ordi- nate your replies with those given in section 1.

3. Criteria of Success or F a i l w e

The purpose of this section is to investigate the conditions for claiming success in teaching the subjects referred to above.

3.1 Criteria of overall success: For each of the groups of subjects referred to in 2.1 to 2.5, indicate separately what percentage of pupils receiving in- struction in these subjects must succeed in assimilating them (in the sense defined below) for you to describe the instruction as having been successful. (For instance: if you are in favour of a subdivision which places future mathe- matics and physics in a single sub-group, and if in 2.31 you have listed an introduction to group theory, your reply should be along the following lines: ‘The result will be regarded as successful if at least 50% of the pupils succeed, in the sense defines below, in understanding the theory in question’).

Criteria of individual success: For the subjects referred to above, this sub-section investigates the ‘success’ achieved in instructing individual pupils and the criteria by which the authorities can decide whether or not the instruction has been successful.

3.211 In what circumstances will you consider the child to have success- fully mastered the subjects referred to in 2.1 ?

3.212 In what way should the school authorities decide whether the instruction has been successful, in the sense in which you have just defined the term ‘success’ ? If possible, give one or two examples of what you consider to be typical examination questions that could be used for this purpose.

Should the control methods which have just described be applied during (or a t the end of) the child’s schooling only, or should they also be applied latter to a statistical sample of adults who have received this instruc- tion, so as to determine whether they are still in possession of this knowledge for use in their daily lives ?

3.2

3.213

Questionnaire on the Teaching of Mathematics 51

3.221 The same question as 3.211 in relation to 2.2. 3.222 The same question as 3.212 in relation to 2.2. 3.223 What criteria do you suggest for determining whether the sub-

jects taught have in fact contributed to the purpose assigned by you in 2.23 to the acquisition of this knowledge? (If, for example, you suggested that the language of the theory of sets should be taught to the pupil in order to fami- liarize him with the spirit of contemporary mathematics, state here what criteria you would use to decide whether the pupil who succeeded in learning this language, as taught. has likewise acquired ‘familiarity with the spirit of contemporary mathematics’.)

Same question as 3.211 in relation to 2.3. (Please be explicit. If, for example, you included an introduction to infinitesimal calculus in your list in 2.3, state here whether the objective in view is, for instance, a knowledge of the formal techniques of derivation and integration, an understanding of the mathematical concept of limit, etc.)

3.231

3.232 Same question as 3.212 in relation to 2.3. 3.233 Is the following difinition of ‘success’ (in relation to the subjects

covered in 2.3) equivalent to the one you have just given ? Definition: the results are ‘successful‘ if the university can safely assume

that the subjects listed in 2.3 are mastered without having to repeat them. (N.B. ‘without having to repeat them’ naturally means at the same level as a t school. The university can, of course, revert to these subjects in order to consider them in a more detailed context, just as the university returns to arithmetic in courses on the theory of numbers or mathematical logic without, however, repeating the primary arithmetic course.)

If you have listed subjects under 2.41, indicate here how you propose to decide whether the assimilation of these subjects has in fact helped to ‘train the mind’ of the pupil.

If you listed subjects under 2.5, please discuss them in the same manner as above.

3.24

3.25

4. Criteria of Effeectivefiess

4.11 For each of the sub-group of the subdivision envisaged in section 1, state explicitly whether the proposals you have made in section 2 imply an increase in the number of hours a t present devoted in your country to mathe- matics teaching. Where necessary, indicate the extent of the increase. If possible, also indicate for each of the sub-groups, what percentage of the total time devoted to secondary studies by pupils should be taken up by mathe- matics.

52 A. Wittenberg

4.12 If you envisage an increase in the number of hours for all or some of the sub-groups, should this increase be achieved in its entirety by a corre- sponding increase in the total number of hours of study by the pupils 1

4.13 If your reply to the preceding question is no for a t least one of the sub-groups, state how you propose to make necessary time available.

(Be specific. Do not say, for example, that ‘a lot of useless subjects are taught’, but ‘I suggest that geography teaching should be eliminated’ or ‘I propose that only the geography of our own country be taught, and that the rest be eliminated’.)

4.2 The questions in this sub-section refer solely to the subjects listed in 2.3:

4.21 These subjects, if not taught a t school, have to be taught a t the uni- versity itself. Assuming that the teaching a t both school and university is fully effective, what is the approximate relationship between the time to saved by the university if the students arrive knowing the subjects in question, and the time tl required by the school to teach these subjects ?

4.22 Is the time t o independent of, or affected by, the way in which non- mathematical subjects are taught a t school, and in particular by the teaching of ‘arts’ subjects (in the broad sense)? In other words, can the effective teaching of the latter subjects reduce the time t o ?

If your reply to this question is yes, what is your reply to question 4.21, assuming that the school uses the time tl for effective instruction in well selected non-mathematical subjects.

4.24 Assuming the optimum choice of subjects C. C equals the full range of non-mathematical subjects that can be taught in the time tl. Do you consider that the saving time to justifies sacrificing C ?

4.25 If your reply to the preceding question is yes, state whether you were chiefly influenced by the importance of the saving to or by the unimportance of teaching the pupil C.

4.23

5. Teaching in Practice

5.11 For each of the areas of knowledge listed in 2, specify whether the following statement is valid: ‘These subjects can usefully be taught only by a teacher whose own training included the theoretical context of the subjects in question’. (For instance, elementary group theory could usefully be taught only by teachers whose university training included the study of group theory.) It will be noted that the converse of the statement is as follows: ‘These subjects can usefully be taught by teachers who have received an ad hoc training aimed specifically a t the instruction which they themselves will dis- pense’. (The standard practice of American reform groups.)

Questionnaire on the Teaching of Mathematics 53

5.12 In the light of the reply you have just given, state which of these sub- jects you would suggest should be immediately introduced into teaching, and which of them you would suggest should not be introduced until qualified teachers are available (according to your definition in 2.11).

5.211 Do you consider that teaching should proceed to more theoretical or more abstract developments after concrete and or intuitive preparation ?

5.212 Do you consider that teaching should seek to make the pupil re- discover for himself the mathematical ideas, concepts, methods and facts that it seeks to impart to him (the ‘heuristic’ method, or teaching by ‘rediscovery’) ?

Do you consider that teaching should introduce concepts (such as the ‘group’ concept) and methods (such as the ‘language of the theory of sets’) only to the extent that their necessity or usefulness can be grasped by the pupil himself ?

5.221 Is the range of subjects listed by you in 2 compatible with the time required to fulfil your recommendations in 5.21 ?

5.222 Are the criteria indicated by you in 3.2 sufficient to determine whether the teaching conforms to your recommendations in 5.21 (i.e. to determine, where necessary, whether the pupil understands the intuitive basis or justification for the mathematics that he ‘knows’) ? If your reply is no, how should the criteria be modified ?

Will the teachers providing instruction in accordance with your recommendations in 5.1 be in a position to follow your recommendations in 5.21 ?

5.213

5.223

6. Findings and Problems

6.1 In the light of the replies you have given to the above questions, please state explicitly:

6.11 Which of your recommendations in 1 and 2 do you suggest should immediately be put into practice ?

6.12 Which of these recommendations do you suggest should be put into practice as and when suitable teachers become available ?

6.131 Which of these recommendations, in your opinion, should be re- garded merely as ‘working hypotheses’ for future thinking and experimenta- tion, it being understood that you do not suggest their immediate application ?

In the case of these latter recommendation, specify exactly the nature of the ‘thinking and experimentation’ you have in mind.

Please indicate, a t least in general terms, how you envisage the future of the reform movement in mathematics teaching. In particular, state whether you regard present procedures as entirely satisfactory (i.e. proposals for re- form drawn up conferences of ‘experts’, selected and convened by organiza-

6.132

6.2

54 E. Emery

tions like OECD; by and textbooks based on those proposals, written by groups of secondary school teachers) ? If necessary, indicate the changes you would like to see in these procedures.

6.3 State here whether or not you agree with the ideas expressed in the editorial.

6.4 Please add any other observations, not covered by the questionnaire, that you would like to make in connexion with this survey.

A. Wittenberg t Toronto

B. RBponses ii l’enquete sur l’enseignement des mathematiques B. Replies on the Teaching of Mathematics

U N E C O N T R I B U T I O N AU DeBAT1)

par ERIC EMERY, La Chaux-de-Fonds

Problemes de compktences

Avant d‘entrer dans le vif du sujet, il me parait utile et honn&te de dCfinir en quelques mots les limites de mes compCtences. Je crois, en effet, que seule une Cquipe de travail largement reprksentative est susceptible, en matihe d‘enseignernent des mathkmatiques, de proposer d’une manibre exhaustive des conclusions pertinentes et nuancCes ; cette Cquipe devrait comprendre des mathCmaticiens Cminents, bien sQr, mais Cgalement des utilisateurs des math& matiques (physiciens, ingknieurs, biologistes, statisticiens ...) ainsi que des logi- ciens, des psychologues, des pkdagogues de tous les niveaux, sans oublier les philosophes. Alors que dire, sinon que ma contributiorc ne peut conserver un caracthe d’authenticitk que si elle se place dClibCrCment dans des limites res- treintes et qu’elle se prCsente sous la forme d’un simple tkmoignage; tCmoi- gnage d u n professeur qui a connu pendant une quinzaine d’annCes les exi- gences des enseignements du premier cycle (11 i 15 ans) et du second cycle (16 A 19 ans) ainsi que, dix i douze semestres durant, les problbmes pCdago- giques que posent les cours du soir (adultes de tous Ages et de formations trbs hCtCrogbnes). Je voudrais done que les rCflexions qui suivent prennent essen- tiellement une tournure A la fois informative et interrogative, qu’elles soient offertes B la critique comme 616ments de dCbat, et rien de plus, car la zone

1) Texte rkdigt? en janvier 1966.